%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU752^1 : 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 : n006.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 09:36:54 AM UTC 2026
% Result : Theorem 6.63s 2.38s
% Output : Refutation 6.63s
% Verified :
% SZS Type : Refutation
% Derivation depth : 23
% Number of leaves : 21
% Syntax : Number of formulae : 124 ( 28 unt; 0 typ; 4 def)
% Number of atoms : 5607 (1481 equ; 0 cnn)
% Maximal formula atoms : 273 ( 45 avg)
% Number of connectives : 3603 ( 214 ~; 163 |; 904 &;1170 @)
% ( 22 <=>;1130 =>; 0 <=; 0 <~>)
% Maximal formula depth : 283 ( 20 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 485 ( 482 usr; 296 con; 0-4 aty)
% Number of variables : 256 ( 0 ^; 182 !; 74 ?; 256 :)
% 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_320,type,
sP0: $i > $i > $i > $i > $o ).
thf(func_def_326,type,
sP6: $o > $i > $i > $i > $o ).
thf(func_def_327,type,
sP7: $i > $i > $i > $o ).
thf(func_def_328,type,
sP8: $i > $i > $o ).
thf(func_def_329,type,
sP9: $i > $o ).
thf(func_def_330,type,
sP10: $i > $i > $o ).
thf(func_def_331,type,
sP11: $i > $i > $o ).
thf(func_def_333,type,
sK13: $i > $i > $i ).
thf(func_def_340,type,
sK20: $i > ( $i > $o ) > $i ).
thf(func_def_341,type,
sK21: $i > $o ).
thf(func_def_345,type,
sK25: ( $i > $o ) > $i > $i ).
thf(func_def_347,type,
sK27: $i > $o ).
thf(func_def_361,type,
sK41: $i > $i > $i > $i > $i ).
thf(func_def_362,type,
sK42: $i > $i > $i > $i > $i ).
thf(func_def_363,type,
sK43: $i > $i > $i > $i > $i ).
thf(func_def_364,type,
sK44: $i > $i > $i > $i > $i ).
thf(func_def_366,type,
sK46: $i > $o ).
thf(func_def_373,type,
sK53: $i > $i > $o ).
thf(func_def_376,type,
sK56: $i > $i ).
thf(func_def_379,type,
sK59: ( $i > $i ) > $i > $i > $i ).
thf(func_def_382,type,
sK62: $i > $i ).
thf(func_def_384,type,
sK64: $i > $i > $o ).
thf(func_def_388,type,
sK68: $i > $i > $i ).
thf(func_def_416,type,
sK96: ( $i > $o ) > $i > $i ).
thf(func_def_418,type,
sK98: $i > $o ).
thf(func_def_422,type,
sK102: $i > $i > $o ).
thf(func_def_442,type,
sK122: $i > $o ).
thf(func_def_445,type,
sK125: $i > ( $i > $o ) > $i ).
thf(func_def_460,type,
sK140: $i > $i > $o ).
thf(func_def_501,type,
sK181: $i > $i ).
thf(func_def_512,type,
sK192: $i > $i > $i ).
thf(func_def_513,type,
sK193: $i > $i > $i ).
thf(func_def_518,type,
sK198: $i > $i > $i > $i > $i ).
thf(func_def_519,type,
sK199: $i > $i > $i > $i > $i ).
thf(func_def_528,type,
sK208: $i > $o ).
thf(func_def_531,type,
sK211: $i > $i ).
thf(func_def_537,type,
sK217: $i > ( $i > $o ) > $i ).
thf(func_def_538,type,
sK218: $i > $o ).
thf(func_def_544,type,
sK224: $i > ( $i > $i ) > $i > $i ).
thf(func_def_546,type,
sK226: $i > $i ).
thf(func_def_549,type,
sK229: $i > $o ).
thf(func_def_557,type,
sK237: $i > $i > $i > $i ).
thf(func_def_558,type,
sK238: $i > $i > $i > $i ).
thf(func_def_562,type,
sK242: $i > $o ).
thf(func_def_564,type,
sK244: $i > ( $i > $o ) > $i ).
thf(func_def_576,type,
sK256: $i > $o ).
thf(func_def_578,type,
sK258: ( $i > $o ) > $i > $i ).
thf(func_def_579,type,
sK259: ( $i > $o ) > $i > $i ).
thf(func_def_580,type,
sK260: $i > $o ).
thf(func_def_583,type,
sK263: $i > $i ).
thf(func_def_585,type,
sK265: $i > $i ).
thf(func_def_590,type,
sK270: $i > $i > $o ).
thf(func_def_591,type,
sK271: $i > $o ).
thf(func_def_602,type,
sK282: $i > $i > $i > $i > $i ).
thf(func_def_606,type,
sK286: $i > $o ).
thf(func_def_608,type,
sK288: ( $i > $o ) > $i > $i ).
thf(func_def_626,type,
sK306: $i > $i > $o ).
thf(func_def_628,type,
sK308: $i > $o ).
thf(func_def_631,type,
sK311: $i > ( $i > $o ) > $i ).
thf(func_def_632,type,
sK312: $i > ( $i > $o ) > $i ).
thf(func_def_642,type,
sK322: $i > $o ).
thf(func_def_644,type,
sK324: ( $i > $o ) > $i > $i ).
thf(func_def_646,type,
sK326: $i > $i ).
thf(func_def_655,type,
sK335: $i > $o ).
thf(func_def_659,type,
sK339: $i > $i ).
thf(func_def_661,type,
sK341: $i > ( $i > $i ) > $i > $i ).
thf(func_def_662,type,
sK342: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_663,type,
sK343: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_666,type,
sK346: $i > $o ).
thf(func_def_667,type,
sK347: $i > $o ).
thf(func_def_677,type,
sK357: $i > $i > $i > $i > $i ).
thf(func_def_690,type,
sK370: $i > $i > $i ).
thf(func_def_698,type,
sK378: $i > $i > $i ).
thf(func_def_705,type,
sK385: $i > $o ).
thf(func_def_707,type,
sK387: ( $i > $o ) > $i ).
thf(func_def_708,type,
sK388: ( $i > $o ) > $i ).
thf(func_def_729,type,
sK409: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_730,type,
sK410: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_731,type,
sK411: $i > $o ).
thf(func_def_732,type,
sK412: $i > $o ).
thf(func_def_740,type,
sK420: $i > $o ).
thf(func_def_743,type,
sK423: $o > $i > $i > $i ).
thf(func_def_757,type,
sK437: ( $i > $o ) > $i ).
thf(func_def_758,type,
sK438: $i > $o ).
thf(func_def_759,type,
sK439: $i > $i ).
thf(func_def_760,type,
sK440: $i > $i > $i > $i ).
thf(func_def_764,type,
sK444: $i > $i > $i > $i ).
thf(func_def_765,type,
sK445: $i > $i > $i > $i ).
thf(func_def_770,type,
sK450: $i > $o ).
thf(func_def_803,type,
sK483: $i > $o ).
thf(func_def_804,type,
sK484: $i > $i ).
thf(func_def_805,type,
sK485: ( $i > $o ) > $i ).
thf(func_def_806,type,
sK486: $i > $o ).
thf(func_def_838,type,
sK518: $i > $i > $o ).
thf(func_def_843,type,
sK523: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_844,type,
sK524: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_848,type,
sK528: $i > $o ).
thf(func_def_853,type,
sK533: $i > $i > $i ).
thf(func_def_865,type,
sK545: $i > $i > $i ).
thf(func_def_870,type,
sK550: $i > $o ).
thf(func_def_895,type,
sK575: $i > $i > ( $i > $i ) > $i ).
thf(func_def_896,type,
sK576: $i > $i ).
thf(func_def_903,type,
sK583: $i > $o ).
thf(func_def_908,type,
sK588: ( $i > $i ) > $i > $i > $i ).
thf(func_def_911,type,
sK591: $i > $i ).
thf(func_def_949,type,
sK629: $i > $o ).
thf(func_def_950,type,
sK630: $i > $o ).
thf(func_def_951,type,
sK631: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_952,type,
sK632: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_975,type,
sK655: $i > $o ).
thf(func_def_986,type,
sK666: $i > $o ).
thf(func_def_987,type,
sK667: $i > $o ).
thf(func_def_988,type,
sK668: ( $i > $o ) > $i ).
thf(func_def_989,type,
sK669: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_990,type,
sK670: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_994,type,
sK674: $i > $o ).
thf(func_def_1002,type,
sK682: $i > $i > $i ).
thf(func_def_1003,type,
sK683: $i > $i > $i ).
thf(func_def_1004,type,
sK684: $i > $i > $i ).
thf(func_def_1005,type,
sK685: $i > $i > $i ).
thf(func_def_1006,type,
sK686: $i > $i > $i ).
thf(func_def_1007,type,
sK687: $i > $i > $i > $i ).
thf(func_def_1008,type,
sK688: $i > $i ).
thf(func_def_1009,type,
sK689: $i > $i ).
thf(func_def_1010,type,
sK690: $i > $i ).
thf(func_def_1011,type,
sK691: $i > $i ).
thf(func_def_1012,type,
sK692: $i > $i > $i ).
thf(func_def_1013,type,
sK693: $i > $i > $i > $i ).
thf(func_def_1014,type,
sK694: $i > $i > $i > $i ).
thf(func_def_1015,type,
sK695: $i > $i > $i ).
thf(func_def_1016,type,
sK696: $i > $i > $i ).
thf(func_def_1017,type,
sK697: $i > $i ).
thf(func_def_1019,type,
sK699: $i > $i ).
thf(func_def_1020,type,
sK700: $i > $i ).
thf(func_def_1062,type,
sK742: $i > $i > $i ).
thf(func_def_1066,type,
sK746: $i > $i > $o ).
thf(func_def_1067,type,
sK747: $i > $i ).
thf(func_def_1068,type,
sK748: $i > $i ).
thf(func_def_1069,type,
sK749: ( $i > $i > $o ) > $i > $i ).
thf(func_def_1070,type,
sK750: ( $i > $i > $o ) > $i > $i ).
thf(func_def_1071,type,
sK751: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1088,type,
sK768: $i > $i > $i > $i > $i ).
thf(func_def_1093,type,
sK773: ( $i > $o ) > $i > $i ).
thf(func_def_1095,type,
sK775: $i > $o ).
thf(func_def_1097,type,
sK777: $i > $i > $i ).
thf(func_def_1105,type,
sK785: $i > $i > $i > $i ).
thf(func_def_1136,type,
sK816: $i > $o ).
thf(func_def_1139,type,
sK819: $i > $i > ( $i > $o ) > $i ).
thf(func_def_1142,type,
sK822: $i > $i ).
thf(func_def_1144,type,
sK824: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1168,type,
sK848: $i > $i ).
thf(func_def_1187,type,
ph867:
!>[X0: $tType] : X0 ).
thf(f7897,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4365,f4584,f4607,f5395,f7896]) ).
thf(f7896,plain,
( ~ spl866_29
| ~ spl866_7 ),
inference(avatar_split_clause,[status(thm)],[f4480,f4238,f4511]) ).
thf(f4511,definition,
( spl866_29
<=> ( $true
= ( in @ sK293 @ sK292 ) ) ),
introduced(definition,[new_symbols(naming,[spl866_29])],[avatar_definition]) ).
thf(f4238,definition,
( spl866_7
<=> ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK292 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl866_7])],[avatar_definition]) ).
thf(f4480,plain,
( ( $true
!= ( in @ sK293 @ sK292 ) )
| ~ spl866_7 ),
inference(trivial_inequality_removal,[status(thm)],[f4479]) ).
thf(f4479,plain,
( ( $true
!= ( in @ sK293 @ sK292 ) )
| ( $true != $true )
| ~ spl866_7 ),
inference(forward_demodulation,[status(thm)],[f4478,f3051]) ).
thf(f3051,plain,
setminusER = $true,
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f1635,plain,
( ( dpsetconstrERa = $true )
& ( vacuousDall = $true )
& ( inIntersectImpInUnion = $true )
& ( complementUnionInPowersetComplement = $true )
& ( binunionTE = $true )
& ( notdallE = $true )
& ( upairsetIL = $true )
& ( infuncsetfunc = $true )
& ( prop2setE = $true )
& ( apProp = $true )
& ( subsetemptysetimpeq = $true )
& ( setadjoinE = $true )
& ( setukpairinjR = $true )
& ( prop2set2propI = $true )
& ( notequalI1 = $true )
& ( omega__Cong = $true )
& ( binintersectSubset3 = $true )
& ( ex1E1 = $true )
& ( cartprodmempair = $true )
& ( complementSubsetComplementIntersect = $true )
& ( kfstpairEq = $true )
& ( setadjoinOr = $true )
& ( binunionRsub = $true )
& ( emptysetE = $true )
& ( setminusEL = $true )
& ( funcGraphProp4 = $true )
& ( powersetAx = $true )
& ( eqbreln = $true )
& ( doubleComplementE1 = $true )
& ( binintersectSubset4 = $true )
& ( foundationAx = $true )
& ( beta2 = $true )
& ( complementImpComplementIntersect = $true )
& ( iffalseProp1 = $true )
& ( contrasubsetT1 = $true )
& ( binunionEcases = $true )
& ( noeltsimpempty = $true )
& ( symdiffI2 = $true )
& ( upairsetIR = $true )
& ( cartprodpairmemEL = $true )
& ( setbeta = $true )
& ( subPowSU = $true )
& ( ex1E2 = $true )
& ( powersetE1 = $true )
& ( contrasubsetT = $true )
& ( symdiffIneg1 = $true )
& ( setukpairinjR11 = $true )
& ( binunionLsub = $true )
& ( doubleComplementSub1 = $true )
& ( setextT = $true )
& ( iffalseProp2 = $true )
& ( ksndsingleton = $true )
& ( upairsubunion = $true )
& ( binunionTILcontra = $true )
& ( setadjoinSub2 = $true )
& ( prop2setI = $true )
& ( iftrueorfalse = $true )
& ( singletonsubset = $true )
& ( omega0Ax = $true )
& ( setminusLsub = $true )
& ( symdiffI1 = $true )
& ( quantDeMorgan1 = $true )
& ( iffalse = $true )
& ( setminusSubset1 = $true )
& ( lamProp = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectT_lem = $true )
& ( complementT_lem = $true )
& ( ap2p = $true )
& ( binunionE = $true )
& ( complementTI1 = $true )
& ( exuI2 = $true )
& ( binintersectTERcontra = $true )
& ( emptyset__Cong = $true )
& ( setunionE = $true )
& ( binintersectSubset1 = $true )
& ( setOfPairsIsBReln = $true )
& ( setunionE2 = $true )
& ( setukpairinjL = $true )
& ( doubleComplementI1 = $true )
& ( funcext2 = $true )
& ( powersetTE1 = $true )
& ( lamp = $true )
& ( dpsetconstrEL1 = $true )
& ( setukpairinjR2 = $true )
& ( setext = $true )
& ( beta1 = $true )
& ( demorgan2a2 = $true )
& ( exuE1 = $true )
& ( powersetTI1 = $true )
& ( omegaSAx = $true )
& ( setunionsingleton2 = $true )
& ( omegaIndAx = $true )
& ( sepInPowerset = $true )
& ( kpairp = $true )
& ( dpsetconstrI = $true )
& ( setukpairinjR1 = $true )
& ( funcGraphProp2 = $true )
& ( eqimpsubset1 = $true )
& ( setminusILneg = $true )
& ( eqinunit = $true )
& ( ifp = $true )
& ( ksndpairEq = $true )
& ( setadjoinIL = $true )
& ( ap2apEq1 = $true )
& ( lam2p = $true )
& ( kfstsingleton = $true )
& ( subset2powerset = $true )
& ( binintersectSubset2 = $true )
& ( singletoninpowerset = $true )
& ( subsetRefl = $true )
& ( subbreln = $true )
& ( setunionsingleton = $true )
& ( setminusER = $true )
& ( subsetTI = $true )
& ( dsetconstrI = $true )
& ( upairset2IR = $true )
& ( iftrue = $true )
& ( theprop = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( eta2 = $true )
& ( ubforcartprodlem1 = $true )
& ( setminusI = $true )
& ( singletonsuniq = $true )
& ( upairsetE = $true )
& ( dpsetconstrSub = $true )
& ( binintersectER = $true )
& ( setminusIRneg = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinSub = $true )
& ( exuI3 = $true )
& ( setminusELneg = $true )
& ( complementTcontraSubset = $true )
& ( app = $true )
& ( inIntersectImpInUnion2 = $true )
& ( nonemptyI1 = $true )
& ( eta1 = $true )
& ( doubleComplementSub2 = $true )
& ( dpsetconstrER = $true )
& ( iftrueProp1 = $true )
& ( binunionTEcontra = $true )
& ( setextAx = $true )
& ( emptysetimpfalse = $true )
& ( singletoninpowunion = $true )
& ( funcGraphProp3 = $true )
& ( setunionsingleton1 = $true )
& ( replAx = $true )
& ( singletonprop = $true )
& ( setukpairinjL1 = $true )
& ( setukpairIL = $true )
& ( powerset__Cong = $true )
& ( setextsub = $true )
& ( inPowerset = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodsndpairEq = $true )
& ( sepSubset = $true )
& ( upairequniteq = $true )
& ( exuE3e = $true )
& ( powersetI1 = $true )
& ( disjointsetsI1 = $true )
& ( subsetI2 = $true )
& ( iftrueProp2 = $true )
& ( brelnall1 = $true )
& ( binintersectEL = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( cartprodmempaircEq = $true )
& ( nonemptyImpWitness = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( complementTnotintersectT = $true )
& ( brelnall2 = $true )
& ( quantDeMorgan3 = $true )
& ( funcGraphProp1 = $true )
& ( binunionTIRcontra = $true )
& ( setoftrueEq = $true )
& ( complementTE1 = $true )
& ( symdiffE = $true )
& ( exuI1 = $true )
& ( notdexE = $true )
& ( singletonsswitch = $true )
& ( dsetconstrEL = $true )
& ( ifSingleton = $true )
& ( subsetE2 = $true )
& ( setminusSubset2 = $true )
& ( secondinupair = $true )
& ( setunionAx = $true )
& ( contrasubsetT2 = $true )
& ( nonemptyE1 = $true )
& ( lam2lamEq = $true )
& ( funcextLem = $true )
& ( powersetE = $true )
& ( emptysetAx = $true )
& ( theeq = $true )
& ( emptyinunitempty = $true )
& ( exuE3u = $true )
& ( emptyI = $true )
& ( inCongP = $true )
& ( nonemptyI = $true )
& ( cartprodsndin = $true )
& ( emptyE1 = $true )
& ( binunionIL = $true )
& ( setunionI = $true )
& ( descrp = $true )
& ( notinemptyset = $true )
& ( setadjoin__Cong = $true )
& ( notinsingleton = $true )
& ( binunionT_lem = $true )
& ( wellorderingAx = $true )
& ( notsubsetI = $true )
& ( ex1I = $true )
& ( binintersectLsub = $true )
& ( cartprodpairmemER = $true )
& ( exu__Cong = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( powersetsubset = $true )
& ( cartprodpairsurjEq = $true )
& ( notequalI2 = $true )
& ( kpairsurjEq = $true )
& ( doubleComplementEq = $true )
& ( setminusERneg = $true )
& ( binintersectI = $true )
& ( powersetI = $true )
& ( upairinpowunion = $true )
& ( funcImageSingleton = $true )
& ( symdiffIneg2 = $true )
& ( $true
= ( in @ sK293 @ sK290 ) )
& ( $true
= ( in @ sK293 @ ( binunion @ ( setminus @ sK290 @ sK291 ) @ ( setminus @ sK290 @ sK292 ) ) ) )
& ( ( in @ sK293 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ sK292 ) ) )
!= $true )
& ( ( in @ sK292 @ ( powerset @ sK290 ) )
= $true )
& ( $true
= ( in @ sK291 @ ( powerset @ sK290 ) ) )
& ( setadjoinAx = $true )
& ( descr__Cong = $true )
& ( emptysetsubset = $true )
& ( ap2apEq2 = $true )
& ( binintersectSubset5 = $true )
& ( subsetE = $true )
& ( upairset2E = $true )
& ( quantDeMorgan4 = $true )
& ( subsetI1 = $true )
& ( funcinfuncset = $true )
& ( subsetTrans = $true )
& ( emptyinPowerset = $true )
& ( ex1I2 = $true )
& ( bs114d = $true )
& ( ubforcartprodlem2 = $true )
& ( kpairiskpair = $true )
& ( exuE2 = $true )
& ( demorgan1a = $true )
& ( uniqinunit = $true )
& ( in__Cong = $true )
& ( exuEu = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setukpairIR = $true )
& ( powersetT_lem = $true )
& ( setukpairinjL2 = $true )
& ( dsetconstrER = $true )
& ( funcext = $true )
& ( contraSubsetComplement = $true )
& ( setunion__Cong = $true )
& ( contrasubsetT3 = $true )
& ( setminusT_lem = $true )
& ( cartprodfstpairEq = $true )
& ( dsetconstr__Cong = $true )
& ( binunionIR = $true )
& ( eqimpsubset2 = $true )
& ( cartprodfstin = $true )
& ( binintersectTELcontra = $true )
& ( cartprodmempair1 = $true )
& ( setukpairinjR12 = $true )
& ( binintersectRsub = $true )
& ( setadjoinIR = $true )
& ( cartprodpairin = $true )
& ( demorgan2a1 = $true )
& ( emptyInPowerset = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK290,sK291,sK292,sK293])],[f1021,f1634,f1633,f1632]) ).
thf(f1632,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( $true
= ( in @ X3 @ ( binunion @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ X2 ) ) ) )
& ( ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
!= $true ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
=> ( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK290 ) )
& ( ( in @ X3 @ ( binunion @ ( setminus @ sK290 @ sK291 ) @ ( setminus @ sK290 @ X2 ) ) )
= $true )
& ( $true
!= ( in @ X3 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ X2 ) ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK290 ) ) ) )
& ( $true
= ( in @ sK291 @ ( powerset @ sK290 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1633,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK290 ) )
& ( ( in @ X3 @ ( binunion @ ( setminus @ sK290 @ sK291 ) @ ( setminus @ sK290 @ X2 ) ) )
= $true )
& ( $true
!= ( in @ X3 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ X2 ) ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK290 ) ) ) )
=> ( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK290 ) )
& ( ( in @ X3 @ ( binunion @ ( setminus @ sK290 @ sK291 ) @ ( setminus @ sK290 @ sK292 ) ) )
= $true )
& ( ( in @ X3 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ sK292 ) ) )
!= $true ) )
& ( ( in @ sK292 @ ( powerset @ sK290 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1634,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK290 ) )
& ( ( in @ X3 @ ( binunion @ ( setminus @ sK290 @ sK291 ) @ ( setminus @ sK290 @ sK292 ) ) )
= $true )
& ( ( in @ X3 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ sK292 ) ) )
!= $true ) )
=> ( ( $true
= ( in @ sK293 @ sK290 ) )
& ( $true
= ( in @ sK293 @ ( binunion @ ( setminus @ sK290 @ sK291 ) @ ( setminus @ sK290 @ sK292 ) ) ) )
& ( ( in @ sK293 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ sK292 ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1021,plain,
( ( dpsetconstrERa = $true )
& ( vacuousDall = $true )
& ( inIntersectImpInUnion = $true )
& ( complementUnionInPowersetComplement = $true )
& ( binunionTE = $true )
& ( notdallE = $true )
& ( upairsetIL = $true )
& ( infuncsetfunc = $true )
& ( prop2setE = $true )
& ( apProp = $true )
& ( subsetemptysetimpeq = $true )
& ( setadjoinE = $true )
& ( setukpairinjR = $true )
& ( prop2set2propI = $true )
& ( notequalI1 = $true )
& ( omega__Cong = $true )
& ( binintersectSubset3 = $true )
& ( ex1E1 = $true )
& ( cartprodmempair = $true )
& ( complementSubsetComplementIntersect = $true )
& ( kfstpairEq = $true )
& ( setadjoinOr = $true )
& ( binunionRsub = $true )
& ( emptysetE = $true )
& ( setminusEL = $true )
& ( funcGraphProp4 = $true )
& ( powersetAx = $true )
& ( eqbreln = $true )
& ( doubleComplementE1 = $true )
& ( binintersectSubset4 = $true )
& ( foundationAx = $true )
& ( beta2 = $true )
& ( complementImpComplementIntersect = $true )
& ( iffalseProp1 = $true )
& ( contrasubsetT1 = $true )
& ( binunionEcases = $true )
& ( noeltsimpempty = $true )
& ( symdiffI2 = $true )
& ( upairsetIR = $true )
& ( cartprodpairmemEL = $true )
& ( setbeta = $true )
& ( subPowSU = $true )
& ( ex1E2 = $true )
& ( powersetE1 = $true )
& ( contrasubsetT = $true )
& ( symdiffIneg1 = $true )
& ( setukpairinjR11 = $true )
& ( binunionLsub = $true )
& ( doubleComplementSub1 = $true )
& ( setextT = $true )
& ( iffalseProp2 = $true )
& ( ksndsingleton = $true )
& ( upairsubunion = $true )
& ( binunionTILcontra = $true )
& ( setadjoinSub2 = $true )
& ( prop2setI = $true )
& ( iftrueorfalse = $true )
& ( singletonsubset = $true )
& ( omega0Ax = $true )
& ( setminusLsub = $true )
& ( symdiffI1 = $true )
& ( quantDeMorgan1 = $true )
& ( iffalse = $true )
& ( setminusSubset1 = $true )
& ( lamProp = $true )
& ( quantDeMorgan2 = $true )
& ( binintersectT_lem = $true )
& ( complementT_lem = $true )
& ( ap2p = $true )
& ( binunionE = $true )
& ( complementTI1 = $true )
& ( exuI2 = $true )
& ( binintersectTERcontra = $true )
& ( emptyset__Cong = $true )
& ( setunionE = $true )
& ( binintersectSubset1 = $true )
& ( setOfPairsIsBReln = $true )
& ( setunionE2 = $true )
& ( setukpairinjL = $true )
& ( doubleComplementI1 = $true )
& ( funcext2 = $true )
& ( powersetTE1 = $true )
& ( lamp = $true )
& ( dpsetconstrEL1 = $true )
& ( setukpairinjR2 = $true )
& ( setext = $true )
& ( beta1 = $true )
& ( demorgan2a2 = $true )
& ( exuE1 = $true )
& ( powersetTI1 = $true )
& ( omegaSAx = $true )
& ( setunionsingleton2 = $true )
& ( omegaIndAx = $true )
& ( sepInPowerset = $true )
& ( kpairp = $true )
& ( dpsetconstrI = $true )
& ( setukpairinjR1 = $true )
& ( funcGraphProp2 = $true )
& ( eqimpsubset1 = $true )
& ( setminusILneg = $true )
& ( eqinunit = $true )
& ( ifp = $true )
& ( ksndpairEq = $true )
& ( setadjoinIL = $true )
& ( ap2apEq1 = $true )
& ( lam2p = $true )
& ( kfstsingleton = $true )
& ( subset2powerset = $true )
& ( binintersectSubset2 = $true )
& ( singletoninpowerset = $true )
& ( subsetRefl = $true )
& ( subbreln = $true )
& ( setunionsingleton = $true )
& ( setminusER = $true )
& ( subsetTI = $true )
& ( dsetconstrI = $true )
& ( upairset2IR = $true )
& ( iftrue = $true )
& ( theprop = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( eta2 = $true )
& ( ubforcartprodlem1 = $true )
& ( setminusI = $true )
& ( singletonsuniq = $true )
& ( upairsetE = $true )
& ( dpsetconstrSub = $true )
& ( binintersectER = $true )
& ( setminusIRneg = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinSub = $true )
& ( exuI3 = $true )
& ( setminusELneg = $true )
& ( complementTcontraSubset = $true )
& ( app = $true )
& ( inIntersectImpInUnion2 = $true )
& ( nonemptyI1 = $true )
& ( eta1 = $true )
& ( doubleComplementSub2 = $true )
& ( dpsetconstrER = $true )
& ( iftrueProp1 = $true )
& ( binunionTEcontra = $true )
& ( setextAx = $true )
& ( emptysetimpfalse = $true )
& ( singletoninpowunion = $true )
& ( funcGraphProp3 = $true )
& ( setunionsingleton1 = $true )
& ( replAx = $true )
& ( singletonprop = $true )
& ( setukpairinjL1 = $true )
& ( setukpairIL = $true )
& ( powerset__Cong = $true )
& ( setextsub = $true )
& ( inPowerset = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodsndpairEq = $true )
& ( sepSubset = $true )
& ( upairequniteq = $true )
& ( exuE3e = $true )
& ( powersetI1 = $true )
& ( disjointsetsI1 = $true )
& ( subsetI2 = $true )
& ( iftrueProp2 = $true )
& ( brelnall1 = $true )
& ( binintersectEL = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( cartprodmempaircEq = $true )
& ( nonemptyImpWitness = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( complementTnotintersectT = $true )
& ( brelnall2 = $true )
& ( quantDeMorgan3 = $true )
& ( funcGraphProp1 = $true )
& ( binunionTIRcontra = $true )
& ( setoftrueEq = $true )
& ( complementTE1 = $true )
& ( symdiffE = $true )
& ( exuI1 = $true )
& ( notdexE = $true )
& ( singletonsswitch = $true )
& ( dsetconstrEL = $true )
& ( ifSingleton = $true )
& ( subsetE2 = $true )
& ( setminusSubset2 = $true )
& ( secondinupair = $true )
& ( setunionAx = $true )
& ( contrasubsetT2 = $true )
& ( nonemptyE1 = $true )
& ( lam2lamEq = $true )
& ( funcextLem = $true )
& ( powersetE = $true )
& ( emptysetAx = $true )
& ( theeq = $true )
& ( emptyinunitempty = $true )
& ( exuE3u = $true )
& ( emptyI = $true )
& ( inCongP = $true )
& ( nonemptyI = $true )
& ( cartprodsndin = $true )
& ( emptyE1 = $true )
& ( binunionIL = $true )
& ( setunionI = $true )
& ( descrp = $true )
& ( notinemptyset = $true )
& ( setadjoin__Cong = $true )
& ( notinsingleton = $true )
& ( binunionT_lem = $true )
& ( wellorderingAx = $true )
& ( notsubsetI = $true )
& ( ex1I = $true )
& ( binintersectLsub = $true )
& ( cartprodpairmemER = $true )
& ( exu__Cong = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( powersetsubset = $true )
& ( cartprodpairsurjEq = $true )
& ( notequalI2 = $true )
& ( kpairsurjEq = $true )
& ( doubleComplementEq = $true )
& ( setminusERneg = $true )
& ( binintersectI = $true )
& ( powersetI = $true )
& ( upairinpowunion = $true )
& ( funcImageSingleton = $true )
& ( symdiffIneg2 = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( $true
= ( in @ X3 @ ( binunion @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ X2 ) ) ) )
& ( ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
!= $true ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
& ( setadjoinAx = $true )
& ( descr__Cong = $true )
& ( emptysetsubset = $true )
& ( ap2apEq2 = $true )
& ( binintersectSubset5 = $true )
& ( subsetE = $true )
& ( upairset2E = $true )
& ( quantDeMorgan4 = $true )
& ( subsetI1 = $true )
& ( funcinfuncset = $true )
& ( subsetTrans = $true )
& ( emptyinPowerset = $true )
& ( ex1I2 = $true )
& ( bs114d = $true )
& ( ubforcartprodlem2 = $true )
& ( kpairiskpair = $true )
& ( exuE2 = $true )
& ( demorgan1a = $true )
& ( uniqinunit = $true )
& ( in__Cong = $true )
& ( exuEu = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setukpairIR = $true )
& ( powersetT_lem = $true )
& ( setukpairinjL2 = $true )
& ( dsetconstrER = $true )
& ( funcext = $true )
& ( contraSubsetComplement = $true )
& ( setunion__Cong = $true )
& ( contrasubsetT3 = $true )
& ( setminusT_lem = $true )
& ( cartprodfstpairEq = $true )
& ( dsetconstr__Cong = $true )
& ( binunionIR = $true )
& ( eqimpsubset2 = $true )
& ( cartprodfstin = $true )
& ( binintersectTELcontra = $true )
& ( cartprodmempair1 = $true )
& ( setukpairinjR12 = $true )
& ( binintersectRsub = $true )
& ( setadjoinIR = $true )
& ( cartprodpairin = $true )
& ( demorgan2a1 = $true )
& ( emptyInPowerset = $true ) ),
inference(flattening,[status(thm)],[f1020]) ).
thf(f1020,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
!= $true )
& ( $true
= ( in @ X3 @ ( binunion @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ X2 ) ) ) )
& ( $true
= ( in @ X3 @ X0 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( 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)],[f459]) ).
thf(f459,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 )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
=> ( ( $true
= ( in @ X3 @ ( binunion @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ X2 ) ) ) )
=> ( ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f458]) ).
thf(f458,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
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ( in @ X3 @ ( binunion @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ X2 ) ) )
=> ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f274]) ).
thf(f274,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
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ ( binunion @ ( setminus @ X3 @ X11 ) @ ( setminus @ X3 @ X16 ) ) )
=> ( in @ X1 @ ( setminus @ X3 @ ( binintersect @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f273]) ).
thf(f273,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
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ ( binunion @ ( setminus @ X3 @ X11 ) @ ( setminus @ X3 @ X16 ) ) )
=> ( in @ X1 @ ( setminus @ X3 @ ( binintersect @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',demorgan1b) ).
thf(f4478,plain,
( ( setminusER != $true )
| ( $true
!= ( in @ sK293 @ sK292 ) )
| ~ spl866_7 ),
inference(trivial_inequality_removal,[status(thm)],[f4466]) ).
thf(f4466,plain,
( ( $true != $true )
| ( setminusER != $true )
| ( $true
!= ( in @ sK293 @ sK292 ) )
| ~ spl866_7 ),
inference(superposition,[status(thm)],[f3409,f4239]) ).
thf(f4239,plain,
( ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK292 ) ) )
| ~ spl866_7 ),
inference(avatar_component_clause,[status(thm)],[f4238]) ).
thf(f3409,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( setminus @ X3 @ X5 ) ) )
| ( $true
!= ( in @ X4 @ X5 ) )
| ( setminusER != $true ) ),
inference(cnf_transformation,[status(thm)],[f1900]) ).
thf(f1900,plain,
( ( ( setminusER = $true )
| ( ( $true
= ( in @ sK467 @ ( setminus @ sK466 @ sK468 ) ) )
& ( $true
= ( in @ sK467 @ sK468 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( setminus @ X3 @ X5 ) ) )
| ( $true
!= ( in @ X4 @ X5 ) ) )
| ( setminusER != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK466,sK467,sK468])],[f1898,f1899]) ).
thf(f1899,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( setminus @ X0 @ X2 ) ) )
& ( $true
= ( in @ X1 @ X2 ) ) )
=> ( ( $true
= ( in @ sK467 @ ( setminus @ sK466 @ sK468 ) ) )
& ( $true
= ( in @ sK467 @ sK468 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1898,plain,
( ( ( setminusER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( setminus @ X0 @ X2 ) ) )
& ( $true
= ( in @ X1 @ X2 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( setminus @ X3 @ X5 ) ) )
| ( $true
!= ( in @ X4 @ X5 ) ) )
| ( setminusER != $true ) ) ),
inference(rectify,[status(thm)],[f1897]) ).
thf(f1897,plain,
( ( ( setminusER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( setminus @ X0 @ X2 ) ) )
& ( $true
= ( in @ X1 @ X2 ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ ( setminus @ X0 @ X2 ) ) )
| ( $true
!= ( in @ X1 @ X2 ) ) )
| ( setminusER != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1024]) ).
thf(f1024,plain,
( ( setminusER = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ ( setminus @ X0 @ X2 ) ) )
| ( $true
!= ( in @ X1 @ X2 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f857]) ).
thf(f857,plain,
( ( setminusER = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( setminus @ X0 @ X2 ) ) )
=> ( $true
!= ( in @ X1 @ X2 ) ) ) ),
inference(flattening,[status(thm)],[f650]) ).
thf(f650,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( setminus @ X0 @ X2 ) ) )
=> ( $true
!= ( in @ X1 @ X2 ) ) )
<=> ( setminusER = $true ) ),
inference(fool_elimination,[status(thm)],[f649]) ).
thf(f649,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X0 @ X2 ) )
=> ~ ( in @ X1 @ X2 ) )
= setminusER ),
inference(rectify,[status(thm)],[f122]) ).
thf(f122,axiom,
! [X3: $i,X1: $i,X4: $i] :
( ( ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ~ ( in @ X1 @ X4 ) )
= setminusER ),
file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',setminusER) ).
thf(f5395,plain,
~ spl866_9,
inference(avatar_split_clause,[status(thm)],[f5394,f4249]) ).
thf(f4249,definition,
( spl866_9
<=> ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl866_9])],[avatar_definition]) ).
thf(f5394,plain,
( $true
!= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f5393]) ).
thf(f5393,plain,
( ( $true
!= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f5392,f2936]) ).
thf(f2936,plain,
( $true
= ( in @ sK291 @ ( powerset @ sK290 ) ) ),
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f5392,plain,
( ( $true
!= ( in @ sK291 @ ( powerset @ sK290 ) ) )
| ( $true
!= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f5391]) ).
thf(f5391,plain,
( ( $true
!= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) )
| ( $true
!= ( in @ sK291 @ ( powerset @ sK290 ) ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f5390,f2937]) ).
thf(f2937,plain,
( ( in @ sK292 @ ( powerset @ sK290 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f5390,plain,
( ( ( in @ sK292 @ ( powerset @ sK290 ) )
!= $true )
| ( $true
!= ( in @ sK291 @ ( powerset @ sK290 ) ) )
| ( $true
!= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f5389]) ).
thf(f5389,plain,
( ( ( in @ sK292 @ ( powerset @ sK290 ) )
!= $true )
| ( $true
!= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) )
| ( $true
!= ( in @ sK291 @ ( powerset @ sK290 ) ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f5354,f3145]) ).
thf(f3145,plain,
complementSubsetComplementIntersect = $true,
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f5354,plain,
( ( complementSubsetComplementIntersect != $true )
| ( $true
!= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) )
| ( $true
!= ( in @ sK291 @ ( powerset @ sK290 ) ) )
| ( ( in @ sK292 @ ( powerset @ sK290 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f5330]) ).
thf(f5330,plain,
( ( ( in @ sK292 @ ( powerset @ sK290 ) )
!= $true )
| ( $true
!= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) )
| ( complementSubsetComplementIntersect != $true )
| ( $true
!= ( in @ sK291 @ ( powerset @ sK290 ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f4304,f2739]) ).
thf(f2739,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ ( setminus @ X4 @ X3 ) @ ( setminus @ X4 @ ( binintersect @ X3 @ X5 ) ) )
= $true )
| ( complementSubsetComplementIntersect != $true )
| ( ( in @ X3 @ ( powerset @ X4 ) )
!= $true )
| ( $true
!= ( in @ X5 @ ( powerset @ X4 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1467]) ).
thf(f1467,plain,
( ( ( complementSubsetComplementIntersect = $true )
| ( ( $true
!= ( subset @ ( setminus @ sK183 @ sK182 ) @ ( setminus @ sK183 @ ( binintersect @ sK182 @ sK184 ) ) ) )
& ( $true
= ( in @ sK184 @ ( powerset @ sK183 ) ) )
& ( $true
= ( in @ sK182 @ ( powerset @ sK183 ) ) ) ) )
& ( ! [X3: $i,X4: $i] :
( ! [X5: $i] :
( ( ( subset @ ( setminus @ X4 @ X3 ) @ ( setminus @ X4 @ ( binintersect @ X3 @ X5 ) ) )
= $true )
| ( $true
!= ( in @ X5 @ ( powerset @ X4 ) ) ) )
| ( ( in @ X3 @ ( powerset @ X4 ) )
!= $true ) )
| ( complementSubsetComplementIntersect != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK182,sK183,sK184])],[f1464,f1466,f1465]) ).
thf(f1465,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( subset @ ( setminus @ X1 @ X0 ) @ ( setminus @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
=> ( ? [X2: $i] :
( ( $true
!= ( subset @ ( setminus @ sK183 @ sK182 ) @ ( setminus @ sK183 @ ( binintersect @ sK182 @ X2 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK183 ) ) ) )
& ( $true
= ( in @ sK182 @ ( powerset @ sK183 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1466,plain,
( ? [X2: $i] :
( ( $true
!= ( subset @ ( setminus @ sK183 @ sK182 ) @ ( setminus @ sK183 @ ( binintersect @ sK182 @ X2 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK183 ) ) ) )
=> ( ( $true
!= ( subset @ ( setminus @ sK183 @ sK182 ) @ ( setminus @ sK183 @ ( binintersect @ sK182 @ sK184 ) ) ) )
& ( $true
= ( in @ sK184 @ ( powerset @ sK183 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1464,plain,
( ( ( complementSubsetComplementIntersect = $true )
| ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( subset @ ( setminus @ X1 @ X0 ) @ ( setminus @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) ) )
& ( ! [X3: $i,X4: $i] :
( ! [X5: $i] :
( ( ( subset @ ( setminus @ X4 @ X3 ) @ ( setminus @ X4 @ ( binintersect @ X3 @ X5 ) ) )
= $true )
| ( $true
!= ( in @ X5 @ ( powerset @ X4 ) ) ) )
| ( ( in @ X3 @ ( powerset @ X4 ) )
!= $true ) )
| ( complementSubsetComplementIntersect != $true ) ) ),
inference(rectify,[status(thm)],[f1463]) ).
thf(f1463,plain,
( ( ( complementSubsetComplementIntersect = $true )
| ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( subset @ ( setminus @ X1 @ X0 ) @ ( setminus @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( subset @ ( setminus @ X1 @ X0 ) @ ( setminus @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
| ( $true
!= ( in @ X2 @ ( powerset @ X1 ) ) ) )
| ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) ) )
| ( complementSubsetComplementIntersect != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1025]) ).
thf(f1025,plain,
( ( complementSubsetComplementIntersect = $true )
<=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( subset @ ( setminus @ X1 @ X0 ) @ ( setminus @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
| ( $true
!= ( in @ X2 @ ( powerset @ X1 ) ) ) )
| ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f323]) ).
thf(f323,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( $true
= ( subset @ ( setminus @ X1 @ X0 ) @ ( setminus @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) ) )
<=> ( complementSubsetComplementIntersect = $true ) ),
inference(fool_elimination,[status(thm)],[f322]) ).
thf(f322,plain,
( complementSubsetComplementIntersect
= ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ ( setminus @ X1 @ X0 ) @ ( setminus @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f255]) ).
thf(f255,axiom,
( complementSubsetComplementIntersect
= ( ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( subset @ ( setminus @ X3 @ X11 ) @ ( setminus @ X3 @ ( binintersect @ X11 @ X16 ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',complementSubsetComplementIntersect) ).
thf(f4304,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ sK292 ) ) ) )
| ( ( in @ sK293 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4303]) ).
thf(f4303,plain,
! [X0: $i] :
( ( $true != $true )
| ( $true
!= ( subset @ X0 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ sK292 ) ) ) )
| ( ( in @ sK293 @ X0 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f4223,f2983]) ).
thf(f2983,plain,
subsetE2 = $true,
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f4223,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ sK292 ) ) ) )
| ( ( in @ sK293 @ X0 )
!= $true )
| ( subsetE2 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4220]) ).
thf(f4220,plain,
! [X0: $i] :
( ( subsetE2 != $true )
| ( $true != $true )
| ( $true
!= ( subset @ X0 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ sK292 ) ) ) )
| ( ( in @ sK293 @ X0 )
!= $true ) ),
inference(superposition,[status(thm)],[f2938,f2680]) ).
thf(f2680,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
| ( ( subset @ X2 @ X1 )
!= $true )
| ( ( in @ X0 @ X2 )
!= $true )
| ( subsetE2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1397]) ).
thf(f1397,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( in @ X0 @ X2 )
!= $true ) )
| ( subsetE2 != $true ) )
& ( ( subsetE2 = $true )
| ( ( $true
= ( subset @ sK136 @ sK135 ) )
& ( $true
!= ( in @ sK134 @ sK135 ) )
& ( $true
= ( in @ sK134 @ sK136 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK134,sK135,sK136])],[f1395,f1396]) ).
thf(f1396,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X4 )
= $true )
& ( $true
!= ( in @ X3 @ X4 ) )
& ( $true
= ( in @ X3 @ X5 ) ) )
=> ( ( $true
= ( subset @ sK136 @ sK135 ) )
& ( $true
!= ( in @ sK134 @ sK135 ) )
& ( $true
= ( in @ sK134 @ sK136 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1395,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( in @ X0 @ X2 )
!= $true ) )
| ( subsetE2 != $true ) )
& ( ( subsetE2 = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X4 )
= $true )
& ( $true
!= ( in @ X3 @ X4 ) )
& ( $true
= ( in @ X3 @ X5 ) ) ) ) ),
inference(rectify,[status(thm)],[f1394]) ).
thf(f1394,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( in @ X0 @ X2 )
!= $true ) )
| ( subsetE2 != $true ) )
& ( ( subsetE2 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X1 )
= $true )
& ( $true
!= ( in @ X0 @ X1 ) )
& ( ( in @ X0 @ X2 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1045]) ).
thf(f1045,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( in @ X0 @ X2 )
!= $true ) )
<=> ( subsetE2 = $true ) ),
inference(flattening,[status(thm)],[f1044]) ).
thf(f1044,plain,
( ( subsetE2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( subset @ X2 @ X1 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f839]) ).
thf(f839,plain,
( ( subsetE2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X1 )
= $true )
=> ( ( $true
!= ( in @ X0 @ X1 ) )
=> ( ( in @ X0 @ X2 )
!= $true ) ) ) ),
inference(flattening,[status(thm)],[f493]) ).
thf(f493,plain,
( ( subsetE2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X1 )
= $true )
=> ( ( $true
!= ( in @ X0 @ X1 ) )
=> ( ( in @ X0 @ X2 )
!= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f492]) ).
thf(f492,plain,
( subsetE2
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ~ ( in @ X0 @ X1 )
=> ~ ( in @ X0 @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f84]) ).
thf(f84,axiom,
( subsetE2
= ( ! [X1: $i,X4: $i,X3: $i] :
( ( subset @ X3 @ X4 )
=> ( ~ ( in @ X1 @ X4 )
=> ~ ( in @ X1 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',subsetE2) ).
thf(f2938,plain,
( ( in @ sK293 @ ( setminus @ sK290 @ ( binintersect @ sK291 @ sK292 ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f4607,plain,
( spl866_29
| ~ spl866_10 ),
inference(avatar_split_clause,[status(thm)],[f4606,f4277,f4511]) ).
thf(f4277,definition,
( spl866_10
<=> ( ( in @ sK293 @ ( binintersect @ sK291 @ sK292 ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl866_10])],[avatar_definition]) ).
thf(f4606,plain,
( ( $true
= ( in @ sK293 @ sK292 ) )
| ~ spl866_10 ),
inference(trivial_inequality_removal,[status(thm)],[f4605]) ).
thf(f4605,plain,
( ( $true
= ( in @ sK293 @ sK292 ) )
| ( $true != $true )
| ~ spl866_10 ),
inference(forward_demodulation,[status(thm)],[f4597,f3038]) ).
thf(f3038,plain,
binintersectER = $true,
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f4597,plain,
( ( $true
= ( in @ sK293 @ sK292 ) )
| ( binintersectER != $true )
| ~ spl866_10 ),
inference(trivial_inequality_removal,[status(thm)],[f4588]) ).
thf(f4588,plain,
( ( binintersectER != $true )
| ( $true != $true )
| ( $true
= ( in @ sK293 @ sK292 ) )
| ~ spl866_10 ),
inference(superposition,[status(thm)],[f3459,f4279]) ).
thf(f4279,plain,
( ( ( in @ sK293 @ ( binintersect @ sK291 @ sK292 ) )
= $true )
| ~ spl866_10 ),
inference(avatar_component_clause,[status(thm)],[f4277]) ).
thf(f3459,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ ( binintersect @ X3 @ X5 ) )
!= $true )
| ( binintersectER != $true )
| ( $true
= ( in @ X4 @ X5 ) ) ),
inference(cnf_transformation,[status(thm)],[f1966]) ).
thf(f1966,plain,
( ( ( binintersectER = $true )
| ( ( $true
!= ( in @ sK504 @ sK505 ) )
& ( $true
= ( in @ sK504 @ ( binintersect @ sK503 @ sK505 ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X4 @ X5 ) )
| ( ( in @ X4 @ ( binintersect @ X3 @ X5 ) )
!= $true ) )
| ( binintersectER != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK503,sK504,sK505])],[f1964,f1965]) ).
thf(f1965,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X2 ) )
& ( $true
= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
=> ( ( $true
!= ( in @ sK504 @ sK505 ) )
& ( $true
= ( in @ sK504 @ ( binintersect @ sK503 @ sK505 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1964,plain,
( ( ( binintersectER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X2 ) )
& ( $true
= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X4 @ X5 ) )
| ( ( in @ X4 @ ( binintersect @ X3 @ X5 ) )
!= $true ) )
| ( binintersectER != $true ) ) ),
inference(rectify,[status(thm)],[f1963]) ).
thf(f1963,plain,
( ( ( binintersectER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X2 ) )
& ( $true
= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X2 ) )
| ( $true
!= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
| ( binintersectER != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1161]) ).
thf(f1161,plain,
( ( binintersectER = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X2 ) )
| ( $true
!= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f519]) ).
thf(f519,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( binintersect @ X0 @ X2 ) ) )
=> ( $true
= ( in @ X1 @ X2 ) ) )
<=> ( binintersectER = $true ) ),
inference(fool_elimination,[status(thm)],[f518]) ).
thf(f518,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binintersect @ X0 @ X2 ) )
=> ( in @ X1 @ X2 ) )
= binintersectER ),
inference(rectify,[status(thm)],[f114]) ).
thf(f114,axiom,
! [X3: $i,X1: $i,X4: $i] :
( ( ( in @ X1 @ ( binintersect @ X3 @ X4 ) )
=> ( in @ X1 @ X4 ) )
= binintersectER ),
file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',binintersectER) ).
thf(f4584,plain,
spl866_10,
inference(avatar_split_clause,[status(thm)],[f4269,f4277]) ).
thf(f4269,plain,
( ( in @ sK293 @ ( binintersect @ sK291 @ sK292 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f4268]) ).
thf(f4268,plain,
( ( $true != $true )
| ( ( in @ sK293 @ ( binintersect @ sK291 @ sK292 ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f4267,f3042]) ).
thf(f3042,plain,
setminusI = $true,
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f4267,plain,
( ( setminusI != $true )
| ( ( in @ sK293 @ ( binintersect @ sK291 @ sK292 ) )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4266]) ).
thf(f4266,plain,
( ( setminusI != $true )
| ( ( in @ sK293 @ ( binintersect @ sK291 @ sK292 ) )
= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4228,f2940]) ).
thf(f2940,plain,
( $true
= ( in @ sK293 @ sK290 ) ),
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f4228,plain,
( ( $true
!= ( in @ sK293 @ sK290 ) )
| ( ( in @ sK293 @ ( binintersect @ sK291 @ sK292 ) )
= $true )
| ( setminusI != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4215]) ).
thf(f4215,plain,
( ( $true
!= ( in @ sK293 @ sK290 ) )
| ( setminusI != $true )
| ( $true != $true )
| ( ( in @ sK293 @ ( binintersect @ sK291 @ sK292 ) )
= $true ) ),
inference(superposition,[status(thm)],[f2938,f2659]) ).
thf(f2659,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setminus @ X2 @ X1 ) ) )
| ( setminusI != $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( in @ X0 @ X2 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1373]) ).
thf(f1373,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( setminus @ X2 @ X1 ) ) ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ( ( $true
= ( in @ sK117 @ sK119 ) )
& ( ( in @ sK117 @ sK118 )
!= $true )
& ( ( in @ sK117 @ ( setminus @ sK119 @ sK118 ) )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK117,sK118,sK119])],[f1371,f1372]) ).
thf(f1372,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ X5 ) )
& ( $true
!= ( in @ X3 @ X4 ) )
& ( ( in @ X3 @ ( setminus @ X5 @ X4 ) )
!= $true ) )
=> ( ( $true
= ( in @ sK117 @ sK119 ) )
& ( ( in @ sK117 @ sK118 )
!= $true )
& ( ( in @ sK117 @ ( setminus @ sK119 @ sK118 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1371,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( setminus @ X2 @ X1 ) ) ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ X5 ) )
& ( $true
!= ( in @ X3 @ X4 ) )
& ( ( in @ X3 @ ( setminus @ X5 @ X4 ) )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1370]) ).
thf(f1370,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( setminus @ X2 @ X1 ) ) ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
& ( $true
!= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ ( setminus @ X2 @ X1 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1124]) ).
thf(f1124,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( setminus @ X2 @ X1 ) ) ) )
<=> ( setminusI = $true ) ),
inference(flattening,[status(thm)],[f1123]) ).
thf(f1123,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( setminus @ X2 @ X1 ) ) )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( in @ X0 @ X2 )
!= $true ) )
<=> ( setminusI = $true ) ),
inference(ennf_transformation,[status(thm)],[f868]) ).
thf(f868,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
=> ( ( $true
!= ( in @ X0 @ X1 ) )
=> ( $true
= ( in @ X0 @ ( setminus @ X2 @ X1 ) ) ) ) )
<=> ( setminusI = $true ) ),
inference(flattening,[status(thm)],[f815]) ).
thf(f815,plain,
( ( setminusI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
=> ( ( $true
!= ( in @ X0 @ X1 ) )
=> ( $true
= ( in @ X0 @ ( setminus @ X2 @ X1 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f814]) ).
thf(f814,plain,
( setminusI
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ( ~ ( in @ X0 @ X1 )
=> ( in @ X0 @ ( setminus @ X2 @ X1 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f120]) ).
thf(f120,axiom,
( setminusI
= ( ! [X1: $i,X4: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ X4 )
=> ( in @ X1 @ ( setminus @ X3 @ X4 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',setminusI) ).
thf(f4365,plain,
( spl866_9
| spl866_7 ),
inference(avatar_split_clause,[status(thm)],[f4364,f4238,f4249]) ).
thf(f4364,plain,
( ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK292 ) ) )
| ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4363]) ).
thf(f4363,plain,
( ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK292 ) ) )
| ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4323,f3095]) ).
thf(f3095,plain,
binunionE = $true,
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f4323,plain,
( ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) )
| ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK292 ) ) )
| ( binunionE != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4316]) ).
thf(f4316,plain,
( ( binunionE != $true )
| ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK292 ) ) )
| ( $true != $true )
| ( $true
= ( in @ sK293 @ ( setminus @ sK290 @ sK291 ) ) ) ),
inference(superposition,[status(thm)],[f3257,f2939]) ).
thf(f2939,plain,
( $true
= ( in @ sK293 @ ( binunion @ ( setminus @ sK290 @ sK291 ) @ ( setminus @ sK290 @ sK292 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f3257,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X3 @ ( binunion @ X5 @ X4 ) ) )
| ( $true
= ( in @ X3 @ X4 ) )
| ( $true
= ( in @ X3 @ X5 ) )
| ( binunionE != $true ) ),
inference(cnf_transformation,[status(thm)],[f1739]) ).
thf(f1739,plain,
( ( ( binunionE = $true )
| ( ( $true
!= ( in @ sK367 @ sK368 ) )
& ( ( in @ sK367 @ ( binunion @ sK369 @ sK368 ) )
= $true )
& ( ( in @ sK367 @ sK369 )
!= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ X4 ) )
| ( $true
!= ( in @ X3 @ ( binunion @ X5 @ X4 ) ) )
| ( $true
= ( in @ X3 @ X5 ) ) )
| ( binunionE != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK367,sK368,sK369])],[f1737,f1738]) ).
thf(f1738,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
& ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
& ( ( in @ X0 @ X2 )
!= $true ) )
=> ( ( $true
!= ( in @ sK367 @ sK368 ) )
& ( ( in @ sK367 @ ( binunion @ sK369 @ sK368 ) )
= $true )
& ( ( in @ sK367 @ sK369 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1737,plain,
( ( ( binunionE = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
& ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
& ( ( in @ X0 @ X2 )
!= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ X4 ) )
| ( $true
!= ( in @ X3 @ ( binunion @ X5 @ X4 ) ) )
| ( $true
= ( in @ X3 @ X5 ) ) )
| ( binunionE != $true ) ) ),
inference(rectify,[status(thm)],[f1736]) ).
thf(f1736,plain,
( ( ( binunionE = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
& ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
& ( ( in @ X0 @ X2 )
!= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
| ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
!= $true )
| ( ( in @ X0 @ X2 )
= $true ) )
| ( binunionE != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1118]) ).
thf(f1118,plain,
( ( binunionE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
| ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
!= $true )
| ( ( in @ X0 @ X2 )
= $true ) ) ),
inference(flattening,[status(thm)],[f1117]) ).
thf(f1117,plain,
( ( binunionE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f664]) ).
thf(f664,plain,
( ( binunionE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
= $true )
=> ( ( ( in @ X0 @ X2 )
= $true )
| ( $true
= ( in @ X0 @ X1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f663]) ).
thf(f663,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binunion @ X2 @ X1 ) )
=> ( ( in @ X0 @ X1 )
| ( in @ X0 @ X2 ) ) )
= binunionE ),
inference(rectify,[status(thm)],[f105]) ).
thf(f105,axiom,
! [X1: $i,X4: $i,X3: $i] :
( ( ( in @ X1 @ ( binunion @ X3 @ X4 ) )
=> ( ( in @ X1 @ X4 )
| ( in @ X1 @ X3 ) ) )
= binunionE ),
file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',binunionE) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU752^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.10 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.12/0.30 % Computer : n006.cluster.edu
% 0.12/0.30 % Model : x86_64 x86_64
% 0.12/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.30 % Memory : 8042.1875MB
% 0.12/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.30 % CPULimit : 300
% 0.12/0.30 % WCLimit : 300
% 0.12/0.30 % DateTime : Tue Feb 24 12:32:45 EST 2026
% 0.12/0.30 % CPUTime :
% 0.12/0.30 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.39 ---- Original DTF file ---
% 0.17/0.39 thf(spec,logic,$$dhol).
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 % File : SEU752^1 : TPTP v9.2.1. Released v3.7.0.
% 0.17/0.39 % Domain : Set Theory
% 0.17/0.39 % Problem : Typed Set Theory - Laws for Typed Sets - DeMorgan Laws
% 0.17/0.39 % Version : Especial.
% 0.17/0.39 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.17/0.39 % (! x:i.in x A -> in x (binunion (setminus A X) (setminus A Y)) ->
% 0.17/0.39 % in x (setminus A (binintersect X Y)))))
% 0.17/0.39
% 0.17/0.39 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.17/0.39 % Source : [Bro08]
% 0.17/0.39 % Names : ZFC254g [Bro08]
% 0.17/0.39
% 0.17/0.39 % Status : Theorem
% 0.17/0.39 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% 0.17/0.39 % Syntax : Number of formulae : 579 ( 272 unt; 306 typ; 272 def)
% 0.17/0.39 % Number of atoms : 1605 ( 398 equ; 0 cnn)
% 0.17/0.39 % Maximal formula atoms : 273 ( 5 avg)
% 0.17/0.39 % Number of connectives : 3330 ( 73 ~; 12 |; 44 &;2386 @)
% 0.17/0.39 % ( 14 <=>; 801 =>; 0 <=; 0 <~>)
% 0.17/0.39 % Maximal formula depth : 282 ( 2 avg)
% 0.17/0.39 % Number of types : 2 ( 0 usr)
% 0.17/0.39 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.17/0.39 % Number of symbols : 309 ( 306 usr; 272 con; 0-4 aty)
% 0.17/0.39 % Number of variables : 926 ( 74 ^; 817 !; 35 ?; 926 :)
% 0.17/0.39 % SPC : TH0_THM_EQU_NAR
% 0.17/0.39
% 0.17/0.39 % Comments : http://mathgate.info/detsetitem.php?id=313
% 0.17/0.39 %------------------------------------------------------------------------------
% 0.17/0.39 thf(in_type,type,
% 0.17/0.39 in: $i > $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(exu_type,type,
% 0.17/0.39 exu: ( $i > $o ) > $o ).
% 0.17/0.39
% 0.17/0.39 thf(exu,definition,
% 0.17/0.39 ( exu
% 0.17/0.39 = ( ^ [Xphi: $i > $o] :
% 0.17/0.39 ? [Xx: $i] :
% 0.17/0.39 ( ( Xphi @ Xx )
% 0.17/0.39 & ! [Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xy )
% 0.17/0.39 => ( Xx = Xy ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setextAx_type,type,
% 0.17/0.39 setextAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setextAx,definition,
% 0.17/0.39 ( setextAx
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 <=> ( in @ Xx @ B ) )
% 0.17/0.39 => ( A = B ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyset_type,type,
% 0.17/0.39 emptyset: $i ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetAx_type,type,
% 0.17/0.39 emptysetAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetAx,definition,
% 0.17/0.39 ( emptysetAx
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoin_type,type,
% 0.17/0.39 setadjoin: $i > $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinAx_type,type,
% 0.17/0.39 setadjoinAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinAx,definition,
% 0.17/0.39 ( setadjoinAx
% 0.17/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.39 <=> ( ( Xy = Xx )
% 0.17/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(powerset_type,type,
% 0.17/0.39 powerset: $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(powersetAx_type,type,
% 0.17/0.39 powersetAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(powersetAx,definition,
% 0.17/0.39 ( powersetAx
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.39 <=> ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setunion_type,type,
% 0.17/0.39 setunion: $i > $i ).
% 0.17/0.39
% 0.17/0.39 thf(setunionAx_type,type,
% 0.17/0.39 setunionAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setunionAx,definition,
% 0.17/0.39 ( setunionAx
% 0.17/0.39 = ( ! [A: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.39 <=> ? [B: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 & ( in @ B @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(omega_type,type,
% 0.17/0.39 omega: $i ).
% 0.17/0.39
% 0.17/0.39 thf(omega0Ax_type,type,
% 0.17/0.39 omega0Ax: $o ).
% 0.17/0.39
% 0.17/0.39 thf(omega0Ax,definition,
% 0.17/0.39 ( omega0Ax
% 0.17/0.39 = ( in @ emptyset @ omega ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(omegaSAx_type,type,
% 0.17/0.39 omegaSAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(omegaSAx,definition,
% 0.17/0.39 ( omegaSAx
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ omega )
% 0.17/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(omegaIndAx_type,type,
% 0.17/0.39 omegaIndAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(omegaIndAx,definition,
% 0.17/0.39 ( omegaIndAx
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ( ( in @ emptyset @ A )
% 0.17/0.39 & ! [Xx: $i] :
% 0.17/0.39 ( ( ( in @ Xx @ omega )
% 0.17/0.39 & ( in @ Xx @ A ) )
% 0.17/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ omega )
% 0.17/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(replAx_type,type,
% 0.17/0.39 replAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(replAx,definition,
% 0.17/0.39 ( replAx
% 0.17/0.39 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( exu
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.17/0.39 => ? [B: $i] :
% 0.17/0.39 ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 <=> ? [Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ A )
% 0.17/0.39 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(foundationAx_type,type,
% 0.17/0.39 foundationAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(foundationAx,definition,
% 0.17/0.39 ( foundationAx
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.39 => ? [B: $i] :
% 0.17/0.39 ( ( in @ B @ A )
% 0.17/0.39 & ~ ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(wellorderingAx_type,type,
% 0.17/0.39 wellorderingAx: $o ).
% 0.17/0.39
% 0.17/0.39 thf(wellorderingAx,definition,
% 0.17/0.39 ( wellorderingAx
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ? [B: $i] :
% 0.17/0.39 ( ! [C: $i] :
% 0.17/0.39 ( ( in @ C @ B )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ C )
% 0.17/0.39 => ( in @ Xx @ A ) ) )
% 0.17/0.39 & ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( ( in @ Xx @ A )
% 0.17/0.39 & ( in @ Xy @ A ) )
% 0.17/0.39 => ( ! [C: $i] :
% 0.17/0.39 ( ( in @ C @ B )
% 0.17/0.39 => ( ( in @ Xx @ C )
% 0.17/0.39 <=> ( in @ Xy @ C ) ) )
% 0.17/0.39 => ( Xx = Xy ) ) )
% 0.17/0.39 & ! [C: $i,D: $i] :
% 0.17/0.39 ( ( ( in @ C @ B )
% 0.17/0.39 & ( in @ D @ B ) )
% 0.17/0.39 => ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ C )
% 0.17/0.39 => ( in @ Xx @ D ) )
% 0.17/0.39 | ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ D )
% 0.17/0.39 => ( in @ Xx @ C ) ) ) )
% 0.17/0.39 & ! [C: $i] :
% 0.17/0.39 ( ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ C )
% 0.17/0.39 => ( in @ Xx @ A ) )
% 0.17/0.39 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.17/0.39 => ? [D: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ D @ B )
% 0.17/0.39 & ( in @ Xx @ C )
% 0.17/0.39 & ~ ? [Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ D )
% 0.17/0.39 & ( in @ Xy @ C ) )
% 0.17/0.39 & ! [E: $i] :
% 0.17/0.39 ( ( in @ E @ B )
% 0.17/0.39 => ( ! [Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ E )
% 0.17/0.39 => ( in @ Xy @ D ) )
% 0.17/0.39 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(descr_type,type,
% 0.17/0.39 descr: ( $i > $o ) > $i ).
% 0.17/0.39
% 0.17/0.39 thf(descrp_type,type,
% 0.17/0.39 descrp: $o ).
% 0.17/0.39
% 0.17/0.39 thf(descrp,definition,
% 0.17/0.39 ( descrp
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ( Xphi
% 0.17/0.39 @ ( descr
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstr_type,type,
% 0.17/0.39 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrI_type,type,
% 0.17/0.39 dsetconstrI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrI,definition,
% 0.17/0.39 ( dsetconstrI
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( ( Xphi @ Xx )
% 0.17/0.39 => ( in @ Xx
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrEL_type,type,
% 0.17/0.39 dsetconstrEL: $o ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrEL,definition,
% 0.17/0.39 ( dsetconstrEL
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrER_type,type,
% 0.17/0.39 dsetconstrER: $o ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstrER,definition,
% 0.17/0.39 ( dsetconstrER
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuE1_type,type,
% 0.17/0.39 exuE1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuE1,definition,
% 0.17/0.39 ( exuE1
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ( ( Xphi @ Xx )
% 0.17/0.39 & ! [Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xy )
% 0.17/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(prop2set_type,type,
% 0.17/0.39 prop2set: $o > $i ).
% 0.17/0.39
% 0.17/0.39 thf(prop2setE_type,type,
% 0.17/0.39 prop2setE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(prop2setE,definition,
% 0.17/0.39 ( prop2setE
% 0.17/0.39 = ( ! [Xphi: $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.17/0.39 => Xphi ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetE_type,type,
% 0.17/0.39 emptysetE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetE,definition,
% 0.17/0.39 ( emptysetE
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ emptyset )
% 0.17/0.39 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetimpfalse_type,type,
% 0.17/0.39 emptysetimpfalse: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetimpfalse,definition,
% 0.17/0.39 ( emptysetimpfalse
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ emptyset )
% 0.17/0.39 => $false ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notinemptyset_type,type,
% 0.17/0.39 notinemptyset: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notinemptyset,definition,
% 0.17/0.39 ( notinemptyset
% 0.17/0.39 = ( ! [Xx: $i] :
% 0.17/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuE3e_type,type,
% 0.17/0.39 exuE3e: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuE3e,definition,
% 0.17/0.39 ( exuE3e
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setext_type,type,
% 0.17/0.39 setext: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setext,definition,
% 0.17/0.39 ( setext
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( in @ Xx @ B ) )
% 0.17/0.39 => ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( in @ Xx @ A ) )
% 0.17/0.39 => ( A = B ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyI_type,type,
% 0.17/0.39 emptyI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyI,definition,
% 0.17/0.39 ( emptyI
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ~ ( in @ Xx @ A )
% 0.17/0.39 => ( A = emptyset ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(noeltsimpempty_type,type,
% 0.17/0.39 noeltsimpempty: $o ).
% 0.17/0.39
% 0.17/0.39 thf(noeltsimpempty,definition,
% 0.17/0.39 ( noeltsimpempty
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ~ ( in @ Xx @ A )
% 0.17/0.39 => ( A = emptyset ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setbeta_type,type,
% 0.17/0.39 setbeta: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setbeta,definition,
% 0.17/0.39 ( setbeta
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( ( in @ Xx
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.39 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonempty_type,type,
% 0.17/0.39 nonempty: $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonempty,definition,
% 0.17/0.39 ( nonempty
% 0.17/0.39 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyE1_type,type,
% 0.17/0.39 nonemptyE1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyE1,definition,
% 0.17/0.39 ( nonemptyE1
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ( nonempty @ A )
% 0.17/0.39 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyI_type,type,
% 0.17/0.39 nonemptyI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyI,definition,
% 0.17/0.39 ( nonemptyI
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( ( Xphi @ Xx )
% 0.17/0.39 => ( nonempty
% 0.17/0.39 @ ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyI1_type,type,
% 0.17/0.39 nonemptyI1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyI1,definition,
% 0.17/0.39 ( nonemptyI1
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.39 => ( nonempty @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinIL_type,type,
% 0.17/0.39 setadjoinIL: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinIL,definition,
% 0.17/0.39 ( setadjoinIL
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyinunitempty_type,type,
% 0.17/0.39 emptyinunitempty: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyinunitempty,definition,
% 0.17/0.39 ( emptyinunitempty
% 0.17/0.39 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinIR_type,type,
% 0.17/0.39 setadjoinIR: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinIR,definition,
% 0.17/0.39 ( setadjoinIR
% 0.17/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ A )
% 0.17/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinE_type,type,
% 0.17/0.39 setadjoinE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinE,definition,
% 0.17/0.39 ( setadjoinE
% 0.17/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.39 => ! [Xphi: $o] :
% 0.17/0.39 ( ( ( Xy = Xx )
% 0.17/0.39 => Xphi )
% 0.17/0.39 => ( ( ( in @ Xy @ A )
% 0.17/0.39 => Xphi )
% 0.17/0.39 => Xphi ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinOr_type,type,
% 0.17/0.39 setadjoinOr: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinOr,definition,
% 0.17/0.39 ( setadjoinOr
% 0.17/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.39 => ( ( Xy = Xx )
% 0.17/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setoftrueEq_type,type,
% 0.17/0.39 setoftrueEq: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setoftrueEq,definition,
% 0.17/0.39 ( setoftrueEq
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xx: $i] : $true )
% 0.17/0.39 = A ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(powersetI_type,type,
% 0.17/0.39 powersetI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(powersetI,definition,
% 0.17/0.39 ( powersetI
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( in @ Xx @ A ) )
% 0.17/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyinPowerset_type,type,
% 0.17/0.39 emptyinPowerset: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyinPowerset,definition,
% 0.17/0.39 ( emptyinPowerset
% 0.17/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyInPowerset_type,type,
% 0.17/0.39 emptyInPowerset: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyInPowerset,definition,
% 0.17/0.39 ( emptyInPowerset
% 0.17/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(powersetE_type,type,
% 0.17/0.39 powersetE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(powersetE,definition,
% 0.17/0.39 ( powersetE
% 0.17/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.39 => ( ( in @ Xx @ B )
% 0.17/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setunionI_type,type,
% 0.17/0.39 setunionI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setunionI,definition,
% 0.17/0.39 ( setunionI
% 0.17/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( ( in @ B @ A )
% 0.17/0.39 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setunionE_type,type,
% 0.17/0.39 setunionE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setunionE,definition,
% 0.17/0.39 ( setunionE
% 0.17/0.39 = ( ! [A: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.39 => ! [Xphi: $o] :
% 0.17/0.39 ( ! [B: $i] :
% 0.17/0.39 ( ( in @ Xx @ B )
% 0.17/0.39 => ( ( in @ B @ A )
% 0.17/0.39 => Xphi ) )
% 0.17/0.39 => Xphi ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(subPowSU_type,type,
% 0.17/0.39 subPowSU: $o ).
% 0.17/0.39
% 0.17/0.39 thf(subPowSU,definition,
% 0.17/0.39 ( subPowSU
% 0.17/0.39 = ( ! [A: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuE2_type,type,
% 0.17/0.39 exuE2: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuE2,definition,
% 0.17/0.39 ( exuE2
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ! [Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xy )
% 0.17/0.39 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyImpWitness_type,type,
% 0.17/0.39 nonemptyImpWitness: $o ).
% 0.17/0.39
% 0.17/0.39 thf(nonemptyImpWitness,definition,
% 0.17/0.39 ( nonemptyImpWitness
% 0.17/0.39 = ( ! [A: $i] :
% 0.17/0.39 ( ( nonempty @ A )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & $true ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(uniqinunit_type,type,
% 0.17/0.39 uniqinunit: $o ).
% 0.17/0.39
% 0.17/0.39 thf(uniqinunit,definition,
% 0.17/0.39 ( uniqinunit
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notinsingleton_type,type,
% 0.17/0.39 notinsingleton: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notinsingleton,definition,
% 0.17/0.39 ( notinsingleton
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xx != Xy )
% 0.17/0.39 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(eqinunit_type,type,
% 0.17/0.39 eqinunit: $o ).
% 0.17/0.39
% 0.17/0.39 thf(eqinunit,definition,
% 0.17/0.39 ( eqinunit
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xx = Xy )
% 0.17/0.39 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(singletonsswitch_type,type,
% 0.17/0.39 singletonsswitch: $o ).
% 0.17/0.39
% 0.17/0.39 thf(singletonsswitch,definition,
% 0.17/0.39 ( singletonsswitch
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetE_type,type,
% 0.17/0.39 upairsetE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetE,definition,
% 0.17/0.39 ( upairsetE
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.39 => ( ( Xz = Xx )
% 0.17/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetIL_type,type,
% 0.17/0.39 upairsetIL: $o ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetIL,definition,
% 0.17/0.39 ( upairsetIL
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetIR_type,type,
% 0.17/0.39 upairsetIR: $o ).
% 0.17/0.39
% 0.17/0.39 thf(upairsetIR,definition,
% 0.17/0.39 ( upairsetIR
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyE1_type,type,
% 0.17/0.39 emptyE1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyE1,definition,
% 0.17/0.39 ( emptyE1
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ( Xphi @ Xx ) )
% 0.17/0.39 => ( ( ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 = emptyset )
% 0.17/0.39 => $false ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(vacuousDall_type,type,
% 0.17/0.39 vacuousDall: $o ).
% 0.17/0.39
% 0.17/0.39 thf(vacuousDall,definition,
% 0.17/0.39 ( vacuousDall
% 0.17/0.39 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ emptyset )
% 0.17/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan1_type,type,
% 0.17/0.39 quantDeMorgan1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan1,definition,
% 0.17/0.39 ( quantDeMorgan1
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ~ ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan2_type,type,
% 0.17/0.39 quantDeMorgan2: $o ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan2,definition,
% 0.17/0.39 ( quantDeMorgan2
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ~ ( Xphi @ Xx ) )
% 0.17/0.39 => ~ ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan3_type,type,
% 0.17/0.39 quantDeMorgan3: $o ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan3,definition,
% 0.17/0.39 ( quantDeMorgan3
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ~ ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ( Xphi @ Xx ) )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan4_type,type,
% 0.17/0.39 quantDeMorgan4: $o ).
% 0.17/0.39
% 0.17/0.39 thf(quantDeMorgan4,definition,
% 0.17/0.39 ( quantDeMorgan4
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ~ ( Xphi @ Xx ) )
% 0.17/0.39 => ~ ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(prop2setI_type,type,
% 0.17/0.39 prop2setI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(prop2setI,definition,
% 0.17/0.39 ( prop2setI
% 0.17/0.39 = ( ! [Xphi: $o] :
% 0.17/0.39 ( Xphi
% 0.17/0.39 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(set2prop_type,type,
% 0.17/0.39 set2prop: $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(prop2set2propI_type,type,
% 0.17/0.39 prop2set2propI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(prop2set2propI,definition,
% 0.17/0.39 ( prop2set2propI
% 0.17/0.39 = ( ! [Xphi: $o] :
% 0.17/0.39 ( Xphi
% 0.17/0.39 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notdexE_type,type,
% 0.17/0.39 notdexE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notdexE,definition,
% 0.17/0.39 ( notdexE
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ~ ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ( Xphi @ Xx ) )
% 0.17/0.39 => ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notdallE_type,type,
% 0.17/0.39 notdallE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notdallE,definition,
% 0.17/0.39 ( notdallE
% 0.17/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.39 ( ~ ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( Xphi @ Xx ) )
% 0.17/0.39 => ? [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuI1_type,type,
% 0.17/0.39 exuI1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuI1,definition,
% 0.17/0.39 ( exuI1
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ? [Xx: $i] :
% 0.17/0.39 ( ( Xphi @ Xx )
% 0.17/0.39 & ! [Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xy )
% 0.17/0.39 => ( Xx = Xy ) ) )
% 0.17/0.39 => ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuI3_type,type,
% 0.17/0.39 exuI3: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuI3,definition,
% 0.17/0.39 ( exuI3
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.17/0.39 => ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xx )
% 0.17/0.39 => ( ( Xphi @ Xy )
% 0.17/0.39 => ( Xx = Xy ) ) )
% 0.17/0.39 => ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuI2_type,type,
% 0.17/0.39 exuI2: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuI2,definition,
% 0.17/0.39 ( exuI2
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ? [Xx: $i] :
% 0.17/0.39 ! [Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xy )
% 0.17/0.39 <=> ( Xy = Xx ) )
% 0.17/0.39 => ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(inCongP_type,type,
% 0.17/0.39 inCongP: $o ).
% 0.17/0.39
% 0.17/0.39 thf(inCongP,definition,
% 0.17/0.39 ( inCongP
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( A = B )
% 0.17/0.39 => ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xx = Xy )
% 0.17/0.39 => ( ( in @ Xx @ A )
% 0.17/0.39 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(in__Cong_type,type,
% 0.17/0.39 in__Cong: $o ).
% 0.17/0.39
% 0.17/0.39 thf(in__Cong,definition,
% 0.17/0.39 ( in__Cong
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( A = B )
% 0.17/0.39 => ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xx = Xy )
% 0.17/0.39 => ( ( in @ Xx @ A )
% 0.17/0.39 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuE3u_type,type,
% 0.17/0.39 exuE3u: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuE3u,definition,
% 0.17/0.39 ( exuE3u
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xx )
% 0.17/0.39 => ( ( Xphi @ Xy )
% 0.17/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exu__Cong_type,type,
% 0.17/0.39 exu__Cong: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exu__Cong,definition,
% 0.17/0.39 ( exu__Cong
% 0.17/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xx = Xy )
% 0.17/0.39 => ( ( Xphi @ Xx )
% 0.17/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.39 => ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 <=> ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptyset__Cong_type,type,
% 0.17/0.39 emptyset__Cong: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptyset__Cong,definition,
% 0.17/0.39 ( emptyset__Cong
% 0.17/0.39 = ( emptyset = emptyset ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoin__Cong_type,type,
% 0.17/0.39 setadjoin__Cong: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoin__Cong,definition,
% 0.17/0.39 ( setadjoin__Cong
% 0.17/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xx = Xy )
% 0.17/0.39 => ! [Xz: $i,Xu: $i] :
% 0.17/0.39 ( ( Xz = Xu )
% 0.17/0.39 => ( ( setadjoin @ Xx @ Xz )
% 0.17/0.39 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(powerset__Cong_type,type,
% 0.17/0.39 powerset__Cong: $o ).
% 0.17/0.39
% 0.17/0.39 thf(powerset__Cong,definition,
% 0.17/0.39 ( powerset__Cong
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( A = B )
% 0.17/0.39 => ( ( powerset @ A )
% 0.17/0.39 = ( powerset @ B ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setunion__Cong_type,type,
% 0.17/0.39 setunion__Cong: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setunion__Cong,definition,
% 0.17/0.39 ( setunion__Cong
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( A = B )
% 0.17/0.39 => ( ( setunion @ A )
% 0.17/0.39 = ( setunion @ B ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(omega__Cong_type,type,
% 0.17/0.39 omega__Cong: $o ).
% 0.17/0.39
% 0.17/0.39 thf(omega__Cong,definition,
% 0.17/0.39 ( omega__Cong
% 0.17/0.39 = ( omega = omega ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(exuEu_type,type,
% 0.17/0.39 exuEu: $o ).
% 0.17/0.39
% 0.17/0.39 thf(exuEu,definition,
% 0.17/0.39 ( exuEu
% 0.17/0.39 = ( ! [Xphi: $i > $o] :
% 0.17/0.39 ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xphi @ Xx )
% 0.17/0.39 => ( ( Xphi @ Xy )
% 0.17/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(descr__Cong_type,type,
% 0.17/0.39 descr__Cong: $o ).
% 0.17/0.39
% 0.17/0.39 thf(descr__Cong,definition,
% 0.17/0.39 ( descr__Cong
% 0.17/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.39 ( ( Xx = Xy )
% 0.17/0.39 => ( ( Xphi @ Xx )
% 0.17/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.39 => ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 => ( ( exu
% 0.17/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.17/0.39 => ( ( descr
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 = ( descr
% 0.17/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstr__Cong_type,type,
% 0.17/0.39 dsetconstr__Cong: $o ).
% 0.17/0.39
% 0.17/0.39 thf(dsetconstr__Cong,definition,
% 0.17/0.39 ( dsetconstr__Cong
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( A = B )
% 0.17/0.39 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ! [Xy: $i] :
% 0.17/0.39 ( ( in @ Xy @ B )
% 0.17/0.39 => ( ( Xx = Xy )
% 0.17/0.39 => ( ( Xphi @ Xx )
% 0.17/0.39 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.17/0.39 => ( ( dsetconstr @ A
% 0.17/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.39 = ( dsetconstr @ B
% 0.17/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(subset_type,type,
% 0.17/0.39 subset: $i > $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(disjoint_type,type,
% 0.17/0.39 disjoint: $i > $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(setsmeet_type,type,
% 0.17/0.39 setsmeet: $i > $i > $o ).
% 0.17/0.39
% 0.17/0.39 thf(subsetI1_type,type,
% 0.17/0.39 subsetI1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(subsetI1,definition,
% 0.17/0.39 ( subsetI1
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( in @ Xx @ B ) )
% 0.17/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(eqimpsubset2_type,type,
% 0.17/0.39 eqimpsubset2: $o ).
% 0.17/0.39
% 0.17/0.39 thf(eqimpsubset2,definition,
% 0.17/0.39 ( eqimpsubset2
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( A = B )
% 0.17/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(eqimpsubset1_type,type,
% 0.17/0.39 eqimpsubset1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(eqimpsubset1,definition,
% 0.17/0.39 ( eqimpsubset1
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ( A = B )
% 0.17/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(subsetI2_type,type,
% 0.17/0.39 subsetI2: $o ).
% 0.17/0.39
% 0.17/0.39 thf(subsetI2,definition,
% 0.17/0.39 ( subsetI2
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ! [Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( in @ Xx @ B ) )
% 0.17/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetsubset_type,type,
% 0.17/0.39 emptysetsubset: $o ).
% 0.17/0.39
% 0.17/0.39 thf(emptysetsubset,definition,
% 0.17/0.39 ( emptysetsubset
% 0.17/0.39 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(subsetE_type,type,
% 0.17/0.39 subsetE: $o ).
% 0.17/0.39
% 0.17/0.39 thf(subsetE,definition,
% 0.17/0.39 ( subsetE
% 0.17/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.39 ( ( subset @ A @ B )
% 0.17/0.39 => ( ( in @ Xx @ A )
% 0.17/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(subsetE2_type,type,
% 0.17/0.39 subsetE2: $o ).
% 0.17/0.39
% 0.17/0.39 thf(subsetE2,definition,
% 0.17/0.39 ( subsetE2
% 0.17/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.39 ( ( subset @ A @ B )
% 0.17/0.39 => ( ~ ( in @ Xx @ B )
% 0.17/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notsubsetI_type,type,
% 0.17/0.39 notsubsetI: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notsubsetI,definition,
% 0.17/0.39 ( notsubsetI
% 0.17/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( ~ ( in @ Xx @ B )
% 0.17/0.39 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notequalI1_type,type,
% 0.17/0.39 notequalI1: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notequalI1,definition,
% 0.17/0.39 ( notequalI1
% 0.17/0.39 = ( ! [A: $i,B: $i] :
% 0.17/0.39 ( ~ ( subset @ A @ B )
% 0.17/0.39 => ( A != B ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(notequalI2_type,type,
% 0.17/0.39 notequalI2: $o ).
% 0.17/0.39
% 0.17/0.39 thf(notequalI2,definition,
% 0.17/0.39 ( notequalI2
% 0.17/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.39 ( ( in @ Xx @ A )
% 0.17/0.39 => ( ~ ( in @ Xx @ B )
% 0.17/0.39 => ( A != B ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(subsetRefl_type,type,
% 0.17/0.39 subsetRefl: $o ).
% 0.17/0.39
% 0.17/0.39 thf(subsetRefl,definition,
% 0.17/0.39 ( subsetRefl
% 0.17/0.39 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(subsetTrans_type,type,
% 0.17/0.39 subsetTrans: $o ).
% 0.17/0.39
% 0.17/0.39 thf(subsetTrans,definition,
% 0.17/0.39 ( subsetTrans
% 0.17/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.39 ( ( subset @ A @ B )
% 0.17/0.39 => ( ( subset @ B @ C )
% 0.17/0.39 => ( subset @ A @ C ) ) ) ) ) ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinSub_type,type,
% 0.17/0.39 setadjoinSub: $o ).
% 0.17/0.39
% 0.17/0.39 thf(setadjoinSub,definition,
% 0.17/0.39 ( setadjoinSub
% 0.17/0.40 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinSub2_type,type,
% 0.17/0.40 setadjoinSub2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setadjoinSub2,definition,
% 0.17/0.40 ( setadjoinSub2
% 0.17/0.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subset2powerset_type,type,
% 0.17/0.40 subset2powerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subset2powerset,definition,
% 0.17/0.40 ( subset2powerset
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setextsub_type,type,
% 0.17/0.40 setextsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setextsub,definition,
% 0.17/0.40 ( setextsub
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( ( subset @ B @ A )
% 0.17/0.40 => ( A = B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subsetemptysetimpeq_type,type,
% 0.17/0.40 subsetemptysetimpeq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetemptysetimpeq,definition,
% 0.17/0.40 ( subsetemptysetimpeq
% 0.17/0.40 = ( ! [A: $i] :
% 0.17/0.40 ( ( subset @ A @ emptyset )
% 0.17/0.40 => ( A = emptyset ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetI1_type,type,
% 0.17/0.40 powersetI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetI1,definition,
% 0.17/0.40 ( powersetI1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ B @ A )
% 0.17/0.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetE1_type,type,
% 0.17/0.40 powersetE1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetE1,definition,
% 0.17/0.40 ( powersetE1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.40 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(inPowerset_type,type,
% 0.17/0.40 inPowerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(inPowerset,definition,
% 0.17/0.40 ( inPowerset
% 0.17/0.40 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetsubset_type,type,
% 0.17/0.40 powersetsubset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetsubset,definition,
% 0.17/0.40 ( powersetsubset
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(sepInPowerset_type,type,
% 0.17/0.40 sepInPowerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(sepInPowerset,definition,
% 0.17/0.40 ( sepInPowerset
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( in
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 @ ( powerset @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(sepSubset_type,type,
% 0.17/0.40 sepSubset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(sepSubset,definition,
% 0.17/0.40 ( sepSubset
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( subset
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunion_type,type,
% 0.17/0.40 binunion: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(binunionIL_type,type,
% 0.17/0.40 binunionIL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionIL,definition,
% 0.17/0.40 ( binunionIL
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairset2IR_type,type,
% 0.17/0.40 upairset2IR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairset2IR,definition,
% 0.17/0.40 ( upairset2IR
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionIR_type,type,
% 0.17/0.40 binunionIR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionIR,definition,
% 0.17/0.40 ( binunionIR
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionEcases_type,type,
% 0.17/0.40 binunionEcases: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionEcases,definition,
% 0.17/0.40 ( binunionEcases
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.17/0.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.40 => ( ( ( in @ Xx @ A )
% 0.17/0.40 => Xphi )
% 0.17/0.40 => ( ( ( in @ Xx @ B )
% 0.17/0.40 => Xphi )
% 0.17/0.40 => Xphi ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionE_type,type,
% 0.17/0.40 binunionE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionE,definition,
% 0.17/0.40 ( binunionE
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.40 => ( ( in @ Xx @ A )
% 0.17/0.40 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionLsub_type,type,
% 0.17/0.40 binunionLsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionLsub,definition,
% 0.17/0.40 ( binunionLsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionRsub_type,type,
% 0.17/0.40 binunionRsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionRsub,definition,
% 0.17/0.40 ( binunionRsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersect_type,type,
% 0.17/0.40 binintersect: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectI_type,type,
% 0.17/0.40 binintersectI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectI,definition,
% 0.17/0.40 ( binintersectI
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset5_type,type,
% 0.17/0.40 binintersectSubset5: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset5,definition,
% 0.17/0.40 ( binintersectSubset5
% 0.17/0.40 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.40 ( ( subset @ C @ A )
% 0.17/0.40 => ( ( subset @ C @ B )
% 0.17/0.40 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectEL_type,type,
% 0.17/0.40 binintersectEL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectEL,definition,
% 0.17/0.40 ( binintersectEL
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectLsub_type,type,
% 0.17/0.40 binintersectLsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectLsub,definition,
% 0.17/0.40 ( binintersectLsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset2_type,type,
% 0.17/0.40 binintersectSubset2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset2,definition,
% 0.17/0.40 ( binintersectSubset2
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( ( binintersect @ A @ B )
% 0.17/0.40 = A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset3_type,type,
% 0.17/0.40 binintersectSubset3: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset3,definition,
% 0.17/0.40 ( binintersectSubset3
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( ( binintersect @ A @ B )
% 0.17/0.40 = B )
% 0.17/0.40 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectER_type,type,
% 0.17/0.40 binintersectER: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectER,definition,
% 0.17/0.40 ( binintersectER
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.17/0.40 => ( in @ Xx @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(disjointsetsI1_type,type,
% 0.17/0.40 disjointsetsI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(disjointsetsI1,definition,
% 0.17/0.40 ( disjointsetsI1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ~ ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( in @ Xx @ B ) )
% 0.17/0.40 => ( ( binintersect @ A @ B )
% 0.17/0.40 = emptyset ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectRsub_type,type,
% 0.17/0.40 binintersectRsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectRsub,definition,
% 0.17/0.40 ( binintersectRsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset4_type,type,
% 0.17/0.40 binintersectSubset4: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset4,definition,
% 0.17/0.40 ( binintersectSubset4
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ B @ A )
% 0.17/0.40 => ( ( binintersect @ A @ B )
% 0.17/0.40 = B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset1_type,type,
% 0.17/0.40 binintersectSubset1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectSubset1,definition,
% 0.17/0.40 ( binintersectSubset1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( ( binintersect @ A @ B )
% 0.17/0.40 = A )
% 0.17/0.40 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(bs114d_type,type,
% 0.17/0.40 bs114d: $o ).
% 0.17/0.40
% 0.17/0.40 thf(bs114d,definition,
% 0.17/0.40 ( bs114d
% 0.17/0.40 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.40 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.17/0.40 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(regular_type,type,
% 0.17/0.40 regular: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminus_type,type,
% 0.17/0.40 setminus: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(setminusI_type,type,
% 0.17/0.40 setminusI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusI,definition,
% 0.17/0.40 ( setminusI
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusEL_type,type,
% 0.17/0.40 setminusEL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusEL,definition,
% 0.17/0.40 ( setminusEL
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusER_type,type,
% 0.17/0.40 setminusER: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusER,definition,
% 0.17/0.40 ( setminusER
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.40 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusSubset2_type,type,
% 0.17/0.40 setminusSubset2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusSubset2,definition,
% 0.17/0.40 ( setminusSubset2
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( subset @ A @ B )
% 0.17/0.40 => ( ( setminus @ A @ B )
% 0.17/0.40 = emptyset ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusERneg_type,type,
% 0.17/0.40 setminusERneg: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusERneg,definition,
% 0.17/0.40 ( setminusERneg
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.40 => ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusELneg_type,type,
% 0.17/0.40 setminusELneg: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusELneg,definition,
% 0.17/0.40 ( setminusELneg
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusILneg_type,type,
% 0.17/0.40 setminusILneg: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusILneg,definition,
% 0.17/0.40 ( setminusILneg
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ A )
% 0.17/0.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusIRneg_type,type,
% 0.17/0.40 setminusIRneg: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusIRneg,definition,
% 0.17/0.40 ( setminusIRneg
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ B )
% 0.17/0.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusLsub_type,type,
% 0.17/0.40 setminusLsub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusLsub,definition,
% 0.17/0.40 ( setminusLsub
% 0.17/0.40 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusSubset1_type,type,
% 0.17/0.40 setminusSubset1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusSubset1,definition,
% 0.17/0.40 ( setminusSubset1
% 0.17/0.40 = ( ! [A: $i,B: $i] :
% 0.17/0.40 ( ( ( setminus @ A @ B )
% 0.17/0.40 = emptyset )
% 0.17/0.40 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiff_type,type,
% 0.17/0.40 symdiff: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffE_type,type,
% 0.17/0.40 symdiffE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffE,definition,
% 0.17/0.40 ( symdiffE
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.17/0.40 => ! [Xphi: $o] :
% 0.17/0.40 ( ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => Xphi ) )
% 0.17/0.40 => ( ( ~ ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ B )
% 0.17/0.40 => Xphi ) )
% 0.17/0.40 => Xphi ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffI1_type,type,
% 0.17/0.40 symdiffI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffI1,definition,
% 0.17/0.40 ( symdiffI1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffI2_type,type,
% 0.17/0.40 symdiffI2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffI2,definition,
% 0.17/0.40 ( symdiffI2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ B )
% 0.17/0.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffIneg1_type,type,
% 0.17/0.40 symdiffIneg1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffIneg1,definition,
% 0.17/0.40 ( symdiffIneg1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ B )
% 0.17/0.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffIneg2_type,type,
% 0.17/0.40 symdiffIneg2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(symdiffIneg2,definition,
% 0.17/0.40 ( symdiffIneg2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ~ ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ B )
% 0.17/0.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iskpair_type,type,
% 0.17/0.40 iskpair: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(secondinupair_type,type,
% 0.17/0.40 secondinupair: $o ).
% 0.17/0.40
% 0.17/0.40 thf(secondinupair,definition,
% 0.17/0.40 ( secondinupair
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairIL_type,type,
% 0.17/0.40 setukpairIL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairIL,definition,
% 0.17/0.40 ( setukpairIL
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairIR_type,type,
% 0.17/0.40 setukpairIR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairIR,definition,
% 0.17/0.40 ( setukpairIR
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kpairiskpair_type,type,
% 0.17/0.40 kpairiskpair: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kpairiskpair,definition,
% 0.17/0.40 ( kpairiskpair
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kpair_type,type,
% 0.17/0.40 kpair: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(kpairp_type,type,
% 0.17/0.40 kpairp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kpairp,definition,
% 0.17/0.40 ( kpairp
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprod_type,type,
% 0.17/0.40 cartprod: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsubset_type,type,
% 0.17/0.40 singletonsubset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsubset,definition,
% 0.17/0.40 ( singletonsubset
% 0.17/0.40 = ( ! [A: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(singletoninpowerset_type,type,
% 0.17/0.40 singletoninpowerset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletoninpowerset,definition,
% 0.17/0.40 ( singletoninpowerset
% 0.17/0.40 = ( ! [A: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(singletoninpowunion_type,type,
% 0.17/0.40 singletoninpowunion: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletoninpowunion,definition,
% 0.17/0.40 ( singletoninpowunion
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairset2E_type,type,
% 0.17/0.40 upairset2E: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairset2E,definition,
% 0.17/0.40 ( upairset2E
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.40 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.40 => ( ( Xz = Xx )
% 0.17/0.40 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairsubunion_type,type,
% 0.17/0.40 upairsubunion: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairsubunion,definition,
% 0.17/0.40 ( upairsubunion
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairinpowunion_type,type,
% 0.17/0.40 upairinpowunion: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairinpowunion,definition,
% 0.17/0.40 ( upairinpowunion
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem1_type,type,
% 0.17/0.40 ubforcartprodlem1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem1,definition,
% 0.17/0.40 ( ubforcartprodlem1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem2_type,type,
% 0.17/0.40 ubforcartprodlem2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem2,definition,
% 0.17/0.40 ( ubforcartprodlem2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem3_type,type,
% 0.17/0.40 ubforcartprodlem3: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ubforcartprodlem3,definition,
% 0.17/0.40 ( ubforcartprodlem3
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairin_type,type,
% 0.17/0.40 cartprodpairin: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairin,definition,
% 0.17/0.40 ( cartprodpairin
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempair1_type,type,
% 0.17/0.40 cartprodmempair1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempair1,definition,
% 0.17/0.40 ( cartprodmempair1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ? [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 & ( Xu
% 0.17/0.40 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempair_type,type,
% 0.17/0.40 cartprodmempair: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempair,definition,
% 0.17/0.40 ( cartprodmempair
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( iskpair @ Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionE2_type,type,
% 0.17/0.40 setunionE2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionE2,definition,
% 0.17/0.40 ( setunionE2
% 0.17/0.40 = ( ! [A: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.40 => ? [X: $i] :
% 0.17/0.40 ( ( in @ X @ A )
% 0.17/0.40 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton1_type,type,
% 0.17/0.40 setunionsingleton1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton1,definition,
% 0.17/0.40 ( setunionsingleton1
% 0.17/0.40 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton2_type,type,
% 0.17/0.40 setunionsingleton2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton2,definition,
% 0.17/0.40 ( setunionsingleton2
% 0.17/0.40 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton_type,type,
% 0.17/0.40 setunionsingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setunionsingleton,definition,
% 0.17/0.40 ( setunionsingleton
% 0.17/0.40 = ( ! [Xx: $i] :
% 0.17/0.40 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.17/0.40 = Xx ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(singleton_type,type,
% 0.17/0.40 singleton: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletonprop_type,type,
% 0.17/0.40 singletonprop: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletonprop,definition,
% 0.17/0.40 ( singletonprop
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) )
% 0.17/0.40 => ( ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1_type,type,
% 0.17/0.40 ex1: $i > ( $i > $o ) > $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1,definition,
% 0.17/0.40 ( ex1
% 0.17/0.40 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( singleton
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1E1_type,type,
% 0.17/0.40 ex1E1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1E1,definition,
% 0.17/0.40 ( ex1E1
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ( ex1 @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1I_type,type,
% 0.17/0.40 ex1I: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1I,definition,
% 0.17/0.40 ( ex1I
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xy = Xx ) ) )
% 0.17/0.40 => ( ex1 @ A
% 0.17/0.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1I2_type,type,
% 0.17/0.40 ex1I2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1I2,definition,
% 0.17/0.40 ( ex1I2
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) )
% 0.17/0.40 => ( ? [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 & ( Xphi @ Xx ) )
% 0.17/0.40 => ( ex1 @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsuniq_type,type,
% 0.17/0.40 singletonsuniq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(singletonsuniq,definition,
% 0.17/0.40 ( singletonsuniq
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( ( setadjoin @ Xx @ emptyset )
% 0.17/0.40 = ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(atmost1p_type,type,
% 0.17/0.40 atmost1p: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(atleast2p_type,type,
% 0.17/0.40 atleast2p: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(atmost2p_type,type,
% 0.17/0.40 atmost2p: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairsetp_type,type,
% 0.17/0.40 upairsetp: $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL1_type,type,
% 0.17/0.40 setukpairinjL1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL1,definition,
% 0.17/0.40 ( setukpairinjL1
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.40 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.40 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kfstsingleton_type,type,
% 0.17/0.40 kfstsingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kfstsingleton,definition,
% 0.17/0.40 ( kfstsingleton
% 0.17/0.40 = ( ! [Xu: $i] :
% 0.17/0.40 ( ( iskpair @ Xu )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ ( setunion @ Xu )
% 0.17/0.40 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(theprop_type,type,
% 0.17/0.40 theprop: $o ).
% 0.17/0.40
% 0.17/0.40 thf(theprop,definition,
% 0.17/0.40 ( theprop
% 0.17/0.40 = ( ! [X: $i] :
% 0.17/0.40 ( ( singleton @ X )
% 0.17/0.40 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kfst_type,type,
% 0.17/0.40 kfst: $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(kfstpairEq_type,type,
% 0.17/0.40 kfstpairEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kfstpairEq,definition,
% 0.17/0.40 ( kfstpairEq
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.17/0.40 = Xx ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodfstin_type,type,
% 0.17/0.40 cartprodfstin: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodfstin,definition,
% 0.17/0.40 ( cartprodfstin
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL2_type,type,
% 0.17/0.40 setukpairinjL2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL2,definition,
% 0.17/0.40 ( setukpairinjL2
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.40 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL_type,type,
% 0.17/0.40 setukpairinjL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjL,definition,
% 0.17/0.40 ( setukpairinjL
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( kpair @ Xx @ Xy )
% 0.17/0.40 = ( kpair @ Xz @ Xu ) )
% 0.17/0.40 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR11_type,type,
% 0.17/0.40 setukpairinjR11: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR11,definition,
% 0.17/0.40 ( setukpairinjR11
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR12_type,type,
% 0.17/0.40 setukpairinjR12: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR12,definition,
% 0.17/0.40 ( setukpairinjR12
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xx = Xy )
% 0.17/0.40 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR1_type,type,
% 0.17/0.40 setukpairinjR1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR1,definition,
% 0.17/0.40 ( setukpairinjR1
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.40 => ( ( Xz = Xu )
% 0.17/0.40 => ( Xy = Xu ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(upairequniteq_type,type,
% 0.17/0.40 upairequniteq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(upairequniteq,definition,
% 0.17/0.40 ( upairequniteq
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.40 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ Xz @ emptyset ) )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR2_type,type,
% 0.17/0.40 setukpairinjR2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR2,definition,
% 0.17/0.40 ( setukpairinjR2
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.40 => ( Xy = Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR_type,type,
% 0.17/0.40 setukpairinjR: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setukpairinjR,definition,
% 0.17/0.40 ( setukpairinjR
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.40 ( ( ( kpair @ Xx @ Xy )
% 0.17/0.40 = ( kpair @ Xz @ Xu ) )
% 0.17/0.40 => ( Xy = Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ksndsingleton_type,type,
% 0.17/0.40 ksndsingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ksndsingleton,definition,
% 0.17/0.40 ( ksndsingleton
% 0.17/0.40 = ( ! [Xu: $i] :
% 0.17/0.40 ( ( iskpair @ Xu )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ ( setunion @ Xu )
% 0.17/0.40 @ ^ [Xx: $i] :
% 0.17/0.40 ( Xu
% 0.17/0.40 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ksnd_type,type,
% 0.17/0.40 ksnd: $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(ksndpairEq_type,type,
% 0.17/0.40 ksndpairEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ksndpairEq,definition,
% 0.17/0.40 ( ksndpairEq
% 0.17/0.40 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.17/0.40 = Xy ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(kpairsurjEq_type,type,
% 0.17/0.40 kpairsurjEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(kpairsurjEq,definition,
% 0.17/0.40 ( kpairsurjEq
% 0.17/0.40 = ( ! [Xu: $i] :
% 0.17/0.40 ( ( iskpair @ Xu )
% 0.17/0.40 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.17/0.40 = Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodsndin_type,type,
% 0.17/0.40 cartprodsndin: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodsndin,definition,
% 0.17/0.40 ( cartprodsndin
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairmemEL_type,type,
% 0.17/0.40 cartprodpairmemEL: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairmemEL,definition,
% 0.17/0.40 ( cartprodpairmemEL
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairmemER_type,type,
% 0.17/0.40 cartprodpairmemER: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairmemER,definition,
% 0.17/0.40 ( cartprodpairmemER
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( in @ Xy @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempaircEq_type,type,
% 0.17/0.40 cartprodmempaircEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodmempaircEq,definition,
% 0.17/0.40 ( cartprodmempaircEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( kpair @ Xx @ Xy )
% 0.17/0.40 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodfstpairEq_type,type,
% 0.17/0.40 cartprodfstpairEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodfstpairEq,definition,
% 0.17/0.40 ( cartprodfstpairEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.17/0.40 = Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodsndpairEq_type,type,
% 0.17/0.40 cartprodsndpairEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodsndpairEq,definition,
% 0.17/0.40 ( cartprodsndpairEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.17/0.40 = Xy ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairsurjEq_type,type,
% 0.17/0.40 cartprodpairsurjEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(cartprodpairsurjEq,definition,
% 0.17/0.40 ( cartprodpairsurjEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.40 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.17/0.40 = Xu ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(breln_type,type,
% 0.17/0.40 breln: $i > $i > $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(breln,definition,
% 0.17/0.40 ( breln
% 0.17/0.40 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstr_type,type,
% 0.17/0.40 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrI_type,type,
% 0.17/0.40 dpsetconstrI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrI,definition,
% 0.17/0.40 ( dpsetconstrI
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( Xphi @ Xx @ Xy )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrSub_type,type,
% 0.17/0.40 dpsetconstrSub: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrSub,definition,
% 0.17/0.40 ( dpsetconstrSub
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.17/0.40 ( subset
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.17/0.40 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setOfPairsIsBReln_type,type,
% 0.17/0.40 setOfPairsIsBReln: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setOfPairsIsBReln,definition,
% 0.17/0.40 ( setOfPairsIsBReln
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.17/0.40 ( breln @ A @ B
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrERa_type,type,
% 0.17/0.40 dpsetconstrERa: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrERa,definition,
% 0.17/0.40 ( dpsetconstrERa
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.40 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrEL1_type,type,
% 0.17/0.40 dpsetconstrEL1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrEL1,definition,
% 0.17/0.40 ( dpsetconstrEL1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.40 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrEL2_type,type,
% 0.17/0.40 dpsetconstrEL2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrEL2,definition,
% 0.17/0.40 ( dpsetconstrEL2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.40 => ( in @ Xy @ B ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrER_type,type,
% 0.17/0.40 dpsetconstrER: $o ).
% 0.17/0.40
% 0.17/0.40 thf(dpsetconstrER,definition,
% 0.17/0.40 ( dpsetconstrER
% 0.17/0.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.40 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(func_type,type,
% 0.17/0.40 func: $i > $i > $i > $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcSet_type,type,
% 0.17/0.40 funcSet: $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(funcImageSingleton_type,type,
% 0.17/0.40 funcImageSingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcImageSingleton,definition,
% 0.17/0.40 ( funcImageSingleton
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ B
% 0.17/0.40 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(apProp_type,type,
% 0.17/0.40 apProp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(apProp,definition,
% 0.17/0.40 ( apProp
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in
% 0.17/0.40 @ ( setunion
% 0.17/0.40 @ ( dsetconstr @ B
% 0.17/0.40 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.17/0.40 @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ap_type,type,
% 0.17/0.40 ap: $i > $i > $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(app_type,type,
% 0.17/0.40 app: $o ).
% 0.17/0.40
% 0.17/0.40 thf(app,definition,
% 0.17/0.40 ( app
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(infuncsetfunc_type,type,
% 0.17/0.40 infuncsetfunc: $o ).
% 0.17/0.40
% 0.17/0.40 thf(infuncsetfunc,definition,
% 0.17/0.40 ( infuncsetfunc
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ap2p_type,type,
% 0.17/0.40 ap2p: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ap2p,definition,
% 0.17/0.40 ( ap2p
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcinfuncset_type,type,
% 0.17/0.40 funcinfuncset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcinfuncset,definition,
% 0.17/0.40 ( funcinfuncset
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(lamProp_type,type,
% 0.17/0.40 lamProp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(lamProp,definition,
% 0.17/0.40 ( lamProp
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ( func @ A @ B
% 0.17/0.40 @ ( dpsetconstr @ A @ B
% 0.17/0.40 @ ^ [Xx: $i,Xy: $i] :
% 0.17/0.40 ( ( Xf @ Xx )
% 0.17/0.40 = Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(lam_type,type,
% 0.17/0.40 lam: $i > $i > ( $i > $i ) > $i ).
% 0.17/0.40
% 0.17/0.40 thf(lamp_type,type,
% 0.17/0.40 lamp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(lamp,definition,
% 0.17/0.40 ( lamp
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ( func @ A @ B
% 0.17/0.40 @ ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(lam2p_type,type,
% 0.17/0.40 lam2p: $o ).
% 0.17/0.40
% 0.17/0.40 thf(lam2p,definition,
% 0.17/0.40 ( lam2p
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ( in
% 0.17/0.40 @ ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.17/0.40 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(brelnall1_type,type,
% 0.17/0.40 brelnall1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(brelnall1,definition,
% 0.17/0.40 ( brelnall1
% 0.17/0.40 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ R )
% 0.17/0.40 => ! [Xphi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.40 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ R )
% 0.17/0.40 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(brelnall2_type,type,
% 0.17/0.40 brelnall2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(brelnall2,definition,
% 0.17/0.40 ( brelnall2
% 0.17/0.40 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ R )
% 0.17/0.40 => ! [Xphi: $i > $o] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.40 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ R )
% 0.17/0.40 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ex1E2_type,type,
% 0.17/0.40 ex1E2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ex1E2,definition,
% 0.17/0.40 ( ex1E2
% 0.17/0.40 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.40 ( ( ex1 @ A
% 0.17/0.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ( Xphi @ Xx )
% 0.17/0.40 => ( ( Xphi @ Xy )
% 0.17/0.40 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp1_type,type,
% 0.17/0.40 funcGraphProp1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp1,definition,
% 0.17/0.40 ( funcGraphProp1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp3_type,type,
% 0.17/0.40 funcGraphProp3: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp3,definition,
% 0.17/0.40 ( funcGraphProp3
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp2_type,type,
% 0.17/0.40 funcGraphProp2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp2,definition,
% 0.17/0.40 ( funcGraphProp2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcextLem_type,type,
% 0.17/0.40 funcextLem: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcextLem,definition,
% 0.17/0.40 ( funcextLem
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xg: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xg )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp4_type,type,
% 0.17/0.40 funcGraphProp4: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcGraphProp4,definition,
% 0.17/0.40 ( funcGraphProp4
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = Xy ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subbreln_type,type,
% 0.17/0.40 subbreln: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subbreln,definition,
% 0.17/0.40 ( subbreln
% 0.17/0.40 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ R )
% 0.17/0.40 => ! [S: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ S )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.40 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(eqbreln_type,type,
% 0.17/0.40 eqbreln: $o ).
% 0.17/0.40
% 0.17/0.40 thf(eqbreln,definition,
% 0.17/0.40 ( eqbreln
% 0.17/0.40 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ R )
% 0.17/0.40 => ! [S: $i] :
% 0.17/0.40 ( ( breln @ A @ B @ S )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ B )
% 0.17/0.40 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.17/0.40 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.17/0.40 => ( R = S ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcext_type,type,
% 0.17/0.40 funcext: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcext,definition,
% 0.17/0.40 ( funcext
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xg: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xg )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.40 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(funcext2_type,type,
% 0.17/0.40 funcext2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(funcext2,definition,
% 0.17/0.40 ( funcext2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xg: $i] :
% 0.17/0.40 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.40 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ap2apEq1_type,type,
% 0.17/0.40 ap2apEq1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ap2apEq1,definition,
% 0.17/0.40 ( ap2apEq1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ap2apEq2_type,type,
% 0.17/0.40 ap2apEq2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ap2apEq2,definition,
% 0.17/0.40 ( ap2apEq2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.40 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(beta1_type,type,
% 0.17/0.40 beta1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(beta1,definition,
% 0.17/0.40 ( beta1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B
% 0.17/0.40 @ ( lam @ A @ B
% 0.17/0.40 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.17/0.40 @ Xx )
% 0.17/0.40 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(eta1_type,type,
% 0.17/0.40 eta1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(eta1,definition,
% 0.17/0.40 ( eta1
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( func @ A @ B @ Xf )
% 0.17/0.40 => ( ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.17/0.40 = Xf ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(lam2lamEq_type,type,
% 0.17/0.40 lam2lamEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(lam2lamEq,definition,
% 0.17/0.40 ( lam2lamEq
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ( ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.17/0.40 = ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(beta2_type,type,
% 0.17/0.40 beta2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(beta2,definition,
% 0.17/0.40 ( beta2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.40 ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( ap @ A @ B
% 0.17/0.40 @ ( lam @ A @ B
% 0.17/0.40 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.17/0.40 @ Xx )
% 0.17/0.40 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(eta2_type,type,
% 0.17/0.40 eta2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(eta2,definition,
% 0.17/0.40 ( eta2
% 0.17/0.40 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.40 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.40 => ( ( lam @ A @ B
% 0.17/0.40 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.17/0.40 = Xf ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iffalseProp1_type,type,
% 0.17/0.40 iffalseProp1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iffalseProp1,definition,
% 0.17/0.40 ( iffalseProp1
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ~ Xphi
% 0.17/0.40 => ( in @ Xy
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iffalseProp2_type,type,
% 0.17/0.40 iffalseProp2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iffalseProp2,definition,
% 0.17/0.40 ( iffalseProp2
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ~ Xphi
% 0.17/0.40 => ( ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) )
% 0.17/0.40 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueProp1_type,type,
% 0.17/0.40 iftrueProp1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueProp1,definition,
% 0.17/0.40 ( iftrueProp1
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( Xphi
% 0.17/0.40 => ( in @ Xx
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueProp2_type,type,
% 0.17/0.40 iftrueProp2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueProp2,definition,
% 0.17/0.40 ( iftrueProp2
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( Xphi
% 0.17/0.40 => ( ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) )
% 0.17/0.40 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(ifSingleton_type,type,
% 0.17/0.40 ifSingleton: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ifSingleton,definition,
% 0.17/0.40 ( ifSingleton
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( singleton
% 0.17/0.40 @ ( dsetconstr @ A
% 0.17/0.40 @ ^ [Xz: $i] :
% 0.17/0.40 ( ( Xphi
% 0.17/0.40 & ( Xz = Xx ) )
% 0.17/0.40 | ( ~ Xphi
% 0.17/0.40 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(if_type,type,
% 0.17/0.40 if: $i > $o > $i > $i > $i ).
% 0.17/0.40
% 0.17/0.40 thf(ifp_type,type,
% 0.17/0.40 ifp: $o ).
% 0.17/0.40
% 0.17/0.40 thf(ifp,definition,
% 0.17/0.40 ( ifp
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(theeq_type,type,
% 0.17/0.40 theeq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(theeq,definition,
% 0.17/0.40 ( theeq
% 0.17/0.40 = ( ! [X: $i] :
% 0.17/0.40 ( ( singleton @ X )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ X )
% 0.17/0.40 => ( ( setunion @ X )
% 0.17/0.40 = Xx ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iftrue_type,type,
% 0.17/0.40 iftrue: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iftrue,definition,
% 0.17/0.40 ( iftrue
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( Xphi
% 0.17/0.40 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.17/0.40 = Xx ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iffalse_type,type,
% 0.17/0.40 iffalse: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iffalse,definition,
% 0.17/0.40 ( iffalse
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( ~ Xphi
% 0.17/0.40 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.17/0.40 = Xy ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueorfalse_type,type,
% 0.17/0.40 iftrueorfalse: $o ).
% 0.17/0.40
% 0.17/0.40 thf(iftrueorfalse,definition,
% 0.17/0.40 ( iftrueorfalse
% 0.17/0.40 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ! [Xy: $i] :
% 0.17/0.40 ( ( in @ Xy @ A )
% 0.17/0.40 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectT_lem_type,type,
% 0.17/0.40 binintersectT_lem: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectT_lem,definition,
% 0.17/0.40 ( binintersectT_lem
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionT_lem_type,type,
% 0.17/0.40 binunionT_lem: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionT_lem,definition,
% 0.17/0.40 ( binunionT_lem
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetT_lem_type,type,
% 0.17/0.40 powersetT_lem: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetT_lem,definition,
% 0.17/0.40 ( powersetT_lem
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setminusT_lem_type,type,
% 0.17/0.40 setminusT_lem: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setminusT_lem,definition,
% 0.17/0.40 ( setminusT_lem
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(complementT_lem_type,type,
% 0.17/0.40 complementT_lem: $o ).
% 0.17/0.40
% 0.17/0.40 thf(complementT_lem,definition,
% 0.17/0.40 ( complementT_lem
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(setextT_type,type,
% 0.17/0.40 setextT: $o ).
% 0.17/0.40
% 0.17/0.40 thf(setextT,definition,
% 0.17/0.40 ( setextT
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ X )
% 0.17/0.40 => ( in @ Xx @ Y ) ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ Y )
% 0.17/0.40 => ( in @ Xx @ X ) ) )
% 0.17/0.40 => ( X = Y ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(subsetTI_type,type,
% 0.17/0.40 subsetTI: $o ).
% 0.17/0.40
% 0.17/0.40 thf(subsetTI,definition,
% 0.17/0.40 ( subsetTI
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ X )
% 0.17/0.40 => ( in @ Xx @ Y ) ) )
% 0.17/0.40 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetTI1_type,type,
% 0.17/0.40 powersetTI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetTI1,definition,
% 0.17/0.40 ( powersetTI1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ X )
% 0.17/0.40 => ( in @ Xx @ Y ) ) )
% 0.17/0.40 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(powersetTE1_type,type,
% 0.17/0.40 powersetTE1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(powersetTE1,definition,
% 0.17/0.40 ( powersetTE1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ X @ ( powerset @ Y ) )
% 0.17/0.40 => ( ( in @ Xx @ X )
% 0.17/0.40 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(complementTI1_type,type,
% 0.17/0.40 complementTI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(complementTI1,definition,
% 0.17/0.40 ( complementTI1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ X )
% 0.17/0.40 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(complementTE1_type,type,
% 0.17/0.40 complementTE1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(complementTE1,definition,
% 0.17/0.40 ( complementTE1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.17/0.40 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectTELcontra_type,type,
% 0.17/0.40 binintersectTELcontra: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectTELcontra,definition,
% 0.17/0.40 ( binintersectTELcontra
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ X )
% 0.17/0.40 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectTERcontra_type,type,
% 0.17/0.40 binintersectTERcontra: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binintersectTERcontra,definition,
% 0.17/0.40 ( binintersectTERcontra
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ Y )
% 0.17/0.40 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(contrasubsetT_type,type,
% 0.17/0.40 contrasubsetT: $o ).
% 0.17/0.40
% 0.17/0.40 thf(contrasubsetT,definition,
% 0.17/0.40 ( contrasubsetT
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.17/0.40 => ( ( in @ Xx @ Y )
% 0.17/0.40 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(contrasubsetT1_type,type,
% 0.17/0.40 contrasubsetT1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(contrasubsetT1,definition,
% 0.17/0.40 ( contrasubsetT1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( subset @ X @ Y )
% 0.17/0.40 => ( ~ ( in @ Xx @ Y )
% 0.17/0.40 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(contrasubsetT2_type,type,
% 0.17/0.40 contrasubsetT2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(contrasubsetT2,definition,
% 0.17/0.40 ( contrasubsetT2
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( ( subset @ X @ Y )
% 0.17/0.40 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(contrasubsetT3_type,type,
% 0.17/0.40 contrasubsetT3: $o ).
% 0.17/0.40
% 0.17/0.40 thf(contrasubsetT3,definition,
% 0.17/0.40 ( contrasubsetT3
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.17/0.40 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementI1_type,type,
% 0.17/0.40 doubleComplementI1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementI1,definition,
% 0.17/0.40 ( doubleComplementI1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ X )
% 0.17/0.40 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementE1_type,type,
% 0.17/0.40 doubleComplementE1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementE1,definition,
% 0.17/0.40 ( doubleComplementE1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.17/0.40 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementSub1_type,type,
% 0.17/0.40 doubleComplementSub1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementSub1,definition,
% 0.17/0.40 ( doubleComplementSub1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementSub2_type,type,
% 0.17/0.40 doubleComplementSub2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementSub2,definition,
% 0.17/0.40 ( doubleComplementSub2
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementEq_type,type,
% 0.17/0.40 doubleComplementEq: $o ).
% 0.17/0.40
% 0.17/0.40 thf(doubleComplementEq,definition,
% 0.17/0.40 ( doubleComplementEq
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ( X
% 0.17/0.40 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(complementTnotintersectT_type,type,
% 0.17/0.40 complementTnotintersectT: $o ).
% 0.17/0.40
% 0.17/0.40 thf(complementTnotintersectT,definition,
% 0.17/0.40 ( complementTnotintersectT
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.17/0.40 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(complementImpComplementIntersect_type,type,
% 0.17/0.40 complementImpComplementIntersect: $o ).
% 0.17/0.40
% 0.17/0.40 thf(complementImpComplementIntersect,definition,
% 0.17/0.40 ( complementImpComplementIntersect
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.17/0.40 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(complementSubsetComplementIntersect_type,type,
% 0.17/0.40 complementSubsetComplementIntersect: $o ).
% 0.17/0.40
% 0.17/0.40 thf(complementSubsetComplementIntersect,definition,
% 0.17/0.40 ( complementSubsetComplementIntersect
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(complementInPowersetComplementIntersect_type,type,
% 0.17/0.40 complementInPowersetComplementIntersect: $o ).
% 0.17/0.40
% 0.17/0.40 thf(complementInPowersetComplementIntersect,definition,
% 0.17/0.40 ( complementInPowersetComplementIntersect
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(contraSubsetComplement_type,type,
% 0.17/0.40 contraSubsetComplement: $o ).
% 0.17/0.40
% 0.17/0.40 thf(contraSubsetComplement,definition,
% 0.17/0.40 ( contraSubsetComplement
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ Y )
% 0.17/0.40 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(complementTcontraSubset_type,type,
% 0.17/0.40 complementTcontraSubset: $o ).
% 0.17/0.40
% 0.17/0.40 thf(complementTcontraSubset,definition,
% 0.17/0.40 ( complementTcontraSubset
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.17/0.40 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionTILcontra_type,type,
% 0.17/0.40 binunionTILcontra: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionTILcontra,definition,
% 0.17/0.40 ( binunionTILcontra
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.17/0.40 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionTIRcontra_type,type,
% 0.17/0.40 binunionTIRcontra: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionTIRcontra,definition,
% 0.17/0.40 ( binunionTIRcontra
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.17/0.40 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(inIntersectImpInUnion_type,type,
% 0.17/0.40 inIntersectImpInUnion: $o ).
% 0.17/0.40
% 0.17/0.40 thf(inIntersectImpInUnion,definition,
% 0.17/0.40 ( inIntersectImpInUnion
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Z: $i] :
% 0.17/0.40 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.17/0.40 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(inIntersectImpInUnion2_type,type,
% 0.17/0.40 inIntersectImpInUnion2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(inIntersectImpInUnion2,definition,
% 0.17/0.40 ( inIntersectImpInUnion2
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Z: $i] :
% 0.17/0.40 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.17/0.40 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(inIntersectImpInIntersectUnions_type,type,
% 0.17/0.40 inIntersectImpInIntersectUnions: $o ).
% 0.17/0.40
% 0.17/0.40 thf(inIntersectImpInIntersectUnions,definition,
% 0.17/0.40 ( inIntersectImpInIntersectUnions
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Z: $i] :
% 0.17/0.40 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.17/0.40 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(intersectInPowersetIntersectUnions_type,type,
% 0.17/0.40 intersectInPowersetIntersectUnions: $o ).
% 0.17/0.40
% 0.17/0.40 thf(intersectInPowersetIntersectUnions,definition,
% 0.17/0.40 ( intersectInPowersetIntersectUnions
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Z: $i] :
% 0.17/0.40 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.40 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(inComplementUnionImpNotIn1_type,type,
% 0.17/0.40 inComplementUnionImpNotIn1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(inComplementUnionImpNotIn1,definition,
% 0.17/0.40 ( inComplementUnionImpNotIn1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.17/0.40 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(inComplementUnionImpInComplement1_type,type,
% 0.17/0.40 inComplementUnionImpInComplement1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(inComplementUnionImpInComplement1,definition,
% 0.17/0.40 ( inComplementUnionImpInComplement1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.17/0.40 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionTE_type,type,
% 0.17/0.40 binunionTE: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionTE,definition,
% 0.17/0.40 ( binunionTE
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xphi: $o,Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.17/0.40 => ( ( ( in @ Xx @ X )
% 0.17/0.40 => Xphi )
% 0.17/0.40 => ( ( ( in @ Xx @ Y )
% 0.17/0.40 => Xphi )
% 0.17/0.40 => Xphi ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(binunionTEcontra_type,type,
% 0.17/0.40 binunionTEcontra: $o ).
% 0.17/0.40
% 0.17/0.40 thf(binunionTEcontra,definition,
% 0.17/0.40 ( binunionTEcontra
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ~ ( in @ Xx @ X )
% 0.17/0.40 => ( ~ ( in @ Xx @ Y )
% 0.17/0.40 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(demorgan2a1_type,type,
% 0.17/0.40 demorgan2a1: $o ).
% 0.17/0.40
% 0.17/0.40 thf(demorgan2a1,definition,
% 0.17/0.40 ( demorgan2a1
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.17/0.40 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(complementUnionInPowersetComplement_type,type,
% 0.17/0.40 complementUnionInPowersetComplement: $o ).
% 0.17/0.40
% 0.17/0.40 thf(complementUnionInPowersetComplement,definition,
% 0.17/0.40 ( complementUnionInPowersetComplement
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(demorgan2a2_type,type,
% 0.17/0.40 demorgan2a2: $o ).
% 0.17/0.40
% 0.17/0.40 thf(demorgan2a2,definition,
% 0.17/0.40 ( demorgan2a2
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.17/0.40 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(demorgan1a_type,type,
% 0.17/0.40 demorgan1a: $o ).
% 0.17/0.40
% 0.17/0.40 thf(demorgan1a,definition,
% 0.17/0.40 ( demorgan1a
% 0.17/0.40 = ( ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.17/0.40 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 thf(demorgan1b,conjecture,
% 0.17/0.40 ( setextAx
% 0.17/0.40 => ( emptysetAx
% 0.17/0.40 => ( setadjoinAx
% 0.17/0.40 => ( powersetAx
% 0.17/0.40 => ( setunionAx
% 0.17/0.40 => ( omega0Ax
% 0.17/0.40 => ( omegaSAx
% 0.17/0.40 => ( omegaIndAx
% 0.17/0.40 => ( replAx
% 0.17/0.40 => ( foundationAx
% 0.17/0.40 => ( wellorderingAx
% 0.17/0.40 => ( descrp
% 0.17/0.40 => ( dsetconstrI
% 0.17/0.40 => ( dsetconstrEL
% 0.17/0.40 => ( dsetconstrER
% 0.17/0.40 => ( exuE1
% 0.17/0.40 => ( prop2setE
% 0.17/0.40 => ( emptysetE
% 0.17/0.40 => ( emptysetimpfalse
% 0.17/0.40 => ( notinemptyset
% 0.17/0.40 => ( exuE3e
% 0.17/0.40 => ( setext
% 0.17/0.40 => ( emptyI
% 0.17/0.40 => ( noeltsimpempty
% 0.17/0.40 => ( setbeta
% 0.17/0.40 => ( nonemptyE1
% 0.17/0.40 => ( nonemptyI
% 0.17/0.40 => ( nonemptyI1
% 0.17/0.40 => ( setadjoinIL
% 0.17/0.40 => ( emptyinunitempty
% 0.17/0.40 => ( setadjoinIR
% 0.17/0.40 => ( setadjoinE
% 0.17/0.40 => ( setadjoinOr
% 0.17/0.40 => ( setoftrueEq
% 0.17/0.40 => ( powersetI
% 0.17/0.40 => ( emptyinPowerset
% 0.17/0.40 => ( emptyInPowerset
% 0.17/0.40 => ( powersetE
% 0.17/0.40 => ( setunionI
% 0.17/0.40 => ( setunionE
% 0.17/0.40 => ( subPowSU
% 0.17/0.40 => ( exuE2
% 0.17/0.40 => ( nonemptyImpWitness
% 0.17/0.40 => ( uniqinunit
% 0.17/0.40 => ( notinsingleton
% 0.17/0.40 => ( eqinunit
% 0.17/0.40 => ( singletonsswitch
% 0.17/0.40 => ( upairsetE
% 0.17/0.40 => ( upairsetIL
% 0.17/0.40 => ( upairsetIR
% 0.17/0.40 => ( emptyE1
% 0.17/0.40 => ( vacuousDall
% 0.17/0.40 => ( quantDeMorgan1
% 0.17/0.40 => ( quantDeMorgan2
% 0.17/0.40 => ( quantDeMorgan3
% 0.17/0.40 => ( quantDeMorgan4
% 0.17/0.40 => ( prop2setI
% 0.17/0.40 => ( prop2set2propI
% 0.17/0.40 => ( notdexE
% 0.17/0.40 => ( notdallE
% 0.17/0.40 => ( exuI1
% 0.17/0.40 => ( exuI3
% 0.17/0.40 => ( exuI2
% 0.17/0.40 => ( inCongP
% 0.17/0.40 => ( in__Cong
% 0.17/0.40 => ( exuE3u
% 0.17/0.40 => ( exu__Cong
% 0.17/0.40 => ( emptyset__Cong
% 0.17/0.40 => ( setadjoin__Cong
% 0.17/0.40 => ( powerset__Cong
% 0.17/0.40 => ( setunion__Cong
% 0.17/0.40 => ( omega__Cong
% 0.17/0.40 => ( exuEu
% 0.17/0.40 => ( descr__Cong
% 0.17/0.40 => ( dsetconstr__Cong
% 0.17/0.40 => ( subsetI1
% 0.17/0.40 => ( eqimpsubset2
% 0.17/0.40 => ( eqimpsubset1
% 0.17/0.40 => ( subsetI2
% 0.17/0.40 => ( emptysetsubset
% 0.17/0.40 => ( subsetE
% 0.17/0.40 => ( subsetE2
% 0.17/0.40 => ( notsubsetI
% 0.17/0.40 => ( notequalI1
% 0.17/0.40 => ( notequalI2
% 0.17/0.40 => ( subsetRefl
% 0.17/0.40 => ( subsetTrans
% 0.17/0.40 => ( setadjoinSub
% 0.17/0.40 => ( setadjoinSub2
% 0.17/0.40 => ( subset2powerset
% 0.17/0.40 => ( setextsub
% 0.17/0.40 => ( subsetemptysetimpeq
% 0.17/0.40 => ( powersetI1
% 0.17/0.40 => ( powersetE1
% 0.17/0.40 => ( inPowerset
% 0.17/0.40 => ( powersetsubset
% 0.17/0.40 => ( sepInPowerset
% 0.17/0.40 => ( sepSubset
% 0.17/0.40 => ( binunionIL
% 0.17/0.40 => ( upairset2IR
% 0.17/0.40 => ( binunionIR
% 0.17/0.40 => ( binunionEcases
% 0.17/0.40 => ( binunionE
% 0.17/0.40 => ( binunionLsub
% 0.17/0.40 => ( binunionRsub
% 0.17/0.40 => ( binintersectI
% 0.17/0.40 => ( binintersectSubset5
% 0.17/0.40 => ( binintersectEL
% 0.17/0.40 => ( binintersectLsub
% 0.17/0.40 => ( binintersectSubset2
% 0.17/0.40 => ( binintersectSubset3
% 0.17/0.40 => ( binintersectER
% 0.17/0.40 => ( disjointsetsI1
% 0.17/0.40 => ( binintersectRsub
% 0.17/0.40 => ( binintersectSubset4
% 0.17/0.40 => ( binintersectSubset1
% 0.17/0.40 => ( bs114d
% 0.17/0.40 => ( setminusI
% 0.17/0.40 => ( setminusEL
% 0.17/0.40 => ( setminusER
% 0.17/0.40 => ( setminusSubset2
% 0.17/0.40 => ( setminusERneg
% 0.17/0.40 => ( setminusELneg
% 0.17/0.40 => ( setminusILneg
% 0.17/0.40 => ( setminusIRneg
% 0.17/0.40 => ( setminusLsub
% 0.17/0.40 => ( setminusSubset1
% 0.17/0.40 => ( symdiffE
% 0.17/0.40 => ( symdiffI1
% 0.17/0.40 => ( symdiffI2
% 0.17/0.40 => ( symdiffIneg1
% 0.17/0.40 => ( symdiffIneg2
% 0.17/0.40 => ( secondinupair
% 0.17/0.40 => ( setukpairIL
% 0.17/0.40 => ( setukpairIR
% 0.17/0.40 => ( kpairiskpair
% 0.17/0.40 => ( kpairp
% 0.17/0.40 => ( singletonsubset
% 0.17/0.40 => ( singletoninpowerset
% 0.17/0.40 => ( singletoninpowunion
% 0.17/0.40 => ( upairset2E
% 0.17/0.40 => ( upairsubunion
% 0.17/0.40 => ( upairinpowunion
% 0.17/0.40 => ( ubforcartprodlem1
% 0.17/0.40 => ( ubforcartprodlem2
% 0.17/0.40 => ( ubforcartprodlem3
% 0.17/0.40 => ( cartprodpairin
% 0.17/0.40 => ( cartprodmempair1
% 0.17/0.40 => ( cartprodmempair
% 0.17/0.40 => ( setunionE2
% 0.17/0.40 => ( setunionsingleton1
% 0.17/0.40 => ( setunionsingleton2
% 0.17/0.40 => ( setunionsingleton
% 0.17/0.40 => ( singletonprop
% 0.17/0.40 => ( ex1E1
% 0.17/0.40 => ( ex1I
% 0.17/0.40 => ( ex1I2
% 0.17/0.40 => ( singletonsuniq
% 0.17/0.40 => ( setukpairinjL1
% 0.17/0.40 => ( kfstsingleton
% 0.17/0.40 => ( theprop
% 0.17/0.40 => ( kfstpairEq
% 0.17/0.40 => ( cartprodfstin
% 0.17/0.40 => ( setukpairinjL2
% 0.17/0.40 => ( setukpairinjL
% 0.17/0.40 => ( setukpairinjR11
% 0.17/0.40 => ( setukpairinjR12
% 0.17/0.40 => ( setukpairinjR1
% 0.17/0.40 => ( upairequniteq
% 0.17/0.40 => ( setukpairinjR2
% 0.17/0.40 => ( setukpairinjR
% 0.17/0.40 => ( ksndsingleton
% 0.17/0.40 => ( ksndpairEq
% 0.17/0.40 => ( kpairsurjEq
% 0.17/0.40 => ( cartprodsndin
% 0.17/0.40 => ( cartprodpairmemEL
% 0.17/0.40 => ( cartprodpairmemER
% 0.17/0.40 => ( cartprodmempaircEq
% 0.17/0.40 => ( cartprodfstpairEq
% 0.17/0.40 => ( cartprodsndpairEq
% 0.17/0.40 => ( cartprodpairsurjEq
% 0.17/0.40 => ( dpsetconstrI
% 0.17/0.40 => ( dpsetconstrSub
% 0.17/0.40 => ( setOfPairsIsBReln
% 0.17/0.40 => ( dpsetconstrERa
% 0.17/0.40 => ( dpsetconstrEL1
% 0.17/0.40 => ( dpsetconstrEL2
% 0.17/0.40 => ( dpsetconstrER
% 0.17/0.40 => ( funcImageSingleton
% 0.17/0.40 => ( apProp
% 0.17/0.40 => ( app
% 0.17/0.40 => ( infuncsetfunc
% 0.17/0.40 => ( ap2p
% 0.17/0.40 => ( funcinfuncset
% 0.17/0.40 => ( lamProp
% 0.17/0.40 => ( lamp
% 0.17/0.40 => ( lam2p
% 0.17/0.40 => ( brelnall1
% 0.17/0.40 => ( brelnall2
% 0.17/0.40 => ( ex1E2
% 0.17/0.40 => ( funcGraphProp1
% 0.17/0.40 => ( funcGraphProp3
% 0.17/0.40 => ( funcGraphProp2
% 0.17/0.40 => ( funcextLem
% 0.17/0.40 => ( funcGraphProp4
% 0.17/0.40 => ( subbreln
% 0.17/0.40 => ( eqbreln
% 0.17/0.40 => ( funcext
% 0.17/0.40 => ( funcext2
% 0.17/0.40 => ( ap2apEq1
% 0.17/0.40 => ( ap2apEq2
% 0.17/0.40 => ( beta1
% 0.17/0.40 => ( eta1
% 0.17/0.40 => ( lam2lamEq
% 0.17/0.40 => ( beta2
% 0.17/0.40 => ( eta2
% 0.17/0.40 => ( iffalseProp1
% 0.17/0.40 => ( iffalseProp2
% 0.17/0.40 => ( iftrueProp1
% 0.17/0.40 => ( iftrueProp2
% 0.17/0.40 => ( ifSingleton
% 0.17/0.40 => ( ifp
% 0.17/0.40 => ( theeq
% 0.17/0.40 => ( iftrue
% 0.17/0.40 => ( iffalse
% 0.17/0.40 => ( iftrueorfalse
% 0.17/0.40 => ( binintersectT_lem
% 0.17/0.40 => ( binunionT_lem
% 0.17/0.40 => ( powersetT_lem
% 0.17/0.40 => ( setminusT_lem
% 0.17/0.40 => ( complementT_lem
% 0.17/0.40 => ( setextT
% 0.17/0.40 => ( subsetTI
% 0.17/0.40 => ( powersetTI1
% 0.17/0.40 => ( powersetTE1
% 0.17/0.40 => ( complementTI1
% 0.17/0.40 => ( complementTE1
% 0.17/0.40 => ( binintersectTELcontra
% 0.17/0.40 => ( binintersectTERcontra
% 0.17/0.40 => ( contrasubsetT
% 0.17/0.40 => ( contrasubsetT1
% 0.17/0.40 => ( contrasubsetT2
% 0.17/0.40 => ( contrasubsetT3
% 0.17/0.40 => ( doubleComplementI1
% 0.17/0.40 => ( doubleComplementE1
% 0.17/0.40 => ( doubleComplementSub1
% 0.17/0.40 => ( doubleComplementSub2
% 0.17/0.40 => ( doubleComplementEq
% 0.17/0.40 => ( complementTnotintersectT
% 0.17/0.40 => ( complementImpComplementIntersect
% 0.17/0.40 => ( complementSubsetComplementIntersect
% 0.17/0.40 => ( complementInPowersetComplementIntersect
% 0.17/0.40 => ( contraSubsetComplement
% 0.17/0.40 => ( complementTcontraSubset
% 0.17/0.40 => ( binunionTILcontra
% 0.17/0.40 => ( binunionTIRcontra
% 0.17/0.40 => ( inIntersectImpInUnion
% 0.17/0.40 => ( inIntersectImpInUnion2
% 0.17/0.40 => ( inIntersectImpInIntersectUnions
% 0.17/0.40 => ( intersectInPowersetIntersectUnions
% 0.17/0.40 => ( inComplementUnionImpNotIn1
% 0.17/0.40 => ( inComplementUnionImpInComplement1
% 0.17/0.40 => ( binunionTE
% 0.17/0.40 => ( binunionTEcontra
% 0.17/0.40 => ( demorgan2a1
% 0.17/0.40 => ( complementUnionInPowersetComplement
% 0.17/0.40 => ( demorgan2a2
% 0.17/0.40 => ( demorgan1a
% 0.17/0.40 => ! [A: $i,X: $i] :
% 0.17/0.40 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.40 => ! [Y: $i] :
% 0.17/0.40 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.40 => ! [Xx: $i] :
% 0.17/0.40 ( ( in @ Xx @ A )
% 0.17/0.40 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.17/0.40 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.40
% 0.17/0.40 %------------------------------------------------------------------------------
% 0.17/0.40 ------------------------
% 3.36/1.73 ---- Embedded in THF ---
% 3.36/1.73 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.36/1.73 %%% Generated on Tue Feb 24 12:32:46 EST 2026
% 3.36/1.73 %%% using '$$dhol' embedding, version 1.3.0.
% 3.36/1.73 %%% Logic specification used:
% 3.36/1.73 %%% thf(spec, logic, $$dhol).
% 3.36/1.73
% 3.36/1.73 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2DTF13645.p
% See solution above
% 3.36/1.73 ------------------------
% 3.36/1.79 ---- Cleaned THF ---
% 3.36/1.79 thf(in_type,type,
% 3.36/1.79 in: $i > $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(exu_type,type,
% 3.36/1.79 exu: ( $i > $o ) > $o ).
% 3.36/1.79
% 3.36/1.79 thf(exu,definition,
% 3.36/1.79 ( exu
% 3.36/1.79 = ( ^ [Xphi: $i > $o] :
% 3.36/1.79 ? [Xx: $i] :
% 3.36/1.79 ( ( Xphi @ Xx )
% 3.36/1.79 & ! [Xy: $i] :
% 3.36/1.79 ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setextAx_type,type,
% 3.36/1.79 setextAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setextAx,definition,
% 3.36/1.79 ( setextAx
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 <=> ( in @ Xx @ B ) )
% 3.36/1.79 => ( A = B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptyset_type,type,
% 3.36/1.79 emptyset: $i ).
% 3.36/1.79
% 3.36/1.79 thf(emptysetAx_type,type,
% 3.36/1.79 emptysetAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptysetAx,definition,
% 3.36/1.79 ( emptysetAx
% 3.36/1.79 = ( ! [Xx: $i] :
% 3.36/1.79 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoin_type,type,
% 3.36/1.79 setadjoin: $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinAx_type,type,
% 3.36/1.79 setadjoinAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinAx,definition,
% 3.36/1.79 ( setadjoinAx
% 3.36/1.79 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.36/1.79 <=> ( ( Xy = Xx )
% 3.36/1.79 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powerset_type,type,
% 3.36/1.79 powerset: $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(powersetAx_type,type,
% 3.36/1.79 powersetAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powersetAx,definition,
% 3.36/1.79 ( powersetAx
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( in @ B @ ( powerset @ A ) )
% 3.36/1.79 <=> ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setunion_type,type,
% 3.36/1.79 setunion: $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(setunionAx_type,type,
% 3.36/1.79 setunionAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setunionAx,definition,
% 3.36/1.79 ( setunionAx
% 3.36/1.79 = ( ! [A: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( setunion @ A ) )
% 3.36/1.79 <=> ? [B: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 & ( in @ B @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(omega_type,type,
% 3.36/1.79 omega: $i ).
% 3.36/1.79
% 3.36/1.79 thf(omega0Ax_type,type,
% 3.36/1.79 omega0Ax: $o ).
% 3.36/1.79
% 3.36/1.79 thf(omega0Ax,definition,
% 3.36/1.79 ( omega0Ax
% 3.36/1.79 = ( in @ emptyset @ omega ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(omegaSAx_type,type,
% 3.36/1.79 omegaSAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(omegaSAx,definition,
% 3.36/1.79 ( omegaSAx
% 3.36/1.79 = ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ omega )
% 3.36/1.79 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(omegaIndAx_type,type,
% 3.36/1.79 omegaIndAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(omegaIndAx,definition,
% 3.36/1.79 ( omegaIndAx
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ( ( ( in @ emptyset @ A )
% 3.36/1.79 & ! [Xx: $i] :
% 3.36/1.79 ( ( ( in @ Xx @ omega )
% 3.36/1.79 & ( in @ Xx @ A ) )
% 3.36/1.79 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ omega )
% 3.36/1.79 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(replAx_type,type,
% 3.36/1.79 replAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(replAx,definition,
% 3.36/1.79 ( replAx
% 3.36/1.79 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( exu
% 3.36/1.79 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.36/1.79 => ? [B: $i] :
% 3.36/1.79 ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 <=> ? [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(foundationAx_type,type,
% 3.36/1.79 foundationAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(foundationAx,definition,
% 3.36/1.79 ( foundationAx
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.36/1.79 => ? [B: $i] :
% 3.36/1.79 ( ( in @ B @ A )
% 3.36/1.79 & ~ ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(wellorderingAx_type,type,
% 3.36/1.79 wellorderingAx: $o ).
% 3.36/1.79
% 3.36/1.79 thf(wellorderingAx,definition,
% 3.36/1.79 ( wellorderingAx
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ? [B: $i] :
% 3.36/1.79 ( ! [C: $i] :
% 3.36/1.79 ( ( in @ C @ B )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ C )
% 3.36/1.79 => ( in @ Xx @ A ) ) )
% 3.36/1.79 & ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( ( in @ Xx @ A )
% 3.36/1.79 & ( in @ Xy @ A ) )
% 3.36/1.79 => ( ! [C: $i] :
% 3.36/1.79 ( ( in @ C @ B )
% 3.36/1.79 => ( ( in @ Xx @ C )
% 3.36/1.79 <=> ( in @ Xy @ C ) ) )
% 3.36/1.79 => ( Xx = Xy ) ) )
% 3.36/1.79 & ! [C: $i,D: $i] :
% 3.36/1.79 ( ( ( in @ C @ B )
% 3.36/1.79 & ( in @ D @ B ) )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ C )
% 3.36/1.79 => ( in @ Xx @ D ) )
% 3.36/1.79 | ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ D )
% 3.36/1.79 => ( in @ Xx @ C ) ) ) )
% 3.36/1.79 & ! [C: $i] :
% 3.36/1.79 ( ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ C )
% 3.36/1.79 => ( in @ Xx @ A ) )
% 3.36/1.79 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.36/1.79 => ? [D: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ D @ B )
% 3.36/1.79 & ( in @ Xx @ C )
% 3.36/1.79 & ~ ? [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ D )
% 3.36/1.79 & ( in @ Xy @ C ) )
% 3.36/1.79 & ! [E: $i] :
% 3.36/1.79 ( ( in @ E @ B )
% 3.36/1.79 => ( ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ E )
% 3.36/1.79 => ( in @ Xy @ D ) )
% 3.36/1.79 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(descr_type,type,
% 3.36/1.79 descr: ( $i > $o ) > $i ).
% 3.36/1.79
% 3.36/1.79 thf(descrp_type,type,
% 3.36/1.79 descrp: $o ).
% 3.36/1.79
% 3.36/1.79 thf(descrp,definition,
% 3.36/1.79 ( descrp
% 3.36/1.79 = ( ! [Xphi: $i > $o] :
% 3.36/1.79 ( ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 => ( Xphi
% 3.36/1.79 @ ( descr
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dsetconstr_type,type,
% 3.36/1.79 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.36/1.79
% 3.36/1.79 thf(dsetconstrI_type,type,
% 3.36/1.79 dsetconstrI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dsetconstrI,definition,
% 3.36/1.79 ( dsetconstrI
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( Xphi @ Xx )
% 3.36/1.79 => ( in @ Xx
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dsetconstrEL_type,type,
% 3.36/1.79 dsetconstrEL: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dsetconstrEL,definition,
% 3.36/1.79 ( dsetconstrEL
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.36/1.79 => ( in @ Xx @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dsetconstrER_type,type,
% 3.36/1.79 dsetconstrER: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dsetconstrER,definition,
% 3.36/1.79 ( dsetconstrER
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.36/1.79 => ( Xphi @ Xx ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(exuE1_type,type,
% 3.36/1.79 exuE1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(exuE1,definition,
% 3.36/1.79 ( exuE1
% 3.36/1.79 = ( ! [Xphi: $i > $o] :
% 3.36/1.79 ( ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 => ? [Xx: $i] :
% 3.36/1.79 ( ( Xphi @ Xx )
% 3.36/1.79 & ! [Xy: $i] :
% 3.36/1.79 ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(prop2set_type,type,
% 3.36/1.79 prop2set: $o > $i ).
% 3.36/1.79
% 3.36/1.79 thf(prop2setE_type,type,
% 3.36/1.79 prop2setE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(prop2setE,definition,
% 3.36/1.79 ( prop2setE
% 3.36/1.79 = ( ! [Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.36/1.79 => Xphi ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptysetE_type,type,
% 3.36/1.79 emptysetE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptysetE,definition,
% 3.36/1.79 ( emptysetE
% 3.36/1.79 = ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ emptyset )
% 3.36/1.79 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptysetimpfalse_type,type,
% 3.36/1.79 emptysetimpfalse: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptysetimpfalse,definition,
% 3.36/1.79 ( emptysetimpfalse
% 3.36/1.79 = ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ emptyset )
% 3.36/1.79 => $false ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(notinemptyset_type,type,
% 3.36/1.79 notinemptyset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(notinemptyset,definition,
% 3.36/1.79 ( notinemptyset
% 3.36/1.79 = ( ! [Xx: $i] :
% 3.36/1.79 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(exuE3e_type,type,
% 3.36/1.79 exuE3e: $o ).
% 3.36/1.79
% 3.36/1.79 thf(exuE3e,definition,
% 3.36/1.79 ( exuE3e
% 3.36/1.79 = ( ! [Xphi: $i > $o] :
% 3.36/1.79 ( ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setext_type,type,
% 3.36/1.79 setext: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setext,definition,
% 3.36/1.79 ( setext
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ Xx @ B ) )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 => ( in @ Xx @ A ) )
% 3.36/1.79 => ( A = B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptyI_type,type,
% 3.36/1.79 emptyI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptyI,definition,
% 3.36/1.79 ( emptyI
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ~ ( in @ Xx @ A )
% 3.36/1.79 => ( A = emptyset ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(noeltsimpempty_type,type,
% 3.36/1.79 noeltsimpempty: $o ).
% 3.36/1.79
% 3.36/1.79 thf(noeltsimpempty,definition,
% 3.36/1.79 ( noeltsimpempty
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ~ ( in @ Xx @ A )
% 3.36/1.79 => ( A = emptyset ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setbeta_type,type,
% 3.36/1.79 setbeta: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setbeta,definition,
% 3.36/1.79 ( setbeta
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.36/1.79 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(nonempty_type,type,
% 3.36/1.79 nonempty: $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(nonempty,definition,
% 3.36/1.79 ( nonempty
% 3.36/1.79 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(nonemptyE1_type,type,
% 3.36/1.79 nonemptyE1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(nonemptyE1,definition,
% 3.36/1.79 ( nonemptyE1
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ( ( nonempty @ A )
% 3.36/1.79 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(nonemptyI_type,type,
% 3.36/1.79 nonemptyI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(nonemptyI,definition,
% 3.36/1.79 ( nonemptyI
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( Xphi @ Xx )
% 3.36/1.79 => ( nonempty
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(nonemptyI1_type,type,
% 3.36/1.79 nonemptyI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(nonemptyI1,definition,
% 3.36/1.79 ( nonemptyI1
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.36/1.79 => ( nonempty @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinIL_type,type,
% 3.36/1.79 setadjoinIL: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinIL,definition,
% 3.36/1.79 ( setadjoinIL
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptyinunitempty_type,type,
% 3.36/1.79 emptyinunitempty: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptyinunitempty,definition,
% 3.36/1.79 ( emptyinunitempty
% 3.36/1.79 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinIR_type,type,
% 3.36/1.79 setadjoinIR: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinIR,definition,
% 3.36/1.79 ( setadjoinIR
% 3.36/1.79 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinE_type,type,
% 3.36/1.79 setadjoinE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinE,definition,
% 3.36/1.79 ( setadjoinE
% 3.36/1.79 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.36/1.79 => ! [Xphi: $o] :
% 3.36/1.79 ( ( ( Xy = Xx )
% 3.36/1.79 => Xphi )
% 3.36/1.79 => ( ( ( in @ Xy @ A )
% 3.36/1.79 => Xphi )
% 3.36/1.79 => Xphi ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinOr_type,type,
% 3.36/1.79 setadjoinOr: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinOr,definition,
% 3.36/1.79 ( setadjoinOr
% 3.36/1.79 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.36/1.79 => ( ( Xy = Xx )
% 3.36/1.79 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setoftrueEq_type,type,
% 3.36/1.79 setoftrueEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setoftrueEq,definition,
% 3.36/1.79 ( setoftrueEq
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ( ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xx: $i] : $true )
% 3.36/1.79 = A ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powersetI_type,type,
% 3.36/1.79 powersetI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powersetI,definition,
% 3.36/1.79 ( powersetI
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 => ( in @ Xx @ A ) )
% 3.36/1.79 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptyinPowerset_type,type,
% 3.36/1.79 emptyinPowerset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptyinPowerset,definition,
% 3.36/1.79 ( emptyinPowerset
% 3.36/1.79 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptyInPowerset_type,type,
% 3.36/1.79 emptyInPowerset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptyInPowerset,definition,
% 3.36/1.79 ( emptyInPowerset
% 3.36/1.79 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powersetE_type,type,
% 3.36/1.79 powersetE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powersetE,definition,
% 3.36/1.79 ( powersetE
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ B @ ( powerset @ A ) )
% 3.36/1.79 => ( ( in @ Xx @ B )
% 3.36/1.79 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setunionI_type,type,
% 3.36/1.79 setunionI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setunionI,definition,
% 3.36/1.79 ( setunionI
% 3.36/1.79 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 => ( ( in @ B @ A )
% 3.36/1.79 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setunionE_type,type,
% 3.36/1.79 setunionE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setunionE,definition,
% 3.36/1.79 ( setunionE
% 3.36/1.79 = ( ! [A: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( setunion @ A ) )
% 3.36/1.79 => ! [Xphi: $o] :
% 3.36/1.79 ( ! [B: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 => ( ( in @ B @ A )
% 3.36/1.79 => Xphi ) )
% 3.36/1.79 => Xphi ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subPowSU_type,type,
% 3.36/1.79 subPowSU: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subPowSU,definition,
% 3.36/1.79 ( subPowSU
% 3.36/1.79 = ( ! [A: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(exuE2_type,type,
% 3.36/1.79 exuE2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(exuE2,definition,
% 3.36/1.79 ( exuE2
% 3.36/1.79 = ( ! [Xphi: $i > $o] :
% 3.36/1.79 ( ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 => ? [Xx: $i] :
% 3.36/1.79 ! [Xy: $i] :
% 3.36/1.79 ( ( Xphi @ Xy )
% 3.36/1.79 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(nonemptyImpWitness_type,type,
% 3.36/1.79 nonemptyImpWitness: $o ).
% 3.36/1.79
% 3.36/1.79 thf(nonemptyImpWitness,definition,
% 3.36/1.79 ( nonemptyImpWitness
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ( ( nonempty @ A )
% 3.36/1.79 => ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & $true ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(uniqinunit_type,type,
% 3.36/1.79 uniqinunit: $o ).
% 3.36/1.79
% 3.36/1.79 thf(uniqinunit,definition,
% 3.36/1.79 ( uniqinunit
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(notinsingleton_type,type,
% 3.36/1.79 notinsingleton: $o ).
% 3.36/1.79
% 3.36/1.79 thf(notinsingleton,definition,
% 3.36/1.79 ( notinsingleton
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xx != Xy )
% 3.36/1.79 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(eqinunit_type,type,
% 3.36/1.79 eqinunit: $o ).
% 3.36/1.79
% 3.36/1.79 thf(eqinunit,definition,
% 3.36/1.79 ( eqinunit
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xx = Xy )
% 3.36/1.79 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(singletonsswitch_type,type,
% 3.36/1.79 singletonsswitch: $o ).
% 3.36/1.79
% 3.36/1.79 thf(singletonsswitch,definition,
% 3.36/1.79 ( singletonsswitch
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.36/1.79 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(upairsetE_type,type,
% 3.36/1.79 upairsetE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(upairsetE,definition,
% 3.36/1.79 ( upairsetE
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.36/1.79 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.36/1.79 => ( ( Xz = Xx )
% 3.36/1.79 | ( Xz = Xy ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(upairsetIL_type,type,
% 3.36/1.79 upairsetIL: $o ).
% 3.36/1.79
% 3.36/1.79 thf(upairsetIL,definition,
% 3.36/1.79 ( upairsetIL
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(upairsetIR_type,type,
% 3.36/1.79 upairsetIR: $o ).
% 3.36/1.79
% 3.36/1.79 thf(upairsetIR,definition,
% 3.36/1.79 ( upairsetIR
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptyE1_type,type,
% 3.36/1.79 emptyE1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptyE1,definition,
% 3.36/1.79 ( emptyE1
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ( Xphi @ Xx ) )
% 3.36/1.79 => ( ( ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 = emptyset )
% 3.36/1.79 => $false ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(vacuousDall_type,type,
% 3.36/1.79 vacuousDall: $o ).
% 3.36/1.79
% 3.36/1.79 thf(vacuousDall,definition,
% 3.36/1.79 ( vacuousDall
% 3.36/1.79 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ emptyset )
% 3.36/1.79 => ( Xphi @ Xx ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(quantDeMorgan1_type,type,
% 3.36/1.79 quantDeMorgan1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(quantDeMorgan1,definition,
% 3.36/1.79 ( quantDeMorgan1
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ~ ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( Xphi @ Xx ) )
% 3.36/1.79 => ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(quantDeMorgan2_type,type,
% 3.36/1.79 quantDeMorgan2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(quantDeMorgan2,definition,
% 3.36/1.79 ( quantDeMorgan2
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ~ ( Xphi @ Xx ) )
% 3.36/1.79 => ~ ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(quantDeMorgan3_type,type,
% 3.36/1.79 quantDeMorgan3: $o ).
% 3.36/1.79
% 3.36/1.79 thf(quantDeMorgan3,definition,
% 3.36/1.79 ( quantDeMorgan3
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ~ ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ( Xphi @ Xx ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(quantDeMorgan4_type,type,
% 3.36/1.79 quantDeMorgan4: $o ).
% 3.36/1.79
% 3.36/1.79 thf(quantDeMorgan4,definition,
% 3.36/1.79 ( quantDeMorgan4
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ~ ( Xphi @ Xx ) )
% 3.36/1.79 => ~ ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(prop2setI_type,type,
% 3.36/1.79 prop2setI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(prop2setI,definition,
% 3.36/1.79 ( prop2setI
% 3.36/1.79 = ( ! [Xphi: $o] :
% 3.36/1.79 ( Xphi
% 3.36/1.79 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(set2prop_type,type,
% 3.36/1.79 set2prop: $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(prop2set2propI_type,type,
% 3.36/1.79 prop2set2propI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(prop2set2propI,definition,
% 3.36/1.79 ( prop2set2propI
% 3.36/1.79 = ( ! [Xphi: $o] :
% 3.36/1.79 ( Xphi
% 3.36/1.79 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(notdexE_type,type,
% 3.36/1.79 notdexE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(notdexE,definition,
% 3.36/1.79 ( notdexE
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ~ ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ( Xphi @ Xx ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(notdallE_type,type,
% 3.36/1.79 notdallE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(notdallE,definition,
% 3.36/1.79 ( notdallE
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ~ ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( Xphi @ Xx ) )
% 3.36/1.79 => ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(exuI1_type,type,
% 3.36/1.79 exuI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(exuI1,definition,
% 3.36/1.79 ( exuI1
% 3.36/1.79 = ( ! [Xphi: $i > $o] :
% 3.36/1.79 ( ? [Xx: $i] :
% 3.36/1.79 ( ( Xphi @ Xx )
% 3.36/1.79 & ! [Xy: $i] :
% 3.36/1.79 ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xx = Xy ) ) )
% 3.36/1.79 => ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(exuI3_type,type,
% 3.36/1.79 exuI3: $o ).
% 3.36/1.79
% 3.36/1.79 thf(exuI3,definition,
% 3.36/1.79 ( exuI3
% 3.36/1.79 = ( ! [Xphi: $i > $o] :
% 3.36/1.79 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.36/1.79 => ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xphi @ Xx )
% 3.36/1.79 => ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xx = Xy ) ) )
% 3.36/1.79 => ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(exuI2_type,type,
% 3.36/1.79 exuI2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(exuI2,definition,
% 3.36/1.79 ( exuI2
% 3.36/1.79 = ( ! [Xphi: $i > $o] :
% 3.36/1.79 ( ? [Xx: $i] :
% 3.36/1.79 ! [Xy: $i] :
% 3.36/1.79 ( ( Xphi @ Xy )
% 3.36/1.79 <=> ( Xy = Xx ) )
% 3.36/1.79 => ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(inCongP_type,type,
% 3.36/1.79 inCongP: $o ).
% 3.36/1.79
% 3.36/1.79 thf(inCongP,definition,
% 3.36/1.79 ( inCongP
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( A = B )
% 3.36/1.79 => ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xx = Xy )
% 3.36/1.79 => ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(in__Cong_type,type,
% 3.36/1.79 in__Cong: $o ).
% 3.36/1.79
% 3.36/1.79 thf(in__Cong,definition,
% 3.36/1.79 ( in__Cong
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( A = B )
% 3.36/1.79 => ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xx = Xy )
% 3.36/1.79 => ( ( in @ Xx @ A )
% 3.36/1.79 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(exuE3u_type,type,
% 3.36/1.79 exuE3u: $o ).
% 3.36/1.79
% 3.36/1.79 thf(exuE3u,definition,
% 3.36/1.79 ( exuE3u
% 3.36/1.79 = ( ! [Xphi: $i > $o] :
% 3.36/1.79 ( ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 => ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xphi @ Xx )
% 3.36/1.79 => ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(exu__Cong_type,type,
% 3.36/1.79 exu__Cong: $o ).
% 3.36/1.79
% 3.36/1.79 thf(exu__Cong,definition,
% 3.36/1.79 ( exu__Cong
% 3.36/1.79 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.36/1.79 ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xx = Xy )
% 3.36/1.79 => ( ( Xphi @ Xx )
% 3.36/1.79 <=> ( Xpsi @ Xy ) ) )
% 3.36/1.79 => ( ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 <=> ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptyset__Cong_type,type,
% 3.36/1.79 emptyset__Cong: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptyset__Cong,definition,
% 3.36/1.79 ( emptyset__Cong
% 3.36/1.79 = ( emptyset = emptyset ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoin__Cong_type,type,
% 3.36/1.79 setadjoin__Cong: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoin__Cong,definition,
% 3.36/1.79 ( setadjoin__Cong
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xx = Xy )
% 3.36/1.79 => ! [Xz: $i,Xu: $i] :
% 3.36/1.79 ( ( Xz = Xu )
% 3.36/1.79 => ( ( setadjoin @ Xx @ Xz )
% 3.36/1.79 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powerset__Cong_type,type,
% 3.36/1.79 powerset__Cong: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powerset__Cong,definition,
% 3.36/1.79 ( powerset__Cong
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( A = B )
% 3.36/1.79 => ( ( powerset @ A )
% 3.36/1.79 = ( powerset @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setunion__Cong_type,type,
% 3.36/1.79 setunion__Cong: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setunion__Cong,definition,
% 3.36/1.79 ( setunion__Cong
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( A = B )
% 3.36/1.79 => ( ( setunion @ A )
% 3.36/1.79 = ( setunion @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(omega__Cong_type,type,
% 3.36/1.79 omega__Cong: $o ).
% 3.36/1.79
% 3.36/1.79 thf(omega__Cong,definition,
% 3.36/1.79 ( omega__Cong
% 3.36/1.79 = ( omega = omega ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(exuEu_type,type,
% 3.36/1.79 exuEu: $o ).
% 3.36/1.79
% 3.36/1.79 thf(exuEu,definition,
% 3.36/1.79 ( exuEu
% 3.36/1.79 = ( ! [Xphi: $i > $o] :
% 3.36/1.79 ( ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 => ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xphi @ Xx )
% 3.36/1.79 => ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(descr__Cong_type,type,
% 3.36/1.79 descr__Cong: $o ).
% 3.36/1.79
% 3.36/1.79 thf(descr__Cong,definition,
% 3.36/1.79 ( descr__Cong
% 3.36/1.79 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.36/1.79 ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xx = Xy )
% 3.36/1.79 => ( ( Xphi @ Xx )
% 3.36/1.79 <=> ( Xpsi @ Xy ) ) )
% 3.36/1.79 => ( ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 => ( ( exu
% 3.36/1.79 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.36/1.79 => ( ( descr
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 = ( descr
% 3.36/1.79 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dsetconstr__Cong_type,type,
% 3.36/1.79 dsetconstr__Cong: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dsetconstr__Cong,definition,
% 3.36/1.79 ( dsetconstr__Cong
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( A = B )
% 3.36/1.79 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( Xx = Xy )
% 3.36/1.79 => ( ( Xphi @ Xx )
% 3.36/1.79 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.36/1.79 => ( ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 = ( dsetconstr @ B
% 3.36/1.79 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subset_type,type,
% 3.36/1.79 subset: $i > $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(disjoint_type,type,
% 3.36/1.79 disjoint: $i > $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(setsmeet_type,type,
% 3.36/1.79 setsmeet: $i > $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(subsetI1_type,type,
% 3.36/1.79 subsetI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subsetI1,definition,
% 3.36/1.79 ( subsetI1
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ Xx @ B ) )
% 3.36/1.79 => ( subset @ A @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(eqimpsubset2_type,type,
% 3.36/1.79 eqimpsubset2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(eqimpsubset2,definition,
% 3.36/1.79 ( eqimpsubset2
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( A = B )
% 3.36/1.79 => ( subset @ B @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(eqimpsubset1_type,type,
% 3.36/1.79 eqimpsubset1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(eqimpsubset1,definition,
% 3.36/1.79 ( eqimpsubset1
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( A = B )
% 3.36/1.79 => ( subset @ A @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subsetI2_type,type,
% 3.36/1.79 subsetI2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subsetI2,definition,
% 3.36/1.79 ( subsetI2
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ Xx @ B ) )
% 3.36/1.79 => ( subset @ A @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(emptysetsubset_type,type,
% 3.36/1.79 emptysetsubset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(emptysetsubset,definition,
% 3.36/1.79 ( emptysetsubset
% 3.36/1.79 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subsetE_type,type,
% 3.36/1.79 subsetE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subsetE,definition,
% 3.36/1.79 ( subsetE
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( subset @ A @ B )
% 3.36/1.79 => ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subsetE2_type,type,
% 3.36/1.79 subsetE2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subsetE2,definition,
% 3.36/1.79 ( subsetE2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( subset @ A @ B )
% 3.36/1.79 => ( ~ ( in @ Xx @ B )
% 3.36/1.79 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(notsubsetI_type,type,
% 3.36/1.79 notsubsetI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(notsubsetI,definition,
% 3.36/1.79 ( notsubsetI
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ~ ( in @ Xx @ B )
% 3.36/1.79 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(notequalI1_type,type,
% 3.36/1.79 notequalI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(notequalI1,definition,
% 3.36/1.79 ( notequalI1
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ~ ( subset @ A @ B )
% 3.36/1.79 => ( A != B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(notequalI2_type,type,
% 3.36/1.79 notequalI2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(notequalI2,definition,
% 3.36/1.79 ( notequalI2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ~ ( in @ Xx @ B )
% 3.36/1.79 => ( A != B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subsetRefl_type,type,
% 3.36/1.79 subsetRefl: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subsetRefl,definition,
% 3.36/1.79 ( subsetRefl
% 3.36/1.79 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subsetTrans_type,type,
% 3.36/1.79 subsetTrans: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subsetTrans,definition,
% 3.36/1.79 ( subsetTrans
% 3.36/1.79 = ( ! [A: $i,B: $i,C: $i] :
% 3.36/1.79 ( ( subset @ A @ B )
% 3.36/1.79 => ( ( subset @ B @ C )
% 3.36/1.79 => ( subset @ A @ C ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinSub_type,type,
% 3.36/1.79 setadjoinSub: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinSub,definition,
% 3.36/1.79 ( setadjoinSub
% 3.36/1.79 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinSub2_type,type,
% 3.36/1.79 setadjoinSub2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setadjoinSub2,definition,
% 3.36/1.79 ( setadjoinSub2
% 3.36/1.79 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.36/1.79 ( ( subset @ A @ B )
% 3.36/1.79 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subset2powerset_type,type,
% 3.36/1.79 subset2powerset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subset2powerset,definition,
% 3.36/1.79 ( subset2powerset
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( subset @ A @ B )
% 3.36/1.79 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setextsub_type,type,
% 3.36/1.79 setextsub: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setextsub,definition,
% 3.36/1.79 ( setextsub
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( subset @ A @ B )
% 3.36/1.79 => ( ( subset @ B @ A )
% 3.36/1.79 => ( A = B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subsetemptysetimpeq_type,type,
% 3.36/1.79 subsetemptysetimpeq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subsetemptysetimpeq,definition,
% 3.36/1.79 ( subsetemptysetimpeq
% 3.36/1.79 = ( ! [A: $i] :
% 3.36/1.79 ( ( subset @ A @ emptyset )
% 3.36/1.79 => ( A = emptyset ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powersetI1_type,type,
% 3.36/1.79 powersetI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powersetI1,definition,
% 3.36/1.79 ( powersetI1
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( subset @ B @ A )
% 3.36/1.79 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powersetE1_type,type,
% 3.36/1.79 powersetE1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powersetE1,definition,
% 3.36/1.79 ( powersetE1
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( in @ B @ ( powerset @ A ) )
% 3.36/1.79 => ( subset @ B @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(inPowerset_type,type,
% 3.36/1.79 inPowerset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(inPowerset,definition,
% 3.36/1.79 ( inPowerset
% 3.36/1.79 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powersetsubset_type,type,
% 3.36/1.79 powersetsubset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powersetsubset,definition,
% 3.36/1.79 ( powersetsubset
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( subset @ A @ B )
% 3.36/1.79 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(sepInPowerset_type,type,
% 3.36/1.79 sepInPowerset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(sepInPowerset,definition,
% 3.36/1.79 ( sepInPowerset
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( in
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 @ ( powerset @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(sepSubset_type,type,
% 3.36/1.79 sepSubset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(sepSubset,definition,
% 3.36/1.79 ( sepSubset
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( subset
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 @ A ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binunion_type,type,
% 3.36/1.79 binunion: $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(binunionIL_type,type,
% 3.36/1.79 binunionIL: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binunionIL,definition,
% 3.36/1.79 ( binunionIL
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(upairset2IR_type,type,
% 3.36/1.79 upairset2IR: $o ).
% 3.36/1.79
% 3.36/1.79 thf(upairset2IR,definition,
% 3.36/1.79 ( upairset2IR
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binunionIR_type,type,
% 3.36/1.79 binunionIR: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binunionIR,definition,
% 3.36/1.79 ( binunionIR
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binunionEcases_type,type,
% 3.36/1.79 binunionEcases: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binunionEcases,definition,
% 3.36/1.79 ( binunionEcases
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.36/1.79 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.36/1.79 => ( ( ( in @ Xx @ A )
% 3.36/1.79 => Xphi )
% 3.36/1.79 => ( ( ( in @ Xx @ B )
% 3.36/1.79 => Xphi )
% 3.36/1.79 => Xphi ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binunionE_type,type,
% 3.36/1.79 binunionE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binunionE,definition,
% 3.36/1.79 ( binunionE
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.36/1.79 => ( ( in @ Xx @ A )
% 3.36/1.79 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binunionLsub_type,type,
% 3.36/1.79 binunionLsub: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binunionLsub,definition,
% 3.36/1.79 ( binunionLsub
% 3.36/1.79 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binunionRsub_type,type,
% 3.36/1.79 binunionRsub: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binunionRsub,definition,
% 3.36/1.79 ( binunionRsub
% 3.36/1.79 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersect_type,type,
% 3.36/1.79 binintersect: $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectI_type,type,
% 3.36/1.79 binintersectI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectI,definition,
% 3.36/1.79 ( binintersectI
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ B )
% 3.36/1.79 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset5_type,type,
% 3.36/1.79 binintersectSubset5: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset5,definition,
% 3.36/1.79 ( binintersectSubset5
% 3.36/1.79 = ( ! [A: $i,B: $i,C: $i] :
% 3.36/1.79 ( ( subset @ C @ A )
% 3.36/1.79 => ( ( subset @ C @ B )
% 3.36/1.79 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectEL_type,type,
% 3.36/1.79 binintersectEL: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectEL,definition,
% 3.36/1.79 ( binintersectEL
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.36/1.79 => ( in @ Xx @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectLsub_type,type,
% 3.36/1.79 binintersectLsub: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectLsub,definition,
% 3.36/1.79 ( binintersectLsub
% 3.36/1.79 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset2_type,type,
% 3.36/1.79 binintersectSubset2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset2,definition,
% 3.36/1.79 ( binintersectSubset2
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( subset @ A @ B )
% 3.36/1.79 => ( ( binintersect @ A @ B )
% 3.36/1.79 = A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset3_type,type,
% 3.36/1.79 binintersectSubset3: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset3,definition,
% 3.36/1.79 ( binintersectSubset3
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( ( binintersect @ A @ B )
% 3.36/1.79 = B )
% 3.36/1.79 => ( subset @ B @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectER_type,type,
% 3.36/1.79 binintersectER: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectER,definition,
% 3.36/1.79 ( binintersectER
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.36/1.79 => ( in @ Xx @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(disjointsetsI1_type,type,
% 3.36/1.79 disjointsetsI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(disjointsetsI1,definition,
% 3.36/1.79 ( disjointsetsI1
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ~ ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ( in @ Xx @ B ) )
% 3.36/1.79 => ( ( binintersect @ A @ B )
% 3.36/1.79 = emptyset ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectRsub_type,type,
% 3.36/1.79 binintersectRsub: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectRsub,definition,
% 3.36/1.79 ( binintersectRsub
% 3.36/1.79 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset4_type,type,
% 3.36/1.79 binintersectSubset4: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset4,definition,
% 3.36/1.79 ( binintersectSubset4
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( subset @ B @ A )
% 3.36/1.79 => ( ( binintersect @ A @ B )
% 3.36/1.79 = B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset1_type,type,
% 3.36/1.79 binintersectSubset1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectSubset1,definition,
% 3.36/1.79 ( binintersectSubset1
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( ( binintersect @ A @ B )
% 3.36/1.79 = A )
% 3.36/1.79 => ( subset @ A @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(bs114d_type,type,
% 3.36/1.79 bs114d: $o ).
% 3.36/1.79
% 3.36/1.79 thf(bs114d,definition,
% 3.36/1.79 ( bs114d
% 3.36/1.79 = ( ! [A: $i,B: $i,C: $i] :
% 3.36/1.79 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.36/1.79 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(regular_type,type,
% 3.36/1.79 regular: $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminus_type,type,
% 3.36/1.79 setminus: $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(setminusI_type,type,
% 3.36/1.79 setminusI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusI,definition,
% 3.36/1.79 ( setminusI
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ~ ( in @ Xx @ B )
% 3.36/1.79 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusEL_type,type,
% 3.36/1.79 setminusEL: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusEL,definition,
% 3.36/1.79 ( setminusEL
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.36/1.79 => ( in @ Xx @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusER_type,type,
% 3.36/1.79 setminusER: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusER,definition,
% 3.36/1.79 ( setminusER
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.36/1.79 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusSubset2_type,type,
% 3.36/1.79 setminusSubset2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusSubset2,definition,
% 3.36/1.79 ( setminusSubset2
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( subset @ A @ B )
% 3.36/1.79 => ( ( setminus @ A @ B )
% 3.36/1.79 = emptyset ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusERneg_type,type,
% 3.36/1.79 setminusERneg: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusERneg,definition,
% 3.36/1.79 ( setminusERneg
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.36/1.79 => ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusELneg_type,type,
% 3.36/1.79 setminusELneg: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusELneg,definition,
% 3.36/1.79 ( setminusELneg
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.36/1.79 => ( ~ ( in @ Xx @ B )
% 3.36/1.79 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusILneg_type,type,
% 3.36/1.79 setminusILneg: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusILneg,definition,
% 3.36/1.79 ( setminusILneg
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ~ ( in @ Xx @ A )
% 3.36/1.79 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusIRneg_type,type,
% 3.36/1.79 setminusIRneg: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusIRneg,definition,
% 3.36/1.79 ( setminusIRneg
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ B )
% 3.36/1.79 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusLsub_type,type,
% 3.36/1.79 setminusLsub: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusLsub,definition,
% 3.36/1.79 ( setminusLsub
% 3.36/1.79 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusSubset1_type,type,
% 3.36/1.79 setminusSubset1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusSubset1,definition,
% 3.36/1.79 ( setminusSubset1
% 3.36/1.79 = ( ! [A: $i,B: $i] :
% 3.36/1.79 ( ( ( setminus @ A @ B )
% 3.36/1.79 = emptyset )
% 3.36/1.79 => ( subset @ A @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(symdiff_type,type,
% 3.36/1.79 symdiff: $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffE_type,type,
% 3.36/1.79 symdiffE: $o ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffE,definition,
% 3.36/1.79 ( symdiffE
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.36/1.79 => ! [Xphi: $o] :
% 3.36/1.79 ( ( ( in @ Xx @ A )
% 3.36/1.79 => ( ~ ( in @ Xx @ B )
% 3.36/1.79 => Xphi ) )
% 3.36/1.79 => ( ( ~ ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ B )
% 3.36/1.79 => Xphi ) )
% 3.36/1.79 => Xphi ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffI1_type,type,
% 3.36/1.79 symdiffI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffI1,definition,
% 3.36/1.79 ( symdiffI1
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ~ ( in @ Xx @ B )
% 3.36/1.79 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffI2_type,type,
% 3.36/1.79 symdiffI2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffI2,definition,
% 3.36/1.79 ( symdiffI2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ~ ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ B )
% 3.36/1.79 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffIneg1_type,type,
% 3.36/1.79 symdiffIneg1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffIneg1,definition,
% 3.36/1.79 ( symdiffIneg1
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ B )
% 3.36/1.79 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffIneg2_type,type,
% 3.36/1.79 symdiffIneg2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(symdiffIneg2,definition,
% 3.36/1.79 ( symdiffIneg2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ~ ( in @ Xx @ A )
% 3.36/1.79 => ( ~ ( in @ Xx @ B )
% 3.36/1.79 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(iskpair_type,type,
% 3.36/1.79 iskpair: $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(secondinupair_type,type,
% 3.36/1.79 secondinupair: $o ).
% 3.36/1.79
% 3.36/1.79 thf(secondinupair,definition,
% 3.36/1.79 ( secondinupair
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairIL_type,type,
% 3.36/1.79 setukpairIL: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairIL,definition,
% 3.36/1.79 ( setukpairIL
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairIR_type,type,
% 3.36/1.79 setukpairIR: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairIR,definition,
% 3.36/1.79 ( setukpairIR
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(kpairiskpair_type,type,
% 3.36/1.79 kpairiskpair: $o ).
% 3.36/1.79
% 3.36/1.79 thf(kpairiskpair,definition,
% 3.36/1.79 ( kpairiskpair
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(kpair_type,type,
% 3.36/1.79 kpair: $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(kpairp_type,type,
% 3.36/1.79 kpairp: $o ).
% 3.36/1.79
% 3.36/1.79 thf(kpairp,definition,
% 3.36/1.79 ( kpairp
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprod_type,type,
% 3.36/1.79 cartprod: $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(singletonsubset_type,type,
% 3.36/1.79 singletonsubset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(singletonsubset,definition,
% 3.36/1.79 ( singletonsubset
% 3.36/1.79 = ( ! [A: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(singletoninpowerset_type,type,
% 3.36/1.79 singletoninpowerset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(singletoninpowerset,definition,
% 3.36/1.79 ( singletoninpowerset
% 3.36/1.79 = ( ! [A: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(singletoninpowunion_type,type,
% 3.36/1.79 singletoninpowunion: $o ).
% 3.36/1.79
% 3.36/1.79 thf(singletoninpowunion,definition,
% 3.36/1.79 ( singletoninpowunion
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(upairset2E_type,type,
% 3.36/1.79 upairset2E: $o ).
% 3.36/1.79
% 3.36/1.79 thf(upairset2E,definition,
% 3.36/1.79 ( upairset2E
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.36/1.79 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.36/1.79 => ( ( Xz = Xx )
% 3.36/1.79 | ( Xz = Xy ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(upairsubunion_type,type,
% 3.36/1.79 upairsubunion: $o ).
% 3.36/1.79
% 3.36/1.79 thf(upairsubunion,definition,
% 3.36/1.79 ( upairsubunion
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(upairinpowunion_type,type,
% 3.36/1.79 upairinpowunion: $o ).
% 3.36/1.79
% 3.36/1.79 thf(upairinpowunion,definition,
% 3.36/1.79 ( upairinpowunion
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ubforcartprodlem1_type,type,
% 3.36/1.79 ubforcartprodlem1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ubforcartprodlem1,definition,
% 3.36/1.79 ( ubforcartprodlem1
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ubforcartprodlem2_type,type,
% 3.36/1.79 ubforcartprodlem2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ubforcartprodlem2,definition,
% 3.36/1.79 ( ubforcartprodlem2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ubforcartprodlem3_type,type,
% 3.36/1.79 ubforcartprodlem3: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ubforcartprodlem3,definition,
% 3.36/1.79 ( ubforcartprodlem3
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodpairin_type,type,
% 3.36/1.79 cartprodpairin: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodpairin,definition,
% 3.36/1.79 ( cartprodpairin
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodmempair1_type,type,
% 3.36/1.79 cartprodmempair1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodmempair1,definition,
% 3.36/1.79 ( cartprodmempair1
% 3.36/1.79 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.36/1.79 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.36/1.79 => ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ? [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 & ( Xu
% 3.36/1.79 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodmempair_type,type,
% 3.36/1.79 cartprodmempair: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodmempair,definition,
% 3.36/1.79 ( cartprodmempair
% 3.36/1.79 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.36/1.79 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.36/1.79 => ( iskpair @ Xu ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setunionE2_type,type,
% 3.36/1.79 setunionE2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setunionE2,definition,
% 3.36/1.79 ( setunionE2
% 3.36/1.79 = ( ! [A: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ ( setunion @ A ) )
% 3.36/1.79 => ? [X: $i] :
% 3.36/1.79 ( ( in @ X @ A )
% 3.36/1.79 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setunionsingleton1_type,type,
% 3.36/1.79 setunionsingleton1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setunionsingleton1,definition,
% 3.36/1.79 ( setunionsingleton1
% 3.36/1.79 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setunionsingleton2_type,type,
% 3.36/1.79 setunionsingleton2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setunionsingleton2,definition,
% 3.36/1.79 ( setunionsingleton2
% 3.36/1.79 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setunionsingleton_type,type,
% 3.36/1.79 setunionsingleton: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setunionsingleton,definition,
% 3.36/1.79 ( setunionsingleton
% 3.36/1.79 = ( ! [Xx: $i] :
% 3.36/1.79 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.36/1.79 = Xx ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(singleton_type,type,
% 3.36/1.79 singleton: $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(singletonprop_type,type,
% 3.36/1.79 singletonprop: $o ).
% 3.36/1.79
% 3.36/1.79 thf(singletonprop,definition,
% 3.36/1.79 ( singletonprop
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( ( Xphi @ Xx )
% 3.36/1.79 => ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) )
% 3.36/1.79 => ( ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ( Xphi @ Xx ) )
% 3.36/1.79 => ( singleton
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ex1_type,type,
% 3.36/1.79 ex1: $i > ( $i > $o ) > $o ).
% 3.36/1.79
% 3.36/1.79 thf(ex1,definition,
% 3.36/1.79 ( ex1
% 3.36/1.79 = ( ^ [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( singleton
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ex1E1_type,type,
% 3.36/1.79 ex1E1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ex1E1,definition,
% 3.36/1.79 ( ex1E1
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ( ex1 @ A
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 => ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ex1I_type,type,
% 3.36/1.79 ex1I: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ex1I,definition,
% 3.36/1.79 ( ex1I
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( Xphi @ Xx )
% 3.36/1.79 => ( ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xy = Xx ) ) )
% 3.36/1.79 => ( ex1 @ A
% 3.36/1.79 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ex1I2_type,type,
% 3.36/1.79 ex1I2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ex1I2,definition,
% 3.36/1.79 ( ex1I2
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( ( Xphi @ Xx )
% 3.36/1.79 => ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) )
% 3.36/1.79 => ( ? [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 & ( Xphi @ Xx ) )
% 3.36/1.79 => ( ex1 @ A
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(singletonsuniq_type,type,
% 3.36/1.79 singletonsuniq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(singletonsuniq,definition,
% 3.36/1.79 ( singletonsuniq
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( ( setadjoin @ Xx @ emptyset )
% 3.36/1.79 = ( setadjoin @ Xy @ emptyset ) )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(atmost1p_type,type,
% 3.36/1.79 atmost1p: $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(atleast2p_type,type,
% 3.36/1.79 atleast2p: $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(atmost2p_type,type,
% 3.36/1.79 atmost2p: $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(upairsetp_type,type,
% 3.36/1.79 upairsetp: $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjL1_type,type,
% 3.36/1.79 setukpairinjL1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjL1,definition,
% 3.36/1.79 ( setukpairinjL1
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.36/1.79 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.36/1.79 => ( Xx = Xz ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(kfstsingleton_type,type,
% 3.36/1.79 kfstsingleton: $o ).
% 3.36/1.79
% 3.36/1.79 thf(kfstsingleton,definition,
% 3.36/1.79 ( kfstsingleton
% 3.36/1.79 = ( ! [Xu: $i] :
% 3.36/1.79 ( ( iskpair @ Xu )
% 3.36/1.79 => ( singleton
% 3.36/1.79 @ ( dsetconstr @ ( setunion @ Xu )
% 3.36/1.79 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(theprop_type,type,
% 3.36/1.79 theprop: $o ).
% 3.36/1.79
% 3.36/1.79 thf(theprop,definition,
% 3.36/1.79 ( theprop
% 3.36/1.79 = ( ! [X: $i] :
% 3.36/1.79 ( ( singleton @ X )
% 3.36/1.79 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(kfst_type,type,
% 3.36/1.79 kfst: $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(kfstpairEq_type,type,
% 3.36/1.79 kfstpairEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(kfstpairEq,definition,
% 3.36/1.79 ( kfstpairEq
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.36/1.79 = Xx ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodfstin_type,type,
% 3.36/1.79 cartprodfstin: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodfstin,definition,
% 3.36/1.79 ( cartprodfstin
% 3.36/1.79 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.36/1.79 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.36/1.79 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjL2_type,type,
% 3.36/1.79 setukpairinjL2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjL2,definition,
% 3.36/1.79 ( setukpairinjL2
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.36/1.79 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.36/1.79 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.36/1.79 => ( Xx = Xz ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjL_type,type,
% 3.36/1.79 setukpairinjL: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjL,definition,
% 3.36/1.79 ( setukpairinjL
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.36/1.79 ( ( ( kpair @ Xx @ Xy )
% 3.36/1.79 = ( kpair @ Xz @ Xu ) )
% 3.36/1.79 => ( Xx = Xz ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR11_type,type,
% 3.36/1.79 setukpairinjR11: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR11,definition,
% 3.36/1.79 ( setukpairinjR11
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xx = Xy )
% 3.36/1.79 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.36/1.79 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR12_type,type,
% 3.36/1.79 setukpairinjR12: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR12,definition,
% 3.36/1.79 ( setukpairinjR12
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xx = Xy )
% 3.36/1.79 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.36/1.79 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR1_type,type,
% 3.36/1.79 setukpairinjR1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR1,definition,
% 3.36/1.79 ( setukpairinjR1
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.36/1.79 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.36/1.79 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.36/1.79 => ( ( Xz = Xu )
% 3.36/1.79 => ( Xy = Xu ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(upairequniteq_type,type,
% 3.36/1.79 upairequniteq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(upairequniteq,definition,
% 3.36/1.79 ( upairequniteq
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.36/1.79 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.36/1.79 = ( setadjoin @ Xz @ emptyset ) )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR2_type,type,
% 3.36/1.79 setukpairinjR2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR2,definition,
% 3.36/1.79 ( setukpairinjR2
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.36/1.79 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.36/1.79 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.36/1.79 => ( Xy = Xu ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR_type,type,
% 3.36/1.79 setukpairinjR: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setukpairinjR,definition,
% 3.36/1.79 ( setukpairinjR
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.36/1.79 ( ( ( kpair @ Xx @ Xy )
% 3.36/1.79 = ( kpair @ Xz @ Xu ) )
% 3.36/1.79 => ( Xy = Xu ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ksndsingleton_type,type,
% 3.36/1.79 ksndsingleton: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ksndsingleton,definition,
% 3.36/1.79 ( ksndsingleton
% 3.36/1.79 = ( ! [Xu: $i] :
% 3.36/1.79 ( ( iskpair @ Xu )
% 3.36/1.79 => ( singleton
% 3.36/1.79 @ ( dsetconstr @ ( setunion @ Xu )
% 3.36/1.79 @ ^ [Xx: $i] :
% 3.36/1.79 ( Xu
% 3.36/1.79 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ksnd_type,type,
% 3.36/1.79 ksnd: $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(ksndpairEq_type,type,
% 3.36/1.79 ksndpairEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ksndpairEq,definition,
% 3.36/1.79 ( ksndpairEq
% 3.36/1.79 = ( ! [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.36/1.79 = Xy ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(kpairsurjEq_type,type,
% 3.36/1.79 kpairsurjEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(kpairsurjEq,definition,
% 3.36/1.79 ( kpairsurjEq
% 3.36/1.79 = ( ! [Xu: $i] :
% 3.36/1.79 ( ( iskpair @ Xu )
% 3.36/1.79 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.36/1.79 = Xu ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodsndin_type,type,
% 3.36/1.79 cartprodsndin: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodsndin,definition,
% 3.36/1.79 ( cartprodsndin
% 3.36/1.79 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.36/1.79 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.36/1.79 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodpairmemEL_type,type,
% 3.36/1.79 cartprodpairmemEL: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodpairmemEL,definition,
% 3.36/1.79 ( cartprodpairmemEL
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.36/1.79 => ( in @ Xx @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodpairmemER_type,type,
% 3.36/1.79 cartprodpairmemER: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodpairmemER,definition,
% 3.36/1.79 ( cartprodpairmemER
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.36/1.79 => ( in @ Xy @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodmempaircEq_type,type,
% 3.36/1.79 cartprodmempaircEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodmempaircEq,definition,
% 3.36/1.79 ( cartprodmempaircEq
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( kpair @ Xx @ Xy )
% 3.36/1.79 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodfstpairEq_type,type,
% 3.36/1.79 cartprodfstpairEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodfstpairEq,definition,
% 3.36/1.79 ( cartprodfstpairEq
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.36/1.79 = Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodsndpairEq_type,type,
% 3.36/1.79 cartprodsndpairEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodsndpairEq,definition,
% 3.36/1.79 ( cartprodsndpairEq
% 3.36/1.79 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.36/1.79 = Xy ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodpairsurjEq_type,type,
% 3.36/1.79 cartprodpairsurjEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(cartprodpairsurjEq,definition,
% 3.36/1.79 ( cartprodpairsurjEq
% 3.36/1.79 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.36/1.79 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.36/1.79 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.36/1.79 = Xu ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(breln_type,type,
% 3.36/1.79 breln: $i > $i > $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(breln,definition,
% 3.36/1.79 ( breln
% 3.36/1.79 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstr_type,type,
% 3.36/1.79 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrI_type,type,
% 3.36/1.79 dpsetconstrI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrI,definition,
% 3.36/1.79 ( dpsetconstrI
% 3.36/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( Xphi @ Xx @ Xy )
% 3.36/1.79 => ( in @ ( kpair @ Xx @ Xy )
% 3.36/1.79 @ ( dpsetconstr @ A @ B
% 3.36/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrSub_type,type,
% 3.36/1.79 dpsetconstrSub: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrSub,definition,
% 3.36/1.79 ( dpsetconstrSub
% 3.36/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.36/1.79 ( subset
% 3.36/1.79 @ ( dpsetconstr @ A @ B
% 3.36/1.79 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.36/1.79 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setOfPairsIsBReln_type,type,
% 3.36/1.79 setOfPairsIsBReln: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setOfPairsIsBReln,definition,
% 3.36/1.79 ( setOfPairsIsBReln
% 3.36/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.36/1.79 ( breln @ A @ B
% 3.36/1.79 @ ( dpsetconstr @ A @ B
% 3.36/1.79 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrERa_type,type,
% 3.36/1.79 dpsetconstrERa: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrERa,definition,
% 3.36/1.79 ( dpsetconstrERa
% 3.36/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.36/1.79 @ ( dpsetconstr @ A @ B
% 3.36/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.36/1.79 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrEL1_type,type,
% 3.36/1.79 dpsetconstrEL1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrEL1,definition,
% 3.36/1.79 ( dpsetconstrEL1
% 3.36/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ ( kpair @ Xx @ Xy )
% 3.36/1.79 @ ( dpsetconstr @ A @ B
% 3.36/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.36/1.79 => ( in @ Xx @ A ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrEL2_type,type,
% 3.36/1.79 dpsetconstrEL2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrEL2,definition,
% 3.36/1.79 ( dpsetconstrEL2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ ( kpair @ Xx @ Xy )
% 3.36/1.79 @ ( dpsetconstr @ A @ B
% 3.36/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.36/1.79 => ( in @ Xy @ B ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrER_type,type,
% 3.36/1.79 dpsetconstrER: $o ).
% 3.36/1.79
% 3.36/1.79 thf(dpsetconstrER,definition,
% 3.36/1.79 ( dpsetconstrER
% 3.36/1.79 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( in @ ( kpair @ Xx @ Xy )
% 3.36/1.79 @ ( dpsetconstr @ A @ B
% 3.36/1.79 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.36/1.79 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(func_type,type,
% 3.36/1.79 func: $i > $i > $i > $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcSet_type,type,
% 3.36/1.79 funcSet: $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(funcImageSingleton_type,type,
% 3.36/1.79 funcImageSingleton: $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcImageSingleton,definition,
% 3.36/1.79 ( funcImageSingleton
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( singleton
% 3.36/1.79 @ ( dsetconstr @ B
% 3.36/1.79 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(apProp_type,type,
% 3.36/1.79 apProp: $o ).
% 3.36/1.79
% 3.36/1.79 thf(apProp,definition,
% 3.36/1.79 ( apProp
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in
% 3.36/1.79 @ ( setunion
% 3.36/1.79 @ ( dsetconstr @ B
% 3.36/1.79 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.36/1.79 @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ap_type,type,
% 3.36/1.79 ap: $i > $i > $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(app_type,type,
% 3.36/1.79 app: $o ).
% 3.36/1.79
% 3.36/1.79 thf(app,definition,
% 3.36/1.79 ( app
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(infuncsetfunc_type,type,
% 3.36/1.79 infuncsetfunc: $o ).
% 3.36/1.79
% 3.36/1.79 thf(infuncsetfunc,definition,
% 3.36/1.79 ( infuncsetfunc
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.36/1.79 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ap2p_type,type,
% 3.36/1.79 ap2p: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ap2p,definition,
% 3.36/1.79 ( ap2p
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(funcinfuncset_type,type,
% 3.36/1.79 funcinfuncset: $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcinfuncset,definition,
% 3.36/1.79 ( funcinfuncset
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(lamProp_type,type,
% 3.36/1.79 lamProp: $o ).
% 3.36/1.79
% 3.36/1.79 thf(lamProp,definition,
% 3.36/1.79 ( lamProp
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.36/1.79 => ( func @ A @ B
% 3.36/1.79 @ ( dpsetconstr @ A @ B
% 3.36/1.79 @ ^ [Xx: $i,Xy: $i] :
% 3.36/1.79 ( ( Xf @ Xx )
% 3.36/1.79 = Xy ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(lam_type,type,
% 3.36/1.79 lam: $i > $i > ( $i > $i ) > $i ).
% 3.36/1.79
% 3.36/1.79 thf(lamp_type,type,
% 3.36/1.79 lamp: $o ).
% 3.36/1.79
% 3.36/1.79 thf(lamp,definition,
% 3.36/1.79 ( lamp
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.36/1.79 => ( func @ A @ B
% 3.36/1.79 @ ( lam @ A @ B
% 3.36/1.79 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(lam2p_type,type,
% 3.36/1.79 lam2p: $o ).
% 3.36/1.79
% 3.36/1.79 thf(lam2p,definition,
% 3.36/1.79 ( lam2p
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.36/1.79 => ( in
% 3.36/1.79 @ ( lam @ A @ B
% 3.36/1.79 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.36/1.79 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(brelnall1_type,type,
% 3.36/1.79 brelnall1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(brelnall1,definition,
% 3.36/1.79 ( brelnall1
% 3.36/1.79 = ( ! [A: $i,B: $i,R: $i] :
% 3.36/1.79 ( ( breln @ A @ B @ R )
% 3.36/1.79 => ! [Xphi: $i > $o] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.36/1.79 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ R )
% 3.36/1.79 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(brelnall2_type,type,
% 3.36/1.79 brelnall2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(brelnall2,definition,
% 3.36/1.79 ( brelnall2
% 3.36/1.79 = ( ! [A: $i,B: $i,R: $i] :
% 3.36/1.79 ( ( breln @ A @ B @ R )
% 3.36/1.79 => ! [Xphi: $i > $o] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.36/1.79 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ R )
% 3.36/1.79 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ex1E2_type,type,
% 3.36/1.79 ex1E2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ex1E2,definition,
% 3.36/1.79 ( ex1E2
% 3.36/1.79 = ( ! [A: $i,Xphi: $i > $o] :
% 3.36/1.79 ( ( ex1 @ A
% 3.36/1.79 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( ( Xphi @ Xx )
% 3.36/1.79 => ( ( Xphi @ Xy )
% 3.36/1.79 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(funcGraphProp1_type,type,
% 3.36/1.79 funcGraphProp1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcGraphProp1,definition,
% 3.36/1.79 ( funcGraphProp1
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(funcGraphProp3_type,type,
% 3.36/1.79 funcGraphProp3: $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcGraphProp3,definition,
% 3.36/1.79 ( funcGraphProp3
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(funcGraphProp2_type,type,
% 3.36/1.79 funcGraphProp2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcGraphProp2,definition,
% 3.36/1.79 ( funcGraphProp2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.36/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.36/1.79 = Xy ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(funcextLem_type,type,
% 3.36/1.79 funcextLem: $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcextLem,definition,
% 3.36/1.79 ( funcextLem
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ! [Xg: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xg )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.36/1.79 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.36/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(funcGraphProp4_type,type,
% 3.36/1.79 funcGraphProp4: $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcGraphProp4,definition,
% 3.36/1.79 ( funcGraphProp4
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.36/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.36/1.79 = Xy ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subbreln_type,type,
% 3.36/1.79 subbreln: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subbreln,definition,
% 3.36/1.79 ( subbreln
% 3.36/1.79 = ( ! [A: $i,B: $i,R: $i] :
% 3.36/1.79 ( ( breln @ A @ B @ R )
% 3.36/1.79 => ! [S: $i] :
% 3.36/1.79 ( ( breln @ A @ B @ S )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.36/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.36/1.79 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(eqbreln_type,type,
% 3.36/1.79 eqbreln: $o ).
% 3.36/1.79
% 3.36/1.79 thf(eqbreln,definition,
% 3.36/1.79 ( eqbreln
% 3.36/1.79 = ( ! [A: $i,B: $i,R: $i] :
% 3.36/1.79 ( ( breln @ A @ B @ R )
% 3.36/1.79 => ! [S: $i] :
% 3.36/1.79 ( ( breln @ A @ B @ S )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.36/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ B )
% 3.36/1.79 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.36/1.79 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.36/1.79 => ( R = S ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(funcext_type,type,
% 3.36/1.79 funcext: $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcext,definition,
% 3.36/1.79 ( funcext
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ! [Xg: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xg )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.36/1.79 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.36/1.79 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(funcext2_type,type,
% 3.36/1.79 funcext2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(funcext2,definition,
% 3.36/1.79 ( funcext2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.36/1.79 => ! [Xg: $i] :
% 3.36/1.79 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.36/1.79 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.36/1.79 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ap2apEq1_type,type,
% 3.36/1.79 ap2apEq1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ap2apEq1,definition,
% 3.36/1.79 ( ap2apEq1
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.36/1.79 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ap2apEq2_type,type,
% 3.36/1.79 ap2apEq2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ap2apEq2,definition,
% 3.36/1.79 ( ap2apEq2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.36/1.79 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(beta1_type,type,
% 3.36/1.79 beta1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(beta1,definition,
% 3.36/1.79 ( beta1
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( ap @ A @ B
% 3.36/1.79 @ ( lam @ A @ B
% 3.36/1.79 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.36/1.79 @ Xx )
% 3.36/1.79 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(eta1_type,type,
% 3.36/1.79 eta1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(eta1,definition,
% 3.36/1.79 ( eta1
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( func @ A @ B @ Xf )
% 3.36/1.79 => ( ( lam @ A @ B
% 3.36/1.79 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.36/1.79 = Xf ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(lam2lamEq_type,type,
% 3.36/1.79 lam2lamEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(lam2lamEq,definition,
% 3.36/1.79 ( lam2lamEq
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.36/1.79 => ( ( lam @ A @ B
% 3.36/1.79 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.36/1.79 = ( lam @ A @ B
% 3.36/1.79 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(beta2_type,type,
% 3.36/1.79 beta2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(beta2,definition,
% 3.36/1.79 ( beta2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.36/1.79 ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( ap @ A @ B
% 3.36/1.79 @ ( lam @ A @ B
% 3.36/1.79 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.36/1.79 @ Xx )
% 3.36/1.79 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(eta2_type,type,
% 3.36/1.79 eta2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(eta2,definition,
% 3.36/1.79 ( eta2
% 3.36/1.79 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.36/1.79 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.36/1.79 => ( ( lam @ A @ B
% 3.36/1.79 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.36/1.79 = Xf ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(iffalseProp1_type,type,
% 3.36/1.79 iffalseProp1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(iffalseProp1,definition,
% 3.36/1.79 ( iffalseProp1
% 3.36/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( ~ Xphi
% 3.36/1.79 => ( in @ Xy
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xz: $i] :
% 3.36/1.79 ( ( Xphi
% 3.36/1.79 & ( Xz = Xx ) )
% 3.36/1.79 | ( ~ Xphi
% 3.36/1.79 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(iffalseProp2_type,type,
% 3.36/1.79 iffalseProp2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(iffalseProp2,definition,
% 3.36/1.79 ( iffalseProp2
% 3.36/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( ~ Xphi
% 3.36/1.79 => ( ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xz: $i] :
% 3.36/1.79 ( ( Xphi
% 3.36/1.79 & ( Xz = Xx ) )
% 3.36/1.79 | ( ~ Xphi
% 3.36/1.79 & ( Xz = Xy ) ) ) )
% 3.36/1.79 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(iftrueProp1_type,type,
% 3.36/1.79 iftrueProp1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(iftrueProp1,definition,
% 3.36/1.79 ( iftrueProp1
% 3.36/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( Xphi
% 3.36/1.79 => ( in @ Xx
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xz: $i] :
% 3.36/1.79 ( ( Xphi
% 3.36/1.79 & ( Xz = Xx ) )
% 3.36/1.79 | ( ~ Xphi
% 3.36/1.79 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(iftrueProp2_type,type,
% 3.36/1.79 iftrueProp2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(iftrueProp2,definition,
% 3.36/1.79 ( iftrueProp2
% 3.36/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( Xphi
% 3.36/1.79 => ( ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xz: $i] :
% 3.36/1.79 ( ( Xphi
% 3.36/1.79 & ( Xz = Xx ) )
% 3.36/1.79 | ( ~ Xphi
% 3.36/1.79 & ( Xz = Xy ) ) ) )
% 3.36/1.79 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(ifSingleton_type,type,
% 3.36/1.79 ifSingleton: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ifSingleton,definition,
% 3.36/1.79 ( ifSingleton
% 3.36/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( singleton
% 3.36/1.79 @ ( dsetconstr @ A
% 3.36/1.79 @ ^ [Xz: $i] :
% 3.36/1.79 ( ( Xphi
% 3.36/1.79 & ( Xz = Xx ) )
% 3.36/1.79 | ( ~ Xphi
% 3.36/1.79 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(if_type,type,
% 3.36/1.79 if: $i > $o > $i > $i > $i ).
% 3.36/1.79
% 3.36/1.79 thf(ifp_type,type,
% 3.36/1.79 ifp: $o ).
% 3.36/1.79
% 3.36/1.79 thf(ifp,definition,
% 3.36/1.79 ( ifp
% 3.36/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(theeq_type,type,
% 3.36/1.79 theeq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(theeq,definition,
% 3.36/1.79 ( theeq
% 3.36/1.79 = ( ! [X: $i] :
% 3.36/1.79 ( ( singleton @ X )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ X )
% 3.36/1.79 => ( ( setunion @ X )
% 3.36/1.79 = Xx ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(iftrue_type,type,
% 3.36/1.79 iftrue: $o ).
% 3.36/1.79
% 3.36/1.79 thf(iftrue,definition,
% 3.36/1.79 ( iftrue
% 3.36/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( Xphi
% 3.36/1.79 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.36/1.79 = Xx ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(iffalse_type,type,
% 3.36/1.79 iffalse: $o ).
% 3.36/1.79
% 3.36/1.79 thf(iffalse,definition,
% 3.36/1.79 ( iffalse
% 3.36/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( ~ Xphi
% 3.36/1.79 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.36/1.79 = Xy ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(iftrueorfalse_type,type,
% 3.36/1.79 iftrueorfalse: $o ).
% 3.36/1.79
% 3.36/1.79 thf(iftrueorfalse,definition,
% 3.36/1.79 ( iftrueorfalse
% 3.36/1.79 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ! [Xy: $i] :
% 3.36/1.79 ( ( in @ Xy @ A )
% 3.36/1.79 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectT_lem_type,type,
% 3.36/1.79 binintersectT_lem: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectT_lem,definition,
% 3.36/1.79 ( binintersectT_lem
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binunionT_lem_type,type,
% 3.36/1.79 binunionT_lem: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binunionT_lem,definition,
% 3.36/1.79 ( binunionT_lem
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powersetT_lem_type,type,
% 3.36/1.79 powersetT_lem: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powersetT_lem,definition,
% 3.36/1.79 ( powersetT_lem
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setminusT_lem_type,type,
% 3.36/1.79 setminusT_lem: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setminusT_lem,definition,
% 3.36/1.79 ( setminusT_lem
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(complementT_lem_type,type,
% 3.36/1.79 complementT_lem: $o ).
% 3.36/1.79
% 3.36/1.79 thf(complementT_lem,definition,
% 3.36/1.79 ( complementT_lem
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(setextT_type,type,
% 3.36/1.79 setextT: $o ).
% 3.36/1.79
% 3.36/1.79 thf(setextT,definition,
% 3.36/1.79 ( setextT
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ X )
% 3.36/1.79 => ( in @ Xx @ Y ) ) )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ Y )
% 3.36/1.79 => ( in @ Xx @ X ) ) )
% 3.36/1.79 => ( X = Y ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(subsetTI_type,type,
% 3.36/1.79 subsetTI: $o ).
% 3.36/1.79
% 3.36/1.79 thf(subsetTI,definition,
% 3.36/1.79 ( subsetTI
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ X )
% 3.36/1.79 => ( in @ Xx @ Y ) ) )
% 3.36/1.79 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powersetTI1_type,type,
% 3.36/1.79 powersetTI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powersetTI1,definition,
% 3.36/1.79 ( powersetTI1
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ( ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ X )
% 3.36/1.79 => ( in @ Xx @ Y ) ) )
% 3.36/1.79 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(powersetTE1_type,type,
% 3.36/1.79 powersetTE1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(powersetTE1,definition,
% 3.36/1.79 ( powersetTE1
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ X @ ( powerset @ Y ) )
% 3.36/1.79 => ( ( in @ Xx @ X )
% 3.36/1.79 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(complementTI1_type,type,
% 3.36/1.79 complementTI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(complementTI1,definition,
% 3.36/1.79 ( complementTI1
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ X )
% 3.36/1.79 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(complementTE1_type,type,
% 3.36/1.79 complementTE1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(complementTE1,definition,
% 3.36/1.79 ( complementTE1
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.36/1.79 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectTELcontra_type,type,
% 3.36/1.79 binintersectTELcontra: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectTELcontra,definition,
% 3.36/1.79 ( binintersectTELcontra
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ~ ( in @ Xx @ X )
% 3.36/1.79 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectTERcontra_type,type,
% 3.36/1.79 binintersectTERcontra: $o ).
% 3.36/1.79
% 3.36/1.79 thf(binintersectTERcontra,definition,
% 3.36/1.79 ( binintersectTERcontra
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ~ ( in @ Xx @ Y )
% 3.36/1.79 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(contrasubsetT_type,type,
% 3.36/1.79 contrasubsetT: $o ).
% 3.36/1.79
% 3.36/1.79 thf(contrasubsetT,definition,
% 3.36/1.79 ( contrasubsetT
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.36/1.79 => ( ( in @ Xx @ Y )
% 3.36/1.79 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(contrasubsetT1_type,type,
% 3.36/1.79 contrasubsetT1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(contrasubsetT1,definition,
% 3.36/1.79 ( contrasubsetT1
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( subset @ X @ Y )
% 3.36/1.79 => ( ~ ( in @ Xx @ Y )
% 3.36/1.79 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(contrasubsetT2_type,type,
% 3.36/1.79 contrasubsetT2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(contrasubsetT2,definition,
% 3.36/1.79 ( contrasubsetT2
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ( ( subset @ X @ Y )
% 3.36/1.79 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(contrasubsetT3_type,type,
% 3.36/1.79 contrasubsetT3: $o ).
% 3.36/1.79
% 3.36/1.79 thf(contrasubsetT3,definition,
% 3.36/1.79 ( contrasubsetT3
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.36/1.79 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementI1_type,type,
% 3.36/1.79 doubleComplementI1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementI1,definition,
% 3.36/1.79 ( doubleComplementI1
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ X )
% 3.36/1.79 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementE1_type,type,
% 3.36/1.79 doubleComplementE1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementE1,definition,
% 3.36/1.79 ( doubleComplementE1
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.36/1.79 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementSub1_type,type,
% 3.36/1.79 doubleComplementSub1: $o ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementSub1,definition,
% 3.36/1.79 ( doubleComplementSub1
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementSub2_type,type,
% 3.36/1.79 doubleComplementSub2: $o ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementSub2,definition,
% 3.36/1.79 ( doubleComplementSub2
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementEq_type,type,
% 3.36/1.79 doubleComplementEq: $o ).
% 3.36/1.79
% 3.36/1.79 thf(doubleComplementEq,definition,
% 3.36/1.79 ( doubleComplementEq
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ( X
% 3.36/1.79 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(complementTnotintersectT_type,type,
% 3.36/1.79 complementTnotintersectT: $o ).
% 3.36/1.79
% 3.36/1.79 thf(complementTnotintersectT,definition,
% 3.36/1.79 ( complementTnotintersectT
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.36/1.79 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(complementImpComplementIntersect_type,type,
% 3.36/1.79 complementImpComplementIntersect: $o ).
% 3.36/1.79
% 3.36/1.79 thf(complementImpComplementIntersect,definition,
% 3.36/1.79 ( complementImpComplementIntersect
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ! [Xx: $i] :
% 3.36/1.79 ( ( in @ Xx @ A )
% 3.36/1.79 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.36/1.79 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(complementSubsetComplementIntersect_type,type,
% 3.36/1.79 complementSubsetComplementIntersect: $o ).
% 3.36/1.79
% 3.36/1.79 thf(complementSubsetComplementIntersect,definition,
% 3.36/1.79 ( complementSubsetComplementIntersect
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.79 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.36/1.79
% 3.36/1.79 thf(complementInPowersetComplementIntersect_type,type,
% 3.36/1.79 complementInPowersetComplementIntersect: $o ).
% 3.36/1.79
% 3.36/1.79 thf(complementInPowersetComplementIntersect,definition,
% 3.36/1.79 ( complementInPowersetComplementIntersect
% 3.36/1.79 = ( ! [A: $i,X: $i] :
% 3.36/1.79 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.79 => ! [Y: $i] :
% 3.36/1.79 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(contraSubsetComplement_type,type,
% 3.36/1.80 contraSubsetComplement: $o ).
% 3.36/1.80
% 3.36/1.80 thf(contraSubsetComplement,definition,
% 3.36/1.80 ( contraSubsetComplement
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ Y )
% 3.36/1.80 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(complementTcontraSubset_type,type,
% 3.36/1.80 complementTcontraSubset: $o ).
% 3.36/1.80
% 3.36/1.80 thf(complementTcontraSubset,definition,
% 3.36/1.80 ( complementTcontraSubset
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.36/1.80 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(binunionTILcontra_type,type,
% 3.36/1.80 binunionTILcontra: $o ).
% 3.36/1.80
% 3.36/1.80 thf(binunionTILcontra,definition,
% 3.36/1.80 ( binunionTILcontra
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.36/1.80 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(binunionTIRcontra_type,type,
% 3.36/1.80 binunionTIRcontra: $o ).
% 3.36/1.80
% 3.36/1.80 thf(binunionTIRcontra,definition,
% 3.36/1.80 ( binunionTIRcontra
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.36/1.80 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(inIntersectImpInUnion_type,type,
% 3.36/1.80 inIntersectImpInUnion: $o ).
% 3.36/1.80
% 3.36/1.80 thf(inIntersectImpInUnion,definition,
% 3.36/1.80 ( inIntersectImpInUnion
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Z: $i] :
% 3.36/1.80 ( ( in @ Z @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.36/1.80 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(inIntersectImpInUnion2_type,type,
% 3.36/1.80 inIntersectImpInUnion2: $o ).
% 3.36/1.80
% 3.36/1.80 thf(inIntersectImpInUnion2,definition,
% 3.36/1.80 ( inIntersectImpInUnion2
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Z: $i] :
% 3.36/1.80 ( ( in @ Z @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.36/1.80 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(inIntersectImpInIntersectUnions_type,type,
% 3.36/1.80 inIntersectImpInIntersectUnions: $o ).
% 3.36/1.80
% 3.36/1.80 thf(inIntersectImpInIntersectUnions,definition,
% 3.36/1.80 ( inIntersectImpInIntersectUnions
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Z: $i] :
% 3.36/1.80 ( ( in @ Z @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.36/1.80 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(intersectInPowersetIntersectUnions_type,type,
% 3.36/1.80 intersectInPowersetIntersectUnions: $o ).
% 3.36/1.80
% 3.36/1.80 thf(intersectInPowersetIntersectUnions,definition,
% 3.36/1.80 ( intersectInPowersetIntersectUnions
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Z: $i] :
% 3.36/1.80 ( ( in @ Z @ ( powerset @ A ) )
% 3.36/1.80 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(inComplementUnionImpNotIn1_type,type,
% 3.36/1.80 inComplementUnionImpNotIn1: $o ).
% 3.36/1.80
% 3.36/1.80 thf(inComplementUnionImpNotIn1,definition,
% 3.36/1.80 ( inComplementUnionImpNotIn1
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.36/1.80 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(inComplementUnionImpInComplement1_type,type,
% 3.36/1.80 inComplementUnionImpInComplement1: $o ).
% 3.36/1.80
% 3.36/1.80 thf(inComplementUnionImpInComplement1,definition,
% 3.36/1.80 ( inComplementUnionImpInComplement1
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.36/1.80 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(binunionTE_type,type,
% 3.36/1.80 binunionTE: $o ).
% 3.36/1.80
% 3.36/1.80 thf(binunionTE,definition,
% 3.36/1.80 ( binunionTE
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xphi: $o,Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.36/1.80 => ( ( ( in @ Xx @ X )
% 3.36/1.80 => Xphi )
% 3.36/1.80 => ( ( ( in @ Xx @ Y )
% 3.36/1.80 => Xphi )
% 3.36/1.80 => Xphi ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(binunionTEcontra_type,type,
% 3.36/1.80 binunionTEcontra: $o ).
% 3.36/1.80
% 3.36/1.80 thf(binunionTEcontra,definition,
% 3.36/1.80 ( binunionTEcontra
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ~ ( in @ Xx @ X )
% 3.36/1.80 => ( ~ ( in @ Xx @ Y )
% 3.36/1.80 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(demorgan2a1_type,type,
% 3.36/1.80 demorgan2a1: $o ).
% 3.36/1.80
% 3.36/1.80 thf(demorgan2a1,definition,
% 3.36/1.80 ( demorgan2a1
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.36/1.80 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(complementUnionInPowersetComplement_type,type,
% 3.36/1.80 complementUnionInPowersetComplement: $o ).
% 3.36/1.80
% 3.36/1.80 thf(complementUnionInPowersetComplement,definition,
% 3.36/1.80 ( complementUnionInPowersetComplement
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(demorgan2a2_type,type,
% 3.36/1.80 demorgan2a2: $o ).
% 3.36/1.80
% 3.36/1.80 thf(demorgan2a2,definition,
% 3.36/1.80 ( demorgan2a2
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.36/1.80 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(demorgan1a_type,type,
% 3.36/1.80 demorgan1a: $o ).
% 3.36/1.80
% 3.36/1.80 thf(demorgan1a,definition,
% 3.36/1.80 ( demorgan1a
% 3.36/1.80 = ( ! [A: $i,X: $i] :
% 3.36/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.36/1.80 => ! [Y: $i] :
% 3.36/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.36/1.80 => ! [Xx: $i] :
% 3.36/1.80 ( ( in @ Xx @ A )
% 3.36/1.80 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.36/1.80 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.36/1.80
% 3.36/1.80 thf(demorgan1b,conjecture,
% 3.36/1.80 ( setextAx
% 3.36/1.80 => ( emptysetAx
% 3.36/1.80 => ( setadjoinAx
% 3.36/1.80 => ( powersetAx
% 3.36/1.80 => ( setunionAx
% 3.36/1.80 => ( omega0Ax
% 3.36/1.80 => ( omegaSAx
% 3.36/1.80 => ( omegaIndAx
% 3.36/1.80 => ( replAx
% 3.36/1.80 => ( foundationAx
% 3.36/1.80 => ( wellorderingAx
% 3.36/1.80 => ( descrp
% 3.36/1.80 => ( dsetconstrI
% 3.36/1.80 => ( dsetconstrEL
% 3.36/1.80 => ( dsetconstrER
% 3.36/1.80 => ( exuE1
% 3.36/1.80 => ( prop2setE
% 3.36/1.80 => ( emptysetE
% 3.36/1.80 => ( emptysetimpfalse
% 3.36/1.80 => ( notinemptyset
% 3.36/1.80 => ( exuE3e
% 3.36/1.80 => ( setext
% 3.36/1.80 => ( emptyI
% 3.36/1.80 => ( noeltsimpempty
% 3.36/1.80 => ( setbeta
% 3.36/1.80 => ( nonemptyE1
% 3.36/1.80 => ( nonemptyI
% 3.36/1.80 => ( nonemptyI1
% 3.36/1.80 => ( setadjoinIL
% 3.36/1.80 => ( emptyinunitempty
% 3.36/1.80 => ( setadjoinIR
% 3.36/1.80 => ( setadjoinE
% 3.36/1.80 => ( setadjoinOr
% 3.36/1.80 => ( setoftrueEq
% 3.36/1.80 => ( powersetI
% 3.36/1.80 => ( emptyinPowerset
% 3.36/1.80 => ( emptyInPowerset
% 3.36/1.80 => ( powersetE
% 3.36/1.80 => ( setunionI
% 3.36/1.80 => ( setunionE
% 3.36/1.80 => ( subPowSU
% 3.36/1.80 => ( exuE2
% 3.36/1.80 => ( nonemptyImpWitness
% 3.36/1.80 => ( uniqinunit
% 3.36/1.80 => ( notinsingleton
% 3.36/1.80 => ( eqinunit
% 3.36/1.80 => ( singletonsswitch
% 3.36/1.80 => ( upairsetE
% 3.36/1.80 => ( upairsetIL
% 3.36/1.80 => ( upairsetIR
% 3.36/1.80 => ( emptyE1
% 3.36/1.80 => ( vacuousDall
% 3.36/1.80 => ( quantDeMorgan1
% 3.36/1.80 => ( quantDeMorgan2
% 3.36/1.80 => ( quantDeMorgan3
% 3.36/1.80 => ( quantDeMorgan4
% 3.36/1.80 => ( prop2setI
% 3.36/1.80 => ( prop2set2propI
% 3.36/1.80 => ( notdexE
% 3.36/1.80 => ( notdallE
% 3.36/1.80 => ( exuI1
% 3.36/1.80 => ( exuI3
% 3.36/1.80 => ( exuI2
% 3.36/1.80 => ( inCongP
% 3.36/1.80 => ( in__Cong
% 3.36/1.80 => ( exuE3u
% 3.36/1.80 => ( exu__Cong
% 3.36/1.80 => ( emptyset__Cong
% 3.36/1.80 => ( setadjoin__Cong
% 3.36/1.80 => ( powerset__Cong
% 3.36/1.80 => ( setunion__Cong
% 3.36/1.80 => ( omega__Cong
% 3.36/1.80 => ( exuEu
% 3.36/1.80 => ( descr__Cong
% 3.36/1.80 => ( dsetconstr__Cong
% 3.36/1.80 => ( subsetI1
% 3.36/1.80 => ( eqimpsubset2
% 3.36/1.80 => ( eqimpsubset1
% 3.36/1.80 => ( subsetI2
% 3.36/1.80 => ( emptysetsubset
% 3.36/1.80 => ( subsetE
% 3.36/1.80 => ( subsetE2
% 3.36/1.80 => ( notsubsetI
% 3.36/1.80 => ( notequalI1
% 3.36/1.80 => ( notequalI2
% 3.36/1.80 => ( subsetRefl
% 3.36/1.80 => ( subsetTrans
% 3.36/1.80 => ( setadjoinSub
% 3.36/1.80 => ( setadjoinSub2
% 3.36/1.80 => ( subset2powerset
% 3.36/1.80 => ( setextsub
% 3.36/1.80 => ( subsetemptysetimpeq
% 3.36/1.80 => ( powersetI1
% 3.36/1.80 => ( powersetE1
% 3.36/1.80 => ( inPowerset
% 3.36/1.80 => ( powersetsubset
% 3.36/1.80 => ( sepInPowerset
% 3.36/1.80 => ( sepSubset
% 3.36/1.80 => ( binunionIL
% 3.36/1.80 => ( upairset2IR
% 3.36/1.80 => ( binunionIR
% 3.36/1.80 => ( binunionEcases
% 3.36/1.80 => ( binunionE
% 3.36/1.80 => ( binunionLsub
% 3.36/1.80 => ( binunionRsub
% 3.36/1.80 => ( binintersectI
% 3.36/1.80 => ( binintersectSubset5
% 3.36/1.80 => ( binintersectEL
% 3.36/1.80 => ( binintersectLsub
% 3.36/1.80 => ( binintersectSubset2
% 3.36/1.80 => ( binintersectSubset3
% 3.36/1.80 => ( binintersectER
% 3.36/1.80 => ( disjointsetsI1
% 3.36/1.80 => ( binintersectRsub
% 3.36/1.80 => ( binintersectSubset4
% 3.36/1.80 => ( binintersectSubset1
% 3.36/1.80 => ( bs114d
% 3.36/1.80 => ( setminusI
% 3.36/1.80 => ( setminusEL
% 3.36/1.80 => ( setminusER
% 3.36/1.80 => ( setminusSubset2
% 3.36/1.80 => ( setminusERneg
% 3.36/1.80 => ( setminusELneg
% 3.36/1.80 => ( setminusILneg
% 3.36/1.80 => ( setminusIRneg
% 3.36/1.80 => ( setminusLsub
% 3.36/1.80 => ( setminusSubset1
% 3.36/1.80 => ( symdiffE
% 3.36/1.80 => ( symdiffI1
% 3.36/1.80 => ( symdiffI2
% 3.36/1.80 => ( symdiffIneg1
% 3.36/1.80 => ( symdiffIneg2
% 3.36/1.80 => ( secondinupair
% 3.36/1.80 => ( setukpairIL
% 3.36/1.80 => ( setukpairIR
% 3.36/1.80 => ( kpairiskpair
% 3.36/1.80 => ( kpairp
% 3.36/1.80 => ( singletonsubset
% 3.36/1.80 => ( singletoninpowerset
% 3.36/1.80 => ( singletoninpowunion
% 3.36/1.80 => ( upairset2E
% 3.36/1.80 => ( upairsubunion
% 3.36/1.80 => ( upairinpowunion
% 3.36/1.80 => ( ubforcartprodlem1
% 3.36/1.80 => ( ubforcartprodlem2
% 3.47/1.80 => ( ubforcartprodlem3
% 3.47/1.80 => ( cartprodpairin
% 3.47/1.80 => ( cartprodmempair1
% 3.47/1.80 => ( cartprodmempair
% 3.47/1.80 => ( setunionE2
% 3.47/1.80 => ( setunionsingleton1
% 3.47/1.80 => ( setunionsingleton2
% 3.47/1.80 => ( setunionsingleton
% 3.47/1.80 => ( singletonprop
% 3.47/1.80 => ( ex1E1
% 3.47/1.80 => ( ex1I
% 3.47/1.80 => ( ex1I2
% 3.47/1.80 => ( singletonsuniq
% 3.47/1.80 => ( setukpairinjL1
% 3.47/1.80 => ( kfstsingleton
% 3.47/1.80 => ( theprop
% 3.47/1.80 => ( kfstpairEq
% 3.47/1.80 => ( cartprodfstin
% 3.47/1.80 => ( setukpairinjL2
% 3.47/1.80 => ( setukpairinjL
% 3.47/1.80 => ( setukpairinjR11
% 3.47/1.80 => ( setukpairinjR12
% 3.47/1.80 => ( setukpairinjR1
% 3.47/1.80 => ( upairequniteq
% 3.47/1.80 => ( setukpairinjR2
% 3.47/1.80 => ( setukpairinjR
% 3.47/1.80 => ( ksndsingleton
% 3.47/1.80 => ( ksndpairEq
% 3.47/1.80 => ( kpairsurjEq
% 3.47/1.80 => ( cartprodsndin
% 3.47/1.80 => ( cartprodpairmemEL
% 3.47/1.80 => ( cartprodpairmemER
% 3.47/1.80 => ( cartprodmempaircEq
% 3.47/1.80 => ( cartprodfstpairEq
% 3.47/1.80 => ( cartprodsndpairEq
% 3.47/1.80 => ( cartprodpairsurjEq
% 3.47/1.80 => ( dpsetconstrI
% 3.47/1.80 => ( dpsetconstrSub
% 3.47/1.80 => ( setOfPairsIsBReln
% 3.47/1.80 => ( dpsetconstrERa
% 3.47/1.80 => ( dpsetconstrEL1
% 3.47/1.80 => ( dpsetconstrEL2
% 3.47/1.80 => ( dpsetconstrER
% 3.47/1.80 => ( funcImageSingleton
% 3.47/1.80 => ( apProp
% 3.47/1.80 => ( app
% 3.47/1.80 => ( infuncsetfunc
% 3.47/1.80 => ( ap2p
% 3.47/1.80 => ( funcinfuncset
% 3.47/1.80 => ( lamProp
% 3.47/1.80 => ( lamp
% 3.47/1.80 => ( lam2p
% 3.47/1.80 => ( brelnall1
% 3.47/1.80 => ( brelnall2
% 3.47/1.80 => ( ex1E2
% 3.47/1.80 => ( funcGraphProp1
% 3.47/1.80 => ( funcGraphProp3
% 3.47/1.80 => ( funcGraphProp2
% 3.47/1.80 => ( funcextLem
% 3.47/1.80 => ( funcGraphProp4
% 3.47/1.80 => ( subbreln
% 3.47/1.80 => ( eqbreln
% 3.47/1.80 => ( funcext
% 3.47/1.80 => ( funcext2
% 3.47/1.80 => ( ap2apEq1
% 3.47/1.80 => ( ap2apEq2
% 3.47/1.80 => ( beta1
% 3.47/1.80 => ( eta1
% 3.47/1.80 => ( lam2lamEq
% 3.47/1.80 => ( beta2
% 3.47/1.80 => ( eta2
% 3.47/1.80 => ( iffalseProp1
% 3.47/1.80 => ( iffalseProp2
% 3.47/1.80 => ( iftrueProp1
% 3.47/1.80 => ( iftrueProp2
% 3.47/1.80 => ( ifSingleton
% 3.47/1.80 => ( ifp
% 3.47/1.80 => ( theeq
% 3.47/1.80 => ( iftrue
% 3.47/1.80 => ( iffalse
% 3.47/1.80 => ( iftrueorfalse
% 3.47/1.80 => ( binintersectT_lem
% 3.47/1.80 => ( binunionT_lem
% 3.47/1.80 => ( powersetT_lem
% 3.47/1.80 => ( setminusT_lem
% 3.47/1.80 => ( complementT_lem
% 3.47/1.80 => ( setextT
% 3.47/1.80 => ( subsetTI
% 3.47/1.80 => ( powersetTI1
% 3.47/1.80 => ( powersetTE1
% 3.47/1.80 => ( complementTI1
% 3.47/1.80 => ( complementTE1
% 3.47/1.80 => ( binintersectTELcontra
% 3.47/1.80 => ( binintersectTERcontra
% 3.47/1.80 => ( contrasubsetT
% 3.47/1.80 => ( contrasubsetT1
% 3.47/1.80 => ( contrasubsetT2
% 3.47/1.80 => ( contrasubsetT3
% 3.47/1.80 => ( doubleComplementI1
% 3.47/1.80 => ( doubleComplementE1
% 3.47/1.80 => ( doubleComplementSub1
% 3.47/1.80 => ( doubleComplementSub2
% 3.47/1.80 => ( doubleComplementEq
% 3.47/1.80 => ( complementTnotintersectT
% 3.47/1.80 => ( complementImpComplementIntersect
% 3.47/1.80 => ( complementSubsetComplementIntersect
% 3.47/1.80 => ( complementInPowersetComplementIntersect
% 3.47/1.80 => ( contraSubsetComplement
% 3.47/1.80 => ( complementTcontraSubset
% 3.47/1.80 => ( binunionTILcontra
% 3.47/1.80 => ( binunionTIRcontra
% 3.47/1.80 => ( inIntersectImpInUnion
% 3.47/1.80 => ( inIntersectImpInUnion2
% 3.47/1.80 => ( inIntersectImpInIntersectUnions
% 3.47/1.80 => ( intersectInPowersetIntersectUnions
% 3.47/1.80 => ( inComplementUnionImpNotIn1
% 3.47/1.80 => ( inComplementUnionImpInComplement1
% 3.47/1.80 => ( binunionTE
% 3.47/1.80 => ( binunionTEcontra
% 3.47/1.80 => ( demorgan2a1
% 3.47/1.80 => ( complementUnionInPowersetComplement
% 3.47/1.80 => ( demorgan2a2
% 3.47/1.80 => ( demorgan1a
% 3.47/1.80 => ! [A: $i,X: $i] :
% 3.47/1.80 ( ( in @ X @ ( powerset @ A ) )
% 3.47/1.80 => ! [Y: $i] :
% 3.47/1.80 ( ( in @ Y @ ( powerset @ A ) )
% 3.47/1.80 => ! [Xx: $i] :
% 3.47/1.80 ( ( in @ Xx @ A )
% 3.47/1.80 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.47/1.80 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.47/1.80 ------------------------
% 3.47/1.84 % (13750)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_13645 for (2999ds/275Mi)
% 3.47/1.84 % (13747)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_13645 for (2999ds/27Mi)
% 3.47/1.84 % (13751)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/18Mi)
% 3.47/1.84 % (13749)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_13645 for (2999ds/2Mi)
% 3.47/1.84 % (13746)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_13645 for (2999ds/4Mi)
% 3.47/1.84 % (13745)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/183Mi)
% 3.47/1.84 % (13749)Instruction limit reached!
% 3.47/1.84 % (13749)------------------------------
% 3.47/1.84 % (13749)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.84 % (13749)Termination reason: Unknown
% 3.47/1.84 % (13749)Termination phase: shuffling
% 3.47/1.84
% 3.47/1.84 % (13749)Memory used [KB]: 1535
% 3.47/1.84 % (13749)Time elapsed: 0.004 s
% 3.47/1.84 % (13749)Instructions burned: 3 (million)
% 3.47/1.84 % (13749)------------------------------
% 3.47/1.84 % (13749)------------------------------
% 3.47/1.84 % (13746)Instruction limit reached!
% 3.47/1.84 % (13746)------------------------------
% 3.47/1.84 % (13746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.84 % (13746)Termination reason: Unknown
% 3.47/1.84 % (13746)Termination phase: shuffling
% 3.47/1.84
% 3.47/1.84 % (13746)Memory used [KB]: 1535
% 3.47/1.84 % (13746)Time elapsed: 0.005 s
% 3.47/1.84 % (13746)Instructions burned: 5 (million)
% 3.47/1.84 % (13746)------------------------------
% 3.47/1.84 % (13746)------------------------------
% 3.47/1.84 % (13748)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/2Mi)
% 3.47/1.84 % (13748)Instruction limit reached!
% 3.47/1.84 % (13748)------------------------------
% 3.47/1.84 % (13748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.84 % (13748)Termination reason: Unknown
% 3.47/1.84 % (13748)Termination phase: shuffling
% 3.47/1.84
% 3.47/1.84 % (13748)Memory used [KB]: 1407
% 3.47/1.84 % (13748)Time elapsed: 0.003 s
% 3.47/1.84 % (13748)Instructions burned: 2 (million)
% 3.47/1.84 % (13748)------------------------------
% 3.47/1.84 % (13748)------------------------------
% 3.47/1.85 % (13751)Instruction limit reached!
% 3.47/1.85 % (13751)------------------------------
% 3.47/1.85 % (13751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.85 % (13751)Termination reason: Unknown
% 3.47/1.85 % (13751)Termination phase: shuffling
% 3.47/1.85
% 3.47/1.85 % (13751)Memory used [KB]: 1791
% 3.47/1.85 % (13751)Time elapsed: 0.011 s
% 3.47/1.85 % (13751)Instructions burned: 18 (million)
% 3.47/1.85 % (13751)------------------------------
% 3.47/1.85 % (13751)------------------------------
% 3.47/1.85 % (13747)Instruction limit reached!
% 3.47/1.85 % (13747)------------------------------
% 3.47/1.85 % (13747)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.85 % (13747)Termination reason: Unknown
% 3.47/1.85 % (13747)Termination phase: shuffling
% 3.47/1.85
% 3.47/1.85 % (13747)Memory used [KB]: 2046
% 3.47/1.85 % (13747)Time elapsed: 0.014 s
% 3.47/1.85 % (13747)Instructions burned: 27 (million)
% 3.47/1.85 % (13747)------------------------------
% 3.47/1.85 % (13747)------------------------------
% 3.47/1.86 % (13752)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/3Mi)
% 3.47/1.86 % (13753)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_13645 for (2999ds/37Mi)
% 3.47/1.86 % (13752)Instruction limit reached!
% 3.47/1.86 % (13752)------------------------------
% 3.47/1.86 % (13752)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.86 % (13752)Termination reason: Unknown
% 3.47/1.86 % (13752)Termination phase: shuffling
% 3.47/1.86
% 3.47/1.86 % (13752)Memory used [KB]: 1535
% 3.47/1.86 % (13752)Time elapsed: 0.003 s
% 3.47/1.86 % (13752)Instructions burned: 3 (million)
% 3.47/1.86 % (13752)------------------------------
% 3.47/1.86 % (13752)------------------------------
% 3.47/1.86 % (13754)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_13645 for (2999ds/15Mi)
% 3.47/1.86 % (13755)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_13645 for (2999ds/3Mi)
% 3.47/1.87 % (13755)Instruction limit reached!
% 3.47/1.87 % (13755)------------------------------
% 3.47/1.87 % (13755)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.87 % (13755)Termination reason: Unknown
% 3.47/1.87 % (13755)Termination phase: shuffling
% 3.47/1.87
% 3.47/1.87 % (13755)Memory used [KB]: 1407
% 3.47/1.87 % (13755)Time elapsed: 0.003 s
% 3.47/1.87 % (13755)Instructions burned: 3 (million)
% 3.47/1.87 % (13755)------------------------------
% 3.47/1.87 % (13755)------------------------------
% 3.47/1.87 % (13756)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_13645 for (2999ds/1041Mi)
% 3.47/1.87 % (13754)Instruction limit reached!
% 3.47/1.87 % (13754)------------------------------
% 3.47/1.87 % (13754)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.87 % (13754)Termination reason: Unknown
% 3.47/1.87 % (13754)Termination phase: shuffling
% 3.47/1.87
% 3.47/1.87 % (13754)Memory used [KB]: 1791
% 3.47/1.87 % (13754)Time elapsed: 0.010 s
% 3.47/1.87 % (13754)Instructions burned: 15 (million)
% 3.47/1.87 % (13754)------------------------------
% 3.47/1.87 % (13754)------------------------------
% 3.47/1.87 % (13757)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/7Mi)
% 3.47/1.88 % (13753)Instruction limit reached!
% 3.47/1.88 % (13753)------------------------------
% 3.47/1.88 % (13753)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.88 % (13753)Termination reason: Unknown
% 3.47/1.88 % (13753)Termination phase: shuffling
% 3.47/1.88
% 3.47/1.88 % (13753)Memory used [KB]: 2302
% 3.47/1.88 % (13753)Time elapsed: 0.019 s
% 3.47/1.88 % (13753)Instructions burned: 38 (million)
% 3.47/1.88 % (13753)------------------------------
% 3.47/1.88 % (13753)------------------------------
% 3.47/1.88 % (13757)Instruction limit reached!
% 3.47/1.88 % (13757)------------------------------
% 3.47/1.88 % (13757)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.88 % (13757)Termination reason: Unknown
% 3.47/1.88 % (13757)Termination phase: shuffling
% 3.47/1.88
% 3.47/1.88 % (13757)Memory used [KB]: 1663
% 3.47/1.88 % (13757)Time elapsed: 0.005 s
% 3.47/1.88 % (13757)Instructions burned: 7 (million)
% 3.47/1.88 % (13757)------------------------------
% 3.47/1.88 % (13757)------------------------------
% 3.47/1.88 % (13758)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/16Mi)
% 3.47/1.89 % (13761)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/7Mi)
% 3.47/1.89 % (13759)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_13645 for (2999ds/3Mi)
% 3.47/1.89 % (13758)Instruction limit reached!
% 3.47/1.89 % (13758)------------------------------
% 3.47/1.89 % (13758)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.89 % (13758)Termination reason: Unknown
% 3.47/1.89 % (13758)Termination phase: shuffling
% 3.47/1.89
% 3.47/1.89 % (13758)Memory used [KB]: 1791
% 3.47/1.89 % (13758)Time elapsed: 0.010 s
% 3.47/1.89 % (13758)Instructions burned: 17 (million)
% 3.47/1.89 % (13758)------------------------------
% 3.47/1.89 % (13758)------------------------------
% 3.47/1.89 % (13761)Instruction limit reached!
% 3.47/1.89 % (13761)------------------------------
% 3.47/1.89 % (13761)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.89 % (13761)Termination reason: Unknown
% 3.47/1.89 % (13761)Termination phase: shuffling
% 3.47/1.89
% 3.47/1.89 % (13761)Memory used [KB]: 1535
% 3.47/1.89 % (13761)Time elapsed: 0.004 s
% 3.47/1.89 % (13761)Instructions burned: 7 (million)
% 3.47/1.89 % (13761)------------------------------
% 3.47/1.89 % (13761)------------------------------
% 3.47/1.89 % (13759)Instruction limit reached!
% 3.47/1.89 % (13759)------------------------------
% 3.47/1.89 % (13759)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.89 % (13759)Termination reason: Unknown
% 3.47/1.89 % (13759)Termination phase: shuffling
% 3.47/1.89
% 3.47/1.89 % (13759)Memory used [KB]: 1407
% 3.47/1.89 % (13759)Time elapsed: 0.003 s
% 3.47/1.89 % (13759)Instructions burned: 3 (million)
% 3.47/1.89 % (13759)------------------------------
% 3.47/1.89 % (13759)------------------------------
% 3.47/1.89 % (13760)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_13645 for (2999ds/3Mi)
% 3.47/1.89 % (13760)Instruction limit reached!
% 3.47/1.89 % (13760)------------------------------
% 3.47/1.89 % (13760)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.89 % (13760)Termination reason: Unknown
% 3.47/1.89 % (13760)Termination phase: shuffling
% 3.47/1.89
% 3.47/1.89 % (13760)Memory used [KB]: 1535
% 3.47/1.89 % (13760)Time elapsed: 0.003 s
% 3.47/1.89 % (13760)Instructions burned: 3 (million)
% 3.47/1.89 % (13760)------------------------------
% 3.47/1.89 % (13760)------------------------------
% 3.48/1.91 % (13762)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/3Mi)
% 3.48/1.91 % (13763)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/4Mi)
% 3.48/1.91 % (13762)Instruction limit reached!
% 3.48/1.91 % (13762)------------------------------
% 3.48/1.91 % (13762)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.91 % (13765)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/710Mi)
% 3.48/1.91 % (13762)Termination reason: Unknown
% 3.48/1.91 % (13762)Termination phase: shuffling
% 3.48/1.91
% 3.48/1.91 % (13762)Memory used [KB]: 1535
% 3.48/1.91 % (13762)Time elapsed: 0.004 s
% 3.48/1.91 % (13762)Instructions burned: 3 (million)
% 3.48/1.91 % (13762)------------------------------
% 3.48/1.91 % (13762)------------------------------
% 3.48/1.91 % (13763)Instruction limit reached!
% 3.48/1.91 % (13763)------------------------------
% 3.48/1.91 % (13763)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.91 % (13763)Termination reason: Unknown
% 3.48/1.91 % (13763)Termination phase: shuffling
% 3.48/1.91
% 3.48/1.91 % (13763)Memory used [KB]: 1535
% 3.48/1.91 % (13763)Time elapsed: 0.004 s
% 3.48/1.91 % (13763)Instructions burned: 4 (million)
% 3.48/1.91 % (13763)------------------------------
% 3.48/1.91 % (13763)------------------------------
% 3.48/1.91 % (13764)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13645 for (2999ds/18Mi)
% 3.48/1.92 % (13764)Instruction limit reached!
% 3.48/1.92 % (13764)------------------------------
% 3.48/1.92 % (13764)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.92 % (13764)Termination reason: Unknown
% 3.48/1.92 % (13764)Termination phase: shuffling
% 3.48/1.92
% 3.48/1.92 % (13764)Memory used [KB]: 1791
% 3.48/1.92 % (13764)Time elapsed: 0.011 s
% 3.48/1.92 % (13764)Instructions burned: 18 (million)
% 3.48/1.92 % (13764)------------------------------
% 3.48/1.92 % (13764)------------------------------
% 3.48/1.92 % (13745)Instruction limit reached!
% 3.48/1.92 % (13745)------------------------------
% 3.48/1.92 % (13745)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.92 % (13745)Termination reason: Unknown
% 3.48/1.92 % (13745)Termination phase: Saturation
% 3.48/1.92
% 3.48/1.92 % (13745)Memory used [KB]: 7419
% 3.48/1.92 % (13745)Time elapsed: 0.083 s
% 3.48/1.92 % (13745)Instructions burned: 184 (million)
% 3.48/1.92 % (13745)------------------------------
% 3.48/1.92 % (13745)------------------------------
% 3.48/1.93 % (13766)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13645 for (2998ds/6Mi)
% 3.48/1.93 % (13767)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_13645 for (2998ds/902Mi)
% 3.48/1.93 % (13766)Instruction limit reached!
% 3.48/1.93 % (13766)------------------------------
% 3.48/1.93 % (13766)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.93 % (13766)Termination reason: Unknown
% 3.48/1.93 % (13766)Termination phase: shuffling
% 3.48/1.93
% 3.48/1.93 % (13766)Memory used [KB]: 1535
% 3.48/1.93 % (13766)Time elapsed: 0.005 s
% 3.48/1.93 % (13766)Instructions burned: 7 (million)
% 3.48/1.93 % (13766)------------------------------
% 3.48/1.93 % (13766)------------------------------
% 3.48/1.94 % (13768)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_13645 for (2998ds/21Mi)
% 3.48/1.94 % (13769)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13645 for (2998ds/5Mi)
% 3.48/1.94 % (13769)Instruction limit reached!
% 3.48/1.94 % (13769)------------------------------
% 3.48/1.94 % (13769)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.94 % (13769)Termination reason: Unknown
% 3.48/1.94 % (13769)Termination phase: shuffling
% 3.48/1.94
% 3.48/1.94 % (13769)Memory used [KB]: 1535
% 3.48/1.94 % (13769)Time elapsed: 0.004 s
% 3.48/1.94 % (13769)Instructions burned: 6 (million)
% 3.48/1.94 % (13769)------------------------------
% 3.48/1.94 % (13769)------------------------------
% 3.48/1.95 % (13770)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_13645 for (2998ds/6Mi)
% 3.48/1.95 % (13768)Instruction limit reached!
% 3.48/1.95 % (13768)------------------------------
% 3.48/1.95 % (13768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.95 % (13768)Termination reason: Unknown
% 3.48/1.95 % (13768)Termination phase: shuffling
% 3.48/1.95
% 3.48/1.95 % (13768)Memory used [KB]: 1791
% 3.48/1.95 % (13768)Time elapsed: 0.012 s
% 3.48/1.95 % (13768)Instructions burned: 21 (million)
% 3.48/1.95 % (13768)------------------------------
% 3.48/1.95 % (13768)------------------------------
% 3.48/1.95 % (13770)Instruction limit reached!
% 3.48/1.95 % (13770)------------------------------
% 3.48/1.95 % (13770)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.95 % (13770)Termination reason: Unknown
% 3.48/1.95 % (13770)Termination phase: shuffling
% 3.48/1.95
% 3.48/1.95 % (13770)Memory used [KB]: 1535
% 3.48/1.95 % (13770)Time elapsed: 0.005 s
% 3.48/1.95 % (13770)Instructions burned: 7 (million)
% 3.48/1.95 % (13770)------------------------------
% 3.48/1.95 % (13770)------------------------------
% 3.48/1.96 % (13771)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13645 for (2998ds/377Mi)
% 3.48/1.96 % (13772)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_13645 for (2998ds/779Mi)
% 3.48/1.96 % (13773)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_13645 for (2998ds/19Mi)
% 3.48/1.97 % (13750)Instruction limit reached!
% 3.48/1.97 % (13750)------------------------------
% 3.48/1.97 % (13750)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.97 % (13750)Termination reason: Unknown
% 3.48/1.97 % (13750)Termination phase: Saturation
% 3.48/1.97
% 3.48/1.97 % (13750)Memory used [KB]: 9850
% 3.48/1.97 % (13750)Time elapsed: 0.135 s
% 3.48/1.97 % (13750)Instructions burned: 276 (million)
% 3.48/1.97 % (13750)------------------------------
% 3.48/1.97 % (13750)------------------------------
% 3.48/1.97 % (13773)Instruction limit reached!
% 3.48/1.97 % (13773)------------------------------
% 3.48/1.97 % (13773)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/1.97 % (13773)Termination reason: Unknown
% 3.48/1.97 % (13773)Termination phase: shuffling
% 3.48/1.97
% 3.48/1.97 % (13773)Memory used [KB]: 1791
% 3.48/1.97 % (13773)Time elapsed: 0.010 s
% 3.48/1.97 % (13773)Instructions burned: 20 (million)
% 3.48/1.97 % (13773)------------------------------
% 3.48/1.97 % (13773)------------------------------
% 3.48/1.99 % (13774)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13645 for (2998ds/879Mi)
% 3.48/1.99 % (13775)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_13645 for (2998ds/17Mi)
% 3.48/2.00 % (13775)Instruction limit reached!
% 3.48/2.00 % (13775)------------------------------
% 3.48/2.00 % (13775)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/2.00 % (13775)Termination reason: Unknown
% 3.48/2.00 % (13775)Termination phase: shuffling
% 3.48/2.00
% 3.48/2.00 % (13775)Memory used [KB]: 1791
% 3.48/2.00 % (13775)Time elapsed: 0.010 s
% 3.48/2.00 % (13775)Instructions burned: 17 (million)
% 3.48/2.00 % (13775)------------------------------
% 3.48/2.00 % (13775)------------------------------
% 3.48/2.02 % (13776)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13645 for (2997ds/3Mi)
% 3.48/2.02 % (13776)Instruction limit reached!
% 3.48/2.02 % (13776)------------------------------
% 3.48/2.02 % (13776)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.48/2.02 % (13776)Termination reason: Unknown
% 3.48/2.02 % (13776)Termination phase: shuffling
% 3.48/2.02
% 3.48/2.02 % (13776)Memory used [KB]: 1535
% 3.48/2.02 % (13776)Time elapsed: 0.004 s
% 3.48/2.02 % (13776)Instructions burned: 3 (million)
% 3.48/2.02 % (13776)------------------------------
% 3.48/2.02 % (13776)------------------------------
% 4.80/2.03 % (13777)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_13645 for (2997ds/30Mi)
% 4.80/2.05 % (13777)Instruction limit reached!
% 4.80/2.05 % (13777)------------------------------
% 4.80/2.05 % (13777)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.80/2.05 % (13777)Termination reason: Unknown
% 4.80/2.05 % (13777)Termination phase: shuffling
% 4.80/2.05
% 4.80/2.05 % (13777)Memory used [KB]: 2046
% 4.80/2.05 % (13777)Time elapsed: 0.016 s
% 4.80/2.05 % (13777)Instructions burned: 30 (million)
% 4.80/2.05 % (13777)------------------------------
% 4.80/2.05 % (13777)------------------------------
% 4.80/2.06 % (13778)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_13645 for (2997ds/127Mi)
% 4.80/2.12 % (13778)Instruction limit reached!
% 4.80/2.12 % (13778)------------------------------
% 4.80/2.12 % (13778)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.80/2.12 % (13778)Termination reason: Unknown
% 4.80/2.12 % (13778)Termination phase: Function definition elimination
% 4.80/2.12
% 4.80/2.12 % (13778)Memory used [KB]: 2686
% 4.80/2.12 % (13778)Time elapsed: 0.059 s
% 4.80/2.12 % (13778)Instructions burned: 128 (million)
% 4.80/2.12 % (13778)------------------------------
% 4.80/2.12 % (13778)------------------------------
% 5.52/2.13 % (13771)Instruction limit reached!
% 5.52/2.13 % (13771)------------------------------
% 5.52/2.13 % (13771)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.52/2.13 % (13771)Termination reason: Unknown
% 5.52/2.13 % (13771)Termination phase: Saturation
% 5.52/2.13
% 5.52/2.13 % (13771)Memory used [KB]: 9338
% 5.52/2.13 % (13771)Time elapsed: 0.173 s
% 5.52/2.13 % (13771)Instructions burned: 378 (million)
% 5.52/2.13 % (13771)------------------------------
% 5.52/2.13 % (13771)------------------------------
% 5.52/2.14 % (13779)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_13645 for (2996ds/100Mi)
% 5.52/2.15 % (13780)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_13645 for (2996ds/3Mi)
% 5.52/2.15 % (13780)Instruction limit reached!
% 5.52/2.15 % (13780)------------------------------
% 5.52/2.15 % (13780)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.52/2.15 % (13780)Termination reason: Unknown
% 5.52/2.15 % (13780)Termination phase: shuffling
% 5.52/2.15
% 5.52/2.15 % (13780)Memory used [KB]: 1535
% 5.52/2.15 % (13780)Time elapsed: 0.003 s
% 5.52/2.15 % (13780)Instructions burned: 3 (million)
% 5.52/2.15 % (13780)------------------------------
% 5.52/2.15 % (13780)------------------------------
% 5.52/2.16 % (13781)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_13645 for (2996ds/20Mi)
% 5.52/2.17 % (13781)Instruction limit reached!
% 5.52/2.17 % (13781)------------------------------
% 5.52/2.17 % (13781)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.52/2.17 % (13781)Termination reason: Unknown
% 5.52/2.17 % (13781)Termination phase: shuffling
% 5.52/2.17
% 5.52/2.17 % (13781)Memory used [KB]: 1791
% 5.52/2.17 % (13781)Time elapsed: 0.012 s
% 5.52/2.17 % (13781)Instructions burned: 20 (million)
% 5.52/2.17 % (13781)------------------------------
% 5.52/2.17 % (13781)------------------------------
% 5.52/2.18 % (13779)Instruction limit reached!
% 5.52/2.18 % (13779)------------------------------
% 5.52/2.18 % (13779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.52/2.18 % (13779)Termination reason: Unknown
% 5.52/2.18 % (13779)Termination phase: Property scanning
% 5.52/2.18
% 5.52/2.18 % (13779)Memory used [KB]: 2686
% 5.52/2.18 % (13779)Time elapsed: 0.047 s
% 5.52/2.18 % (13779)Instructions burned: 100 (million)
% 5.52/2.18 % (13779)------------------------------
% 5.52/2.18 % (13779)------------------------------
% 5.52/2.19 % (13782)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_13645 for (2996ds/86Mi)
% 6.00/2.20 % (13783)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_13645 for (2996ds/107Mi)
% 6.00/2.23 % (13782)Instruction limit reached!
% 6.00/2.23 % (13782)------------------------------
% 6.00/2.23 % (13782)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.00/2.23 % (13782)Termination reason: Unknown
% 6.00/2.23 % (13782)Termination phase: Property scanning
% 6.00/2.23
% 6.00/2.23 % (13782)Memory used [KB]: 2686
% 6.00/2.23 % (13782)Time elapsed: 0.041 s
% 6.00/2.23 % (13782)Instructions burned: 88 (million)
% 6.00/2.23 % (13782)------------------------------
% 6.00/2.23 % (13782)------------------------------
% 6.00/2.25 % (13784)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_13645 for (2995ds/39Mi)
% 6.00/2.25 % (13783)Instruction limit reached!
% 6.00/2.25 % (13783)------------------------------
% 6.00/2.25 % (13783)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.00/2.25 % (13783)Termination reason: Unknown
% 6.00/2.25 % (13783)Termination phase: Property scanning
% 6.00/2.25
% 6.00/2.25 % (13783)Memory used [KB]: 2686
% 6.00/2.25 % (13783)Time elapsed: 0.049 s
% 6.00/2.25 % (13783)Instructions burned: 108 (million)
% 6.00/2.25 % (13783)------------------------------
% 6.00/2.25 % (13783)------------------------------
% 6.00/2.27 % (13784)Instruction limit reached!
% 6.00/2.27 % (13784)------------------------------
% 6.00/2.27 % (13784)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.00/2.27 % (13784)Termination reason: Unknown
% 6.00/2.27 % (13784)Termination phase: shuffling
% 6.00/2.27
% 6.00/2.27 % (13784)Memory used [KB]: 2302
% 6.00/2.27 % (13784)Time elapsed: 0.020 s
% 6.00/2.27 % (13784)Instructions burned: 40 (million)
% 6.00/2.27 % (13784)------------------------------
% 6.00/2.27 % (13784)------------------------------
% 6.00/2.27 % (13785)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_13645 for (2995ds/448Mi)
% 6.00/2.27 % (13765)Instruction limit reached!
% 6.00/2.27 % (13765)------------------------------
% 6.00/2.27 % (13765)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.00/2.27 % (13765)Termination reason: Unknown
% 6.00/2.27 % (13765)Termination phase: Saturation
% 6.00/2.27
% 6.00/2.27 % (13765)Memory used [KB]: 10618
% 6.00/2.27 % (13765)Time elapsed: 0.360 s
% 6.00/2.27 % (13765)Instructions burned: 710 (million)
% 6.00/2.27 % (13765)------------------------------
% 6.00/2.27 % (13765)------------------------------
% 6.00/2.27 % (13756)Instruction limit reached!
% 6.00/2.27 % (13756)------------------------------
% 6.00/2.27 % (13756)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.00/2.27 % (13756)Termination reason: Unknown
% 6.00/2.27 % (13756)Termination phase: Saturation
% 6.00/2.27
% 6.00/2.27 % (13756)Memory used [KB]: 15223
% 6.00/2.27 % (13756)Time elapsed: 0.406 s
% 6.00/2.27 % (13756)Instructions burned: 1043 (million)
% 6.00/2.27 % (13756)------------------------------
% 6.00/2.27 % (13756)------------------------------
% 6.00/2.27 % (13772)Instruction limit reached!
% 6.00/2.27 % (13772)------------------------------
% 6.00/2.27 % (13772)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.00/2.27 % (13772)Termination reason: Unknown
% 6.00/2.27 % (13772)Termination phase: Saturation
% 6.00/2.27
% 6.00/2.27 % (13772)Memory used [KB]: 7547
% 6.00/2.27 % (13772)Time elapsed: 0.312 s
% 6.00/2.27 % (13772)Instructions burned: 779 (million)
% 6.00/2.27 % (13772)------------------------------
% 6.00/2.27 % (13772)------------------------------
% 6.00/2.28 % (13786)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_13645 for (2995ds/46Mi)
% 6.00/2.28 % (13787)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_13645 for (2995ds/98Mi)
% 6.00/2.29 % (13788)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_13645 for (2995ds/507Mi)
% 6.00/2.29 % (13789)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_13645 for (2995ds/149Mi)
% 6.63/2.30 % (13786)Instruction limit reached!
% 6.63/2.30 % (13786)------------------------------
% 6.63/2.30 % (13786)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.63/2.30 % (13786)Termination reason: Unknown
% 6.63/2.30 % (13786)Termination phase: shuffling
% 6.63/2.30
% 6.63/2.30 % (13786)Memory used [KB]: 2430
% 6.63/2.30 % (13786)Time elapsed: 0.023 s
% 6.63/2.30 % (13786)Instructions burned: 46 (million)
% 6.63/2.30 % (13786)------------------------------
% 6.63/2.30 % (13786)------------------------------
% 6.63/2.32 % (13790)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_13645 for (2994ds/39Mi)
% 6.63/2.33 % (13787)Instruction limit reached!
% 6.63/2.33 % (13787)------------------------------
% 6.63/2.33 % (13787)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.63/2.33 % (13787)Termination reason: Unknown
% 6.63/2.33 % (13787)Termination phase: Property scanning
% 6.63/2.33
% 6.63/2.33 % (13787)Memory used [KB]: 2558
% 6.63/2.33 % (13787)Time elapsed: 0.046 s
% 6.63/2.33 % (13787)Instructions burned: 99 (million)
% 6.63/2.33 % (13787)------------------------------
% 6.63/2.33 % (13787)------------------------------
% 6.63/2.34 % (13790)Instruction limit reached!
% 6.63/2.34 % (13790)------------------------------
% 6.63/2.34 % (13790)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.63/2.34 % (13790)Termination reason: Unknown
% 6.63/2.34 % (13790)Termination phase: shuffling
% 6.63/2.34
% 6.63/2.34 % (13790)Memory used [KB]: 2302
% 6.63/2.34 % (13790)Time elapsed: 0.020 s
% 6.63/2.34 % (13790)Instructions burned: 40 (million)
% 6.63/2.34 % (13790)------------------------------
% 6.63/2.34 % (13790)------------------------------
% 6.63/2.35 % (13791)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13645 for (2994ds/6Mi)
% 6.63/2.35 % (13791)Instruction limit reached!
% 6.63/2.35 % (13791)------------------------------
% 6.63/2.35 % (13791)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.63/2.35 % (13791)Termination reason: Unknown
% 6.63/2.35 % (13791)Termination phase: shuffling
% 6.63/2.35
% 6.63/2.35 % (13791)Memory used [KB]: 1535
% 6.63/2.35 % (13791)Time elapsed: 0.005 s
% 6.63/2.35 % (13791)Instructions burned: 7 (million)
% 6.63/2.35 % (13791)------------------------------
% 6.63/2.35 % (13791)------------------------------
% 6.63/2.36 % (13792)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_13645 for (2994ds/1087Mi)
% 6.63/2.36 % (13789)Instruction limit reached!
% 6.63/2.36 % (13789)------------------------------
% 6.63/2.36 % (13789)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.63/2.36 % (13789)Termination reason: Unknown
% 6.63/2.36 % (13789)Termination phase: Twee Goal Transformation
% 6.63/2.36
% 6.63/2.36 % (13789)Memory used [KB]: 3326
% 6.63/2.36 % (13789)Time elapsed: 0.069 s
% 6.63/2.36 % (13789)Instructions burned: 149 (million)
% 6.63/2.36 % (13789)------------------------------
% 6.63/2.36 % (13789)------------------------------
% 6.63/2.37 % (13793)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_13645 for (2994ds/47Mi)
% 6.63/2.37 % (13767)First to succeed.
% 6.63/2.37 % (13794)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_13645 for (2994ds/44Mi)
% 6.63/2.38 % (13767)Refutation found. Thanks to Tanya!
% 6.63/2.38 % SZS status Theorem for DTF2THF_13645
% 6.63/2.38 % SZS output start Proof for DTF2THF_13645
% 6.63/2.38 thf(func_def_0, type, in: $i > $i > $o).
% 6.63/2.38 thf(func_def_1, type, exu: ($i > $o) > $o).
% 6.63/2.38 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 6.63/2.38 thf(func_def_8, type, powerset: $i > $i).
% 6.63/2.38 thf(func_def_10, type, setunion: $i > $i).
% 6.63/2.38 thf(func_def_19, type, descr: ($i > $o) > $i).
% 6.63/2.38 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 6.63/2.38 thf(func_def_26, type, prop2set: $o > $i).
% 6.63/2.38 thf(func_def_36, type, nonempty: $i > $o).
% 6.63/2.38 thf(func_def_69, type, set2prop: $i > $o).
% 6.63/2.38 thf(func_def_88, type, subset: $i > $i > $o).
% 6.63/2.38 thf(func_def_89, type, disjoint: $i > $i > $o).
% 6.63/2.38 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 6.63/2.38 thf(func_def_114, type, binunion: $i > $i > $i).
% 6.63/2.38 thf(func_def_122, type, binintersect: $i > $i > $i).
% 6.63/2.38 thf(func_def_135, type, regular: $i > $o).
% 6.63/2.38 thf(func_def_136, type, setminus: $i > $i > $i).
% 6.63/2.38 thf(func_def_147, type, symdiff: $i > $i > $i).
% 6.63/2.38 thf(func_def_153, type, iskpair: $i > $o).
% 6.63/2.38 thf(func_def_158, type, kpair: $i > $i > $i).
% 6.63/2.38 thf(func_def_160, type, cartprod: $i > $i > $i).
% 6.63/2.38 thf(func_def_177, type, singleton: $i > $o).
% 6.63/2.38 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 6.63/2.38 thf(func_def_184, type, atmost1p: $i > $o).
% 6.63/2.38 thf(func_def_185, type, atleast2p: $i > $o).
% 6.63/2.38 thf(func_def_186, type, atmost2p: $i > $o).
% 6.63/2.38 thf(func_def_187, type, upairsetp: $i > $o).
% 6.63/2.38 thf(func_def_191, type, kfst: $i > $i).
% 6.63/2.38 thf(func_def_203, type, ksnd: $i > $i).
% 6.63/2.38 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 6.63/2.38 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 6.63/2.38 thf(func_def_222, type, func: $i > $i > $i > $o).
% 6.63/2.38 thf(func_def_223, type, funcSet: $i > $i > $i).
% 6.63/2.38 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 6.63/2.38 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 6.63/2.38 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 6.63/2.38 thf(func_def_320, type, sP0: $i > $i > $i > $i > $o).
% 6.63/2.38 thf(func_def_326, type, sP6: $o > $i > $i > $i > $o).
% 6.63/2.38 thf(func_def_327, type, sP7: $i > $i > $i > $o).
% 6.63/2.38 thf(func_def_328, type, sP8: $i > $i > $o).
% 6.63/2.38 thf(func_def_329, type, sP9: $i > $o).
% 6.63/2.38 thf(func_def_330, type, sP10: $i > $i > $o).
% 6.63/2.38 thf(func_def_331, type, sP11: $i > $i > $o).
% 6.63/2.38 thf(func_def_333, type, sK13: $i > $i > $i).
% 6.63/2.38 thf(func_def_340, type, sK20: $i > ($i > $o) > $i).
% 6.63/2.38 thf(func_def_341, type, sK21: $i > $o).
% 6.63/2.38 thf(func_def_345, type, sK25: ($i > $o) > $i > $i).
% 6.63/2.38 thf(func_def_347, type, sK27: $i > $o).
% 6.63/2.38 thf(func_def_361, type, sK41: $i > $i > $i > $i > $i).
% 6.63/2.38 thf(func_def_362, type, sK42: $i > $i > $i > $i > $i).
% 6.63/2.38 thf(func_def_363, type, sK43: $i > $i > $i > $i > $i).
% 6.63/2.38 thf(func_def_364, type, sK44: $i > $i > $i > $i > $i).
% 6.63/2.38 thf(func_def_366, type, sK46: $i > $o).
% 6.63/2.38 thf(func_def_373, type, sK53: $i > $i > $o).
% 6.63/2.38 thf(func_def_376, type, sK56: $i > $i).
% 6.63/2.38 thf(func_def_379, type, sK59: ($i > $i) > $i > $i > $i).
% 6.63/2.38 thf(func_def_382, type, sK62: $i > $i).
% 6.63/2.38 thf(func_def_384, type, sK64: $i > $i > $o).
% 6.63/2.38 thf(func_def_388, type, sK68: $i > $i > $i).
% 6.63/2.38 thf(func_def_416, type, sK96: ($i > $o) > $i > $i).
% 6.63/2.38 thf(func_def_418, type, sK98: $i > $o).
% 6.63/2.38 thf(func_def_422, type, sK102: $i > $i > $o).
% 6.63/2.38 thf(func_def_442, type, sK122: $i > $o).
% 6.63/2.39 thf(func_def_445, type, sK125: $i > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_460, type, sK140: $i > $i > $o).
% 6.63/2.39 thf(func_def_501, type, sK181: $i > $i).
% 6.63/2.39 thf(func_def_512, type, sK192: $i > $i > $i).
% 6.63/2.39 thf(func_def_513, type, sK193: $i > $i > $i).
% 6.63/2.39 thf(func_def_518, type, sK198: $i > $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_519, type, sK199: $i > $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_528, type, sK208: $i > $o).
% 6.63/2.39 thf(func_def_531, type, sK211: $i > $i).
% 6.63/2.39 thf(func_def_537, type, sK217: $i > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_538, type, sK218: $i > $o).
% 6.63/2.39 thf(func_def_544, type, sK224: $i > ($i > $i) > $i > $i).
% 6.63/2.39 thf(func_def_546, type, sK226: $i > $i).
% 6.63/2.39 thf(func_def_549, type, sK229: $i > $o).
% 6.63/2.39 thf(func_def_557, type, sK237: $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_558, type, sK238: $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_562, type, sK242: $i > $o).
% 6.63/2.39 thf(func_def_564, type, sK244: $i > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_576, type, sK256: $i > $o).
% 6.63/2.39 thf(func_def_578, type, sK258: ($i > $o) > $i > $i).
% 6.63/2.39 thf(func_def_579, type, sK259: ($i > $o) > $i > $i).
% 6.63/2.39 thf(func_def_580, type, sK260: $i > $o).
% 6.63/2.39 thf(func_def_583, type, sK263: $i > $i).
% 6.63/2.39 thf(func_def_585, type, sK265: $i > $i).
% 6.63/2.39 thf(func_def_590, type, sK270: $i > $i > $o).
% 6.63/2.39 thf(func_def_591, type, sK271: $i > $o).
% 6.63/2.39 thf(func_def_602, type, sK282: $i > $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_606, type, sK286: $i > $o).
% 6.63/2.39 thf(func_def_608, type, sK288: ($i > $o) > $i > $i).
% 6.63/2.39 thf(func_def_626, type, sK306: $i > $i > $o).
% 6.63/2.39 thf(func_def_628, type, sK308: $i > $o).
% 6.63/2.39 thf(func_def_631, type, sK311: $i > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_632, type, sK312: $i > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_642, type, sK322: $i > $o).
% 6.63/2.39 thf(func_def_644, type, sK324: ($i > $o) > $i > $i).
% 6.63/2.39 thf(func_def_646, type, sK326: $i > $i).
% 6.63/2.39 thf(func_def_655, type, sK335: $i > $o).
% 6.63/2.39 thf(func_def_659, type, sK339: $i > $i).
% 6.63/2.39 thf(func_def_661, type, sK341: $i > ($i > $i) > $i > $i).
% 6.63/2.39 thf(func_def_662, type, sK342: ($i > $o) > ($i > $o) > $i > $i > $i).
% 6.63/2.39 thf(func_def_663, type, sK343: ($i > $o) > ($i > $o) > $i > $i > $i).
% 6.63/2.39 thf(func_def_666, type, sK346: $i > $o).
% 6.63/2.39 thf(func_def_667, type, sK347: $i > $o).
% 6.63/2.39 thf(func_def_677, type, sK357: $i > $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_690, type, sK370: $i > $i > $i).
% 6.63/2.39 thf(func_def_698, type, sK378: $i > $i > $i).
% 6.63/2.39 thf(func_def_705, type, sK385: $i > $o).
% 6.63/2.39 thf(func_def_707, type, sK387: ($i > $o) > $i).
% 6.63/2.39 thf(func_def_708, type, sK388: ($i > $o) > $i).
% 6.63/2.39 thf(func_def_729, type, sK409: ($i > $o) > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_730, type, sK410: ($i > $o) > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_731, type, sK411: $i > $o).
% 6.63/2.39 thf(func_def_732, type, sK412: $i > $o).
% 6.63/2.39 thf(func_def_740, type, sK420: $i > $o).
% 6.63/2.39 thf(func_def_743, type, sK423: $o > $i > $i > $i).
% 6.63/2.39 thf(func_def_757, type, sK437: ($i > $o) > $i).
% 6.63/2.39 thf(func_def_758, type, sK438: $i > $o).
% 6.63/2.39 thf(func_def_759, type, sK439: $i > $i).
% 6.63/2.39 thf(func_def_760, type, sK440: $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_764, type, sK444: $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_765, type, sK445: $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_770, type, sK450: $i > $o).
% 6.63/2.39 thf(func_def_803, type, sK483: $i > $o).
% 6.63/2.39 thf(func_def_804, type, sK484: $i > $i).
% 6.63/2.39 thf(func_def_805, type, sK485: ($i > $o) > $i).
% 6.63/2.39 thf(func_def_806, type, sK486: $i > $o).
% 6.63/2.39 thf(func_def_838, type, sK518: $i > $i > $o).
% 6.63/2.39 thf(func_def_843, type, sK523: ($i > $o) > $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_844, type, sK524: ($i > $o) > $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_848, type, sK528: $i > $o).
% 6.63/2.39 thf(func_def_853, type, sK533: $i > $i > $i).
% 6.63/2.39 thf(func_def_865, type, sK545: $i > $i > $i).
% 6.63/2.39 thf(func_def_870, type, sK550: $i > $o).
% 6.63/2.39 thf(func_def_895, type, sK575: $i > $i > ($i > $i) > $i).
% 6.63/2.39 thf(func_def_896, type, sK576: $i > $i).
% 6.63/2.39 thf(func_def_903, type, sK583: $i > $o).
% 6.63/2.39 thf(func_def_908, type, sK588: ($i > $i) > $i > $i > $i).
% 6.63/2.39 thf(func_def_911, type, sK591: $i > $i).
% 6.63/2.39 thf(func_def_949, type, sK629: $i > $o).
% 6.63/2.39 thf(func_def_950, type, sK630: $i > $o).
% 6.63/2.39 thf(func_def_951, type, sK631: ($i > $o) > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_952, type, sK632: ($i > $o) > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_975, type, sK655: $i > $o).
% 6.63/2.39 thf(func_def_986, type, sK666: $i > $o).
% 6.63/2.39 thf(func_def_987, type, sK667: $i > $o).
% 6.63/2.39 thf(func_def_988, type, sK668: ($i > $o) > $i).
% 6.63/2.39 thf(func_def_989, type, sK669: ($i > $o) > $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_990, type, sK670: ($i > $o) > $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_994, type, sK674: $i > $o).
% 6.63/2.39 thf(func_def_1002, type, sK682: $i > $i > $i).
% 6.63/2.39 thf(func_def_1003, type, sK683: $i > $i > $i).
% 6.63/2.39 thf(func_def_1004, type, sK684: $i > $i > $i).
% 6.63/2.39 thf(func_def_1005, type, sK685: $i > $i > $i).
% 6.63/2.39 thf(func_def_1006, type, sK686: $i > $i > $i).
% 6.63/2.39 thf(func_def_1007, type, sK687: $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_1008, type, sK688: $i > $i).
% 6.63/2.39 thf(func_def_1009, type, sK689: $i > $i).
% 6.63/2.39 thf(func_def_1010, type, sK690: $i > $i).
% 6.63/2.39 thf(func_def_1011, type, sK691: $i > $i).
% 6.63/2.39 thf(func_def_1012, type, sK692: $i > $i > $i).
% 6.63/2.39 thf(func_def_1013, type, sK693: $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_1014, type, sK694: $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_1015, type, sK695: $i > $i > $i).
% 6.63/2.39 thf(func_def_1016, type, sK696: $i > $i > $i).
% 6.63/2.39 thf(func_def_1017, type, sK697: $i > $i).
% 6.63/2.39 thf(func_def_1019, type, sK699: $i > $i).
% 6.63/2.39 thf(func_def_1020, type, sK700: $i > $i).
% 6.63/2.39 thf(func_def_1062, type, sK742: $i > $i > $i).
% 6.63/2.39 thf(func_def_1066, type, sK746: $i > $i > $o).
% 6.63/2.39 thf(func_def_1067, type, sK747: $i > $i).
% 6.63/2.39 thf(func_def_1068, type, sK748: $i > $i).
% 6.63/2.39 thf(func_def_1069, type, sK749: ($i > $i > $o) > $i > $i).
% 6.63/2.39 thf(func_def_1070, type, sK750: ($i > $i > $o) > $i > $i).
% 6.63/2.39 thf(func_def_1071, type, sK751: $i > ($i > $i > $o) > $i > $i).
% 6.63/2.39 thf(func_def_1088, type, sK768: $i > $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_1093, type, sK773: ($i > $o) > $i > $i).
% 6.63/2.39 thf(func_def_1095, type, sK775: $i > $o).
% 6.63/2.39 thf(func_def_1097, type, sK777: $i > $i > $i).
% 6.63/2.39 thf(func_def_1105, type, sK785: $i > $i > $i > $i).
% 6.63/2.39 thf(func_def_1136, type, sK816: $i > $o).
% 6.63/2.39 thf(func_def_1139, type, sK819: $i > $i > ($i > $o) > $i).
% 6.63/2.39 thf(func_def_1142, type, sK822: $i > $i).
% 6.63/2.39 thf(func_def_1144, type, sK824: ($i > $i) > $i > $i > $i).
% 6.63/2.39 thf(func_def_1168, type, sK848: $i > $i).
% 6.63/2.39 thf(func_def_1187, type, ph867: !>[X0: $tType]:(X0)).
% 6.63/2.39 thf(f7897,plain,(
% 6.63/2.39 $false),
% 6.63/2.39 inference(avatar_sat_refutation,[status(thm)],[f4365,f4584,f4607,f5395,f7896])).
% 6.63/2.39 thf(f7896,plain,(
% 6.63/2.39 ~spl866_29 | ~spl866_7),
% 6.63/2.39 inference(avatar_split_clause,[status(thm)],[f4480,f4238,f4511])).
% 6.63/2.39 thf(f4511,definition,(
% 6.63/2.39 spl866_29 <=> ($true = (in @ sK293 @ sK292))),
% 6.63/2.39 introduced(definition,[new_symbols(naming,[spl866_29])],[avatar_definition])).
% 6.63/2.39 thf(f4238,definition,(
% 6.63/2.39 spl866_7 <=> ($true = (in @ sK293 @ (setminus @ sK290 @ sK292)))),
% 6.63/2.39 introduced(definition,[new_symbols(naming,[spl866_7])],[avatar_definition])).
% 6.63/2.39 thf(f4480,plain,(
% 6.63/2.39 ($true != (in @ sK293 @ sK292)) | ~spl866_7),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4479])).
% 6.63/2.39 thf(f4479,plain,(
% 6.63/2.39 ($true != (in @ sK293 @ sK292)) | ($true != $true) | ~spl866_7),
% 6.63/2.39 inference(forward_demodulation,[status(thm)],[f4478,f3051])).
% 6.63/2.39 thf(f3051,plain,(
% 6.63/2.39 (setminusER = $true)),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f1635,plain,(
% 6.63/2.39 (dpsetconstrERa = $true) & (vacuousDall = $true) & (inIntersectImpInUnion = $true) & (complementUnionInPowersetComplement = $true) & (binunionTE = $true) & (notdallE = $true) & (upairsetIL = $true) & (infuncsetfunc = $true) & (prop2setE = $true) & (apProp = $true) & (subsetemptysetimpeq = $true) & (setadjoinE = $true) & (setukpairinjR = $true) & (prop2set2propI = $true) & (notequalI1 = $true) & (omega__Cong = $true) & (binintersectSubset3 = $true) & (ex1E1 = $true) & (cartprodmempair = $true) & (complementSubsetComplementIntersect = $true) & (kfstpairEq = $true) & (setadjoinOr = $true) & (binunionRsub = $true) & (emptysetE = $true) & (setminusEL = $true) & (funcGraphProp4 = $true) & (powersetAx = $true) & (eqbreln = $true) & (doubleComplementE1 = $true) & (binintersectSubset4 = $true) & (foundationAx = $true) & (beta2 = $true) & (complementImpComplementIntersect = $true) & (iffalseProp1 = $true) & (contrasubsetT1 = $true) & (binunionEcases = $true) & (noeltsimpempty = $true) & (symdiffI2 = $true) & (upairsetIR = $true) & (cartprodpairmemEL = $true) & (setbeta = $true) & (subPowSU = $true) & (ex1E2 = $true) & (powersetE1 = $true) & (contrasubsetT = $true) & (symdiffIneg1 = $true) & (setukpairinjR11 = $true) & (binunionLsub = $true) & (doubleComplementSub1 = $true) & (setextT = $true) & (iffalseProp2 = $true) & (ksndsingleton = $true) & (upairsubunion = $true) & (binunionTILcontra = $true) & (setadjoinSub2 = $true) & (prop2setI = $true) & (iftrueorfalse = $true) & (singletonsubset = $true) & (omega0Ax = $true) & (setminusLsub = $true) & (symdiffI1 = $true) & (quantDeMorgan1 = $true) & (iffalse = $true) & (setminusSubset1 = $true) & (lamProp = $true) & (quantDeMorgan2 = $true) & (binintersectT_lem = $true) & (complementT_lem = $true) & (ap2p = $true) & (binunionE = $true) & (complementTI1 = $true) & (exuI2 = $true) & (binintersectTERcontra = $true) & (emptyset__Cong = $true) & (setunionE = $true) & (binintersectSubset1 = $true) & (setOfPairsIsBReln = $true) & (setunionE2 = $true) & (setukpairinjL = $true) & (doubleComplementI1 = $true) & (funcext2 = $true) & (powersetTE1 = $true) & (lamp = $true) & (dpsetconstrEL1 = $true) & (setukpairinjR2 = $true) & (setext = $true) & (beta1 = $true) & (demorgan2a2 = $true) & (exuE1 = $true) & (powersetTI1 = $true) & (omegaSAx = $true) & (setunionsingleton2 = $true) & (omegaIndAx = $true) & (sepInPowerset = $true) & (kpairp = $true) & (dpsetconstrI = $true) & (setukpairinjR1 = $true) & (funcGraphProp2 = $true) & (eqimpsubset1 = $true) & (setminusILneg = $true) & (eqinunit = $true) & (ifp = $true) & (ksndpairEq = $true) & (setadjoinIL = $true) & (ap2apEq1 = $true) & (lam2p = $true) & (kfstsingleton = $true) & (subset2powerset = $true) & (binintersectSubset2 = $true) & (singletoninpowerset = $true) & (subsetRefl = $true) & (subbreln = $true) & (setunionsingleton = $true) & (setminusER = $true) & (subsetTI = $true) & (dsetconstrI = $true) & (upairset2IR = $true) & (iftrue = $true) & (theprop = $true) & (inComplementUnionImpInComplement1 = $true) & (eta2 = $true) & (ubforcartprodlem1 = $true) & (setminusI = $true) & (singletonsuniq = $true) & (upairsetE = $true) & (dpsetconstrSub = $true) & (binintersectER = $true) & (setminusIRneg = $true) & (ubforcartprodlem3 = $true) & (setadjoinSub = $true) & (exuI3 = $true) & (setminusELneg = $true) & (complementTcontraSubset = $true) & (app = $true) & (inIntersectImpInUnion2 = $true) & (nonemptyI1 = $true) & (eta1 = $true) & (doubleComplementSub2 = $true) & (dpsetconstrER = $true) & (iftrueProp1 = $true) & (binunionTEcontra = $true) & (setextAx = $true) & (emptysetimpfalse = $true) & (singletoninpowunion = $true) & (funcGraphProp3 = $true) & (setunionsingleton1 = $true) & (replAx = $true) & (singletonprop = $true) & (setukpairinjL1 = $true) & (setukpairIL = $true) & (powerset__Cong = $true) & (setextsub = $true) & (inPowerset = $true) & (dpsetconstrEL2 = $true) & (cartprodsndpairEq = $true) & (sepSubset = $true) & (upairequniteq = $true) & (exuE3e = $true) & (powersetI1 = $true) & (disjointsetsI1 = $true) & (subsetI2 = $true) & (iftrueProp2 = $true) & (brelnall1 = $true) & (binintersectEL = $true) & (complementInPowersetComplementIntersect = $true) & (cartprodmempaircEq = $true) & (nonemptyImpWitness = $true) & (inIntersectImpInIntersectUnions = $true) & (complementTnotintersectT = $true) & (brelnall2 = $true) & (quantDeMorgan3 = $true) & (funcGraphProp1 = $true) & (binunionTIRcontra = $true) & (setoftrueEq = $true) & (complementTE1 = $true) & (symdiffE = $true) & (exuI1 = $true) & (notdexE = $true) & (singletonsswitch = $true) & (dsetconstrEL = $true) & (ifSingleton = $true) & (subsetE2 = $true) & (setminusSubset2 = $true) & (secondinupair = $true) & (setunionAx = $true) & (contrasubsetT2 = $true) & (nonemptyE1 = $true) & (lam2lamEq = $true) & (funcextLem = $true) & (powersetE = $true) & (emptysetAx = $true) & (theeq = $true) & (emptyinunitempty = $true) & (exuE3u = $true) & (emptyI = $true) & (inCongP = $true) & (nonemptyI = $true) & (cartprodsndin = $true) & (emptyE1 = $true) & (binunionIL = $true) & (setunionI = $true) & (descrp = $true) & (notinemptyset = $true) & (setadjoin__Cong = $true) & (notinsingleton = $true) & (binunionT_lem = $true) & (wellorderingAx = $true) & (notsubsetI = $true) & (ex1I = $true) & (binintersectLsub = $true) & (cartprodpairmemER = $true) & (exu__Cong = $true) & (inComplementUnionImpNotIn1 = $true) & (powersetsubset = $true) & (cartprodpairsurjEq = $true) & (notequalI2 = $true) & (kpairsurjEq = $true) & (doubleComplementEq = $true) & (setminusERneg = $true) & (binintersectI = $true) & (powersetI = $true) & (upairinpowunion = $true) & (funcImageSingleton = $true) & (symdiffIneg2 = $true) & (((($true = (in @ sK293 @ sK290)) & ($true = (in @ sK293 @ (binunion @ (setminus @ sK290 @ sK291) @ (setminus @ sK290 @ sK292)))) & ((in @ sK293 @ (setminus @ sK290 @ (binintersect @ sK291 @ sK292))) != $true)) & ((in @ sK292 @ (powerset @ sK290)) = $true)) & ($true = (in @ sK291 @ (powerset @ sK290)))) & (setadjoinAx = $true) & (descr__Cong = $true) & (emptysetsubset = $true) & (ap2apEq2 = $true) & (binintersectSubset5 = $true) & (subsetE = $true) & (upairset2E = $true) & (quantDeMorgan4 = $true) & (subsetI1 = $true) & (funcinfuncset = $true) & (subsetTrans = $true) & (emptyinPowerset = $true) & (ex1I2 = $true) & (bs114d = $true) & (ubforcartprodlem2 = $true) & (kpairiskpair = $true) & (exuE2 = $true) & (demorgan1a = $true) & (uniqinunit = $true) & (in__Cong = $true) & (exuEu = $true) & (intersectInPowersetIntersectUnions = $true) & (setukpairIR = $true) & (powersetT_lem = $true) & (setukpairinjL2 = $true) & (dsetconstrER = $true) & (funcext = $true) & (contraSubsetComplement = $true) & (setunion__Cong = $true) & (contrasubsetT3 = $true) & (setminusT_lem = $true) & (cartprodfstpairEq = $true) & (dsetconstr__Cong = $true) & (binunionIR = $true) & (eqimpsubset2 = $true) & (cartprodfstin = $true) & (binintersectTELcontra = $true) & (cartprodmempair1 = $true) & (setukpairinjR12 = $true) & (binintersectRsub = $true) & (setadjoinIR = $true) & (cartprodpairin = $true) & (demorgan2a1 = $true) & (emptyInPowerset = $true)),
% 6.63/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK290,sK291,sK292,sK293])],[f1021,f1634,f1633,f1632])).
% 6.63/2.39 thf(f1632,plain,(
% 6.63/2.39 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ X0)) & ($true = (in @ X3 @ (binunion @ (setminus @ X0 @ X1) @ (setminus @ X0 @ X2)))) & ((in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2))) != $true)) & ($true = (in @ X2 @ (powerset @ X0)))) & ((in @ X1 @ (powerset @ X0)) = $true)) => (? [X2 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ sK290)) & ((in @ X3 @ (binunion @ (setminus @ sK290 @ sK291) @ (setminus @ sK290 @ X2))) = $true) & ($true != (in @ X3 @ (setminus @ sK290 @ (binintersect @ sK291 @ X2))))) & ($true = (in @ X2 @ (powerset @ sK290)))) & ($true = (in @ sK291 @ (powerset @ sK290))))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1633,plain,(
% 6.63/2.39 ? [X2 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ sK290)) & ((in @ X3 @ (binunion @ (setminus @ sK290 @ sK291) @ (setminus @ sK290 @ X2))) = $true) & ($true != (in @ X3 @ (setminus @ sK290 @ (binintersect @ sK291 @ X2))))) & ($true = (in @ X2 @ (powerset @ sK290)))) => (? [X3 : $i] : (($true = (in @ X3 @ sK290)) & ((in @ X3 @ (binunion @ (setminus @ sK290 @ sK291) @ (setminus @ sK290 @ sK292))) = $true) & ((in @ X3 @ (setminus @ sK290 @ (binintersect @ sK291 @ sK292))) != $true)) & ((in @ sK292 @ (powerset @ sK290)) = $true))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1634,plain,(
% 6.63/2.39 ? [X3 : $i] : (($true = (in @ X3 @ sK290)) & ((in @ X3 @ (binunion @ (setminus @ sK290 @ sK291) @ (setminus @ sK290 @ sK292))) = $true) & ((in @ X3 @ (setminus @ sK290 @ (binintersect @ sK291 @ sK292))) != $true)) => (($true = (in @ sK293 @ sK290)) & ($true = (in @ sK293 @ (binunion @ (setminus @ sK290 @ sK291) @ (setminus @ sK290 @ sK292)))) & ((in @ sK293 @ (setminus @ sK290 @ (binintersect @ sK291 @ sK292))) != $true))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1021,plain,(
% 6.63/2.39 (dpsetconstrERa = $true) & (vacuousDall = $true) & (inIntersectImpInUnion = $true) & (complementUnionInPowersetComplement = $true) & (binunionTE = $true) & (notdallE = $true) & (upairsetIL = $true) & (infuncsetfunc = $true) & (prop2setE = $true) & (apProp = $true) & (subsetemptysetimpeq = $true) & (setadjoinE = $true) & (setukpairinjR = $true) & (prop2set2propI = $true) & (notequalI1 = $true) & (omega__Cong = $true) & (binintersectSubset3 = $true) & (ex1E1 = $true) & (cartprodmempair = $true) & (complementSubsetComplementIntersect = $true) & (kfstpairEq = $true) & (setadjoinOr = $true) & (binunionRsub = $true) & (emptysetE = $true) & (setminusEL = $true) & (funcGraphProp4 = $true) & (powersetAx = $true) & (eqbreln = $true) & (doubleComplementE1 = $true) & (binintersectSubset4 = $true) & (foundationAx = $true) & (beta2 = $true) & (complementImpComplementIntersect = $true) & (iffalseProp1 = $true) & (contrasubsetT1 = $true) & (binunionEcases = $true) & (noeltsimpempty = $true) & (symdiffI2 = $true) & (upairsetIR = $true) & (cartprodpairmemEL = $true) & (setbeta = $true) & (subPowSU = $true) & (ex1E2 = $true) & (powersetE1 = $true) & (contrasubsetT = $true) & (symdiffIneg1 = $true) & (setukpairinjR11 = $true) & (binunionLsub = $true) & (doubleComplementSub1 = $true) & (setextT = $true) & (iffalseProp2 = $true) & (ksndsingleton = $true) & (upairsubunion = $true) & (binunionTILcontra = $true) & (setadjoinSub2 = $true) & (prop2setI = $true) & (iftrueorfalse = $true) & (singletonsubset = $true) & (omega0Ax = $true) & (setminusLsub = $true) & (symdiffI1 = $true) & (quantDeMorgan1 = $true) & (iffalse = $true) & (setminusSubset1 = $true) & (lamProp = $true) & (quantDeMorgan2 = $true) & (binintersectT_lem = $true) & (complementT_lem = $true) & (ap2p = $true) & (binunionE = $true) & (complementTI1 = $true) & (exuI2 = $true) & (binintersectTERcontra = $true) & (emptyset__Cong = $true) & (setunionE = $true) & (binintersectSubset1 = $true) & (setOfPairsIsBReln = $true) & (setunionE2 = $true) & (setukpairinjL = $true) & (doubleComplementI1 = $true) & (funcext2 = $true) & (powersetTE1 = $true) & (lamp = $true) & (dpsetconstrEL1 = $true) & (setukpairinjR2 = $true) & (setext = $true) & (beta1 = $true) & (demorgan2a2 = $true) & (exuE1 = $true) & (powersetTI1 = $true) & (omegaSAx = $true) & (setunionsingleton2 = $true) & (omegaIndAx = $true) & (sepInPowerset = $true) & (kpairp = $true) & (dpsetconstrI = $true) & (setukpairinjR1 = $true) & (funcGraphProp2 = $true) & (eqimpsubset1 = $true) & (setminusILneg = $true) & (eqinunit = $true) & (ifp = $true) & (ksndpairEq = $true) & (setadjoinIL = $true) & (ap2apEq1 = $true) & (lam2p = $true) & (kfstsingleton = $true) & (subset2powerset = $true) & (binintersectSubset2 = $true) & (singletoninpowerset = $true) & (subsetRefl = $true) & (subbreln = $true) & (setunionsingleton = $true) & (setminusER = $true) & (subsetTI = $true) & (dsetconstrI = $true) & (upairset2IR = $true) & (iftrue = $true) & (theprop = $true) & (inComplementUnionImpInComplement1 = $true) & (eta2 = $true) & (ubforcartprodlem1 = $true) & (setminusI = $true) & (singletonsuniq = $true) & (upairsetE = $true) & (dpsetconstrSub = $true) & (binintersectER = $true) & (setminusIRneg = $true) & (ubforcartprodlem3 = $true) & (setadjoinSub = $true) & (exuI3 = $true) & (setminusELneg = $true) & (complementTcontraSubset = $true) & (app = $true) & (inIntersectImpInUnion2 = $true) & (nonemptyI1 = $true) & (eta1 = $true) & (doubleComplementSub2 = $true) & (dpsetconstrER = $true) & (iftrueProp1 = $true) & (binunionTEcontra = $true) & (setextAx = $true) & (emptysetimpfalse = $true) & (singletoninpowunion = $true) & (funcGraphProp3 = $true) & (setunionsingleton1 = $true) & (replAx = $true) & (singletonprop = $true) & (setukpairinjL1 = $true) & (setukpairIL = $true) & (powerset__Cong = $true) & (setextsub = $true) & (inPowerset = $true) & (dpsetconstrEL2 = $true) & (cartprodsndpairEq = $true) & (sepSubset = $true) & (upairequniteq = $true) & (exuE3e = $true) & (powersetI1 = $true) & (disjointsetsI1 = $true) & (subsetI2 = $true) & (iftrueProp2 = $true) & (brelnall1 = $true) & (binintersectEL = $true) & (complementInPowersetComplementIntersect = $true) & (cartprodmempaircEq = $true) & (nonemptyImpWitness = $true) & (inIntersectImpInIntersectUnions = $true) & (complementTnotintersectT = $true) & (brelnall2 = $true) & (quantDeMorgan3 = $true) & (funcGraphProp1 = $true) & (binunionTIRcontra = $true) & (setoftrueEq = $true) & (complementTE1 = $true) & (symdiffE = $true) & (exuI1 = $true) & (notdexE = $true) & (singletonsswitch = $true) & (dsetconstrEL = $true) & (ifSingleton = $true) & (subsetE2 = $true) & (setminusSubset2 = $true) & (secondinupair = $true) & (setunionAx = $true) & (contrasubsetT2 = $true) & (nonemptyE1 = $true) & (lam2lamEq = $true) & (funcextLem = $true) & (powersetE = $true) & (emptysetAx = $true) & (theeq = $true) & (emptyinunitempty = $true) & (exuE3u = $true) & (emptyI = $true) & (inCongP = $true) & (nonemptyI = $true) & (cartprodsndin = $true) & (emptyE1 = $true) & (binunionIL = $true) & (setunionI = $true) & (descrp = $true) & (notinemptyset = $true) & (setadjoin__Cong = $true) & (notinsingleton = $true) & (binunionT_lem = $true) & (wellorderingAx = $true) & (notsubsetI = $true) & (ex1I = $true) & (binintersectLsub = $true) & (cartprodpairmemER = $true) & (exu__Cong = $true) & (inComplementUnionImpNotIn1 = $true) & (powersetsubset = $true) & (cartprodpairsurjEq = $true) & (notequalI2 = $true) & (kpairsurjEq = $true) & (doubleComplementEq = $true) & (setminusERneg = $true) & (binintersectI = $true) & (powersetI = $true) & (upairinpowunion = $true) & (funcImageSingleton = $true) & (symdiffIneg2 = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (($true = (in @ X3 @ X0)) & ($true = (in @ X3 @ (binunion @ (setminus @ X0 @ X1) @ (setminus @ X0 @ X2)))) & ((in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2))) != $true)) & ($true = (in @ X2 @ (powerset @ X0)))) & ((in @ X1 @ (powerset @ X0)) = $true)) & (setadjoinAx = $true) & (descr__Cong = $true) & (emptysetsubset = $true) & (ap2apEq2 = $true) & (binintersectSubset5 = $true) & (subsetE = $true) & (upairset2E = $true) & (quantDeMorgan4 = $true) & (subsetI1 = $true) & (funcinfuncset = $true) & (subsetTrans = $true) & (emptyinPowerset = $true) & (ex1I2 = $true) & (bs114d = $true) & (ubforcartprodlem2 = $true) & (kpairiskpair = $true) & (exuE2 = $true) & (demorgan1a = $true) & (uniqinunit = $true) & (in__Cong = $true) & (exuEu = $true) & (intersectInPowersetIntersectUnions = $true) & (setukpairIR = $true) & (powersetT_lem = $true) & (setukpairinjL2 = $true) & (dsetconstrER = $true) & (funcext = $true) & (contraSubsetComplement = $true) & (setunion__Cong = $true) & (contrasubsetT3 = $true) & (setminusT_lem = $true) & (cartprodfstpairEq = $true) & (dsetconstr__Cong = $true) & (binunionIR = $true) & (eqimpsubset2 = $true) & (cartprodfstin = $true) & (binintersectTELcontra = $true) & (cartprodmempair1 = $true) & (setukpairinjR12 = $true) & (binintersectRsub = $true) & (setadjoinIR = $true) & (cartprodpairin = $true) & (demorgan2a1 = $true) & (emptyInPowerset = $true)),
% 6.63/2.39 inference(flattening,[status(thm)],[f1020])).
% 6.63/2.39 thf(f1020,plain,(
% 6.63/2.39 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : ((((in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2))) != $true) & ($true = (in @ X3 @ (binunion @ (setminus @ X0 @ X1) @ (setminus @ X0 @ X2))))) & ($true = (in @ X3 @ X0))) & ($true = (in @ X2 @ (powerset @ X0)))) & ((in @ X1 @ (powerset @ X0)) = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (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)),
% 6.63/2.39 inference(ennf_transformation,[status(thm)],[f459])).
% 6.63/2.39 thf(f459,plain,(
% 6.63/2.39 ~((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) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => ! [X3 : $i] : (($true = (in @ X3 @ X0)) => (($true = (in @ X3 @ (binunion @ (setminus @ X0 @ X1) @ (setminus @ X0 @ X2)))) => ((in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2))) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.63/2.39 inference(fool_elimination,[status(thm)],[f458])).
% 6.63/2.39 thf(f458,plain,(
% 6.63/2.39 ~(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 => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ! [X3 : $i] : ((in @ X3 @ X0) => ((in @ X3 @ (binunion @ (setminus @ X0 @ X1) @ (setminus @ X0 @ X2))) => (in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.63/2.39 inference(rectify,[status(thm)],[f274])).
% 6.63/2.39 thf(f274,negated_conjecture,(
% 6.63/2.39 ~(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 => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ (binunion @ (setminus @ X3 @ X11) @ (setminus @ X3 @ X16))) => (in @ X1 @ (setminus @ X3 @ (binintersect @ X11 @ X16)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.63/2.39 inference(negated_conjecture,[status(cth)],[f273])).
% 6.63/2.39 thf(f273,conjecture,(
% 6.63/2.39 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 => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ (binunion @ (setminus @ X3 @ X11) @ (setminus @ X3 @ X16))) => (in @ X1 @ (setminus @ X3 @ (binintersect @ X11 @ X16))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.63/2.39 file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',demorgan1b)).
% 6.63/2.39 thf(f4478,plain,(
% 6.63/2.39 (setminusER != $true) | ($true != (in @ sK293 @ sK292)) | ~spl866_7),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4466])).
% 6.63/2.39 thf(f4466,plain,(
% 6.63/2.39 ($true != $true) | (setminusER != $true) | ($true != (in @ sK293 @ sK292)) | ~spl866_7),
% 6.63/2.39 inference(superposition,[status(thm)],[f3409,f4239])).
% 6.63/2.39 thf(f4239,plain,(
% 6.63/2.39 ($true = (in @ sK293 @ (setminus @ sK290 @ sK292))) | ~spl866_7),
% 6.63/2.39 inference(avatar_component_clause,[status(thm)],[f4238])).
% 6.63/2.39 thf(f3409,plain,(
% 6.63/2.39 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (setminus @ X3 @ X5))) | ($true != (in @ X4 @ X5)) | (setminusER != $true)) )),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1900])).
% 6.63/2.39 thf(f1900,plain,(
% 6.63/2.39 ((setminusER = $true) | (($true = (in @ sK467 @ (setminus @ sK466 @ sK468))) & ($true = (in @ sK467 @ sK468)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (setminus @ X3 @ X5))) | ($true != (in @ X4 @ X5))) | (setminusER != $true))),
% 6.63/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK466,sK467,sK468])],[f1898,f1899])).
% 6.63/2.39 thf(f1899,plain,(
% 6.63/2.39 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (setminus @ X0 @ X2))) & ($true = (in @ X1 @ X2))) => (($true = (in @ sK467 @ (setminus @ sK466 @ sK468))) & ($true = (in @ sK467 @ sK468)))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1898,plain,(
% 6.63/2.39 ((setminusER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (setminus @ X0 @ X2))) & ($true = (in @ X1 @ X2)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (setminus @ X3 @ X5))) | ($true != (in @ X4 @ X5))) | (setminusER != $true))),
% 6.63/2.39 inference(rectify,[status(thm)],[f1897])).
% 6.63/2.39 thf(f1897,plain,(
% 6.63/2.39 ((setminusER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (setminus @ X0 @ X2))) & ($true = (in @ X1 @ X2)))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ (setminus @ X0 @ X2))) | ($true != (in @ X1 @ X2))) | (setminusER != $true))),
% 6.63/2.39 inference(nnf_transformation,[status(thm)],[f1024])).
% 6.63/2.39 thf(f1024,plain,(
% 6.63/2.39 (setminusER = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ (setminus @ X0 @ X2))) | ($true != (in @ X1 @ X2)))),
% 6.63/2.39 inference(ennf_transformation,[status(thm)],[f857])).
% 6.63/2.39 thf(f857,plain,(
% 6.63/2.39 (setminusER = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (setminus @ X0 @ X2))) => ($true != (in @ X1 @ X2)))),
% 6.63/2.39 inference(flattening,[status(thm)],[f650])).
% 6.63/2.39 thf(f650,plain,(
% 6.63/2.39 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (setminus @ X0 @ X2))) => ~($true = (in @ X1 @ X2))) <=> (setminusER = $true)),
% 6.63/2.39 inference(fool_elimination,[status(thm)],[f649])).
% 6.63/2.39 thf(f649,plain,(
% 6.63/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (setminus @ X0 @ X2)) => ~(in @ X1 @ X2)) = setminusER)),
% 6.63/2.39 inference(rectify,[status(thm)],[f122])).
% 6.63/2.39 thf(f122,axiom,(
% 6.63/2.39 (! [X3 : $i,X1 : $i,X4 : $i] : ((in @ X1 @ (setminus @ X3 @ X4)) => ~(in @ X1 @ X4)) = setminusER)),
% 6.63/2.39 file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',setminusER)).
% 6.63/2.39 thf(f5395,plain,(
% 6.63/2.39 ~spl866_9),
% 6.63/2.39 inference(avatar_split_clause,[status(thm)],[f5394,f4249])).
% 6.63/2.39 thf(f4249,definition,(
% 6.63/2.39 spl866_9 <=> ($true = (in @ sK293 @ (setminus @ sK290 @ sK291)))),
% 6.63/2.39 introduced(definition,[new_symbols(naming,[spl866_9])],[avatar_definition])).
% 6.63/2.39 thf(f5394,plain,(
% 6.63/2.39 ($true != (in @ sK293 @ (setminus @ sK290 @ sK291)))),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f5393])).
% 6.63/2.39 thf(f5393,plain,(
% 6.63/2.39 ($true != (in @ sK293 @ (setminus @ sK290 @ sK291))) | ($true != $true)),
% 6.63/2.39 inference(forward_demodulation,[status(thm)],[f5392,f2936])).
% 6.63/2.39 thf(f2936,plain,(
% 6.63/2.39 ($true = (in @ sK291 @ (powerset @ sK290)))),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f5392,plain,(
% 6.63/2.39 ($true != (in @ sK291 @ (powerset @ sK290))) | ($true != (in @ sK293 @ (setminus @ sK290 @ sK291)))),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f5391])).
% 6.63/2.39 thf(f5391,plain,(
% 6.63/2.39 ($true != (in @ sK293 @ (setminus @ sK290 @ sK291))) | ($true != (in @ sK291 @ (powerset @ sK290))) | ($true != $true)),
% 6.63/2.39 inference(forward_demodulation,[status(thm)],[f5390,f2937])).
% 6.63/2.39 thf(f2937,plain,(
% 6.63/2.39 ((in @ sK292 @ (powerset @ sK290)) = $true)),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f5390,plain,(
% 6.63/2.39 ((in @ sK292 @ (powerset @ sK290)) != $true) | ($true != (in @ sK291 @ (powerset @ sK290))) | ($true != (in @ sK293 @ (setminus @ sK290 @ sK291)))),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f5389])).
% 6.63/2.39 thf(f5389,plain,(
% 6.63/2.39 ((in @ sK292 @ (powerset @ sK290)) != $true) | ($true != (in @ sK293 @ (setminus @ sK290 @ sK291))) | ($true != (in @ sK291 @ (powerset @ sK290))) | ($true != $true)),
% 6.63/2.39 inference(forward_demodulation,[status(thm)],[f5354,f3145])).
% 6.63/2.39 thf(f3145,plain,(
% 6.63/2.39 (complementSubsetComplementIntersect = $true)),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f5354,plain,(
% 6.63/2.39 (complementSubsetComplementIntersect != $true) | ($true != (in @ sK293 @ (setminus @ sK290 @ sK291))) | ($true != (in @ sK291 @ (powerset @ sK290))) | ((in @ sK292 @ (powerset @ sK290)) != $true)),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f5330])).
% 6.63/2.39 thf(f5330,plain,(
% 6.63/2.39 ((in @ sK292 @ (powerset @ sK290)) != $true) | ($true != (in @ sK293 @ (setminus @ sK290 @ sK291))) | (complementSubsetComplementIntersect != $true) | ($true != (in @ sK291 @ (powerset @ sK290))) | ($true != $true)),
% 6.63/2.39 inference(superposition,[status(thm)],[f4304,f2739])).
% 6.63/2.39 thf(f2739,plain,(
% 6.63/2.39 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ (setminus @ X4 @ X3) @ (setminus @ X4 @ (binintersect @ X3 @ X5))) = $true) | (complementSubsetComplementIntersect != $true) | ((in @ X3 @ (powerset @ X4)) != $true) | ($true != (in @ X5 @ (powerset @ X4)))) )),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1467])).
% 6.63/2.39 thf(f1467,plain,(
% 6.63/2.39 ((complementSubsetComplementIntersect = $true) | ((($true != (subset @ (setminus @ sK183 @ sK182) @ (setminus @ sK183 @ (binintersect @ sK182 @ sK184)))) & ($true = (in @ sK184 @ (powerset @ sK183)))) & ($true = (in @ sK182 @ (powerset @ sK183))))) & (! [X3 : $i,X4 : $i] : (! [X5 : $i] : (((subset @ (setminus @ X4 @ X3) @ (setminus @ X4 @ (binintersect @ X3 @ X5))) = $true) | ($true != (in @ X5 @ (powerset @ X4)))) | ((in @ X3 @ (powerset @ X4)) != $true)) | (complementSubsetComplementIntersect != $true))),
% 6.63/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK182,sK183,sK184])],[f1464,f1466,f1465])).
% 6.63/2.39 thf(f1465,plain,(
% 6.63/2.39 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true != (subset @ (setminus @ X1 @ X0) @ (setminus @ X1 @ (binintersect @ X0 @ X2)))) & ($true = (in @ X2 @ (powerset @ X1)))) & ($true = (in @ X0 @ (powerset @ X1)))) => (? [X2 : $i] : (($true != (subset @ (setminus @ sK183 @ sK182) @ (setminus @ sK183 @ (binintersect @ sK182 @ X2)))) & ($true = (in @ X2 @ (powerset @ sK183)))) & ($true = (in @ sK182 @ (powerset @ sK183))))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1466,plain,(
% 6.63/2.39 ? [X2 : $i] : (($true != (subset @ (setminus @ sK183 @ sK182) @ (setminus @ sK183 @ (binintersect @ sK182 @ X2)))) & ($true = (in @ X2 @ (powerset @ sK183)))) => (($true != (subset @ (setminus @ sK183 @ sK182) @ (setminus @ sK183 @ (binintersect @ sK182 @ sK184)))) & ($true = (in @ sK184 @ (powerset @ sK183))))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1464,plain,(
% 6.63/2.39 ((complementSubsetComplementIntersect = $true) | ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true != (subset @ (setminus @ X1 @ X0) @ (setminus @ X1 @ (binintersect @ X0 @ X2)))) & ($true = (in @ X2 @ (powerset @ X1)))) & ($true = (in @ X0 @ (powerset @ X1))))) & (! [X3 : $i,X4 : $i] : (! [X5 : $i] : (((subset @ (setminus @ X4 @ X3) @ (setminus @ X4 @ (binintersect @ X3 @ X5))) = $true) | ($true != (in @ X5 @ (powerset @ X4)))) | ((in @ X3 @ (powerset @ X4)) != $true)) | (complementSubsetComplementIntersect != $true))),
% 6.63/2.39 inference(rectify,[status(thm)],[f1463])).
% 6.63/2.39 thf(f1463,plain,(
% 6.63/2.39 ((complementSubsetComplementIntersect = $true) | ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true != (subset @ (setminus @ X1 @ X0) @ (setminus @ X1 @ (binintersect @ X0 @ X2)))) & ($true = (in @ X2 @ (powerset @ X1)))) & ($true = (in @ X0 @ (powerset @ X1))))) & (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (subset @ (setminus @ X1 @ X0) @ (setminus @ X1 @ (binintersect @ X0 @ X2)))) | ($true != (in @ X2 @ (powerset @ X1)))) | ($true != (in @ X0 @ (powerset @ X1)))) | (complementSubsetComplementIntersect != $true))),
% 6.63/2.39 inference(nnf_transformation,[status(thm)],[f1025])).
% 6.63/2.39 thf(f1025,plain,(
% 6.63/2.39 (complementSubsetComplementIntersect = $true) <=> ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (subset @ (setminus @ X1 @ X0) @ (setminus @ X1 @ (binintersect @ X0 @ X2)))) | ($true != (in @ X2 @ (powerset @ X1)))) | ($true != (in @ X0 @ (powerset @ X1))))),
% 6.63/2.39 inference(ennf_transformation,[status(thm)],[f323])).
% 6.63/2.39 thf(f323,plain,(
% 6.63/2.39 ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X1))) => ($true = (subset @ (setminus @ X1 @ X0) @ (setminus @ X1 @ (binintersect @ X0 @ X2)))))) <=> (complementSubsetComplementIntersect = $true)),
% 6.63/2.39 inference(fool_elimination,[status(thm)],[f322])).
% 6.63/2.39 thf(f322,plain,(
% 6.63/2.39 (complementSubsetComplementIntersect = ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X1)) => (subset @ (setminus @ X1 @ X0) @ (setminus @ X1 @ (binintersect @ X0 @ X2))))))),
% 6.63/2.39 inference(rectify,[status(thm)],[f255])).
% 6.63/2.39 thf(f255,axiom,(
% 6.63/2.39 (complementSubsetComplementIntersect = ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (subset @ (setminus @ X3 @ X11) @ (setminus @ X3 @ (binintersect @ X11 @ X16))))))),
% 6.63/2.39 file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',complementSubsetComplementIntersect)).
% 6.63/2.39 thf(f4304,plain,(
% 6.63/2.39 ( ! [X0 : $i] : (($true != (subset @ X0 @ (setminus @ sK290 @ (binintersect @ sK291 @ sK292)))) | ((in @ sK293 @ X0) != $true)) )),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4303])).
% 6.63/2.39 thf(f4303,plain,(
% 6.63/2.39 ( ! [X0 : $i] : (($true != $true) | ($true != (subset @ X0 @ (setminus @ sK290 @ (binintersect @ sK291 @ sK292)))) | ((in @ sK293 @ X0) != $true)) )),
% 6.63/2.39 inference(forward_demodulation,[status(thm)],[f4223,f2983])).
% 6.63/2.39 thf(f2983,plain,(
% 6.63/2.39 (subsetE2 = $true)),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f4223,plain,(
% 6.63/2.39 ( ! [X0 : $i] : (($true != (subset @ X0 @ (setminus @ sK290 @ (binintersect @ sK291 @ sK292)))) | ((in @ sK293 @ X0) != $true) | (subsetE2 != $true)) )),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4220])).
% 6.63/2.39 thf(f4220,plain,(
% 6.63/2.39 ( ! [X0 : $i] : ((subsetE2 != $true) | ($true != $true) | ($true != (subset @ X0 @ (setminus @ sK290 @ (binintersect @ sK291 @ sK292)))) | ((in @ sK293 @ X0) != $true)) )),
% 6.63/2.39 inference(superposition,[status(thm)],[f2938,f2680])).
% 6.63/2.39 thf(f2680,plain,(
% 6.63/2.39 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ X0 @ X1)) | ((subset @ X2 @ X1) != $true) | ((in @ X0 @ X2) != $true) | (subsetE2 != $true)) )),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1397])).
% 6.63/2.39 thf(f1397,plain,(
% 6.63/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X1) != $true) | ($true = (in @ X0 @ X1)) | ((in @ X0 @ X2) != $true)) | (subsetE2 != $true)) & ((subsetE2 = $true) | (($true = (subset @ sK136 @ sK135)) & ($true != (in @ sK134 @ sK135)) & ($true = (in @ sK134 @ sK136))))),
% 6.63/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK134,sK135,sK136])],[f1395,f1396])).
% 6.63/2.39 thf(f1396,plain,(
% 6.63/2.39 ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X4) = $true) & ($true != (in @ X3 @ X4)) & ($true = (in @ X3 @ X5))) => (($true = (subset @ sK136 @ sK135)) & ($true != (in @ sK134 @ sK135)) & ($true = (in @ sK134 @ sK136)))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1395,plain,(
% 6.63/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X1) != $true) | ($true = (in @ X0 @ X1)) | ((in @ X0 @ X2) != $true)) | (subsetE2 != $true)) & ((subsetE2 = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X4) = $true) & ($true != (in @ X3 @ X4)) & ($true = (in @ X3 @ X5))))),
% 6.63/2.39 inference(rectify,[status(thm)],[f1394])).
% 6.63/2.39 thf(f1394,plain,(
% 6.63/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X1) != $true) | ($true = (in @ X0 @ X1)) | ((in @ X0 @ X2) != $true)) | (subsetE2 != $true)) & ((subsetE2 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X1) = $true) & ($true != (in @ X0 @ X1)) & ((in @ X0 @ X2) = $true)))),
% 6.63/2.39 inference(nnf_transformation,[status(thm)],[f1045])).
% 6.63/2.39 thf(f1045,plain,(
% 6.63/2.39 ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X1) != $true) | ($true = (in @ X0 @ X1)) | ((in @ X0 @ X2) != $true)) <=> (subsetE2 = $true)),
% 6.63/2.39 inference(flattening,[status(thm)],[f1044])).
% 6.63/2.39 thf(f1044,plain,(
% 6.63/2.39 (subsetE2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X0 @ X2) != $true) | ($true = (in @ X0 @ X1))) | ((subset @ X2 @ X1) != $true))),
% 6.63/2.39 inference(ennf_transformation,[status(thm)],[f839])).
% 6.63/2.39 thf(f839,plain,(
% 6.63/2.39 (subsetE2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X1) = $true) => (($true != (in @ X0 @ X1)) => ((in @ X0 @ X2) != $true)))),
% 6.63/2.39 inference(flattening,[status(thm)],[f493])).
% 6.63/2.39 thf(f493,plain,(
% 6.63/2.39 (subsetE2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X1) = $true) => (~($true = (in @ X0 @ X1)) => ~((in @ X0 @ X2) = $true)))),
% 6.63/2.39 inference(fool_elimination,[status(thm)],[f492])).
% 6.63/2.39 thf(f492,plain,(
% 6.63/2.39 (subsetE2 = ! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X2 @ X1) => (~(in @ X0 @ X1) => ~(in @ X0 @ X2))))),
% 6.63/2.39 inference(rectify,[status(thm)],[f84])).
% 6.63/2.39 thf(f84,axiom,(
% 6.63/2.39 (subsetE2 = ! [X1 : $i,X4 : $i,X3 : $i] : ((subset @ X3 @ X4) => (~(in @ X1 @ X4) => ~(in @ X1 @ X3))))),
% 6.63/2.39 file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',subsetE2)).
% 6.63/2.39 thf(f2938,plain,(
% 6.63/2.39 ((in @ sK293 @ (setminus @ sK290 @ (binintersect @ sK291 @ sK292))) != $true)),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f4607,plain,(
% 6.63/2.39 spl866_29 | ~spl866_10),
% 6.63/2.39 inference(avatar_split_clause,[status(thm)],[f4606,f4277,f4511])).
% 6.63/2.39 thf(f4277,definition,(
% 6.63/2.39 spl866_10 <=> ((in @ sK293 @ (binintersect @ sK291 @ sK292)) = $true)),
% 6.63/2.39 introduced(definition,[new_symbols(naming,[spl866_10])],[avatar_definition])).
% 6.63/2.39 thf(f4606,plain,(
% 6.63/2.39 ($true = (in @ sK293 @ sK292)) | ~spl866_10),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4605])).
% 6.63/2.39 thf(f4605,plain,(
% 6.63/2.39 ($true = (in @ sK293 @ sK292)) | ($true != $true) | ~spl866_10),
% 6.63/2.39 inference(forward_demodulation,[status(thm)],[f4597,f3038])).
% 6.63/2.39 thf(f3038,plain,(
% 6.63/2.39 (binintersectER = $true)),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f4597,plain,(
% 6.63/2.39 ($true = (in @ sK293 @ sK292)) | (binintersectER != $true) | ~spl866_10),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4588])).
% 6.63/2.39 thf(f4588,plain,(
% 6.63/2.39 (binintersectER != $true) | ($true != $true) | ($true = (in @ sK293 @ sK292)) | ~spl866_10),
% 6.63/2.39 inference(superposition,[status(thm)],[f3459,f4279])).
% 6.63/2.39 thf(f4279,plain,(
% 6.63/2.39 ((in @ sK293 @ (binintersect @ sK291 @ sK292)) = $true) | ~spl866_10),
% 6.63/2.39 inference(avatar_component_clause,[status(thm)],[f4277])).
% 6.63/2.39 thf(f3459,plain,(
% 6.63/2.39 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ (binintersect @ X3 @ X5)) != $true) | (binintersectER != $true) | ($true = (in @ X4 @ X5))) )),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1966])).
% 6.63/2.39 thf(f1966,plain,(
% 6.63/2.39 ((binintersectER = $true) | (($true != (in @ sK504 @ sK505)) & ($true = (in @ sK504 @ (binintersect @ sK503 @ sK505))))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X4 @ X5)) | ((in @ X4 @ (binintersect @ X3 @ X5)) != $true)) | (binintersectER != $true))),
% 6.63/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK503,sK504,sK505])],[f1964,f1965])).
% 6.63/2.39 thf(f1965,plain,(
% 6.63/2.39 ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X2)) & ($true = (in @ X1 @ (binintersect @ X0 @ X2)))) => (($true != (in @ sK504 @ sK505)) & ($true = (in @ sK504 @ (binintersect @ sK503 @ sK505))))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1964,plain,(
% 6.63/2.39 ((binintersectER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X2)) & ($true = (in @ X1 @ (binintersect @ X0 @ X2))))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X4 @ X5)) | ((in @ X4 @ (binintersect @ X3 @ X5)) != $true)) | (binintersectER != $true))),
% 6.63/2.39 inference(rectify,[status(thm)],[f1963])).
% 6.63/2.39 thf(f1963,plain,(
% 6.63/2.39 ((binintersectER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X2)) & ($true = (in @ X1 @ (binintersect @ X0 @ X2))))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X2)) | ($true != (in @ X1 @ (binintersect @ X0 @ X2)))) | (binintersectER != $true))),
% 6.63/2.39 inference(nnf_transformation,[status(thm)],[f1161])).
% 6.63/2.39 thf(f1161,plain,(
% 6.63/2.39 (binintersectER = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X2)) | ($true != (in @ X1 @ (binintersect @ X0 @ X2))))),
% 6.63/2.39 inference(ennf_transformation,[status(thm)],[f519])).
% 6.63/2.39 thf(f519,plain,(
% 6.63/2.39 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (binintersect @ X0 @ X2))) => ($true = (in @ X1 @ X2))) <=> (binintersectER = $true)),
% 6.63/2.39 inference(fool_elimination,[status(thm)],[f518])).
% 6.63/2.39 thf(f518,plain,(
% 6.63/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (binintersect @ X0 @ X2)) => (in @ X1 @ X2)) = binintersectER)),
% 6.63/2.39 inference(rectify,[status(thm)],[f114])).
% 6.63/2.39 thf(f114,axiom,(
% 6.63/2.39 (! [X3 : $i,X1 : $i,X4 : $i] : ((in @ X1 @ (binintersect @ X3 @ X4)) => (in @ X1 @ X4)) = binintersectER)),
% 6.63/2.39 file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',binintersectER)).
% 6.63/2.39 thf(f4584,plain,(
% 6.63/2.39 spl866_10),
% 6.63/2.39 inference(avatar_split_clause,[status(thm)],[f4269,f4277])).
% 6.63/2.39 thf(f4269,plain,(
% 6.63/2.39 ((in @ sK293 @ (binintersect @ sK291 @ sK292)) = $true)),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4268])).
% 6.63/2.39 thf(f4268,plain,(
% 6.63/2.39 ($true != $true) | ((in @ sK293 @ (binintersect @ sK291 @ sK292)) = $true)),
% 6.63/2.39 inference(forward_demodulation,[status(thm)],[f4267,f3042])).
% 6.63/2.39 thf(f3042,plain,(
% 6.63/2.39 (setminusI = $true)),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f4267,plain,(
% 6.63/2.39 (setminusI != $true) | ((in @ sK293 @ (binintersect @ sK291 @ sK292)) = $true)),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4266])).
% 6.63/2.39 thf(f4266,plain,(
% 6.63/2.39 (setminusI != $true) | ((in @ sK293 @ (binintersect @ sK291 @ sK292)) = $true) | ($true != $true)),
% 6.63/2.39 inference(forward_demodulation,[status(thm)],[f4228,f2940])).
% 6.63/2.39 thf(f2940,plain,(
% 6.63/2.39 ($true = (in @ sK293 @ sK290))),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f4228,plain,(
% 6.63/2.39 ($true != (in @ sK293 @ sK290)) | ((in @ sK293 @ (binintersect @ sK291 @ sK292)) = $true) | (setminusI != $true)),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4215])).
% 6.63/2.39 thf(f4215,plain,(
% 6.63/2.39 ($true != (in @ sK293 @ sK290)) | (setminusI != $true) | ($true != $true) | ((in @ sK293 @ (binintersect @ sK291 @ sK292)) = $true)),
% 6.63/2.39 inference(superposition,[status(thm)],[f2938,f2659])).
% 6.63/2.39 thf(f2659,plain,(
% 6.63/2.39 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setminus @ X2 @ X1))) | (setminusI != $true) | ($true = (in @ X0 @ X1)) | ((in @ X0 @ X2) != $true)) )),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1373])).
% 6.63/2.39 thf(f1373,plain,(
% 6.63/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) != $true) | ($true = (in @ X0 @ X1)) | ($true = (in @ X0 @ (setminus @ X2 @ X1)))) | (setminusI != $true)) & ((setminusI = $true) | (($true = (in @ sK117 @ sK119)) & ((in @ sK117 @ sK118) != $true) & ((in @ sK117 @ (setminus @ sK119 @ sK118)) != $true)))),
% 6.63/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK117,sK118,sK119])],[f1371,f1372])).
% 6.63/2.39 thf(f1372,plain,(
% 6.63/2.39 ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ X5)) & ($true != (in @ X3 @ X4)) & ((in @ X3 @ (setminus @ X5 @ X4)) != $true)) => (($true = (in @ sK117 @ sK119)) & ((in @ sK117 @ sK118) != $true) & ((in @ sK117 @ (setminus @ sK119 @ sK118)) != $true))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1371,plain,(
% 6.63/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) != $true) | ($true = (in @ X0 @ X1)) | ($true = (in @ X0 @ (setminus @ X2 @ X1)))) | (setminusI != $true)) & ((setminusI = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ X5)) & ($true != (in @ X3 @ X4)) & ((in @ X3 @ (setminus @ X5 @ X4)) != $true)))),
% 6.63/2.39 inference(rectify,[status(thm)],[f1370])).
% 6.63/2.39 thf(f1370,plain,(
% 6.63/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) != $true) | ($true = (in @ X0 @ X1)) | ($true = (in @ X0 @ (setminus @ X2 @ X1)))) | (setminusI != $true)) & ((setminusI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) = $true) & ($true != (in @ X0 @ X1)) & ($true != (in @ X0 @ (setminus @ X2 @ X1)))))),
% 6.63/2.39 inference(nnf_transformation,[status(thm)],[f1124])).
% 6.63/2.39 thf(f1124,plain,(
% 6.63/2.39 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) != $true) | ($true = (in @ X0 @ X1)) | ($true = (in @ X0 @ (setminus @ X2 @ X1)))) <=> (setminusI = $true)),
% 6.63/2.39 inference(flattening,[status(thm)],[f1123])).
% 6.63/2.39 thf(f1123,plain,(
% 6.63/2.39 ! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X0 @ (setminus @ X2 @ X1))) | ($true = (in @ X0 @ X1))) | ((in @ X0 @ X2) != $true)) <=> (setminusI = $true)),
% 6.63/2.39 inference(ennf_transformation,[status(thm)],[f868])).
% 6.63/2.39 thf(f868,plain,(
% 6.63/2.39 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) = $true) => (($true != (in @ X0 @ X1)) => ($true = (in @ X0 @ (setminus @ X2 @ X1))))) <=> (setminusI = $true)),
% 6.63/2.39 inference(flattening,[status(thm)],[f815])).
% 6.63/2.39 thf(f815,plain,(
% 6.63/2.39 (setminusI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X2) = $true) => (~($true = (in @ X0 @ X1)) => ($true = (in @ X0 @ (setminus @ X2 @ X1)))))),
% 6.63/2.39 inference(fool_elimination,[status(thm)],[f814])).
% 6.63/2.39 thf(f814,plain,(
% 6.63/2.39 (setminusI = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => (~(in @ X0 @ X1) => (in @ X0 @ (setminus @ X2 @ X1)))))),
% 6.63/2.39 inference(rectify,[status(thm)],[f120])).
% 6.63/2.39 thf(f120,axiom,(
% 6.63/2.39 (setminusI = ! [X1 : $i,X4 : $i,X3 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ X4) => (in @ X1 @ (setminus @ X3 @ X4)))))),
% 6.63/2.39 file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',setminusI)).
% 6.63/2.39 thf(f4365,plain,(
% 6.63/2.39 spl866_9 | spl866_7),
% 6.63/2.39 inference(avatar_split_clause,[status(thm)],[f4364,f4238,f4249])).
% 6.63/2.39 thf(f4364,plain,(
% 6.63/2.39 ($true = (in @ sK293 @ (setminus @ sK290 @ sK292))) | ($true = (in @ sK293 @ (setminus @ sK290 @ sK291)))),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4363])).
% 6.63/2.39 thf(f4363,plain,(
% 6.63/2.39 ($true = (in @ sK293 @ (setminus @ sK290 @ sK292))) | ($true = (in @ sK293 @ (setminus @ sK290 @ sK291))) | ($true != $true)),
% 6.63/2.39 inference(forward_demodulation,[status(thm)],[f4323,f3095])).
% 6.63/2.39 thf(f3095,plain,(
% 6.63/2.39 (binunionE = $true)),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f4323,plain,(
% 6.63/2.39 ($true = (in @ sK293 @ (setminus @ sK290 @ sK291))) | ($true = (in @ sK293 @ (setminus @ sK290 @ sK292))) | (binunionE != $true)),
% 6.63/2.39 inference(trivial_inequality_removal,[status(thm)],[f4316])).
% 6.63/2.39 thf(f4316,plain,(
% 6.63/2.39 (binunionE != $true) | ($true = (in @ sK293 @ (setminus @ sK290 @ sK292))) | ($true != $true) | ($true = (in @ sK293 @ (setminus @ sK290 @ sK291)))),
% 6.63/2.39 inference(superposition,[status(thm)],[f3257,f2939])).
% 6.63/2.39 thf(f2939,plain,(
% 6.63/2.39 ($true = (in @ sK293 @ (binunion @ (setminus @ sK290 @ sK291) @ (setminus @ sK290 @ sK292))))),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1635])).
% 6.63/2.39 thf(f3257,plain,(
% 6.63/2.39 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X3 @ (binunion @ X5 @ X4))) | ($true = (in @ X3 @ X4)) | ($true = (in @ X3 @ X5)) | (binunionE != $true)) )),
% 6.63/2.39 inference(cnf_transformation,[status(thm)],[f1739])).
% 6.63/2.39 thf(f1739,plain,(
% 6.63/2.39 ((binunionE = $true) | (($true != (in @ sK367 @ sK368)) & ((in @ sK367 @ (binunion @ sK369 @ sK368)) = $true) & ((in @ sK367 @ sK369) != $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ X4)) | ($true != (in @ X3 @ (binunion @ X5 @ X4))) | ($true = (in @ X3 @ X5))) | (binunionE != $true))),
% 6.63/2.39 inference(skolemisation,[status(esa),new_symbols(skolem,[sK367,sK368,sK369])],[f1737,f1738])).
% 6.63/2.39 thf(f1738,plain,(
% 6.63/2.39 ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) & ((in @ X0 @ (binunion @ X2 @ X1)) = $true) & ((in @ X0 @ X2) != $true)) => (($true != (in @ sK367 @ sK368)) & ((in @ sK367 @ (binunion @ sK369 @ sK368)) = $true) & ((in @ sK367 @ sK369) != $true))),
% 6.63/2.39 introduced(definition,[],[choice_axiom])).
% 6.63/2.39 thf(f1737,plain,(
% 6.63/2.39 ((binunionE = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) & ((in @ X0 @ (binunion @ X2 @ X1)) = $true) & ((in @ X0 @ X2) != $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ X4)) | ($true != (in @ X3 @ (binunion @ X5 @ X4))) | ($true = (in @ X3 @ X5))) | (binunionE != $true))),
% 6.63/2.39 inference(rectify,[status(thm)],[f1736])).
% 6.63/2.39 thf(f1736,plain,(
% 6.63/2.39 ((binunionE = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) & ((in @ X0 @ (binunion @ X2 @ X1)) = $true) & ((in @ X0 @ X2) != $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) | ((in @ X0 @ (binunion @ X2 @ X1)) != $true) | ((in @ X0 @ X2) = $true)) | (binunionE != $true))),
% 6.63/2.39 inference(nnf_transformation,[status(thm)],[f1118])).
% 6.63/2.39 thf(f1118,plain,(
% 6.63/2.39 (binunionE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) | ((in @ X0 @ (binunion @ X2 @ X1)) != $true) | ((in @ X0 @ X2) = $true))),
% 6.63/2.39 inference(flattening,[status(thm)],[f1117])).
% 6.63/2.39 thf(f1117,plain,(
% 6.63/2.39 (binunionE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X0 @ X2) = $true) | ($true = (in @ X0 @ X1))) | ((in @ X0 @ (binunion @ X2 @ X1)) != $true))),
% 6.63/2.39 inference(ennf_transformation,[status(thm)],[f664])).
% 6.63/2.39 thf(f664,plain,(
% 6.63/2.39 (binunionE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binunion @ X2 @ X1)) = $true) => (((in @ X0 @ X2) = $true) | ($true = (in @ X0 @ X1))))),
% 6.63/2.39 inference(fool_elimination,[status(thm)],[f663])).
% 6.63/2.39 thf(f663,plain,(
% 6.63/2.39 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (binunion @ X2 @ X1)) => ((in @ X0 @ X1) | (in @ X0 @ X2))) = binunionE)),
% 6.63/2.39 inference(rectify,[status(thm)],[f105])).
% 6.63/2.39 thf(f105,axiom,(
% 6.63/2.39 (! [X1 : $i,X4 : $i,X3 : $i] : ((in @ X1 @ (binunion @ X3 @ X4)) => ((in @ X1 @ X4) | (in @ X1 @ X3))) = binunionE)),
% 6.63/2.39 file('/export/starexec/sandbox/tmp/tmp.sLgKrezYTG/DTF2THF_13645.p',binunionE)).
% 6.63/2.39 % SZS output end Proof for DTF2THF_13645
% 6.63/2.39 % (13767)------------------------------
% 6.63/2.39 % (13767)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.63/2.39 % (13767)Termination reason: Refutation
% 6.63/2.39
% 6.63/2.39 % (13767)Memory used [KB]: 12665
% 6.63/2.39 % (13767)Time elapsed: 0.457 s
% 6.63/2.39 % (13767)Instructions burned: 762 (million)
% 6.63/2.39 % (13767)------------------------------
% 6.63/2.39 % (13767)------------------------------
% 6.63/2.39 % (13744)Success in time 0.579 s
%------------------------------------------------------------------------------