%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU727^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n004.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 09:36:42 AM UTC 2026
% Result : Theorem 2.79s 1.87s
% Output : Refutation 2.79s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 8
% Syntax : Number of formulae : 52 ( 13 unt; 0 typ; 1 def)
% Number of atoms : 4224 (1121 equ; 0 cnn)
% Maximal formula atoms : 247 ( 81 avg)
% Number of connectives : 2306 ( 79 ~; 51 |; 769 &; 399 @)
% ( 8 <=>;1000 =>; 0 <=; 0 <~>)
% Maximal formula depth : 255 ( 38 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 444 ( 441 usr; 255 con; 0-4 aty)
% Number of variables : 98 ( 0 ^; 69 !; 29 ?; 98 :)
% 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_295,type,
sP0: $i > $i > $i > $i > $o ).
thf(func_def_297,type,
sP2: $i > $i > $o ).
thf(func_def_298,type,
sP3: $i > $o ).
thf(func_def_299,type,
sP4: $i > $i > $o ).
thf(func_def_300,type,
sP5: $i > $i > $o ).
thf(func_def_303,type,
sP8: $i > $i > $i > $o ).
thf(func_def_305,type,
sP10: $i > $o > $i > $i > $o ).
thf(func_def_309,type,
sK14: $i > $i ).
thf(func_def_310,type,
sK15: ( $i > $i ) > $i > $i > $i ).
thf(func_def_317,type,
sK22: $i > $i > $o ).
thf(func_def_333,type,
sK38: $i > $i ).
thf(func_def_340,type,
sK45: $i > $i > $i > $i ).
thf(func_def_344,type,
sK49: $i > $o ).
thf(func_def_352,type,
sK57: $i > $o ).
thf(func_def_357,type,
sK62: $i > $i > $o ).
thf(func_def_389,type,
sK94: $i > $i ).
thf(func_def_391,type,
sK96: ( $i > $i ) > $i > $i > $i ).
thf(func_def_402,type,
sK107: $i > $i ).
thf(func_def_405,type,
sK110: $i > $i ).
thf(func_def_406,type,
sK111: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_407,type,
sK112: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_408,type,
sK113: $i > $o ).
thf(func_def_409,type,
sK114: $i > $o ).
thf(func_def_410,type,
sK115: $i > $o ).
thf(func_def_416,type,
sK121: $i > $i > $i ).
thf(func_def_428,type,
sK133: $i > $i > $i ).
thf(func_def_444,type,
sK149: $i > $i > $o ).
thf(func_def_449,type,
sK154: $i > $i > $i ).
thf(func_def_463,type,
sK168: $i > $i > $i > $i > $i ).
thf(func_def_464,type,
sK169: $i > $i > $i > $i > $i ).
thf(func_def_465,type,
sK170: $i > $i > $i > $i > $i ).
thf(func_def_466,type,
sK171: $i > $i > $i > $i > $i ).
thf(func_def_497,type,
sK202: $i > $i > $o ).
thf(func_def_509,type,
sK214: $i > $i > $i ).
thf(func_def_510,type,
sK215: $i > $i > $i ).
thf(func_def_511,type,
sK216: $i > $i > $i ).
thf(func_def_512,type,
sK217: $i > $i > $i ).
thf(func_def_513,type,
sK218: $i > $i > $i > $i ).
thf(func_def_514,type,
sK219: $i > $i ).
thf(func_def_515,type,
sK220: $i > $i ).
thf(func_def_516,type,
sK221: $i > $i ).
thf(func_def_517,type,
sK222: $i > $i ).
thf(func_def_518,type,
sK223: $i > $i > $i > $i ).
thf(func_def_519,type,
sK224: $i > $i > $i > $i ).
thf(func_def_520,type,
sK225: $i > $i > $i ).
thf(func_def_521,type,
sK226: $i > $i > $i ).
thf(func_def_522,type,
sK227: $i > $i > $i ).
thf(func_def_524,type,
sK229: $i > $i ).
thf(func_def_525,type,
sK230: $i > $i ).
thf(func_def_526,type,
sK231: $i > $i ).
thf(func_def_527,type,
sK232: $i > $i > $i ).
thf(func_def_529,type,
sK234: $i > $i > ( $i > $i ) > $i ).
thf(func_def_530,type,
sK235: $i > $i ).
thf(func_def_556,type,
sK261: $i > $i > $i ).
thf(func_def_576,type,
sK281: $i > $o ).
thf(func_def_578,type,
sK283: ( $i > $o ) > $i > $i ).
thf(func_def_584,type,
sK289: $i > $i ).
thf(func_def_587,type,
sK292: $i > $o ).
thf(func_def_590,type,
sK295: $i > ( $i > $o ) > $i ).
thf(func_def_624,type,
sK329: $i > $i > $i ).
thf(func_def_637,type,
sK342: $i > $o ).
thf(func_def_645,type,
sK350: $i > $o ).
thf(func_def_648,type,
sK353: $i > $o ).
thf(func_def_651,type,
sK356: $i > ( $i > $o ) > $i ).
thf(func_def_652,type,
sK357: $i > ( $i > $o ) > $i ).
thf(func_def_653,type,
sK358: $i > $o ).
thf(func_def_658,type,
sK363: $i > $i > $i ).
thf(func_def_667,type,
sK372: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_668,type,
sK373: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_672,type,
sK377: $i > $o ).
thf(func_def_686,type,
sK391: $i > $i > $o ).
thf(func_def_701,type,
sK406: $i > $i > $i > $i > $i ).
thf(func_def_709,type,
sK414: $i > $i > $o ).
thf(func_def_719,type,
sK424: ( $i > $o ) > $i > $i > $i ).
thf(func_def_722,type,
sK427: $i > $o ).
thf(func_def_732,type,
sK437: $i > $o ).
thf(func_def_751,type,
sK456: $i > $i ).
thf(func_def_754,type,
sK459: $i > $i > ( $i > $i ) > $i ).
thf(func_def_765,type,
sK470: $i > $i > $i > $i > $i ).
thf(func_def_770,type,
sK475: $i > ( $i > $o ) > $i ).
thf(func_def_771,type,
sK476: $i > $o ).
thf(func_def_773,type,
sK478: $i > $o ).
thf(func_def_781,type,
sK486: $i > $i > $i > $i ).
thf(func_def_785,type,
sK490: $i > $i > $i > $i ).
thf(func_def_788,type,
sK493: $i > $i ).
thf(func_def_793,type,
sK498: $i > $i > $i > $i ).
thf(func_def_797,type,
sK502: $i > $o ).
thf(func_def_800,type,
sK505: $i > ( $i > $o ) > $i ).
thf(func_def_806,type,
sK511: $i > $i > $i ).
thf(func_def_809,type,
sK514: $i > $i > $i ).
thf(func_def_810,type,
sK515: $i > $i > $i ).
thf(func_def_813,type,
sK518: $i > $o ).
thf(func_def_816,type,
sK521: $i > ( $i > $o ) > $i ).
thf(func_def_825,type,
sK530: $i > $o ).
thf(func_def_827,type,
sK532: ( $i > $o ) > $i ).
thf(func_def_828,type,
sK533: ( $i > $o ) > $i ).
thf(func_def_831,type,
sK536: $i > $o ).
thf(func_def_833,type,
sK538: ( $i > $o ) > $i > $i ).
thf(func_def_851,type,
sK556: ( $i > $o ) > $i > $i ).
thf(func_def_853,type,
sK558: $i > $o ).
thf(func_def_855,type,
sK560: ( $i > $o ) > $i ).
thf(func_def_856,type,
sK561: $i > $o ).
thf(func_def_857,type,
sK562: $i > $i ).
thf(func_def_859,type,
sK564: $i > $o ).
thf(func_def_861,type,
sK566: ( $i > $o ) > $i > $i ).
thf(func_def_862,type,
sK567: ( $i > $o ) > $i > $i ).
thf(func_def_865,type,
sK570: $o > $i > $i > $i ).
thf(func_def_871,type,
sK576: $i > $o ).
thf(func_def_874,type,
sK579: $i > $i > $o ).
thf(func_def_885,type,
sK590: $i > $o ).
thf(func_def_887,type,
sK592: $i > $i > $i > $i > $i ).
thf(func_def_894,type,
sK599: $i > $i ).
thf(func_def_902,type,
sK607: $i > $o ).
thf(func_def_904,type,
sK609: $i > ( $i > $o ) > $i ).
thf(func_def_905,type,
sK610: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_906,type,
sK611: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_909,type,
sK614: $i > $o ).
thf(func_def_910,type,
sK615: $i > $o ).
thf(func_def_919,type,
sK624: $i > $i ).
thf(func_def_922,type,
sK627: $i > $i > ( $i > $i ) > $i ).
thf(func_def_923,type,
sK628: $i > $i ).
thf(func_def_932,type,
sK637: $i > $i ).
thf(func_def_933,type,
sK638: ( $i > $i ) > $i > $i > $i ).
thf(func_def_951,type,
sK656: ( $i > $o ) > $i ).
thf(func_def_952,type,
sK657: $i > $o ).
thf(func_def_960,type,
sK665: $i > $i > $i > $i ).
thf(func_def_961,type,
sK666: $i > $i > $i > $i ).
thf(func_def_962,type,
sK667: ( $i > $o ) > $i ).
thf(func_def_963,type,
sK668: $i > $o ).
thf(func_def_964,type,
sK669: $i > $i ).
thf(func_def_967,type,
sK672: $i > $o ).
thf(func_def_980,type,
sK685: $i > ( $i > $o ) > $i ).
thf(func_def_981,type,
sK686: $i > $o ).
thf(func_def_991,type,
sK696: $i > $i > $o ).
thf(func_def_993,type,
sK698: $i > $i ).
thf(func_def_994,type,
sK699: $i > $i ).
thf(func_def_995,type,
sK700: $i > ( $i > $i > $o ) > $i ).
thf(func_def_996,type,
sK701: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_997,type,
sK702: $i > ( $i > $i > $o ) > $i ).
thf(func_def_1003,type,
sK708: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1004,type,
sK709: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1005,type,
sK710: $i > $o ).
thf(func_def_1006,type,
sK711: $i > $o ).
thf(func_def_1020,type,
sK725: $i > $i > $i > $i > $i ).
thf(func_def_1021,type,
sK726: $i > $i > $i > $i > $i ).
thf(func_def_1026,type,
sK731: $i > $o ).
thf(func_def_1039,type,
sK744: $i > $i > $i ).
thf(func_def_1050,type,
sK755: $i > $o ).
thf(func_def_1057,type,
sK762: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1058,type,
sK763: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1062,type,
sK767: $i > $o ).
thf(func_def_1068,type,
ph773:
!>[X0: $tType] : X0 ).
thf(f5932,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3868,f5923]) ).
thf(f5923,plain,
~ spl772_11,
inference(avatar_contradiction_clause,[status(thm)],[f5922]) ).
thf(f5922,plain,
( $false
| ~ spl772_11 ),
inference(trivial_inequality_removal,[status(thm)],[f5913]) ).
thf(f5913,plain,
( ( $true != $true )
| ~ spl772_11 ),
inference(superposition,[status(thm)],[f3901,f2790]) ).
thf(f2790,plain,
( ( in @ sK251 @ sK249 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f1443,plain,
( ( setunionsingleton1 = $true )
& ( powersetE = $true )
& ( dpsetconstrEL2 = $true )
& ( ex1I = $true )
& ( brelnall2 = $true )
& ( dpsetconstrI = $true )
& ( ex1I2 = $true )
& ( setukpairIL = $true )
& ( emptyI = $true )
& ( subbreln = $true )
& ( descr__Cong = $true )
& ( exuI2 = $true )
& ( cartprodsndpairEq = $true )
& ( binintersectT_lem = $true )
& ( notequalI2 = $true )
& ( $true
= ( in @ sK249 @ ( powerset @ sK250 ) ) )
& ( ( in @ sK251 @ sK249 )
= $true )
& ( ( in @ sK251 @ ( setminus @ sK250 @ ( setminus @ sK250 @ sK249 ) ) )
!= $true )
& ( ( in @ sK251 @ sK250 )
= $true )
& ( setext = $true )
& ( exuI3 = $true )
& ( emptysetAx = $true )
& ( prop2setE = $true )
& ( nonemptyE1 = $true )
& ( subsetRefl = $true )
& ( binunionEcases = $true )
& ( dpsetconstrEL1 = $true )
& ( setukpairinjR2 = $true )
& ( theprop = $true )
& ( beta2 = $true )
& ( setadjoinSub = $true )
& ( kfstpairEq = $true )
& ( binintersectTERcontra = $true )
& ( powersetTI1 = $true )
& ( singletonsubset = $true )
& ( setminusERneg = $true )
& ( binintersectER = $true )
& ( ifSingleton = $true )
& ( upairinpowunion = $true )
& ( secondinupair = $true )
& ( complementTE1 = $true )
& ( funcext = $true )
& ( ap2apEq1 = $true )
& ( dsetconstrER = $true )
& ( powersetAx = $true )
& ( binunionIR = $true )
& ( setminusLsub = $true )
& ( sepInPowerset = $true )
& ( subsetemptysetimpeq = $true )
& ( cartprodmempaircEq = $true )
& ( inCongP = $true )
& ( eta1 = $true )
& ( cartprodpairmemER = $true )
& ( cartprodsndin = $true )
& ( iftrue = $true )
& ( powersetE1 = $true )
& ( funcextLem = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( infuncsetfunc = $true )
& ( emptyInPowerset = $true )
& ( kfstsingleton = $true )
& ( contrasubsetT1 = $true )
& ( setminusILneg = $true )
& ( subsetTI = $true )
& ( funcGraphProp4 = $true )
& ( cartprodmempair = $true )
& ( setminusIRneg = $true )
& ( setminusELneg = $true )
& ( eqbreln = $true )
& ( subPowSU = $true )
& ( prop2setI = $true )
& ( setextT = $true )
& ( complementTI1 = $true )
& ( iftrueorfalse = $true )
& ( upairsubunion = $true )
& ( iftrueProp1 = $true )
& ( notequalI1 = $true )
& ( emptyE1 = $true )
& ( upairsetE = $true )
& ( singletoninpowerset = $true )
& ( iffalseProp2 = $true )
& ( contrasubsetT3 = $true )
& ( setadjoinIL = $true )
& ( noeltsimpempty = $true )
& ( setadjoinSub2 = $true )
& ( nonemptyImpWitness = $true )
& ( setminusT_lem = $true )
& ( subsetI1 = $true )
& ( upairset2E = $true )
& ( setadjoinIR = $true )
& ( descrp = $true )
& ( quantDeMorgan2 = $true )
& ( ex1E1 = $true )
& ( ifp = $true )
& ( setukpairinjL2 = $true )
& ( setbeta = $true )
& ( setextsub = $true )
& ( subsetI2 = $true )
& ( exuEu = $true )
& ( app = $true )
& ( symdiffIneg1 = $true )
& ( ap2p = $true )
& ( notinemptyset = $true )
& ( singletoninpowunion = $true )
& ( funcGraphProp3 = $true )
& ( eqinunit = $true )
& ( setukpairinjR1 = $true )
& ( bs114d = $true )
& ( upairset2IR = $true )
& ( setadjoin__Cong = $true )
& ( dpsetconstrSub = $true )
& ( wellorderingAx = $true )
& ( binunionRsub = $true )
& ( nonemptyI = $true )
& ( exuE2 = $true )
& ( ksndpairEq = $true )
& ( binintersectTELcontra = $true )
& ( eqimpsubset2 = $true )
& ( kpairiskpair = $true )
& ( powersetTE1 = $true )
& ( singletonsuniq = $true )
& ( nonemptyI1 = $true )
& ( funcGraphProp2 = $true )
& ( singletonprop = $true )
& ( dpsetconstrER = $true )
& ( sepSubset = $true )
& ( cartprodpairsurjEq = $true )
& ( powerset__Cong = $true )
& ( binunionLsub = $true )
& ( notinsingleton = $true )
& ( exuE3e = $true )
& ( subset2powerset = $true )
& ( inPowerset = $true )
& ( setunionAx = $true )
& ( kpairsurjEq = $true )
& ( ap2apEq2 = $true )
& ( disjointsetsI1 = $true )
& ( setextAx = $true )
& ( setOfPairsIsBReln = $true )
& ( singletonsswitch = $true )
& ( exuE3u = $true )
& ( subsetE = $true )
& ( setunion__Cong = $true )
& ( symdiffIneg2 = $true )
& ( powersetI1 = $true )
& ( iffalseProp1 = $true )
& ( ex1E2 = $true )
& ( funcImageSingleton = $true )
& ( setminusEL = $true )
& ( setukpairinjR = $true )
& ( subsetTrans = $true )
& ( funcinfuncset = $true )
& ( setoftrueEq = $true )
& ( setukpairIR = $true )
& ( ubforcartprodlem3 = $true )
& ( setunionsingleton2 = $true )
& ( theeq = $true )
& ( iffalse = $true )
& ( binintersectSubset5 = $true )
& ( dpsetconstrERa = $true )
& ( ubforcartprodlem2 = $true )
& ( contrasubsetT = $true )
& ( setminusER = $true )
& ( emptyinPowerset = $true )
& ( cartprodfstin = $true )
& ( setunionI = $true )
& ( ubforcartprodlem1 = $true )
& ( vacuousDall = $true )
& ( notsubsetI = $true )
& ( dsetconstrI = $true )
& ( in__Cong = $true )
& ( complementT_lem = $true )
& ( upairsetIR = $true )
& ( emptyset__Cong = $true )
& ( upairsetIL = $true )
& ( binunionIL = $true )
& ( binintersectLsub = $true )
& ( kpairp = $true )
& ( quantDeMorgan4 = $true )
& ( exu__Cong = $true )
& ( apProp = $true )
& ( cartprodpairmemEL = $true )
& ( prop2set2propI = $true )
& ( omega__Cong = $true )
& ( omegaSAx = $true )
& ( setukpairinjR11 = $true )
& ( cartprodmempair1 = $true )
& ( replAx = $true )
& ( lam2p = $true )
& ( setunionE = $true )
& ( binintersectSubset4 = $true )
& ( notdallE = $true )
& ( eqimpsubset1 = $true )
& ( emptysetimpfalse = $true )
& ( symdiffE = $true )
& ( foundationAx = $true )
& ( emptysetsubset = $true )
& ( setadjoinOr = $true )
& ( dsetconstr__Cong = $true )
& ( powersetsubset = $true )
& ( beta1 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjR12 = $true )
& ( notdexE = $true )
& ( symdiffI2 = $true )
& ( dsetconstrEL = $true )
& ( emptysetE = $true )
& ( ksndsingleton = $true )
& ( lam2lamEq = $true )
& ( omegaIndAx = $true )
& ( emptyinunitempty = $true )
& ( setukpairinjL1 = $true )
& ( setminusSubset1 = $true )
& ( quantDeMorgan3 = $true )
& ( setminusI = $true )
& ( cartprodpairin = $true )
& ( lamProp = $true )
& ( setunionE2 = $true )
& ( exuI1 = $true )
& ( binintersectRsub = $true )
& ( lamp = $true )
& ( symdiffI1 = $true )
& ( setunionsingleton = $true )
& ( exuE1 = $true )
& ( funcGraphProp1 = $true )
& ( subsetE2 = $true )
& ( powersetI = $true )
& ( funcext2 = $true )
& ( brelnall1 = $true )
& ( setadjoinE = $true )
& ( binintersectEL = $true )
& ( cartprodfstpairEq = $true )
& ( iftrueProp2 = $true )
& ( contrasubsetT2 = $true )
& ( eta2 = $true )
& ( binintersectI = $true )
& ( binunionE = $true )
& ( quantDeMorgan1 = $true )
& ( omega0Ax = $true )
& ( uniqinunit = $true )
& ( powersetT_lem = $true )
& ( setadjoinAx = $true )
& ( binunionT_lem = $true )
& ( setminusSubset2 = $true )
& ( binintersectSubset1 = $true )
& ( upairequniteq = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK249,sK250,sK251])],[f1059,f1442,f1441]) ).
thf(f1441,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X2 @ ( setminus @ X1 @ ( setminus @ X1 @ X0 ) ) )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) ) )
=> ( ( $true
= ( in @ sK249 @ ( powerset @ sK250 ) ) )
& ? [X2: $i] :
( ( ( in @ X2 @ sK249 )
= $true )
& ( ( in @ X2 @ ( setminus @ sK250 @ ( setminus @ sK250 @ sK249 ) ) )
!= $true )
& ( ( in @ X2 @ sK250 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1442,plain,
( ? [X2: $i] :
( ( ( in @ X2 @ sK249 )
= $true )
& ( ( in @ X2 @ ( setminus @ sK250 @ ( setminus @ sK250 @ sK249 ) ) )
!= $true )
& ( ( in @ X2 @ sK250 )
= $true ) )
=> ( ( ( in @ sK251 @ sK249 )
= $true )
& ( ( in @ sK251 @ ( setminus @ sK250 @ ( setminus @ sK250 @ sK249 ) ) )
!= $true )
& ( ( in @ sK251 @ sK250 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1059,plain,
( ( setunionsingleton1 = $true )
& ( powersetE = $true )
& ( dpsetconstrEL2 = $true )
& ( ex1I = $true )
& ( brelnall2 = $true )
& ( dpsetconstrI = $true )
& ( ex1I2 = $true )
& ( setukpairIL = $true )
& ( emptyI = $true )
& ( subbreln = $true )
& ( descr__Cong = $true )
& ( exuI2 = $true )
& ( cartprodsndpairEq = $true )
& ( binintersectT_lem = $true )
& ( notequalI2 = $true )
& ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X2 @ ( setminus @ X1 @ ( setminus @ X1 @ X0 ) ) )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) ) )
& ( setext = $true )
& ( exuI3 = $true )
& ( emptysetAx = $true )
& ( prop2setE = $true )
& ( nonemptyE1 = $true )
& ( subsetRefl = $true )
& ( binunionEcases = $true )
& ( dpsetconstrEL1 = $true )
& ( setukpairinjR2 = $true )
& ( theprop = $true )
& ( beta2 = $true )
& ( setadjoinSub = $true )
& ( kfstpairEq = $true )
& ( binintersectTERcontra = $true )
& ( powersetTI1 = $true )
& ( singletonsubset = $true )
& ( setminusERneg = $true )
& ( binintersectER = $true )
& ( ifSingleton = $true )
& ( upairinpowunion = $true )
& ( secondinupair = $true )
& ( complementTE1 = $true )
& ( funcext = $true )
& ( ap2apEq1 = $true )
& ( dsetconstrER = $true )
& ( powersetAx = $true )
& ( binunionIR = $true )
& ( setminusLsub = $true )
& ( sepInPowerset = $true )
& ( subsetemptysetimpeq = $true )
& ( cartprodmempaircEq = $true )
& ( inCongP = $true )
& ( eta1 = $true )
& ( cartprodpairmemER = $true )
& ( cartprodsndin = $true )
& ( iftrue = $true )
& ( powersetE1 = $true )
& ( funcextLem = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( infuncsetfunc = $true )
& ( emptyInPowerset = $true )
& ( kfstsingleton = $true )
& ( contrasubsetT1 = $true )
& ( setminusILneg = $true )
& ( subsetTI = $true )
& ( funcGraphProp4 = $true )
& ( cartprodmempair = $true )
& ( setminusIRneg = $true )
& ( setminusELneg = $true )
& ( eqbreln = $true )
& ( subPowSU = $true )
& ( prop2setI = $true )
& ( setextT = $true )
& ( complementTI1 = $true )
& ( iftrueorfalse = $true )
& ( upairsubunion = $true )
& ( iftrueProp1 = $true )
& ( notequalI1 = $true )
& ( emptyE1 = $true )
& ( upairsetE = $true )
& ( singletoninpowerset = $true )
& ( iffalseProp2 = $true )
& ( contrasubsetT3 = $true )
& ( setadjoinIL = $true )
& ( noeltsimpempty = $true )
& ( setadjoinSub2 = $true )
& ( nonemptyImpWitness = $true )
& ( setminusT_lem = $true )
& ( subsetI1 = $true )
& ( upairset2E = $true )
& ( setadjoinIR = $true )
& ( descrp = $true )
& ( quantDeMorgan2 = $true )
& ( ex1E1 = $true )
& ( ifp = $true )
& ( setukpairinjL2 = $true )
& ( setbeta = $true )
& ( setextsub = $true )
& ( subsetI2 = $true )
& ( exuEu = $true )
& ( app = $true )
& ( symdiffIneg1 = $true )
& ( ap2p = $true )
& ( notinemptyset = $true )
& ( singletoninpowunion = $true )
& ( funcGraphProp3 = $true )
& ( eqinunit = $true )
& ( setukpairinjR1 = $true )
& ( bs114d = $true )
& ( upairset2IR = $true )
& ( setadjoin__Cong = $true )
& ( dpsetconstrSub = $true )
& ( wellorderingAx = $true )
& ( binunionRsub = $true )
& ( nonemptyI = $true )
& ( exuE2 = $true )
& ( ksndpairEq = $true )
& ( binintersectTELcontra = $true )
& ( eqimpsubset2 = $true )
& ( kpairiskpair = $true )
& ( powersetTE1 = $true )
& ( singletonsuniq = $true )
& ( nonemptyI1 = $true )
& ( funcGraphProp2 = $true )
& ( singletonprop = $true )
& ( dpsetconstrER = $true )
& ( sepSubset = $true )
& ( cartprodpairsurjEq = $true )
& ( powerset__Cong = $true )
& ( binunionLsub = $true )
& ( notinsingleton = $true )
& ( exuE3e = $true )
& ( subset2powerset = $true )
& ( inPowerset = $true )
& ( setunionAx = $true )
& ( kpairsurjEq = $true )
& ( ap2apEq2 = $true )
& ( disjointsetsI1 = $true )
& ( setextAx = $true )
& ( setOfPairsIsBReln = $true )
& ( singletonsswitch = $true )
& ( exuE3u = $true )
& ( subsetE = $true )
& ( setunion__Cong = $true )
& ( symdiffIneg2 = $true )
& ( powersetI1 = $true )
& ( iffalseProp1 = $true )
& ( ex1E2 = $true )
& ( funcImageSingleton = $true )
& ( setminusEL = $true )
& ( setukpairinjR = $true )
& ( subsetTrans = $true )
& ( funcinfuncset = $true )
& ( setoftrueEq = $true )
& ( setukpairIR = $true )
& ( ubforcartprodlem3 = $true )
& ( setunionsingleton2 = $true )
& ( theeq = $true )
& ( iffalse = $true )
& ( binintersectSubset5 = $true )
& ( dpsetconstrERa = $true )
& ( ubforcartprodlem2 = $true )
& ( contrasubsetT = $true )
& ( setminusER = $true )
& ( emptyinPowerset = $true )
& ( cartprodfstin = $true )
& ( setunionI = $true )
& ( ubforcartprodlem1 = $true )
& ( vacuousDall = $true )
& ( notsubsetI = $true )
& ( dsetconstrI = $true )
& ( in__Cong = $true )
& ( complementT_lem = $true )
& ( upairsetIR = $true )
& ( emptyset__Cong = $true )
& ( upairsetIL = $true )
& ( binunionIL = $true )
& ( binintersectLsub = $true )
& ( kpairp = $true )
& ( quantDeMorgan4 = $true )
& ( exu__Cong = $true )
& ( apProp = $true )
& ( cartprodpairmemEL = $true )
& ( prop2set2propI = $true )
& ( omega__Cong = $true )
& ( omegaSAx = $true )
& ( setukpairinjR11 = $true )
& ( cartprodmempair1 = $true )
& ( replAx = $true )
& ( lam2p = $true )
& ( setunionE = $true )
& ( binintersectSubset4 = $true )
& ( notdallE = $true )
& ( eqimpsubset1 = $true )
& ( emptysetimpfalse = $true )
& ( symdiffE = $true )
& ( foundationAx = $true )
& ( emptysetsubset = $true )
& ( setadjoinOr = $true )
& ( dsetconstr__Cong = $true )
& ( powersetsubset = $true )
& ( beta1 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjR12 = $true )
& ( notdexE = $true )
& ( symdiffI2 = $true )
& ( dsetconstrEL = $true )
& ( emptysetE = $true )
& ( ksndsingleton = $true )
& ( lam2lamEq = $true )
& ( omegaIndAx = $true )
& ( emptyinunitempty = $true )
& ( setukpairinjL1 = $true )
& ( setminusSubset1 = $true )
& ( quantDeMorgan3 = $true )
& ( setminusI = $true )
& ( cartprodpairin = $true )
& ( lamProp = $true )
& ( setunionE2 = $true )
& ( exuI1 = $true )
& ( binintersectRsub = $true )
& ( lamp = $true )
& ( symdiffI1 = $true )
& ( setunionsingleton = $true )
& ( exuE1 = $true )
& ( funcGraphProp1 = $true )
& ( subsetE2 = $true )
& ( powersetI = $true )
& ( funcext2 = $true )
& ( brelnall1 = $true )
& ( setadjoinE = $true )
& ( binintersectEL = $true )
& ( cartprodfstpairEq = $true )
& ( iftrueProp2 = $true )
& ( contrasubsetT2 = $true )
& ( eta2 = $true )
& ( binintersectI = $true )
& ( binunionE = $true )
& ( quantDeMorgan1 = $true )
& ( omega0Ax = $true )
& ( uniqinunit = $true )
& ( powersetT_lem = $true )
& ( setadjoinAx = $true )
& ( binunionT_lem = $true )
& ( setminusSubset2 = $true )
& ( binintersectSubset1 = $true )
& ( upairequniteq = $true ) ),
inference(flattening,[status(thm)],[f1058]) ).
thf(f1058,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ ( setminus @ X1 @ ( setminus @ X1 @ X0 ) ) )
!= $true )
& ( ( in @ X2 @ X0 )
= $true )
& ( ( in @ X2 @ X1 )
= $true ) )
& ( ( in @ X0 @ ( powerset @ X1 ) )
= $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)],[f732]) ).
thf(f732,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 )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( ( in @ X2 @ X0 )
= $true )
=> ( ( in @ X2 @ ( setminus @ X1 @ ( setminus @ X1 @ X0 ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f731]) ).
thf(f731,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
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( in @ X2 @ X0 )
=> ( in @ X2 @ ( setminus @ X1 @ ( setminus @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f249]) ).
thf(f249,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ X11 )
=> ( in @ X1 @ ( setminus @ X3 @ ( setminus @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f248]) ).
thf(f248,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ X11 )
=> ( in @ X1 @ ( setminus @ X3 @ ( setminus @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.4HrzPZvLLL/DTF2THF_17732.p',doubleComplementI1) ).
thf(f3901,plain,
( ( ( in @ sK251 @ sK249 )
!= $true )
| ~ spl772_11 ),
inference(trivial_inequality_removal,[status(thm)],[f3900]) ).
thf(f3900,plain,
( ( ( in @ sK251 @ sK249 )
!= $true )
| ( $true != $true )
| ~ spl772_11 ),
inference(forward_demodulation,[status(thm)],[f3893,f2643]) ).
thf(f2643,plain,
setminusER = $true,
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f3893,plain,
( ( ( in @ sK251 @ sK249 )
!= $true )
| ( setminusER != $true )
| ~ spl772_11 ),
inference(trivial_inequality_removal,[status(thm)],[f3882]) ).
thf(f3882,plain,
( ( ( in @ sK251 @ sK249 )
!= $true )
| ( setminusER != $true )
| ( $true != $true )
| ~ spl772_11 ),
inference(superposition,[status(thm)],[f2463,f3813]) ).
thf(f3813,plain,
( ( ( in @ sK251 @ ( setminus @ sK250 @ sK249 ) )
= $true )
| ~ spl772_11 ),
inference(avatar_component_clause,[status(thm)],[f3811]) ).
thf(f3811,definition,
( spl772_11
<=> ( ( in @ sK251 @ ( setminus @ sK250 @ sK249 ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl772_11])],[avatar_definition]) ).
thf(f2463,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true )
| ( setminusER != $true )
| ( ( in @ X1 @ X0 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1332]) ).
thf(f1332,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true ) )
| ( setminusER != $true ) )
& ( ( setminusER = $true )
| ( ( ( in @ sK177 @ sK176 )
= $true )
& ( ( in @ sK177 @ ( setminus @ sK178 @ sK176 ) )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK176,sK177,sK178])],[f1330,f1331]) ).
thf(f1331,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
= $true )
& ( ( in @ X4 @ ( setminus @ X5 @ X3 ) )
= $true ) )
=> ( ( ( in @ sK177 @ sK176 )
= $true )
& ( ( in @ sK177 @ ( setminus @ sK178 @ sK176 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1330,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true ) )
| ( setminusER != $true ) )
& ( ( setminusER = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
= $true )
& ( ( in @ X4 @ ( setminus @ X5 @ X3 ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1329]) ).
thf(f1329,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true ) )
| ( setminusER != $true ) )
& ( ( setminusER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
= $true )
& ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f820]) ).
thf(f820,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true ) )
<=> ( setminusER = $true ) ),
inference(ennf_transformation,[status(thm)],[f757]) ).
thf(f757,plain,
( ( setminusER = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
=> ( ( in @ X1 @ X0 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f401]) ).
thf(f401,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
=> ( ( in @ X1 @ X0 )
!= $true ) )
<=> ( setminusER = $true ) ),
inference(fool_elimination,[status(thm)],[f400]) ).
thf(f400,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
=> ~ ( in @ X1 @ X0 ) )
= setminusER ),
inference(rectify,[status(thm)],[f122]) ).
thf(f122,axiom,
! [X4: $i,X1: $i,X3: $i] :
( ( ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ~ ( in @ X1 @ X4 ) )
= setminusER ),
file('/export/starexec/sandbox2/tmp/tmp.4HrzPZvLLL/DTF2THF_17732.p',setminusER) ).
thf(f3868,plain,
spl772_11,
inference(avatar_split_clause,[status(thm)],[f3805,f3811]) ).
thf(f3805,plain,
( ( in @ sK251 @ ( setminus @ sK250 @ sK249 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3804]) ).
thf(f3804,plain,
( ( ( in @ sK251 @ ( setminus @ sK250 @ sK249 ) )
= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3803,f2788]) ).
thf(f2788,plain,
( ( in @ sK251 @ sK250 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f3803,plain,
( ( ( in @ sK251 @ ( setminus @ sK250 @ sK249 ) )
= $true )
| ( ( in @ sK251 @ sK250 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3802]) ).
thf(f3802,plain,
( ( $true != $true )
| ( ( in @ sK251 @ ( setminus @ sK250 @ sK249 ) )
= $true )
| ( ( in @ sK251 @ sK250 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f3769,f2591]) ).
thf(f2591,plain,
setminusI = $true,
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f3769,plain,
( ( setminusI != $true )
| ( ( in @ sK251 @ ( setminus @ sK250 @ sK249 ) )
= $true )
| ( ( in @ sK251 @ sK250 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3757]) ).
thf(f3757,plain,
( ( setminusI != $true )
| ( ( in @ sK251 @ ( setminus @ sK250 @ sK249 ) )
= $true )
| ( ( in @ sK251 @ sK250 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2789,f2471]) ).
thf(f2471,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
| ( ( in @ X1 @ X2 )
!= $true )
| ( ( in @ X1 @ X0 )
= $true )
| ( setminusI != $true ) ),
inference(cnf_transformation,[status(thm)],[f1340]) ).
thf(f1340,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ( ( ( in @ sK183 @ sK182 )
!= $true )
& ( ( in @ sK183 @ sK184 )
= $true )
& ( ( in @ sK183 @ ( setminus @ sK184 @ sK182 ) )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK182,sK183,sK184])],[f1338,f1339]) ).
thf(f1339,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
!= $true )
& ( ( in @ X4 @ X5 )
= $true )
& ( ( in @ X4 @ ( setminus @ X5 @ X3 ) )
!= $true ) )
=> ( ( ( in @ sK183 @ sK182 )
!= $true )
& ( ( in @ sK183 @ sK184 )
= $true )
& ( ( in @ sK183 @ ( setminus @ sK184 @ sK182 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1338,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
!= $true )
& ( ( in @ X4 @ X5 )
= $true )
& ( ( in @ X4 @ ( setminus @ X5 @ X3 ) )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1337]) ).
thf(f1337,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true ) )
| ( setminusI != $true ) )
& ( ( setminusI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
!= $true )
& ( ( in @ X1 @ X2 )
= $true )
& ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f804]) ).
thf(f804,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true ) )
<=> ( setminusI = $true ) ),
inference(flattening,[status(thm)],[f803]) ).
thf(f803,plain,
( ( setminusI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
| ( ( in @ X1 @ X0 )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f754]) ).
thf(f754,plain,
( ( setminusI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( ( in @ X1 @ X0 )
!= $true )
=> ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true ) ) ) ),
inference(flattening,[status(thm)],[f373]) ).
thf(f373,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( ( in @ X1 @ X0 )
!= $true )
=> ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true ) ) )
<=> ( setminusI = $true ) ),
inference(fool_elimination,[status(thm)],[f372]) ).
thf(f372,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
=> ( ~ ( in @ X1 @ X0 )
=> ( in @ X1 @ ( setminus @ X2 @ X0 ) ) ) )
= setminusI ),
inference(rectify,[status(thm)],[f120]) ).
thf(f120,axiom,
! [X4: $i,X1: $i,X3: $i] :
( ( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ X4 )
=> ( in @ X1 @ ( setminus @ X3 @ X4 ) ) ) )
= setminusI ),
file('/export/starexec/sandbox2/tmp/tmp.4HrzPZvLLL/DTF2THF_17732.p',setminusI) ).
thf(f2789,plain,
( ( in @ sK251 @ ( setminus @ sK250 @ ( setminus @ sK250 @ sK249 ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1443]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU727^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.12/0.32 % Computer : n004.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Tue Feb 24 12:31:56 EST 2026
% 0.12/0.32 % CPUTime :
% 0.12/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.44 ---- Original DTF file ---
% 0.16/0.44 thf(spec,logic,$$dhol).
% 0.16/0.44 %------------------------------------------------------------------------------
% 0.16/0.44 % File : SEU727^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.44 % Domain : Set Theory
% 0.16/0.44 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.16/0.44 % Version : Especial.
% 0.16/0.44 % English : (! A:i.! X:i.in X (powerset A) -> (! x:i.in x A -> in x X ->
% 0.16/0.44 % in x (setminus A (setminus A X))))
% 0.16/0.44
% 0.16/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.44 % Source : [Bro08]
% 0.16/0.44 % Names : ZFC229g [Bro08]
% 0.16/0.44
% 0.16/0.44 % Status : Theorem
% 0.16/0.44 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.33 v7.2.0, 0.25 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, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.16/0.44 % Syntax : Number of formulae : 529 ( 247 unt; 281 typ; 247 def)
% 0.16/0.44 % Number of atoms : 1417 ( 372 equ; 0 cnn)
% 0.16/0.44 % Maximal formula atoms : 247 ( 5 avg)
% 0.16/0.44 % Number of connectives : 2802 ( 64 ~; 12 |; 44 &;1983 @)
% 0.16/0.44 % ( 14 <=>; 685 =>; 0 <=; 0 <~>)
% 0.16/0.44 % Maximal formula depth : 254 ( 2 avg)
% 0.16/0.44 % Number of types : 2 ( 0 usr)
% 0.16/0.44 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.16/0.44 % Number of symbols : 284 ( 281 usr; 247 con; 0-4 aty)
% 0.16/0.44 % Number of variables : 833 ( 74 ^; 724 !; 35 ?; 833 :)
% 0.16/0.44 % SPC : TH0_THM_EQU_NAR
% 0.16/0.44
% 0.16/0.44 % Comments : http://mathgate.info/detsetitem.php?id=287
% 0.16/0.44 %------------------------------------------------------------------------------
% 0.16/0.44 thf(in_type,type,
% 0.16/0.44 in: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(exu_type,type,
% 0.16/0.44 exu: ( $i > $o ) > $o ).
% 0.16/0.44
% 0.16/0.44 thf(exu,definition,
% 0.16/0.44 ( exu
% 0.16/0.44 = ( ^ [Xphi: $i > $o] :
% 0.16/0.44 ? [Xx: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 & ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setextAx_type,type,
% 0.16/0.44 setextAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setextAx,definition,
% 0.16/0.44 ( setextAx
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 <=> ( in @ Xx @ B ) )
% 0.16/0.44 => ( A = B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyset_type,type,
% 0.16/0.44 emptyset: $i ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetAx_type,type,
% 0.16/0.44 emptysetAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetAx,definition,
% 0.16/0.44 ( emptysetAx
% 0.16/0.44 = ( ! [Xx: $i] :
% 0.16/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoin_type,type,
% 0.16/0.44 setadjoin: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinAx_type,type,
% 0.16/0.44 setadjoinAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinAx,definition,
% 0.16/0.44 ( setadjoinAx
% 0.16/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.44 <=> ( ( Xy = Xx )
% 0.16/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powerset_type,type,
% 0.16/0.44 powerset: $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(powersetAx_type,type,
% 0.16/0.44 powersetAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetAx,definition,
% 0.16/0.44 ( powersetAx
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.44 <=> ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunion_type,type,
% 0.16/0.44 setunion: $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(setunionAx_type,type,
% 0.16/0.44 setunionAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionAx,definition,
% 0.16/0.44 ( setunionAx
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.44 <=> ? [B: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(omega_type,type,
% 0.16/0.44 omega: $i ).
% 0.16/0.44
% 0.16/0.44 thf(omega0Ax_type,type,
% 0.16/0.44 omega0Ax: $o ).
% 0.16/0.44
% 0.16/0.44 thf(omega0Ax,definition,
% 0.16/0.44 ( omega0Ax
% 0.16/0.44 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(omegaSAx_type,type,
% 0.16/0.44 omegaSAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(omegaSAx,definition,
% 0.16/0.44 ( omegaSAx
% 0.16/0.44 = ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ omega )
% 0.16/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(omegaIndAx_type,type,
% 0.16/0.44 omegaIndAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(omegaIndAx,definition,
% 0.16/0.44 ( omegaIndAx
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ( ( in @ emptyset @ A )
% 0.16/0.44 & ! [Xx: $i] :
% 0.16/0.44 ( ( ( in @ Xx @ omega )
% 0.16/0.44 & ( in @ Xx @ A ) )
% 0.16/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ omega )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(replAx_type,type,
% 0.16/0.44 replAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(replAx,definition,
% 0.16/0.44 ( replAx
% 0.16/0.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.44 => ? [B: $i] :
% 0.16/0.44 ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 <=> ? [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(foundationAx_type,type,
% 0.16/0.44 foundationAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(foundationAx,definition,
% 0.16/0.44 ( foundationAx
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.44 => ? [B: $i] :
% 0.16/0.44 ( ( in @ B @ A )
% 0.16/0.44 & ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(wellorderingAx_type,type,
% 0.16/0.44 wellorderingAx: $o ).
% 0.16/0.44
% 0.16/0.44 thf(wellorderingAx,definition,
% 0.16/0.44 ( wellorderingAx
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ? [B: $i] :
% 0.16/0.44 ( ! [C: $i] :
% 0.16/0.44 ( ( in @ C @ B )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ C )
% 0.16/0.44 => ( in @ Xx @ A ) ) )
% 0.16/0.44 & ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( ( in @ Xx @ A )
% 0.16/0.44 & ( in @ Xy @ A ) )
% 0.16/0.44 => ( ! [C: $i] :
% 0.16/0.44 ( ( in @ C @ B )
% 0.16/0.44 => ( ( in @ Xx @ C )
% 0.16/0.44 <=> ( in @ Xy @ C ) ) )
% 0.16/0.44 => ( Xx = Xy ) ) )
% 0.16/0.44 & ! [C: $i,D: $i] :
% 0.16/0.44 ( ( ( in @ C @ B )
% 0.16/0.44 & ( in @ D @ B ) )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ C )
% 0.16/0.44 => ( in @ Xx @ D ) )
% 0.16/0.44 | ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ D )
% 0.16/0.44 => ( in @ Xx @ C ) ) ) )
% 0.16/0.44 & ! [C: $i] :
% 0.16/0.44 ( ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ C )
% 0.16/0.44 => ( in @ Xx @ A ) )
% 0.16/0.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.44 => ? [D: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ D @ B )
% 0.16/0.44 & ( in @ Xx @ C )
% 0.16/0.44 & ~ ? [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ D )
% 0.16/0.44 & ( in @ Xy @ C ) )
% 0.16/0.44 & ! [E: $i] :
% 0.16/0.44 ( ( in @ E @ B )
% 0.16/0.44 => ( ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ E )
% 0.16/0.44 => ( in @ Xy @ D ) )
% 0.16/0.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(descr_type,type,
% 0.16/0.44 descr: ( $i > $o ) > $i ).
% 0.16/0.44
% 0.16/0.44 thf(descrp_type,type,
% 0.16/0.44 descrp: $o ).
% 0.16/0.44
% 0.16/0.44 thf(descrp,definition,
% 0.16/0.44 ( descrp
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ( Xphi
% 0.16/0.44 @ ( descr
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstr_type,type,
% 0.16/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstrI_type,type,
% 0.16/0.44 dsetconstrI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstrI,definition,
% 0.16/0.44 ( dsetconstrI
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 => ( in @ Xx
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstrEL_type,type,
% 0.16/0.44 dsetconstrEL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstrEL,definition,
% 0.16/0.44 ( dsetconstrEL
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstrER_type,type,
% 0.16/0.44 dsetconstrER: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstrER,definition,
% 0.16/0.44 ( dsetconstrER
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuE1_type,type,
% 0.16/0.44 exuE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuE1,definition,
% 0.16/0.44 ( exuE1
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 & ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(prop2set_type,type,
% 0.16/0.44 prop2set: $o > $i ).
% 0.16/0.44
% 0.16/0.44 thf(prop2setE_type,type,
% 0.16/0.44 prop2setE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2setE,definition,
% 0.16/0.44 ( prop2setE
% 0.16/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.44 => Xphi ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetE_type,type,
% 0.16/0.44 emptysetE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetE,definition,
% 0.16/0.44 ( emptysetE
% 0.16/0.44 = ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ emptyset )
% 0.16/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetimpfalse_type,type,
% 0.16/0.44 emptysetimpfalse: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetimpfalse,definition,
% 0.16/0.44 ( emptysetimpfalse
% 0.16/0.44 = ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ emptyset )
% 0.16/0.44 => $false ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notinemptyset_type,type,
% 0.16/0.44 notinemptyset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notinemptyset,definition,
% 0.16/0.44 ( notinemptyset
% 0.16/0.44 = ( ! [Xx: $i] :
% 0.16/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuE3e_type,type,
% 0.16/0.44 exuE3e: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuE3e,definition,
% 0.16/0.44 ( exuE3e
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setext_type,type,
% 0.16/0.44 setext: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setext,definition,
% 0.16/0.44 ( setext
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) )
% 0.16/0.44 => ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ A ) )
% 0.16/0.44 => ( A = B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyI_type,type,
% 0.16/0.44 emptyI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptyI,definition,
% 0.16/0.44 ( emptyI
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ~ ( in @ Xx @ A )
% 0.16/0.44 => ( A = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(noeltsimpempty_type,type,
% 0.16/0.44 noeltsimpempty: $o ).
% 0.16/0.44
% 0.16/0.44 thf(noeltsimpempty,definition,
% 0.16/0.44 ( noeltsimpempty
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ~ ( in @ Xx @ A )
% 0.16/0.44 => ( A = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setbeta_type,type,
% 0.16/0.44 setbeta: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setbeta,definition,
% 0.16/0.44 ( setbeta
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(nonempty_type,type,
% 0.16/0.44 nonempty: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(nonempty,definition,
% 0.16/0.44 ( nonempty
% 0.16/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyE1_type,type,
% 0.16/0.44 nonemptyE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyE1,definition,
% 0.16/0.44 ( nonemptyE1
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ( nonempty @ A )
% 0.16/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyI_type,type,
% 0.16/0.44 nonemptyI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyI,definition,
% 0.16/0.44 ( nonemptyI
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 => ( nonempty
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyI1_type,type,
% 0.16/0.44 nonemptyI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyI1,definition,
% 0.16/0.44 ( nonemptyI1
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinIL_type,type,
% 0.16/0.44 setadjoinIL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinIL,definition,
% 0.16/0.44 ( setadjoinIL
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyinunitempty_type,type,
% 0.16/0.44 emptyinunitempty: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptyinunitempty,definition,
% 0.16/0.44 ( emptyinunitempty
% 0.16/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinIR_type,type,
% 0.16/0.44 setadjoinIR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinIR,definition,
% 0.16/0.44 ( setadjoinIR
% 0.16/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinE_type,type,
% 0.16/0.44 setadjoinE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinE,definition,
% 0.16/0.44 ( setadjoinE
% 0.16/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.44 => ! [Xphi: $o] :
% 0.16/0.44 ( ( ( Xy = Xx )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => ( ( ( in @ Xy @ A )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinOr_type,type,
% 0.16/0.44 setadjoinOr: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinOr,definition,
% 0.16/0.44 ( setadjoinOr
% 0.16/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.44 => ( ( Xy = Xx )
% 0.16/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setoftrueEq_type,type,
% 0.16/0.44 setoftrueEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setoftrueEq,definition,
% 0.16/0.44 ( setoftrueEq
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : $true )
% 0.16/0.44 = A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetI_type,type,
% 0.16/0.44 powersetI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetI,definition,
% 0.16/0.44 ( powersetI
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ A ) )
% 0.16/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyinPowerset_type,type,
% 0.16/0.44 emptyinPowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptyinPowerset,definition,
% 0.16/0.44 ( emptyinPowerset
% 0.16/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyInPowerset_type,type,
% 0.16/0.44 emptyInPowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptyInPowerset,definition,
% 0.16/0.44 ( emptyInPowerset
% 0.16/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetE_type,type,
% 0.16/0.44 powersetE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetE,definition,
% 0.16/0.44 ( powersetE
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionI_type,type,
% 0.16/0.44 setunionI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionI,definition,
% 0.16/0.44 ( setunionI
% 0.16/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ( ( in @ B @ A )
% 0.16/0.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionE_type,type,
% 0.16/0.44 setunionE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionE,definition,
% 0.16/0.44 ( setunionE
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.44 => ! [Xphi: $o] :
% 0.16/0.44 ( ! [B: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ( ( in @ B @ A )
% 0.16/0.44 => Xphi ) )
% 0.16/0.44 => Xphi ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subPowSU_type,type,
% 0.16/0.44 subPowSU: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subPowSU,definition,
% 0.16/0.44 ( subPowSU
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuE2_type,type,
% 0.16/0.44 exuE2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuE2,definition,
% 0.16/0.44 ( exuE2
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyImpWitness_type,type,
% 0.16/0.44 nonemptyImpWitness: $o ).
% 0.16/0.44
% 0.16/0.44 thf(nonemptyImpWitness,definition,
% 0.16/0.44 ( nonemptyImpWitness
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ( nonempty @ A )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & $true ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(uniqinunit_type,type,
% 0.16/0.44 uniqinunit: $o ).
% 0.16/0.44
% 0.16/0.44 thf(uniqinunit,definition,
% 0.16/0.44 ( uniqinunit
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notinsingleton_type,type,
% 0.16/0.44 notinsingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notinsingleton,definition,
% 0.16/0.44 ( notinsingleton
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx != Xy )
% 0.16/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqinunit_type,type,
% 0.16/0.44 eqinunit: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqinunit,definition,
% 0.16/0.44 ( eqinunit
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsswitch_type,type,
% 0.16/0.44 singletonsswitch: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsswitch,definition,
% 0.16/0.44 ( singletonsswitch
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetE_type,type,
% 0.16/0.44 upairsetE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetE,definition,
% 0.16/0.44 ( upairsetE
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.44 => ( ( Xz = Xx )
% 0.16/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetIL_type,type,
% 0.16/0.44 upairsetIL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetIL,definition,
% 0.16/0.44 ( upairsetIL
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetIR_type,type,
% 0.16/0.44 upairsetIR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetIR,definition,
% 0.16/0.44 ( upairsetIR
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyE1_type,type,
% 0.16/0.44 emptyE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptyE1,definition,
% 0.16/0.44 ( emptyE1
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ( ( ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 = emptyset )
% 0.16/0.44 => $false ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(vacuousDall_type,type,
% 0.16/0.44 vacuousDall: $o ).
% 0.16/0.44
% 0.16/0.44 thf(vacuousDall,definition,
% 0.16/0.44 ( vacuousDall
% 0.16/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ emptyset )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan1_type,type,
% 0.16/0.44 quantDeMorgan1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan1,definition,
% 0.16/0.44 ( quantDeMorgan1
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan2_type,type,
% 0.16/0.44 quantDeMorgan2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan2,definition,
% 0.16/0.44 ( quantDeMorgan2
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ~ ( Xphi @ Xx ) )
% 0.16/0.44 => ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan3_type,type,
% 0.16/0.44 quantDeMorgan3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan3,definition,
% 0.16/0.44 ( quantDeMorgan3
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan4_type,type,
% 0.16/0.44 quantDeMorgan4: $o ).
% 0.16/0.44
% 0.16/0.44 thf(quantDeMorgan4,definition,
% 0.16/0.44 ( quantDeMorgan4
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ~ ( Xphi @ Xx ) )
% 0.16/0.44 => ~ ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(prop2setI_type,type,
% 0.16/0.44 prop2setI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2setI,definition,
% 0.16/0.44 ( prop2setI
% 0.16/0.44 = ( ! [Xphi: $o] :
% 0.16/0.44 ( Xphi
% 0.16/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(set2prop_type,type,
% 0.16/0.44 set2prop: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2set2propI_type,type,
% 0.16/0.44 prop2set2propI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(prop2set2propI,definition,
% 0.16/0.44 ( prop2set2propI
% 0.16/0.44 = ( ! [Xphi: $o] :
% 0.16/0.44 ( Xphi
% 0.16/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notdexE_type,type,
% 0.16/0.44 notdexE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notdexE,definition,
% 0.16/0.44 ( notdexE
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notdallE_type,type,
% 0.16/0.44 notdallE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notdallE,definition,
% 0.16/0.44 ( notdallE
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ~ ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuI1_type,type,
% 0.16/0.44 exuI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuI1,definition,
% 0.16/0.44 ( exuI1
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 & ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuI3_type,type,
% 0.16/0.44 exuI3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuI3,definition,
% 0.16/0.44 ( exuI3
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuI2_type,type,
% 0.16/0.44 exuI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuI2,definition,
% 0.16/0.44 ( exuI2
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ? [Xx: $i] :
% 0.16/0.44 ! [Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xy )
% 0.16/0.44 <=> ( Xy = Xx ) )
% 0.16/0.44 => ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inCongP_type,type,
% 0.16/0.44 inCongP: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inCongP,definition,
% 0.16/0.44 ( inCongP
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(in__Cong_type,type,
% 0.16/0.44 in__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(in__Cong,definition,
% 0.16/0.44 ( in__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuE3u_type,type,
% 0.16/0.44 exuE3u: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuE3u,definition,
% 0.16/0.44 ( exuE3u
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exu__Cong_type,type,
% 0.16/0.44 exu__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exu__Cong,definition,
% 0.16/0.44 ( exu__Cong
% 0.16/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.44 => ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 <=> ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptyset__Cong_type,type,
% 0.16/0.44 emptyset__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptyset__Cong,definition,
% 0.16/0.44 ( emptyset__Cong
% 0.16/0.44 = ( emptyset = emptyset ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoin__Cong_type,type,
% 0.16/0.44 setadjoin__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoin__Cong,definition,
% 0.16/0.44 ( setadjoin__Cong
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ! [Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( Xz = Xu )
% 0.16/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powerset__Cong_type,type,
% 0.16/0.44 powerset__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powerset__Cong,definition,
% 0.16/0.44 ( powerset__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( ( powerset @ A )
% 0.16/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunion__Cong_type,type,
% 0.16/0.44 setunion__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunion__Cong,definition,
% 0.16/0.44 ( setunion__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( ( setunion @ A )
% 0.16/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(omega__Cong_type,type,
% 0.16/0.44 omega__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(omega__Cong,definition,
% 0.16/0.44 ( omega__Cong
% 0.16/0.44 = ( omega = omega ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(exuEu_type,type,
% 0.16/0.44 exuEu: $o ).
% 0.16/0.44
% 0.16/0.44 thf(exuEu,definition,
% 0.16/0.44 ( exuEu
% 0.16/0.44 = ( ! [Xphi: $i > $o] :
% 0.16/0.44 ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(descr__Cong_type,type,
% 0.16/0.44 descr__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(descr__Cong,definition,
% 0.16/0.44 ( descr__Cong
% 0.16/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.44 => ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ( ( exu
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.44 => ( ( descr
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 = ( descr
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstr__Cong_type,type,
% 0.16/0.44 dsetconstr__Cong: $o ).
% 0.16/0.44
% 0.16/0.44 thf(dsetconstr__Cong,definition,
% 0.16/0.44 ( dsetconstr__Cong
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( Xx = Xy )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.44 => ( ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 = ( dsetconstr @ B
% 0.16/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subset_type,type,
% 0.16/0.44 subset: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(disjoint_type,type,
% 0.16/0.44 disjoint: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(setsmeet_type,type,
% 0.16/0.44 setsmeet: $i > $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI1_type,type,
% 0.16/0.44 subsetI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI1,definition,
% 0.16/0.44 ( subsetI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset2_type,type,
% 0.16/0.44 eqimpsubset2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset2,definition,
% 0.16/0.44 ( eqimpsubset2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset1_type,type,
% 0.16/0.44 eqimpsubset1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(eqimpsubset1,definition,
% 0.16/0.44 ( eqimpsubset1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( A = B )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI2_type,type,
% 0.16/0.44 subsetI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetI2,definition,
% 0.16/0.44 ( subsetI2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetsubset_type,type,
% 0.16/0.44 emptysetsubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(emptysetsubset,definition,
% 0.16/0.44 ( emptysetsubset
% 0.16/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE_type,type,
% 0.16/0.44 subsetE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE,definition,
% 0.16/0.44 ( subsetE
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE2_type,type,
% 0.16/0.44 subsetE2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetE2,definition,
% 0.16/0.44 ( subsetE2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notsubsetI_type,type,
% 0.16/0.44 notsubsetI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notsubsetI,definition,
% 0.16/0.44 ( notsubsetI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI1_type,type,
% 0.16/0.44 notequalI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI1,definition,
% 0.16/0.44 ( notequalI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ~ ( subset @ A @ B )
% 0.16/0.44 => ( A != B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI2_type,type,
% 0.16/0.44 notequalI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(notequalI2,definition,
% 0.16/0.44 ( notequalI2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ( A != B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetRefl_type,type,
% 0.16/0.44 subsetRefl: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetRefl,definition,
% 0.16/0.44 ( subsetRefl
% 0.16/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetTrans_type,type,
% 0.16/0.44 subsetTrans: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetTrans,definition,
% 0.16/0.44 ( subsetTrans
% 0.16/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( subset @ B @ C )
% 0.16/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub_type,type,
% 0.16/0.44 setadjoinSub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub,definition,
% 0.16/0.44 ( setadjoinSub
% 0.16/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub2_type,type,
% 0.16/0.44 setadjoinSub2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setadjoinSub2,definition,
% 0.16/0.44 ( setadjoinSub2
% 0.16/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subset2powerset_type,type,
% 0.16/0.44 subset2powerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subset2powerset,definition,
% 0.16/0.44 ( subset2powerset
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setextsub_type,type,
% 0.16/0.44 setextsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setextsub,definition,
% 0.16/0.44 ( setextsub
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( subset @ B @ A )
% 0.16/0.44 => ( A = B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(subsetemptysetimpeq_type,type,
% 0.16/0.44 subsetemptysetimpeq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(subsetemptysetimpeq,definition,
% 0.16/0.44 ( subsetemptysetimpeq
% 0.16/0.44 = ( ! [A: $i] :
% 0.16/0.44 ( ( subset @ A @ emptyset )
% 0.16/0.44 => ( A = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetI1_type,type,
% 0.16/0.44 powersetI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetI1,definition,
% 0.16/0.44 ( powersetI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ B @ A )
% 0.16/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetE1_type,type,
% 0.16/0.44 powersetE1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetE1,definition,
% 0.16/0.44 ( powersetE1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(inPowerset_type,type,
% 0.16/0.44 inPowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(inPowerset,definition,
% 0.16/0.44 ( inPowerset
% 0.16/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(powersetsubset_type,type,
% 0.16/0.44 powersetsubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(powersetsubset,definition,
% 0.16/0.44 ( powersetsubset
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(sepInPowerset_type,type,
% 0.16/0.44 sepInPowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(sepInPowerset,definition,
% 0.16/0.44 ( sepInPowerset
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( in
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(sepSubset_type,type,
% 0.16/0.44 sepSubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(sepSubset,definition,
% 0.16/0.44 ( sepSubset
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( subset
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunion_type,type,
% 0.16/0.44 binunion: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIL_type,type,
% 0.16/0.44 binunionIL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIL,definition,
% 0.16/0.44 ( binunionIL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2IR_type,type,
% 0.16/0.44 upairset2IR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2IR,definition,
% 0.16/0.44 ( upairset2IR
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIR_type,type,
% 0.16/0.44 binunionIR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionIR,definition,
% 0.16/0.44 ( binunionIR
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionEcases_type,type,
% 0.16/0.44 binunionEcases: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionEcases,definition,
% 0.16/0.44 ( binunionEcases
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.44 => ( ( ( in @ Xx @ A )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => ( ( ( in @ Xx @ B )
% 0.16/0.44 => Xphi )
% 0.16/0.44 => Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionE_type,type,
% 0.16/0.44 binunionE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionE,definition,
% 0.16/0.44 ( binunionE
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionLsub_type,type,
% 0.16/0.44 binunionLsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionLsub,definition,
% 0.16/0.44 ( binunionLsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binunionRsub_type,type,
% 0.16/0.44 binunionRsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binunionRsub,definition,
% 0.16/0.44 ( binunionRsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersect_type,type,
% 0.16/0.44 binintersect: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectI_type,type,
% 0.16/0.44 binintersectI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectI,definition,
% 0.16/0.44 ( binintersectI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset5_type,type,
% 0.16/0.44 binintersectSubset5: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset5,definition,
% 0.16/0.44 ( binintersectSubset5
% 0.16/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.44 ( ( subset @ C @ A )
% 0.16/0.44 => ( ( subset @ C @ B )
% 0.16/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectEL_type,type,
% 0.16/0.44 binintersectEL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectEL,definition,
% 0.16/0.44 ( binintersectEL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectLsub_type,type,
% 0.16/0.44 binintersectLsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectLsub,definition,
% 0.16/0.44 ( binintersectLsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset2_type,type,
% 0.16/0.44 binintersectSubset2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset2,definition,
% 0.16/0.44 ( binintersectSubset2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( binintersect @ A @ B )
% 0.16/0.44 = A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset3_type,type,
% 0.16/0.44 binintersectSubset3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset3,definition,
% 0.16/0.44 ( binintersectSubset3
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( ( binintersect @ A @ B )
% 0.16/0.44 = B )
% 0.16/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectER_type,type,
% 0.16/0.44 binintersectER: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectER,definition,
% 0.16/0.44 ( binintersectER
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(disjointsetsI1_type,type,
% 0.16/0.44 disjointsetsI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(disjointsetsI1,definition,
% 0.16/0.44 ( disjointsetsI1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ~ ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( in @ Xx @ B ) )
% 0.16/0.44 => ( ( binintersect @ A @ B )
% 0.16/0.44 = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectRsub_type,type,
% 0.16/0.44 binintersectRsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectRsub,definition,
% 0.16/0.44 ( binintersectRsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset4_type,type,
% 0.16/0.44 binintersectSubset4: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset4,definition,
% 0.16/0.44 ( binintersectSubset4
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ B @ A )
% 0.16/0.44 => ( ( binintersect @ A @ B )
% 0.16/0.44 = B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset1_type,type,
% 0.16/0.44 binintersectSubset1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(binintersectSubset1,definition,
% 0.16/0.44 ( binintersectSubset1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( ( binintersect @ A @ B )
% 0.16/0.44 = A )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(bs114d_type,type,
% 0.16/0.44 bs114d: $o ).
% 0.16/0.44
% 0.16/0.44 thf(bs114d,definition,
% 0.16/0.44 ( bs114d
% 0.16/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(regular_type,type,
% 0.16/0.44 regular: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminus_type,type,
% 0.16/0.44 setminus: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(setminusI_type,type,
% 0.16/0.44 setminusI: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusI,definition,
% 0.16/0.44 ( setminusI
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusEL_type,type,
% 0.16/0.44 setminusEL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusEL,definition,
% 0.16/0.44 ( setminusEL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusER_type,type,
% 0.16/0.44 setminusER: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusER,definition,
% 0.16/0.44 ( setminusER
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.44 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusSubset2_type,type,
% 0.16/0.44 setminusSubset2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusSubset2,definition,
% 0.16/0.44 ( setminusSubset2
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( subset @ A @ B )
% 0.16/0.44 => ( ( setminus @ A @ B )
% 0.16/0.44 = emptyset ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusERneg_type,type,
% 0.16/0.44 setminusERneg: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusERneg,definition,
% 0.16/0.44 ( setminusERneg
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.44 => ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusELneg_type,type,
% 0.16/0.44 setminusELneg: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusELneg,definition,
% 0.16/0.44 ( setminusELneg
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusILneg_type,type,
% 0.16/0.44 setminusILneg: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusILneg,definition,
% 0.16/0.44 ( setminusILneg
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ A )
% 0.16/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusIRneg_type,type,
% 0.16/0.44 setminusIRneg: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusIRneg,definition,
% 0.16/0.44 ( setminusIRneg
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusLsub_type,type,
% 0.16/0.44 setminusLsub: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusLsub,definition,
% 0.16/0.44 ( setminusLsub
% 0.16/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setminusSubset1_type,type,
% 0.16/0.44 setminusSubset1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setminusSubset1,definition,
% 0.16/0.44 ( setminusSubset1
% 0.16/0.44 = ( ! [A: $i,B: $i] :
% 0.16/0.44 ( ( ( setminus @ A @ B )
% 0.16/0.44 = emptyset )
% 0.16/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiff_type,type,
% 0.16/0.44 symdiff: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffE_type,type,
% 0.16/0.44 symdiffE: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffE,definition,
% 0.16/0.44 ( symdiffE
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.44 => ! [Xphi: $o] :
% 0.16/0.44 ( ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => Xphi ) )
% 0.16/0.44 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => Xphi ) )
% 0.16/0.44 => Xphi ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffI1_type,type,
% 0.16/0.44 symdiffI1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffI1,definition,
% 0.16/0.44 ( symdiffI1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffI2_type,type,
% 0.16/0.44 symdiffI2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffI2,definition,
% 0.16/0.44 ( symdiffI2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffIneg1_type,type,
% 0.16/0.44 symdiffIneg1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffIneg1,definition,
% 0.16/0.44 ( symdiffIneg1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffIneg2_type,type,
% 0.16/0.44 symdiffIneg2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(symdiffIneg2,definition,
% 0.16/0.44 ( symdiffIneg2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ~ ( in @ Xx @ A )
% 0.16/0.44 => ( ~ ( in @ Xx @ B )
% 0.16/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(iskpair_type,type,
% 0.16/0.44 iskpair: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(secondinupair_type,type,
% 0.16/0.44 secondinupair: $o ).
% 0.16/0.44
% 0.16/0.44 thf(secondinupair,definition,
% 0.16/0.44 ( secondinupair
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairIL_type,type,
% 0.16/0.44 setukpairIL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairIL,definition,
% 0.16/0.44 ( setukpairIL
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairIR_type,type,
% 0.16/0.44 setukpairIR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairIR,definition,
% 0.16/0.44 ( setukpairIR
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kpairiskpair_type,type,
% 0.16/0.44 kpairiskpair: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kpairiskpair,definition,
% 0.16/0.44 ( kpairiskpair
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kpair_type,type,
% 0.16/0.44 kpair: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(kpairp_type,type,
% 0.16/0.44 kpairp: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kpairp,definition,
% 0.16/0.44 ( kpairp
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprod_type,type,
% 0.16/0.44 cartprod: $i > $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsubset_type,type,
% 0.16/0.44 singletonsubset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsubset,definition,
% 0.16/0.44 ( singletonsubset
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singletoninpowerset_type,type,
% 0.16/0.44 singletoninpowerset: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletoninpowerset,definition,
% 0.16/0.44 ( singletoninpowerset
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singletoninpowunion_type,type,
% 0.16/0.44 singletoninpowunion: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletoninpowunion,definition,
% 0.16/0.44 ( singletoninpowunion
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2E_type,type,
% 0.16/0.44 upairset2E: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairset2E,definition,
% 0.16/0.44 ( upairset2E
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.44 => ( ( Xz = Xx )
% 0.16/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairsubunion_type,type,
% 0.16/0.44 upairsubunion: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsubunion,definition,
% 0.16/0.44 ( upairsubunion
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairinpowunion_type,type,
% 0.16/0.44 upairinpowunion: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairinpowunion,definition,
% 0.16/0.44 ( upairinpowunion
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem1_type,type,
% 0.16/0.44 ubforcartprodlem1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem1,definition,
% 0.16/0.44 ( ubforcartprodlem1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem2_type,type,
% 0.16/0.44 ubforcartprodlem2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem2,definition,
% 0.16/0.44 ( ubforcartprodlem2
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem3_type,type,
% 0.16/0.44 ubforcartprodlem3: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ubforcartprodlem3,definition,
% 0.16/0.44 ( ubforcartprodlem3
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairin_type,type,
% 0.16/0.44 cartprodpairin: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairin,definition,
% 0.16/0.44 ( cartprodpairin
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempair1_type,type,
% 0.16/0.44 cartprodmempair1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempair1,definition,
% 0.16/0.44 ( cartprodmempair1
% 0.16/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ? [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 & ( Xu
% 0.16/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempair_type,type,
% 0.16/0.44 cartprodmempair: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempair,definition,
% 0.16/0.44 ( cartprodmempair
% 0.16/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( iskpair @ Xu ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionE2_type,type,
% 0.16/0.44 setunionE2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionE2,definition,
% 0.16/0.44 ( setunionE2
% 0.16/0.44 = ( ! [A: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.44 => ? [X: $i] :
% 0.16/0.44 ( ( in @ X @ A )
% 0.16/0.44 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton1_type,type,
% 0.16/0.44 setunionsingleton1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton1,definition,
% 0.16/0.44 ( setunionsingleton1
% 0.16/0.44 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton2_type,type,
% 0.16/0.44 setunionsingleton2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton2,definition,
% 0.16/0.44 ( setunionsingleton2
% 0.16/0.44 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton_type,type,
% 0.16/0.44 setunionsingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setunionsingleton,definition,
% 0.16/0.44 ( setunionsingleton
% 0.16/0.44 = ( ! [Xx: $i] :
% 0.16/0.44 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.16/0.44 = Xx ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singleton_type,type,
% 0.16/0.44 singleton: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonprop_type,type,
% 0.16/0.44 singletonprop: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonprop,definition,
% 0.16/0.44 ( singletonprop
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) )
% 0.16/0.44 => ( ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ( singleton
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ex1_type,type,
% 0.16/0.44 ex1: $i > ( $i > $o ) > $o ).
% 0.16/0.44
% 0.16/0.44 thf(ex1,definition,
% 0.16/0.44 ( ex1
% 0.16/0.44 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( singleton
% 0.16/0.44 @ ( dsetconstr @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ex1E1_type,type,
% 0.16/0.44 ex1E1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ex1E1,definition,
% 0.16/0.44 ( ex1E1
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ( ex1 @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.44 => ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ex1I_type,type,
% 0.16/0.44 ex1I: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ex1I,definition,
% 0.16/0.44 ( ex1I
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 => ( ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xy = Xx ) ) )
% 0.16/0.44 => ( ex1 @ A
% 0.16/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ex1I2_type,type,
% 0.16/0.44 ex1I2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ex1I2,definition,
% 0.16/0.44 ( ex1I2
% 0.16/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.44 ( ! [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ A )
% 0.16/0.44 => ( ( Xphi @ Xx )
% 0.16/0.44 => ( ( Xphi @ Xy )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) )
% 0.16/0.44 => ( ? [Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 & ( Xphi @ Xx ) )
% 0.16/0.44 => ( ex1 @ A
% 0.16/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsuniq_type,type,
% 0.16/0.44 singletonsuniq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(singletonsuniq,definition,
% 0.16/0.44 ( singletonsuniq
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( ( setadjoin @ Xx @ emptyset )
% 0.16/0.44 = ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(atmost1p_type,type,
% 0.16/0.44 atmost1p: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(atleast2p_type,type,
% 0.16/0.44 atleast2p: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(atmost2p_type,type,
% 0.16/0.44 atmost2p: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairsetp_type,type,
% 0.16/0.44 upairsetp: $i > $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL1_type,type,
% 0.16/0.44 setukpairinjL1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL1,definition,
% 0.16/0.44 ( setukpairinjL1
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.44 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kfstsingleton_type,type,
% 0.16/0.44 kfstsingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kfstsingleton,definition,
% 0.16/0.44 ( kfstsingleton
% 0.16/0.44 = ( ! [Xu: $i] :
% 0.16/0.44 ( ( iskpair @ Xu )
% 0.16/0.44 => ( singleton
% 0.16/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.44 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(theprop_type,type,
% 0.16/0.44 theprop: $o ).
% 0.16/0.44
% 0.16/0.44 thf(theprop,definition,
% 0.16/0.44 ( theprop
% 0.16/0.44 = ( ! [X: $i] :
% 0.16/0.44 ( ( singleton @ X )
% 0.16/0.44 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kfst_type,type,
% 0.16/0.44 kfst: $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(kfstpairEq_type,type,
% 0.16/0.44 kfstpairEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kfstpairEq,definition,
% 0.16/0.44 ( kfstpairEq
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.44 = Xx ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodfstin_type,type,
% 0.16/0.44 cartprodfstin: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodfstin,definition,
% 0.16/0.44 ( cartprodfstin
% 0.16/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL2_type,type,
% 0.16/0.44 setukpairinjL2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL2,definition,
% 0.16/0.44 ( setukpairinjL2
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL_type,type,
% 0.16/0.44 setukpairinjL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjL,definition,
% 0.16/0.44 ( setukpairinjL
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.44 = ( kpair @ Xz @ Xu ) )
% 0.16/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR11_type,type,
% 0.16/0.44 setukpairinjR11: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR11,definition,
% 0.16/0.44 ( setukpairinjR11
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR12_type,type,
% 0.16/0.44 setukpairinjR12: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR12,definition,
% 0.16/0.44 ( setukpairinjR12
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( Xx = Xy )
% 0.16/0.44 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR1_type,type,
% 0.16/0.44 setukpairinjR1: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR1,definition,
% 0.16/0.44 ( setukpairinjR1
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.44 => ( ( Xz = Xu )
% 0.16/0.44 => ( Xy = Xu ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(upairequniteq_type,type,
% 0.16/0.44 upairequniteq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(upairequniteq,definition,
% 0.16/0.44 ( upairequniteq
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.44 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ Xz @ emptyset ) )
% 0.16/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR2_type,type,
% 0.16/0.44 setukpairinjR2: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR2,definition,
% 0.16/0.44 ( setukpairinjR2
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR_type,type,
% 0.16/0.44 setukpairinjR: $o ).
% 0.16/0.44
% 0.16/0.44 thf(setukpairinjR,definition,
% 0.16/0.44 ( setukpairinjR
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.44 = ( kpair @ Xz @ Xu ) )
% 0.16/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ksndsingleton_type,type,
% 0.16/0.44 ksndsingleton: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ksndsingleton,definition,
% 0.16/0.44 ( ksndsingleton
% 0.16/0.44 = ( ! [Xu: $i] :
% 0.16/0.44 ( ( iskpair @ Xu )
% 0.16/0.44 => ( singleton
% 0.16/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.44 @ ^ [Xx: $i] :
% 0.16/0.44 ( Xu
% 0.16/0.44 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(ksnd_type,type,
% 0.16/0.44 ksnd: $i > $i ).
% 0.16/0.44
% 0.16/0.44 thf(ksndpairEq_type,type,
% 0.16/0.44 ksndpairEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(ksndpairEq,definition,
% 0.16/0.44 ( ksndpairEq
% 0.16/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.44 = Xy ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(kpairsurjEq_type,type,
% 0.16/0.44 kpairsurjEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(kpairsurjEq,definition,
% 0.16/0.44 ( kpairsurjEq
% 0.16/0.44 = ( ! [Xu: $i] :
% 0.16/0.44 ( ( iskpair @ Xu )
% 0.16/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.44 = Xu ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodsndin_type,type,
% 0.16/0.44 cartprodsndin: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodsndin,definition,
% 0.16/0.44 ( cartprodsndin
% 0.16/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairmemEL_type,type,
% 0.16/0.44 cartprodpairmemEL: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairmemEL,definition,
% 0.16/0.44 ( cartprodpairmemEL
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairmemER_type,type,
% 0.16/0.44 cartprodpairmemER: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodpairmemER,definition,
% 0.16/0.44 ( cartprodpairmemER
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempaircEq_type,type,
% 0.16/0.44 cartprodmempaircEq: $o ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodmempaircEq,definition,
% 0.16/0.44 ( cartprodmempaircEq
% 0.16/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.44 ( ( in @ Xx @ A )
% 0.16/0.44 => ! [Xy: $i] :
% 0.16/0.44 ( ( in @ Xy @ B )
% 0.16/0.44 => ( ( kpair @ Xx @ Xy )
% 0.16/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.44
% 0.16/0.44 thf(cartprodfstpairEq_type,type,
% 0.16/0.44 cartprodfstpairEq: $o ).
% 0.25/0.44
% 0.25/0.44 thf(cartprodfstpairEq,definition,
% 0.25/0.44 ( cartprodfstpairEq
% 0.25/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.25/0.44 = Xx ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(cartprodsndpairEq_type,type,
% 0.25/0.44 cartprodsndpairEq: $o ).
% 0.25/0.44
% 0.25/0.44 thf(cartprodsndpairEq,definition,
% 0.25/0.44 ( cartprodsndpairEq
% 0.25/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.25/0.44 = Xy ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(cartprodpairsurjEq_type,type,
% 0.25/0.44 cartprodpairsurjEq: $o ).
% 0.25/0.44
% 0.25/0.44 thf(cartprodpairsurjEq,definition,
% 0.25/0.44 ( cartprodpairsurjEq
% 0.25/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.25/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.25/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.25/0.44 = Xu ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(breln_type,type,
% 0.25/0.44 breln: $i > $i > $i > $o ).
% 0.25/0.44
% 0.25/0.44 thf(breln,definition,
% 0.25/0.44 ( breln
% 0.25/0.44 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstr_type,type,
% 0.25/0.44 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrI_type,type,
% 0.25/0.44 dpsetconstrI: $o ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrI,definition,
% 0.25/0.44 ( dpsetconstrI
% 0.25/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( Xphi @ Xx @ Xy )
% 0.25/0.44 => ( in @ ( kpair @ Xx @ Xy )
% 0.25/0.44 @ ( dpsetconstr @ A @ B
% 0.25/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrSub_type,type,
% 0.25/0.44 dpsetconstrSub: $o ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrSub,definition,
% 0.25/0.44 ( dpsetconstrSub
% 0.25/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.25/0.44 ( subset
% 0.25/0.44 @ ( dpsetconstr @ A @ B
% 0.25/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.25/0.44 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(setOfPairsIsBReln_type,type,
% 0.25/0.44 setOfPairsIsBReln: $o ).
% 0.25/0.44
% 0.25/0.44 thf(setOfPairsIsBReln,definition,
% 0.25/0.44 ( setOfPairsIsBReln
% 0.25/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.25/0.44 ( breln @ A @ B
% 0.25/0.44 @ ( dpsetconstr @ A @ B
% 0.25/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrERa_type,type,
% 0.25/0.44 dpsetconstrERa: $o ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrERa,definition,
% 0.25/0.44 ( dpsetconstrERa
% 0.25/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.25/0.44 @ ( dpsetconstr @ A @ B
% 0.25/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.25/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrEL1_type,type,
% 0.25/0.44 dpsetconstrEL1: $o ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrEL1,definition,
% 0.25/0.44 ( dpsetconstrEL1
% 0.25/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.25/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.25/0.44 @ ( dpsetconstr @ A @ B
% 0.25/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.25/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrEL2_type,type,
% 0.25/0.44 dpsetconstrEL2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrEL2,definition,
% 0.25/0.44 ( dpsetconstrEL2
% 0.25/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.25/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.25/0.44 @ ( dpsetconstr @ A @ B
% 0.25/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.25/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrER_type,type,
% 0.25/0.44 dpsetconstrER: $o ).
% 0.25/0.44
% 0.25/0.44 thf(dpsetconstrER,definition,
% 0.25/0.44 ( dpsetconstrER
% 0.25/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.25/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.25/0.44 @ ( dpsetconstr @ A @ B
% 0.25/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.25/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(func_type,type,
% 0.25/0.44 func: $i > $i > $i > $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcSet_type,type,
% 0.25/0.44 funcSet: $i > $i > $i ).
% 0.25/0.44
% 0.25/0.44 thf(funcImageSingleton_type,type,
% 0.25/0.44 funcImageSingleton: $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcImageSingleton,definition,
% 0.25/0.44 ( funcImageSingleton
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( singleton
% 0.25/0.44 @ ( dsetconstr @ B
% 0.25/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(apProp_type,type,
% 0.25/0.44 apProp: $o ).
% 0.25/0.44
% 0.25/0.44 thf(apProp,definition,
% 0.25/0.44 ( apProp
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in
% 0.25/0.44 @ ( setunion
% 0.25/0.44 @ ( dsetconstr @ B
% 0.25/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.25/0.44 @ B ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(ap_type,type,
% 0.25/0.44 ap: $i > $i > $i > $i > $i ).
% 0.25/0.44
% 0.25/0.44 thf(app_type,type,
% 0.25/0.44 app: $o ).
% 0.25/0.44
% 0.25/0.44 thf(app,definition,
% 0.25/0.44 ( app
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(infuncsetfunc_type,type,
% 0.25/0.44 infuncsetfunc: $o ).
% 0.25/0.44
% 0.25/0.44 thf(infuncsetfunc,definition,
% 0.25/0.44 ( infuncsetfunc
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.25/0.44 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(ap2p_type,type,
% 0.25/0.44 ap2p: $o ).
% 0.25/0.44
% 0.25/0.44 thf(ap2p,definition,
% 0.25/0.44 ( ap2p
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(funcinfuncset_type,type,
% 0.25/0.44 funcinfuncset: $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcinfuncset,definition,
% 0.25/0.44 ( funcinfuncset
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(lamProp_type,type,
% 0.25/0.44 lamProp: $o ).
% 0.25/0.44
% 0.25/0.44 thf(lamProp,definition,
% 0.25/0.44 ( lamProp
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.25/0.44 ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.25/0.44 => ( func @ A @ B
% 0.25/0.44 @ ( dpsetconstr @ A @ B
% 0.25/0.44 @ ^ [Xx: $i,Xy: $i] :
% 0.25/0.44 ( ( Xf @ Xx )
% 0.25/0.44 = Xy ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(lam_type,type,
% 0.25/0.44 lam: $i > $i > ( $i > $i ) > $i ).
% 0.25/0.44
% 0.25/0.44 thf(lamp_type,type,
% 0.25/0.44 lamp: $o ).
% 0.25/0.44
% 0.25/0.44 thf(lamp,definition,
% 0.25/0.44 ( lamp
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.25/0.44 ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.25/0.44 => ( func @ A @ B
% 0.25/0.44 @ ( lam @ A @ B
% 0.25/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(lam2p_type,type,
% 0.25/0.44 lam2p: $o ).
% 0.25/0.44
% 0.25/0.44 thf(lam2p,definition,
% 0.25/0.44 ( lam2p
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.25/0.44 ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.25/0.44 => ( in
% 0.25/0.44 @ ( lam @ A @ B
% 0.25/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.25/0.44 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(brelnall1_type,type,
% 0.25/0.44 brelnall1: $o ).
% 0.25/0.44
% 0.25/0.44 thf(brelnall1,definition,
% 0.25/0.44 ( brelnall1
% 0.25/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.25/0.44 ( ( breln @ A @ B @ R )
% 0.25/0.44 => ! [Xphi: $i > $o] :
% 0.25/0.44 ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.25/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ R )
% 0.25/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(brelnall2_type,type,
% 0.25/0.44 brelnall2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(brelnall2,definition,
% 0.25/0.44 ( brelnall2
% 0.25/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.25/0.44 ( ( breln @ A @ B @ R )
% 0.25/0.44 => ! [Xphi: $i > $o] :
% 0.25/0.44 ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.25/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ R )
% 0.25/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(ex1E2_type,type,
% 0.25/0.44 ex1E2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(ex1E2,definition,
% 0.25/0.44 ( ex1E2
% 0.25/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.25/0.44 ( ( ex1 @ A
% 0.25/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ A )
% 0.25/0.44 => ( ( Xphi @ Xx )
% 0.25/0.44 => ( ( Xphi @ Xy )
% 0.25/0.44 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(funcGraphProp1_type,type,
% 0.25/0.44 funcGraphProp1: $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcGraphProp1,definition,
% 0.25/0.44 ( funcGraphProp1
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(funcGraphProp3_type,type,
% 0.25/0.44 funcGraphProp3: $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcGraphProp3,definition,
% 0.25/0.44 ( funcGraphProp3
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(funcGraphProp2_type,type,
% 0.25/0.44 funcGraphProp2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcGraphProp2,definition,
% 0.25/0.44 ( funcGraphProp2
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.25/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.25/0.44 = Xy ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(funcextLem_type,type,
% 0.25/0.44 funcextLem: $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcextLem,definition,
% 0.25/0.44 ( funcextLem
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ! [Xg: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xg )
% 0.25/0.44 => ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.25/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.25/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(funcGraphProp4_type,type,
% 0.25/0.44 funcGraphProp4: $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcGraphProp4,definition,
% 0.25/0.44 ( funcGraphProp4
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.25/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.25/0.44 = Xy ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(subbreln_type,type,
% 0.25/0.44 subbreln: $o ).
% 0.25/0.44
% 0.25/0.44 thf(subbreln,definition,
% 0.25/0.44 ( subbreln
% 0.25/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.25/0.44 ( ( breln @ A @ B @ R )
% 0.25/0.44 => ! [S: $i] :
% 0.25/0.44 ( ( breln @ A @ B @ S )
% 0.25/0.44 => ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.25/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.25/0.44 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(eqbreln_type,type,
% 0.25/0.44 eqbreln: $o ).
% 0.25/0.44
% 0.25/0.44 thf(eqbreln,definition,
% 0.25/0.44 ( eqbreln
% 0.25/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.25/0.44 ( ( breln @ A @ B @ R )
% 0.25/0.44 => ! [S: $i] :
% 0.25/0.44 ( ( breln @ A @ B @ S )
% 0.25/0.44 => ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.25/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.25/0.44 => ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ B )
% 0.25/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.25/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.25/0.44 => ( R = S ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(funcext_type,type,
% 0.25/0.44 funcext: $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcext,definition,
% 0.25/0.44 ( funcext
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ! [Xg: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xg )
% 0.25/0.44 => ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.25/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.25/0.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(funcext2_type,type,
% 0.25/0.44 funcext2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(funcext2,definition,
% 0.25/0.44 ( funcext2
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.25/0.44 => ! [Xg: $i] :
% 0.25/0.44 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.25/0.44 => ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.25/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.25/0.44 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(ap2apEq1_type,type,
% 0.25/0.44 ap2apEq1: $o ).
% 0.25/0.44
% 0.25/0.44 thf(ap2apEq1,definition,
% 0.25/0.44 ( ap2apEq1
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.25/0.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(ap2apEq2_type,type,
% 0.25/0.44 ap2apEq2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(ap2apEq2,definition,
% 0.25/0.44 ( ap2apEq2
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.25/0.44 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(beta1_type,type,
% 0.25/0.44 beta1: $o ).
% 0.25/0.44
% 0.25/0.44 thf(beta1,definition,
% 0.25/0.44 ( beta1
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.25/0.44 ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( ( ap @ A @ B
% 0.25/0.44 @ ( lam @ A @ B
% 0.25/0.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.25/0.44 @ Xx )
% 0.25/0.44 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(eta1_type,type,
% 0.25/0.44 eta1: $o ).
% 0.25/0.44
% 0.25/0.44 thf(eta1,definition,
% 0.25/0.44 ( eta1
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( func @ A @ B @ Xf )
% 0.25/0.44 => ( ( lam @ A @ B
% 0.25/0.44 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.25/0.44 = Xf ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(lam2lamEq_type,type,
% 0.25/0.44 lam2lamEq: $o ).
% 0.25/0.44
% 0.25/0.44 thf(lam2lamEq,definition,
% 0.25/0.44 ( lam2lamEq
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.25/0.44 ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.25/0.44 => ( ( lam @ A @ B
% 0.25/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.25/0.44 = ( lam @ A @ B
% 0.25/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(beta2_type,type,
% 0.25/0.44 beta2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(beta2,definition,
% 0.25/0.44 ( beta2
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.25/0.44 ( ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ( ( ap @ A @ B
% 0.25/0.44 @ ( lam @ A @ B
% 0.25/0.44 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.25/0.44 @ Xx )
% 0.25/0.44 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(eta2_type,type,
% 0.25/0.44 eta2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(eta2,definition,
% 0.25/0.44 ( eta2
% 0.25/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.25/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.25/0.44 => ( ( lam @ A @ B
% 0.25/0.44 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.25/0.44 = Xf ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(iffalseProp1_type,type,
% 0.25/0.44 iffalseProp1: $o ).
% 0.25/0.44
% 0.25/0.44 thf(iffalseProp1,definition,
% 0.25/0.44 ( iffalseProp1
% 0.25/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ A )
% 0.25/0.44 => ( ~ Xphi
% 0.25/0.44 => ( in @ Xy
% 0.25/0.44 @ ( dsetconstr @ A
% 0.25/0.44 @ ^ [Xz: $i] :
% 0.25/0.44 ( ( Xphi
% 0.25/0.44 & ( Xz = Xx ) )
% 0.25/0.44 | ( ~ Xphi
% 0.25/0.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(iffalseProp2_type,type,
% 0.25/0.44 iffalseProp2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(iffalseProp2,definition,
% 0.25/0.44 ( iffalseProp2
% 0.25/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ A )
% 0.25/0.44 => ( ~ Xphi
% 0.25/0.44 => ( ( dsetconstr @ A
% 0.25/0.44 @ ^ [Xz: $i] :
% 0.25/0.44 ( ( Xphi
% 0.25/0.44 & ( Xz = Xx ) )
% 0.25/0.44 | ( ~ Xphi
% 0.25/0.44 & ( Xz = Xy ) ) ) )
% 0.25/0.44 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(iftrueProp1_type,type,
% 0.25/0.44 iftrueProp1: $o ).
% 0.25/0.44
% 0.25/0.44 thf(iftrueProp1,definition,
% 0.25/0.44 ( iftrueProp1
% 0.25/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ A )
% 0.25/0.44 => ( Xphi
% 0.25/0.44 => ( in @ Xx
% 0.25/0.44 @ ( dsetconstr @ A
% 0.25/0.44 @ ^ [Xz: $i] :
% 0.25/0.44 ( ( Xphi
% 0.25/0.44 & ( Xz = Xx ) )
% 0.25/0.44 | ( ~ Xphi
% 0.25/0.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(iftrueProp2_type,type,
% 0.25/0.44 iftrueProp2: $o ).
% 0.25/0.44
% 0.25/0.44 thf(iftrueProp2,definition,
% 0.25/0.44 ( iftrueProp2
% 0.25/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ A )
% 0.25/0.44 => ( Xphi
% 0.25/0.44 => ( ( dsetconstr @ A
% 0.25/0.44 @ ^ [Xz: $i] :
% 0.25/0.44 ( ( Xphi
% 0.25/0.44 & ( Xz = Xx ) )
% 0.25/0.44 | ( ~ Xphi
% 0.25/0.44 & ( Xz = Xy ) ) ) )
% 0.25/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(ifSingleton_type,type,
% 0.25/0.44 ifSingleton: $o ).
% 0.25/0.44
% 0.25/0.44 thf(ifSingleton,definition,
% 0.25/0.44 ( ifSingleton
% 0.25/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ A )
% 0.25/0.44 => ( singleton
% 0.25/0.44 @ ( dsetconstr @ A
% 0.25/0.44 @ ^ [Xz: $i] :
% 0.25/0.44 ( ( Xphi
% 0.25/0.44 & ( Xz = Xx ) )
% 0.25/0.44 | ( ~ Xphi
% 0.25/0.44 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(if_type,type,
% 0.25/0.44 if: $i > $o > $i > $i > $i ).
% 0.25/0.44
% 0.25/0.44 thf(ifp_type,type,
% 0.25/0.44 ifp: $o ).
% 0.25/0.44
% 0.25/0.44 thf(ifp,definition,
% 0.25/0.44 ( ifp
% 0.25/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.44 ( ( in @ Xy @ A )
% 0.25/0.44 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(theeq_type,type,
% 0.25/0.44 theeq: $o ).
% 0.25/0.44
% 0.25/0.44 thf(theeq,definition,
% 0.25/0.44 ( theeq
% 0.25/0.44 = ( ! [X: $i] :
% 0.25/0.44 ( ( singleton @ X )
% 0.25/0.44 => ! [Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ X )
% 0.25/0.44 => ( ( setunion @ X )
% 0.25/0.44 = Xx ) ) ) ) ) ).
% 0.25/0.44
% 0.25/0.44 thf(iftrue_type,type,
% 0.25/0.44 iftrue: $o ).
% 0.25/0.44
% 0.25/0.44 thf(iftrue,definition,
% 0.25/0.44 ( iftrue
% 0.25/0.44 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.25/0.44 ( ( in @ Xx @ A )
% 0.25/0.44 => ! [Xy: $i] :
% 0.25/0.45 ( ( in @ Xy @ A )
% 0.25/0.45 => ( Xphi
% 0.25/0.45 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.25/0.45 = Xx ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(iffalse_type,type,
% 0.25/0.45 iffalse: $o ).
% 0.25/0.45
% 0.25/0.45 thf(iffalse,definition,
% 0.25/0.45 ( iffalse
% 0.25/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ! [Xy: $i] :
% 0.25/0.45 ( ( in @ Xy @ A )
% 0.25/0.45 => ( ~ Xphi
% 0.25/0.45 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.25/0.45 = Xy ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(iftrueorfalse_type,type,
% 0.25/0.45 iftrueorfalse: $o ).
% 0.25/0.45
% 0.25/0.45 thf(iftrueorfalse,definition,
% 0.25/0.45 ( iftrueorfalse
% 0.25/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ! [Xy: $i] :
% 0.25/0.45 ( ( in @ Xy @ A )
% 0.25/0.45 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(binintersectT_lem_type,type,
% 0.25/0.45 binintersectT_lem: $o ).
% 0.25/0.45
% 0.25/0.45 thf(binintersectT_lem,definition,
% 0.25/0.45 ( binintersectT_lem
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(binunionT_lem_type,type,
% 0.25/0.45 binunionT_lem: $o ).
% 0.25/0.45
% 0.25/0.45 thf(binunionT_lem,definition,
% 0.25/0.45 ( binunionT_lem
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(powersetT_lem_type,type,
% 0.25/0.45 powersetT_lem: $o ).
% 0.25/0.45
% 0.25/0.45 thf(powersetT_lem,definition,
% 0.25/0.45 ( powersetT_lem
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(setminusT_lem_type,type,
% 0.25/0.45 setminusT_lem: $o ).
% 0.25/0.45
% 0.25/0.45 thf(setminusT_lem,definition,
% 0.25/0.45 ( setminusT_lem
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(complementT_lem_type,type,
% 0.25/0.45 complementT_lem: $o ).
% 0.25/0.45
% 0.25/0.45 thf(complementT_lem,definition,
% 0.25/0.45 ( complementT_lem
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(setextT_type,type,
% 0.25/0.45 setextT: $o ).
% 0.25/0.45
% 0.25/0.45 thf(setextT,definition,
% 0.25/0.45 ( setextT
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ( ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ( in @ Xx @ X )
% 0.25/0.45 => ( in @ Xx @ Y ) ) )
% 0.25/0.45 => ( ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ( in @ Xx @ Y )
% 0.25/0.45 => ( in @ Xx @ X ) ) )
% 0.25/0.45 => ( X = Y ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(subsetTI_type,type,
% 0.25/0.45 subsetTI: $o ).
% 0.25/0.45
% 0.25/0.45 thf(subsetTI,definition,
% 0.25/0.45 ( subsetTI
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ( ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ( in @ Xx @ X )
% 0.25/0.45 => ( in @ Xx @ Y ) ) )
% 0.25/0.45 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(powersetTI1_type,type,
% 0.25/0.45 powersetTI1: $o ).
% 0.25/0.45
% 0.25/0.45 thf(powersetTI1,definition,
% 0.25/0.45 ( powersetTI1
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ( ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ( in @ Xx @ X )
% 0.25/0.45 => ( in @ Xx @ Y ) ) )
% 0.25/0.45 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(powersetTE1_type,type,
% 0.25/0.45 powersetTE1: $o ).
% 0.25/0.45
% 0.25/0.45 thf(powersetTE1,definition,
% 0.25/0.45 ( powersetTE1
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ( in @ X @ ( powerset @ Y ) )
% 0.25/0.45 => ( ( in @ Xx @ X )
% 0.25/0.45 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(complementTI1_type,type,
% 0.25/0.45 complementTI1: $o ).
% 0.25/0.45
% 0.25/0.45 thf(complementTI1,definition,
% 0.25/0.45 ( complementTI1
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ( in @ Xx @ X )
% 0.25/0.45 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(complementTE1_type,type,
% 0.25/0.45 complementTE1: $o ).
% 0.25/0.45
% 0.25/0.45 thf(complementTE1,definition,
% 0.25/0.45 ( complementTE1
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.25/0.45 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(binintersectTELcontra_type,type,
% 0.25/0.45 binintersectTELcontra: $o ).
% 0.25/0.45
% 0.25/0.45 thf(binintersectTELcontra,definition,
% 0.25/0.45 ( binintersectTELcontra
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ~ ( in @ Xx @ X )
% 0.25/0.45 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(binintersectTERcontra_type,type,
% 0.25/0.45 binintersectTERcontra: $o ).
% 0.25/0.45
% 0.25/0.45 thf(binintersectTERcontra,definition,
% 0.25/0.45 ( binintersectTERcontra
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ~ ( in @ Xx @ Y )
% 0.25/0.45 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(contrasubsetT_type,type,
% 0.25/0.45 contrasubsetT: $o ).
% 0.25/0.45
% 0.25/0.45 thf(contrasubsetT,definition,
% 0.25/0.45 ( contrasubsetT
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.25/0.45 => ( ( in @ Xx @ Y )
% 0.25/0.45 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(contrasubsetT1_type,type,
% 0.25/0.45 contrasubsetT1: $o ).
% 0.25/0.45
% 0.25/0.45 thf(contrasubsetT1,definition,
% 0.25/0.45 ( contrasubsetT1
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ( subset @ X @ Y )
% 0.25/0.45 => ( ~ ( in @ Xx @ Y )
% 0.25/0.45 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(contrasubsetT2_type,type,
% 0.25/0.45 contrasubsetT2: $o ).
% 0.25/0.45
% 0.25/0.45 thf(contrasubsetT2,definition,
% 0.25/0.45 ( contrasubsetT2
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ( ( subset @ X @ Y )
% 0.25/0.45 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(contrasubsetT3_type,type,
% 0.25/0.45 contrasubsetT3: $o ).
% 0.25/0.45
% 0.25/0.45 thf(contrasubsetT3,definition,
% 0.25/0.45 ( contrasubsetT3
% 0.25/0.45 = ( ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Y: $i] :
% 0.25/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.25/0.45 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.25/0.45 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 thf(doubleComplementI1,conjecture,
% 0.25/0.45 ( setextAx
% 0.25/0.45 => ( emptysetAx
% 0.25/0.45 => ( setadjoinAx
% 0.25/0.45 => ( powersetAx
% 0.25/0.45 => ( setunionAx
% 0.25/0.45 => ( omega0Ax
% 0.25/0.45 => ( omegaSAx
% 0.25/0.45 => ( omegaIndAx
% 0.25/0.45 => ( replAx
% 0.25/0.45 => ( foundationAx
% 0.25/0.45 => ( wellorderingAx
% 0.25/0.45 => ( descrp
% 0.25/0.45 => ( dsetconstrI
% 0.25/0.45 => ( dsetconstrEL
% 0.25/0.45 => ( dsetconstrER
% 0.25/0.45 => ( exuE1
% 0.25/0.45 => ( prop2setE
% 0.25/0.45 => ( emptysetE
% 0.25/0.45 => ( emptysetimpfalse
% 0.25/0.45 => ( notinemptyset
% 0.25/0.45 => ( exuE3e
% 0.25/0.45 => ( setext
% 0.25/0.45 => ( emptyI
% 0.25/0.45 => ( noeltsimpempty
% 0.25/0.45 => ( setbeta
% 0.25/0.45 => ( nonemptyE1
% 0.25/0.45 => ( nonemptyI
% 0.25/0.45 => ( nonemptyI1
% 0.25/0.45 => ( setadjoinIL
% 0.25/0.45 => ( emptyinunitempty
% 0.25/0.45 => ( setadjoinIR
% 0.25/0.45 => ( setadjoinE
% 0.25/0.45 => ( setadjoinOr
% 0.25/0.45 => ( setoftrueEq
% 0.25/0.45 => ( powersetI
% 0.25/0.45 => ( emptyinPowerset
% 0.25/0.45 => ( emptyInPowerset
% 0.25/0.45 => ( powersetE
% 0.25/0.45 => ( setunionI
% 0.25/0.45 => ( setunionE
% 0.25/0.45 => ( subPowSU
% 0.25/0.45 => ( exuE2
% 0.25/0.45 => ( nonemptyImpWitness
% 0.25/0.45 => ( uniqinunit
% 0.25/0.45 => ( notinsingleton
% 0.25/0.45 => ( eqinunit
% 0.25/0.45 => ( singletonsswitch
% 0.25/0.45 => ( upairsetE
% 0.25/0.45 => ( upairsetIL
% 0.25/0.45 => ( upairsetIR
% 0.25/0.45 => ( emptyE1
% 0.25/0.45 => ( vacuousDall
% 0.25/0.45 => ( quantDeMorgan1
% 0.25/0.45 => ( quantDeMorgan2
% 0.25/0.45 => ( quantDeMorgan3
% 0.25/0.45 => ( quantDeMorgan4
% 0.25/0.45 => ( prop2setI
% 0.25/0.45 => ( prop2set2propI
% 0.25/0.45 => ( notdexE
% 0.25/0.45 => ( notdallE
% 0.25/0.45 => ( exuI1
% 0.25/0.45 => ( exuI3
% 0.25/0.45 => ( exuI2
% 0.25/0.45 => ( inCongP
% 0.25/0.45 => ( in__Cong
% 0.25/0.45 => ( exuE3u
% 0.25/0.45 => ( exu__Cong
% 0.25/0.45 => ( emptyset__Cong
% 0.25/0.45 => ( setadjoin__Cong
% 0.25/0.45 => ( powerset__Cong
% 0.25/0.45 => ( setunion__Cong
% 0.25/0.45 => ( omega__Cong
% 0.25/0.45 => ( exuEu
% 0.25/0.45 => ( descr__Cong
% 0.25/0.45 => ( dsetconstr__Cong
% 0.25/0.45 => ( subsetI1
% 0.25/0.45 => ( eqimpsubset2
% 0.25/0.45 => ( eqimpsubset1
% 0.25/0.45 => ( subsetI2
% 0.25/0.45 => ( emptysetsubset
% 0.25/0.45 => ( subsetE
% 0.25/0.45 => ( subsetE2
% 0.25/0.45 => ( notsubsetI
% 0.25/0.45 => ( notequalI1
% 0.25/0.45 => ( notequalI2
% 0.25/0.45 => ( subsetRefl
% 0.25/0.45 => ( subsetTrans
% 0.25/0.45 => ( setadjoinSub
% 0.25/0.45 => ( setadjoinSub2
% 0.25/0.45 => ( subset2powerset
% 0.25/0.45 => ( setextsub
% 0.25/0.45 => ( subsetemptysetimpeq
% 0.25/0.45 => ( powersetI1
% 0.25/0.45 => ( powersetE1
% 0.25/0.45 => ( inPowerset
% 0.25/0.45 => ( powersetsubset
% 0.25/0.45 => ( sepInPowerset
% 0.25/0.45 => ( sepSubset
% 0.25/0.45 => ( binunionIL
% 0.25/0.45 => ( upairset2IR
% 0.25/0.45 => ( binunionIR
% 0.25/0.45 => ( binunionEcases
% 0.25/0.45 => ( binunionE
% 0.25/0.45 => ( binunionLsub
% 0.25/0.45 => ( binunionRsub
% 0.25/0.45 => ( binintersectI
% 0.25/0.45 => ( binintersectSubset5
% 0.25/0.45 => ( binintersectEL
% 0.25/0.45 => ( binintersectLsub
% 0.25/0.45 => ( binintersectSubset2
% 0.25/0.45 => ( binintersectSubset3
% 0.25/0.45 => ( binintersectER
% 0.25/0.45 => ( disjointsetsI1
% 0.25/0.45 => ( binintersectRsub
% 0.25/0.45 => ( binintersectSubset4
% 0.25/0.45 => ( binintersectSubset1
% 0.25/0.45 => ( bs114d
% 0.25/0.45 => ( setminusI
% 0.25/0.45 => ( setminusEL
% 0.25/0.45 => ( setminusER
% 0.25/0.45 => ( setminusSubset2
% 0.25/0.45 => ( setminusERneg
% 0.25/0.45 => ( setminusELneg
% 0.25/0.45 => ( setminusILneg
% 0.25/0.45 => ( setminusIRneg
% 0.25/0.45 => ( setminusLsub
% 0.25/0.45 => ( setminusSubset1
% 0.25/0.45 => ( symdiffE
% 0.25/0.45 => ( symdiffI1
% 0.25/0.45 => ( symdiffI2
% 0.25/0.45 => ( symdiffIneg1
% 0.25/0.45 => ( symdiffIneg2
% 0.25/0.45 => ( secondinupair
% 0.25/0.45 => ( setukpairIL
% 0.25/0.45 => ( setukpairIR
% 0.25/0.45 => ( kpairiskpair
% 0.25/0.45 => ( kpairp
% 0.25/0.45 => ( singletonsubset
% 0.25/0.45 => ( singletoninpowerset
% 0.25/0.45 => ( singletoninpowunion
% 0.25/0.45 => ( upairset2E
% 0.25/0.45 => ( upairsubunion
% 0.25/0.45 => ( upairinpowunion
% 0.25/0.45 => ( ubforcartprodlem1
% 0.25/0.45 => ( ubforcartprodlem2
% 0.25/0.45 => ( ubforcartprodlem3
% 0.25/0.45 => ( cartprodpairin
% 0.25/0.45 => ( cartprodmempair1
% 0.25/0.45 => ( cartprodmempair
% 0.25/0.45 => ( setunionE2
% 0.25/0.45 => ( setunionsingleton1
% 0.25/0.45 => ( setunionsingleton2
% 0.25/0.45 => ( setunionsingleton
% 0.25/0.45 => ( singletonprop
% 0.25/0.45 => ( ex1E1
% 0.25/0.45 => ( ex1I
% 0.25/0.45 => ( ex1I2
% 0.25/0.45 => ( singletonsuniq
% 0.25/0.45 => ( setukpairinjL1
% 0.25/0.45 => ( kfstsingleton
% 0.25/0.45 => ( theprop
% 0.25/0.45 => ( kfstpairEq
% 0.25/0.45 => ( cartprodfstin
% 0.25/0.45 => ( setukpairinjL2
% 0.25/0.45 => ( setukpairinjL
% 0.25/0.45 => ( setukpairinjR11
% 0.25/0.45 => ( setukpairinjR12
% 0.25/0.45 => ( setukpairinjR1
% 0.25/0.45 => ( upairequniteq
% 0.25/0.45 => ( setukpairinjR2
% 0.25/0.45 => ( setukpairinjR
% 0.25/0.45 => ( ksndsingleton
% 0.25/0.45 => ( ksndpairEq
% 0.25/0.45 => ( kpairsurjEq
% 0.25/0.45 => ( cartprodsndin
% 0.25/0.45 => ( cartprodpairmemEL
% 0.25/0.45 => ( cartprodpairmemER
% 0.25/0.45 => ( cartprodmempaircEq
% 0.25/0.45 => ( cartprodfstpairEq
% 0.25/0.45 => ( cartprodsndpairEq
% 0.25/0.45 => ( cartprodpairsurjEq
% 0.25/0.45 => ( dpsetconstrI
% 0.25/0.45 => ( dpsetconstrSub
% 0.25/0.45 => ( setOfPairsIsBReln
% 0.25/0.45 => ( dpsetconstrERa
% 0.25/0.45 => ( dpsetconstrEL1
% 0.25/0.45 => ( dpsetconstrEL2
% 0.25/0.45 => ( dpsetconstrER
% 0.25/0.45 => ( funcImageSingleton
% 0.25/0.45 => ( apProp
% 0.25/0.45 => ( app
% 0.25/0.45 => ( infuncsetfunc
% 0.25/0.45 => ( ap2p
% 0.25/0.45 => ( funcinfuncset
% 0.25/0.45 => ( lamProp
% 0.25/0.45 => ( lamp
% 0.25/0.45 => ( lam2p
% 0.25/0.45 => ( brelnall1
% 0.25/0.45 => ( brelnall2
% 0.25/0.45 => ( ex1E2
% 0.25/0.45 => ( funcGraphProp1
% 0.25/0.45 => ( funcGraphProp3
% 0.25/0.45 => ( funcGraphProp2
% 0.25/0.45 => ( funcextLem
% 0.25/0.45 => ( funcGraphProp4
% 0.25/0.45 => ( subbreln
% 0.25/0.45 => ( eqbreln
% 0.25/0.45 => ( funcext
% 0.25/0.45 => ( funcext2
% 0.25/0.45 => ( ap2apEq1
% 0.25/0.45 => ( ap2apEq2
% 0.25/0.45 => ( beta1
% 0.25/0.45 => ( eta1
% 0.25/0.45 => ( lam2lamEq
% 0.25/0.45 => ( beta2
% 0.25/0.45 => ( eta2
% 0.25/0.45 => ( iffalseProp1
% 0.25/0.45 => ( iffalseProp2
% 0.25/0.45 => ( iftrueProp1
% 0.25/0.45 => ( iftrueProp2
% 0.25/0.45 => ( ifSingleton
% 0.25/0.45 => ( ifp
% 0.25/0.45 => ( theeq
% 0.25/0.45 => ( iftrue
% 0.25/0.45 => ( iffalse
% 0.25/0.45 => ( iftrueorfalse
% 0.25/0.45 => ( binintersectT_lem
% 0.25/0.45 => ( binunionT_lem
% 0.25/0.45 => ( powersetT_lem
% 0.25/0.45 => ( setminusT_lem
% 0.25/0.45 => ( complementT_lem
% 0.25/0.45 => ( setextT
% 0.25/0.45 => ( subsetTI
% 0.25/0.45 => ( powersetTI1
% 0.25/0.45 => ( powersetTE1
% 0.25/0.45 => ( complementTI1
% 0.25/0.45 => ( complementTE1
% 0.25/0.45 => ( binintersectTELcontra
% 0.25/0.45 => ( binintersectTERcontra
% 0.25/0.45 => ( contrasubsetT
% 0.25/0.45 => ( contrasubsetT1
% 0.25/0.45 => ( contrasubsetT2
% 0.25/0.45 => ( contrasubsetT3
% 0.25/0.45 => ! [A: $i,X: $i] :
% 0.25/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.25/0.45 => ! [Xx: $i] :
% 0.25/0.45 ( ( in @ Xx @ A )
% 0.25/0.45 => ( ( in @ Xx @ X )
% 0.25/0.45 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 %------------------------------------------------------------------------------
% 0.25/0.45 ------------------------
% 2.54/1.48 ---- Embedded in THF ---
% 2.54/1.48 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.54/1.48 %%% Generated on Tue Feb 24 12:31:57 EST 2026
% 2.54/1.48 %%% using '$$dhol' embedding, version 1.3.0.
% 2.54/1.48 %%% Logic specification used:
% 2.54/1.48 %%% thf(spec, logic, $$dhol).
% 2.54/1.48
% 2.54/1.48 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.4HrzPZvLLL/DTF2DTF17732.p
% See solution above
% 2.54/1.48 ------------------------
% 2.77/1.51 ---- Cleaned THF ---
% 2.77/1.51 thf(in_type,type,
% 2.77/1.51 in: $i > $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(exu_type,type,
% 2.77/1.51 exu: ( $i > $o ) > $o ).
% 2.77/1.51
% 2.77/1.51 thf(exu,definition,
% 2.77/1.51 ( exu
% 2.77/1.51 = ( ^ [Xphi: $i > $o] :
% 2.77/1.51 ? [Xx: $i] :
% 2.77/1.51 ( ( Xphi @ Xx )
% 2.77/1.51 & ! [Xy: $i] :
% 2.77/1.51 ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setextAx_type,type,
% 2.77/1.51 setextAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setextAx,definition,
% 2.77/1.51 ( setextAx
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 <=> ( in @ Xx @ B ) )
% 2.77/1.51 => ( A = B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptyset_type,type,
% 2.77/1.51 emptyset: $i ).
% 2.77/1.51
% 2.77/1.51 thf(emptysetAx_type,type,
% 2.77/1.51 emptysetAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptysetAx,definition,
% 2.77/1.51 ( emptysetAx
% 2.77/1.51 = ( ! [Xx: $i] :
% 2.77/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoin_type,type,
% 2.77/1.51 setadjoin: $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinAx_type,type,
% 2.77/1.51 setadjoinAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinAx,definition,
% 2.77/1.51 ( setadjoinAx
% 2.77/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.77/1.51 <=> ( ( Xy = Xx )
% 2.77/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powerset_type,type,
% 2.77/1.51 powerset: $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(powersetAx_type,type,
% 2.77/1.51 powersetAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powersetAx,definition,
% 2.77/1.51 ( powersetAx
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.77/1.51 <=> ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setunion_type,type,
% 2.77/1.51 setunion: $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(setunionAx_type,type,
% 2.77/1.51 setunionAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setunionAx,definition,
% 2.77/1.51 ( setunionAx
% 2.77/1.51 = ( ! [A: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.77/1.51 <=> ? [B: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 & ( in @ B @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(omega_type,type,
% 2.77/1.51 omega: $i ).
% 2.77/1.51
% 2.77/1.51 thf(omega0Ax_type,type,
% 2.77/1.51 omega0Ax: $o ).
% 2.77/1.51
% 2.77/1.51 thf(omega0Ax,definition,
% 2.77/1.51 ( omega0Ax
% 2.77/1.51 = ( in @ emptyset @ omega ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(omegaSAx_type,type,
% 2.77/1.51 omegaSAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(omegaSAx,definition,
% 2.77/1.51 ( omegaSAx
% 2.77/1.51 = ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ omega )
% 2.77/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(omegaIndAx_type,type,
% 2.77/1.51 omegaIndAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(omegaIndAx,definition,
% 2.77/1.51 ( omegaIndAx
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ( ( ( in @ emptyset @ A )
% 2.77/1.51 & ! [Xx: $i] :
% 2.77/1.51 ( ( ( in @ Xx @ omega )
% 2.77/1.51 & ( in @ Xx @ A ) )
% 2.77/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ omega )
% 2.77/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(replAx_type,type,
% 2.77/1.51 replAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(replAx,definition,
% 2.77/1.51 ( replAx
% 2.77/1.51 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( exu
% 2.77/1.51 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.77/1.51 => ? [B: $i] :
% 2.77/1.51 ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 <=> ? [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(foundationAx_type,type,
% 2.77/1.51 foundationAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(foundationAx,definition,
% 2.77/1.51 ( foundationAx
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.77/1.51 => ? [B: $i] :
% 2.77/1.51 ( ( in @ B @ A )
% 2.77/1.51 & ~ ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(wellorderingAx_type,type,
% 2.77/1.51 wellorderingAx: $o ).
% 2.77/1.51
% 2.77/1.51 thf(wellorderingAx,definition,
% 2.77/1.51 ( wellorderingAx
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ? [B: $i] :
% 2.77/1.51 ( ! [C: $i] :
% 2.77/1.51 ( ( in @ C @ B )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ C )
% 2.77/1.51 => ( in @ Xx @ A ) ) )
% 2.77/1.51 & ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( ( in @ Xx @ A )
% 2.77/1.51 & ( in @ Xy @ A ) )
% 2.77/1.51 => ( ! [C: $i] :
% 2.77/1.51 ( ( in @ C @ B )
% 2.77/1.51 => ( ( in @ Xx @ C )
% 2.77/1.51 <=> ( in @ Xy @ C ) ) )
% 2.77/1.51 => ( Xx = Xy ) ) )
% 2.77/1.51 & ! [C: $i,D: $i] :
% 2.77/1.51 ( ( ( in @ C @ B )
% 2.77/1.51 & ( in @ D @ B ) )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ C )
% 2.77/1.51 => ( in @ Xx @ D ) )
% 2.77/1.51 | ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ D )
% 2.77/1.51 => ( in @ Xx @ C ) ) ) )
% 2.77/1.51 & ! [C: $i] :
% 2.77/1.51 ( ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ C )
% 2.77/1.51 => ( in @ Xx @ A ) )
% 2.77/1.51 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.77/1.51 => ? [D: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ D @ B )
% 2.77/1.51 & ( in @ Xx @ C )
% 2.77/1.51 & ~ ? [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ D )
% 2.77/1.51 & ( in @ Xy @ C ) )
% 2.77/1.51 & ! [E: $i] :
% 2.77/1.51 ( ( in @ E @ B )
% 2.77/1.51 => ( ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ E )
% 2.77/1.51 => ( in @ Xy @ D ) )
% 2.77/1.51 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(descr_type,type,
% 2.77/1.51 descr: ( $i > $o ) > $i ).
% 2.77/1.51
% 2.77/1.51 thf(descrp_type,type,
% 2.77/1.51 descrp: $o ).
% 2.77/1.51
% 2.77/1.51 thf(descrp,definition,
% 2.77/1.51 ( descrp
% 2.77/1.51 = ( ! [Xphi: $i > $o] :
% 2.77/1.51 ( ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 => ( Xphi
% 2.77/1.51 @ ( descr
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dsetconstr_type,type,
% 2.77/1.51 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.77/1.51
% 2.77/1.51 thf(dsetconstrI_type,type,
% 2.77/1.51 dsetconstrI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dsetconstrI,definition,
% 2.77/1.51 ( dsetconstrI
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( Xphi @ Xx )
% 2.77/1.51 => ( in @ Xx
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dsetconstrEL_type,type,
% 2.77/1.51 dsetconstrEL: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dsetconstrEL,definition,
% 2.77/1.51 ( dsetconstrEL
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.77/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dsetconstrER_type,type,
% 2.77/1.51 dsetconstrER: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dsetconstrER,definition,
% 2.77/1.51 ( dsetconstrER
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.77/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(exuE1_type,type,
% 2.77/1.51 exuE1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(exuE1,definition,
% 2.77/1.51 ( exuE1
% 2.77/1.51 = ( ! [Xphi: $i > $o] :
% 2.77/1.51 ( ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 => ? [Xx: $i] :
% 2.77/1.51 ( ( Xphi @ Xx )
% 2.77/1.51 & ! [Xy: $i] :
% 2.77/1.51 ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(prop2set_type,type,
% 2.77/1.51 prop2set: $o > $i ).
% 2.77/1.51
% 2.77/1.51 thf(prop2setE_type,type,
% 2.77/1.51 prop2setE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(prop2setE,definition,
% 2.77/1.51 ( prop2setE
% 2.77/1.51 = ( ! [Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.77/1.51 => Xphi ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptysetE_type,type,
% 2.77/1.51 emptysetE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptysetE,definition,
% 2.77/1.51 ( emptysetE
% 2.77/1.51 = ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ emptyset )
% 2.77/1.51 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptysetimpfalse_type,type,
% 2.77/1.51 emptysetimpfalse: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptysetimpfalse,definition,
% 2.77/1.51 ( emptysetimpfalse
% 2.77/1.51 = ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ emptyset )
% 2.77/1.51 => $false ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(notinemptyset_type,type,
% 2.77/1.51 notinemptyset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(notinemptyset,definition,
% 2.77/1.51 ( notinemptyset
% 2.77/1.51 = ( ! [Xx: $i] :
% 2.77/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(exuE3e_type,type,
% 2.77/1.51 exuE3e: $o ).
% 2.77/1.51
% 2.77/1.51 thf(exuE3e,definition,
% 2.77/1.51 ( exuE3e
% 2.77/1.51 = ( ! [Xphi: $i > $o] :
% 2.77/1.51 ( ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setext_type,type,
% 2.77/1.51 setext: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setext,definition,
% 2.77/1.51 ( setext
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ Xx @ B ) )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 => ( in @ Xx @ A ) )
% 2.77/1.51 => ( A = B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptyI_type,type,
% 2.77/1.51 emptyI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptyI,definition,
% 2.77/1.51 ( emptyI
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ~ ( in @ Xx @ A )
% 2.77/1.51 => ( A = emptyset ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(noeltsimpempty_type,type,
% 2.77/1.51 noeltsimpempty: $o ).
% 2.77/1.51
% 2.77/1.51 thf(noeltsimpempty,definition,
% 2.77/1.51 ( noeltsimpempty
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ~ ( in @ Xx @ A )
% 2.77/1.51 => ( A = emptyset ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setbeta_type,type,
% 2.77/1.51 setbeta: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setbeta,definition,
% 2.77/1.51 ( setbeta
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.77/1.51 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(nonempty_type,type,
% 2.77/1.51 nonempty: $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(nonempty,definition,
% 2.77/1.51 ( nonempty
% 2.77/1.51 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(nonemptyE1_type,type,
% 2.77/1.51 nonemptyE1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(nonemptyE1,definition,
% 2.77/1.51 ( nonemptyE1
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ( ( nonempty @ A )
% 2.77/1.51 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(nonemptyI_type,type,
% 2.77/1.51 nonemptyI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(nonemptyI,definition,
% 2.77/1.51 ( nonemptyI
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( Xphi @ Xx )
% 2.77/1.51 => ( nonempty
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(nonemptyI1_type,type,
% 2.77/1.51 nonemptyI1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(nonemptyI1,definition,
% 2.77/1.51 ( nonemptyI1
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.77/1.51 => ( nonempty @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinIL_type,type,
% 2.77/1.51 setadjoinIL: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinIL,definition,
% 2.77/1.51 ( setadjoinIL
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptyinunitempty_type,type,
% 2.77/1.51 emptyinunitempty: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptyinunitempty,definition,
% 2.77/1.51 ( emptyinunitempty
% 2.77/1.51 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinIR_type,type,
% 2.77/1.51 setadjoinIR: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinIR,definition,
% 2.77/1.51 ( setadjoinIR
% 2.77/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinE_type,type,
% 2.77/1.51 setadjoinE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinE,definition,
% 2.77/1.51 ( setadjoinE
% 2.77/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.77/1.51 => ! [Xphi: $o] :
% 2.77/1.51 ( ( ( Xy = Xx )
% 2.77/1.51 => Xphi )
% 2.77/1.51 => ( ( ( in @ Xy @ A )
% 2.77/1.51 => Xphi )
% 2.77/1.51 => Xphi ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinOr_type,type,
% 2.77/1.51 setadjoinOr: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinOr,definition,
% 2.77/1.51 ( setadjoinOr
% 2.77/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.77/1.51 => ( ( Xy = Xx )
% 2.77/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setoftrueEq_type,type,
% 2.77/1.51 setoftrueEq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setoftrueEq,definition,
% 2.77/1.51 ( setoftrueEq
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ( ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xx: $i] : $true )
% 2.77/1.51 = A ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powersetI_type,type,
% 2.77/1.51 powersetI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powersetI,definition,
% 2.77/1.51 ( powersetI
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 => ( in @ Xx @ A ) )
% 2.77/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptyinPowerset_type,type,
% 2.77/1.51 emptyinPowerset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptyinPowerset,definition,
% 2.77/1.51 ( emptyinPowerset
% 2.77/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptyInPowerset_type,type,
% 2.77/1.51 emptyInPowerset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptyInPowerset,definition,
% 2.77/1.51 ( emptyInPowerset
% 2.77/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powersetE_type,type,
% 2.77/1.51 powersetE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powersetE,definition,
% 2.77/1.51 ( powersetE
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.77/1.51 => ( ( in @ Xx @ B )
% 2.77/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setunionI_type,type,
% 2.77/1.51 setunionI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setunionI,definition,
% 2.77/1.51 ( setunionI
% 2.77/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 => ( ( in @ B @ A )
% 2.77/1.51 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setunionE_type,type,
% 2.77/1.51 setunionE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setunionE,definition,
% 2.77/1.51 ( setunionE
% 2.77/1.51 = ( ! [A: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.77/1.51 => ! [Xphi: $o] :
% 2.77/1.51 ( ! [B: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 => ( ( in @ B @ A )
% 2.77/1.51 => Xphi ) )
% 2.77/1.51 => Xphi ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subPowSU_type,type,
% 2.77/1.51 subPowSU: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subPowSU,definition,
% 2.77/1.51 ( subPowSU
% 2.77/1.51 = ( ! [A: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(exuE2_type,type,
% 2.77/1.51 exuE2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(exuE2,definition,
% 2.77/1.51 ( exuE2
% 2.77/1.51 = ( ! [Xphi: $i > $o] :
% 2.77/1.51 ( ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 => ? [Xx: $i] :
% 2.77/1.51 ! [Xy: $i] :
% 2.77/1.51 ( ( Xphi @ Xy )
% 2.77/1.51 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(nonemptyImpWitness_type,type,
% 2.77/1.51 nonemptyImpWitness: $o ).
% 2.77/1.51
% 2.77/1.51 thf(nonemptyImpWitness,definition,
% 2.77/1.51 ( nonemptyImpWitness
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ( ( nonempty @ A )
% 2.77/1.51 => ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & $true ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(uniqinunit_type,type,
% 2.77/1.51 uniqinunit: $o ).
% 2.77/1.51
% 2.77/1.51 thf(uniqinunit,definition,
% 2.77/1.51 ( uniqinunit
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(notinsingleton_type,type,
% 2.77/1.51 notinsingleton: $o ).
% 2.77/1.51
% 2.77/1.51 thf(notinsingleton,definition,
% 2.77/1.51 ( notinsingleton
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xx != Xy )
% 2.77/1.51 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(eqinunit_type,type,
% 2.77/1.51 eqinunit: $o ).
% 2.77/1.51
% 2.77/1.51 thf(eqinunit,definition,
% 2.77/1.51 ( eqinunit
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xx = Xy )
% 2.77/1.51 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(singletonsswitch_type,type,
% 2.77/1.51 singletonsswitch: $o ).
% 2.77/1.51
% 2.77/1.51 thf(singletonsswitch,definition,
% 2.77/1.51 ( singletonsswitch
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(upairsetE_type,type,
% 2.77/1.51 upairsetE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(upairsetE,definition,
% 2.77/1.51 ( upairsetE
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.77/1.51 => ( ( Xz = Xx )
% 2.77/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(upairsetIL_type,type,
% 2.77/1.51 upairsetIL: $o ).
% 2.77/1.51
% 2.77/1.51 thf(upairsetIL,definition,
% 2.77/1.51 ( upairsetIL
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(upairsetIR_type,type,
% 2.77/1.51 upairsetIR: $o ).
% 2.77/1.51
% 2.77/1.51 thf(upairsetIR,definition,
% 2.77/1.51 ( upairsetIR
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptyE1_type,type,
% 2.77/1.51 emptyE1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptyE1,definition,
% 2.77/1.51 ( emptyE1
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ( Xphi @ Xx ) )
% 2.77/1.51 => ( ( ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 = emptyset )
% 2.77/1.51 => $false ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(vacuousDall_type,type,
% 2.77/1.51 vacuousDall: $o ).
% 2.77/1.51
% 2.77/1.51 thf(vacuousDall,definition,
% 2.77/1.51 ( vacuousDall
% 2.77/1.51 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ emptyset )
% 2.77/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(quantDeMorgan1_type,type,
% 2.77/1.51 quantDeMorgan1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(quantDeMorgan1,definition,
% 2.77/1.51 ( quantDeMorgan1
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ~ ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( Xphi @ Xx ) )
% 2.77/1.51 => ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(quantDeMorgan2_type,type,
% 2.77/1.51 quantDeMorgan2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(quantDeMorgan2,definition,
% 2.77/1.51 ( quantDeMorgan2
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ~ ( Xphi @ Xx ) )
% 2.77/1.51 => ~ ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(quantDeMorgan3_type,type,
% 2.77/1.51 quantDeMorgan3: $o ).
% 2.77/1.51
% 2.77/1.51 thf(quantDeMorgan3,definition,
% 2.77/1.51 ( quantDeMorgan3
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ~ ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ( Xphi @ Xx ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(quantDeMorgan4_type,type,
% 2.77/1.51 quantDeMorgan4: $o ).
% 2.77/1.51
% 2.77/1.51 thf(quantDeMorgan4,definition,
% 2.77/1.51 ( quantDeMorgan4
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ~ ( Xphi @ Xx ) )
% 2.77/1.51 => ~ ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(prop2setI_type,type,
% 2.77/1.51 prop2setI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(prop2setI,definition,
% 2.77/1.51 ( prop2setI
% 2.77/1.51 = ( ! [Xphi: $o] :
% 2.77/1.51 ( Xphi
% 2.77/1.51 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(set2prop_type,type,
% 2.77/1.51 set2prop: $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(prop2set2propI_type,type,
% 2.77/1.51 prop2set2propI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(prop2set2propI,definition,
% 2.77/1.51 ( prop2set2propI
% 2.77/1.51 = ( ! [Xphi: $o] :
% 2.77/1.51 ( Xphi
% 2.77/1.51 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(notdexE_type,type,
% 2.77/1.51 notdexE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(notdexE,definition,
% 2.77/1.51 ( notdexE
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ~ ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ( Xphi @ Xx ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(notdallE_type,type,
% 2.77/1.51 notdallE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(notdallE,definition,
% 2.77/1.51 ( notdallE
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ~ ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( Xphi @ Xx ) )
% 2.77/1.51 => ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(exuI1_type,type,
% 2.77/1.51 exuI1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(exuI1,definition,
% 2.77/1.51 ( exuI1
% 2.77/1.51 = ( ! [Xphi: $i > $o] :
% 2.77/1.51 ( ? [Xx: $i] :
% 2.77/1.51 ( ( Xphi @ Xx )
% 2.77/1.51 & ! [Xy: $i] :
% 2.77/1.51 ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xx = Xy ) ) )
% 2.77/1.51 => ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(exuI3_type,type,
% 2.77/1.51 exuI3: $o ).
% 2.77/1.51
% 2.77/1.51 thf(exuI3,definition,
% 2.77/1.51 ( exuI3
% 2.77/1.51 = ( ! [Xphi: $i > $o] :
% 2.77/1.51 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.77/1.51 => ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xphi @ Xx )
% 2.77/1.51 => ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xx = Xy ) ) )
% 2.77/1.51 => ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(exuI2_type,type,
% 2.77/1.51 exuI2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(exuI2,definition,
% 2.77/1.51 ( exuI2
% 2.77/1.51 = ( ! [Xphi: $i > $o] :
% 2.77/1.51 ( ? [Xx: $i] :
% 2.77/1.51 ! [Xy: $i] :
% 2.77/1.51 ( ( Xphi @ Xy )
% 2.77/1.51 <=> ( Xy = Xx ) )
% 2.77/1.51 => ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(inCongP_type,type,
% 2.77/1.51 inCongP: $o ).
% 2.77/1.51
% 2.77/1.51 thf(inCongP,definition,
% 2.77/1.51 ( inCongP
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( A = B )
% 2.77/1.51 => ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xx = Xy )
% 2.77/1.51 => ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(in__Cong_type,type,
% 2.77/1.51 in__Cong: $o ).
% 2.77/1.51
% 2.77/1.51 thf(in__Cong,definition,
% 2.77/1.51 ( in__Cong
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( A = B )
% 2.77/1.51 => ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xx = Xy )
% 2.77/1.51 => ( ( in @ Xx @ A )
% 2.77/1.51 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(exuE3u_type,type,
% 2.77/1.51 exuE3u: $o ).
% 2.77/1.51
% 2.77/1.51 thf(exuE3u,definition,
% 2.77/1.51 ( exuE3u
% 2.77/1.51 = ( ! [Xphi: $i > $o] :
% 2.77/1.51 ( ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 => ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xphi @ Xx )
% 2.77/1.51 => ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(exu__Cong_type,type,
% 2.77/1.51 exu__Cong: $o ).
% 2.77/1.51
% 2.77/1.51 thf(exu__Cong,definition,
% 2.77/1.51 ( exu__Cong
% 2.77/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.77/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xx = Xy )
% 2.77/1.51 => ( ( Xphi @ Xx )
% 2.77/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.77/1.51 => ( ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 <=> ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptyset__Cong_type,type,
% 2.77/1.51 emptyset__Cong: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptyset__Cong,definition,
% 2.77/1.51 ( emptyset__Cong
% 2.77/1.51 = ( emptyset = emptyset ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoin__Cong_type,type,
% 2.77/1.51 setadjoin__Cong: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoin__Cong,definition,
% 2.77/1.51 ( setadjoin__Cong
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xx = Xy )
% 2.77/1.51 => ! [Xz: $i,Xu: $i] :
% 2.77/1.51 ( ( Xz = Xu )
% 2.77/1.51 => ( ( setadjoin @ Xx @ Xz )
% 2.77/1.51 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powerset__Cong_type,type,
% 2.77/1.51 powerset__Cong: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powerset__Cong,definition,
% 2.77/1.51 ( powerset__Cong
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( A = B )
% 2.77/1.51 => ( ( powerset @ A )
% 2.77/1.51 = ( powerset @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setunion__Cong_type,type,
% 2.77/1.51 setunion__Cong: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setunion__Cong,definition,
% 2.77/1.51 ( setunion__Cong
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( A = B )
% 2.77/1.51 => ( ( setunion @ A )
% 2.77/1.51 = ( setunion @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(omega__Cong_type,type,
% 2.77/1.51 omega__Cong: $o ).
% 2.77/1.51
% 2.77/1.51 thf(omega__Cong,definition,
% 2.77/1.51 ( omega__Cong
% 2.77/1.51 = ( omega = omega ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(exuEu_type,type,
% 2.77/1.51 exuEu: $o ).
% 2.77/1.51
% 2.77/1.51 thf(exuEu,definition,
% 2.77/1.51 ( exuEu
% 2.77/1.51 = ( ! [Xphi: $i > $o] :
% 2.77/1.51 ( ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 => ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xphi @ Xx )
% 2.77/1.51 => ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(descr__Cong_type,type,
% 2.77/1.51 descr__Cong: $o ).
% 2.77/1.51
% 2.77/1.51 thf(descr__Cong,definition,
% 2.77/1.51 ( descr__Cong
% 2.77/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.77/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xx = Xy )
% 2.77/1.51 => ( ( Xphi @ Xx )
% 2.77/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.77/1.51 => ( ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 => ( ( exu
% 2.77/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.77/1.51 => ( ( descr
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 = ( descr
% 2.77/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dsetconstr__Cong_type,type,
% 2.77/1.51 dsetconstr__Cong: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dsetconstr__Cong,definition,
% 2.77/1.51 ( dsetconstr__Cong
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( A = B )
% 2.77/1.51 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( Xx = Xy )
% 2.77/1.51 => ( ( Xphi @ Xx )
% 2.77/1.51 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.77/1.51 => ( ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 = ( dsetconstr @ B
% 2.77/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subset_type,type,
% 2.77/1.51 subset: $i > $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(disjoint_type,type,
% 2.77/1.51 disjoint: $i > $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(setsmeet_type,type,
% 2.77/1.51 setsmeet: $i > $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(subsetI1_type,type,
% 2.77/1.51 subsetI1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subsetI1,definition,
% 2.77/1.51 ( subsetI1
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ Xx @ B ) )
% 2.77/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(eqimpsubset2_type,type,
% 2.77/1.51 eqimpsubset2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(eqimpsubset2,definition,
% 2.77/1.51 ( eqimpsubset2
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( A = B )
% 2.77/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(eqimpsubset1_type,type,
% 2.77/1.51 eqimpsubset1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(eqimpsubset1,definition,
% 2.77/1.51 ( eqimpsubset1
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( A = B )
% 2.77/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subsetI2_type,type,
% 2.77/1.51 subsetI2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subsetI2,definition,
% 2.77/1.51 ( subsetI2
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ Xx @ B ) )
% 2.77/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(emptysetsubset_type,type,
% 2.77/1.51 emptysetsubset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(emptysetsubset,definition,
% 2.77/1.51 ( emptysetsubset
% 2.77/1.51 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subsetE_type,type,
% 2.77/1.51 subsetE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subsetE,definition,
% 2.77/1.51 ( subsetE
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( subset @ A @ B )
% 2.77/1.51 => ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subsetE2_type,type,
% 2.77/1.51 subsetE2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subsetE2,definition,
% 2.77/1.51 ( subsetE2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( subset @ A @ B )
% 2.77/1.51 => ( ~ ( in @ Xx @ B )
% 2.77/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(notsubsetI_type,type,
% 2.77/1.51 notsubsetI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(notsubsetI,definition,
% 2.77/1.51 ( notsubsetI
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ~ ( in @ Xx @ B )
% 2.77/1.51 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(notequalI1_type,type,
% 2.77/1.51 notequalI1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(notequalI1,definition,
% 2.77/1.51 ( notequalI1
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ~ ( subset @ A @ B )
% 2.77/1.51 => ( A != B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(notequalI2_type,type,
% 2.77/1.51 notequalI2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(notequalI2,definition,
% 2.77/1.51 ( notequalI2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ~ ( in @ Xx @ B )
% 2.77/1.51 => ( A != B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subsetRefl_type,type,
% 2.77/1.51 subsetRefl: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subsetRefl,definition,
% 2.77/1.51 ( subsetRefl
% 2.77/1.51 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subsetTrans_type,type,
% 2.77/1.51 subsetTrans: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subsetTrans,definition,
% 2.77/1.51 ( subsetTrans
% 2.77/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.77/1.51 ( ( subset @ A @ B )
% 2.77/1.51 => ( ( subset @ B @ C )
% 2.77/1.51 => ( subset @ A @ C ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinSub_type,type,
% 2.77/1.51 setadjoinSub: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinSub,definition,
% 2.77/1.51 ( setadjoinSub
% 2.77/1.51 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinSub2_type,type,
% 2.77/1.51 setadjoinSub2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setadjoinSub2,definition,
% 2.77/1.51 ( setadjoinSub2
% 2.77/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.77/1.51 ( ( subset @ A @ B )
% 2.77/1.51 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subset2powerset_type,type,
% 2.77/1.51 subset2powerset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subset2powerset,definition,
% 2.77/1.51 ( subset2powerset
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( subset @ A @ B )
% 2.77/1.51 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setextsub_type,type,
% 2.77/1.51 setextsub: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setextsub,definition,
% 2.77/1.51 ( setextsub
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( subset @ A @ B )
% 2.77/1.51 => ( ( subset @ B @ A )
% 2.77/1.51 => ( A = B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subsetemptysetimpeq_type,type,
% 2.77/1.51 subsetemptysetimpeq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subsetemptysetimpeq,definition,
% 2.77/1.51 ( subsetemptysetimpeq
% 2.77/1.51 = ( ! [A: $i] :
% 2.77/1.51 ( ( subset @ A @ emptyset )
% 2.77/1.51 => ( A = emptyset ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powersetI1_type,type,
% 2.77/1.51 powersetI1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powersetI1,definition,
% 2.77/1.51 ( powersetI1
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( subset @ B @ A )
% 2.77/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powersetE1_type,type,
% 2.77/1.51 powersetE1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powersetE1,definition,
% 2.77/1.51 ( powersetE1
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.77/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(inPowerset_type,type,
% 2.77/1.51 inPowerset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(inPowerset,definition,
% 2.77/1.51 ( inPowerset
% 2.77/1.51 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powersetsubset_type,type,
% 2.77/1.51 powersetsubset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powersetsubset,definition,
% 2.77/1.51 ( powersetsubset
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( subset @ A @ B )
% 2.77/1.51 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(sepInPowerset_type,type,
% 2.77/1.51 sepInPowerset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(sepInPowerset,definition,
% 2.77/1.51 ( sepInPowerset
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( in
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 @ ( powerset @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(sepSubset_type,type,
% 2.77/1.51 sepSubset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(sepSubset,definition,
% 2.77/1.51 ( sepSubset
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( subset
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 @ A ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binunion_type,type,
% 2.77/1.51 binunion: $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(binunionIL_type,type,
% 2.77/1.51 binunionIL: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binunionIL,definition,
% 2.77/1.51 ( binunionIL
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(upairset2IR_type,type,
% 2.77/1.51 upairset2IR: $o ).
% 2.77/1.51
% 2.77/1.51 thf(upairset2IR,definition,
% 2.77/1.51 ( upairset2IR
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binunionIR_type,type,
% 2.77/1.51 binunionIR: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binunionIR,definition,
% 2.77/1.51 ( binunionIR
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binunionEcases_type,type,
% 2.77/1.51 binunionEcases: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binunionEcases,definition,
% 2.77/1.51 ( binunionEcases
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.77/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.77/1.51 => ( ( ( in @ Xx @ A )
% 2.77/1.51 => Xphi )
% 2.77/1.51 => ( ( ( in @ Xx @ B )
% 2.77/1.51 => Xphi )
% 2.77/1.51 => Xphi ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binunionE_type,type,
% 2.77/1.51 binunionE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binunionE,definition,
% 2.77/1.51 ( binunionE
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.77/1.51 => ( ( in @ Xx @ A )
% 2.77/1.51 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binunionLsub_type,type,
% 2.77/1.51 binunionLsub: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binunionLsub,definition,
% 2.77/1.51 ( binunionLsub
% 2.77/1.51 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binunionRsub_type,type,
% 2.77/1.51 binunionRsub: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binunionRsub,definition,
% 2.77/1.51 ( binunionRsub
% 2.77/1.51 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersect_type,type,
% 2.77/1.51 binintersect: $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectI_type,type,
% 2.77/1.51 binintersectI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectI,definition,
% 2.77/1.51 ( binintersectI
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx @ B )
% 2.77/1.51 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset5_type,type,
% 2.77/1.51 binintersectSubset5: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset5,definition,
% 2.77/1.51 ( binintersectSubset5
% 2.77/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.77/1.51 ( ( subset @ C @ A )
% 2.77/1.51 => ( ( subset @ C @ B )
% 2.77/1.51 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectEL_type,type,
% 2.77/1.51 binintersectEL: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectEL,definition,
% 2.77/1.51 ( binintersectEL
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.77/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectLsub_type,type,
% 2.77/1.51 binintersectLsub: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectLsub,definition,
% 2.77/1.51 ( binintersectLsub
% 2.77/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset2_type,type,
% 2.77/1.51 binintersectSubset2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset2,definition,
% 2.77/1.51 ( binintersectSubset2
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( subset @ A @ B )
% 2.77/1.51 => ( ( binintersect @ A @ B )
% 2.77/1.51 = A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset3_type,type,
% 2.77/1.51 binintersectSubset3: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset3,definition,
% 2.77/1.51 ( binintersectSubset3
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( ( binintersect @ A @ B )
% 2.77/1.51 = B )
% 2.77/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectER_type,type,
% 2.77/1.51 binintersectER: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectER,definition,
% 2.77/1.51 ( binintersectER
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.77/1.51 => ( in @ Xx @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(disjointsetsI1_type,type,
% 2.77/1.51 disjointsetsI1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(disjointsetsI1,definition,
% 2.77/1.51 ( disjointsetsI1
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ~ ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ( in @ Xx @ B ) )
% 2.77/1.51 => ( ( binintersect @ A @ B )
% 2.77/1.51 = emptyset ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectRsub_type,type,
% 2.77/1.51 binintersectRsub: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectRsub,definition,
% 2.77/1.51 ( binintersectRsub
% 2.77/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset4_type,type,
% 2.77/1.51 binintersectSubset4: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset4,definition,
% 2.77/1.51 ( binintersectSubset4
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( subset @ B @ A )
% 2.77/1.51 => ( ( binintersect @ A @ B )
% 2.77/1.51 = B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset1_type,type,
% 2.77/1.51 binintersectSubset1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectSubset1,definition,
% 2.77/1.51 ( binintersectSubset1
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( ( binintersect @ A @ B )
% 2.77/1.51 = A )
% 2.77/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(bs114d_type,type,
% 2.77/1.51 bs114d: $o ).
% 2.77/1.51
% 2.77/1.51 thf(bs114d,definition,
% 2.77/1.51 ( bs114d
% 2.77/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.77/1.51 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.77/1.51 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(regular_type,type,
% 2.77/1.51 regular: $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminus_type,type,
% 2.77/1.51 setminus: $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(setminusI_type,type,
% 2.77/1.51 setminusI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusI,definition,
% 2.77/1.51 ( setminusI
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ~ ( in @ Xx @ B )
% 2.77/1.51 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusEL_type,type,
% 2.77/1.51 setminusEL: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusEL,definition,
% 2.77/1.51 ( setminusEL
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusER_type,type,
% 2.77/1.51 setminusER: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusER,definition,
% 2.77/1.51 ( setminusER
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.51 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusSubset2_type,type,
% 2.77/1.51 setminusSubset2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusSubset2,definition,
% 2.77/1.51 ( setminusSubset2
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( subset @ A @ B )
% 2.77/1.51 => ( ( setminus @ A @ B )
% 2.77/1.51 = emptyset ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusERneg_type,type,
% 2.77/1.51 setminusERneg: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusERneg,definition,
% 2.77/1.51 ( setminusERneg
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.51 => ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusELneg_type,type,
% 2.77/1.51 setminusELneg: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusELneg,definition,
% 2.77/1.51 ( setminusELneg
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.51 => ( ~ ( in @ Xx @ B )
% 2.77/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusILneg_type,type,
% 2.77/1.51 setminusILneg: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusILneg,definition,
% 2.77/1.51 ( setminusILneg
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ~ ( in @ Xx @ A )
% 2.77/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusIRneg_type,type,
% 2.77/1.51 setminusIRneg: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusIRneg,definition,
% 2.77/1.51 ( setminusIRneg
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ B )
% 2.77/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusLsub_type,type,
% 2.77/1.51 setminusLsub: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusLsub,definition,
% 2.77/1.51 ( setminusLsub
% 2.77/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusSubset1_type,type,
% 2.77/1.51 setminusSubset1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusSubset1,definition,
% 2.77/1.51 ( setminusSubset1
% 2.77/1.51 = ( ! [A: $i,B: $i] :
% 2.77/1.51 ( ( ( setminus @ A @ B )
% 2.77/1.51 = emptyset )
% 2.77/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(symdiff_type,type,
% 2.77/1.51 symdiff: $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffE_type,type,
% 2.77/1.51 symdiffE: $o ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffE,definition,
% 2.77/1.51 ( symdiffE
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.77/1.51 => ! [Xphi: $o] :
% 2.77/1.51 ( ( ( in @ Xx @ A )
% 2.77/1.51 => ( ~ ( in @ Xx @ B )
% 2.77/1.51 => Xphi ) )
% 2.77/1.51 => ( ( ~ ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx @ B )
% 2.77/1.51 => Xphi ) )
% 2.77/1.51 => Xphi ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffI1_type,type,
% 2.77/1.51 symdiffI1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffI1,definition,
% 2.77/1.51 ( symdiffI1
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ~ ( in @ Xx @ B )
% 2.77/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffI2_type,type,
% 2.77/1.51 symdiffI2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffI2,definition,
% 2.77/1.51 ( symdiffI2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ~ ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx @ B )
% 2.77/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffIneg1_type,type,
% 2.77/1.51 symdiffIneg1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffIneg1,definition,
% 2.77/1.51 ( symdiffIneg1
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx @ B )
% 2.77/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffIneg2_type,type,
% 2.77/1.51 symdiffIneg2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(symdiffIneg2,definition,
% 2.77/1.51 ( symdiffIneg2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ~ ( in @ Xx @ A )
% 2.77/1.51 => ( ~ ( in @ Xx @ B )
% 2.77/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(iskpair_type,type,
% 2.77/1.51 iskpair: $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(secondinupair_type,type,
% 2.77/1.51 secondinupair: $o ).
% 2.77/1.51
% 2.77/1.51 thf(secondinupair,definition,
% 2.77/1.51 ( secondinupair
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairIL_type,type,
% 2.77/1.51 setukpairIL: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairIL,definition,
% 2.77/1.51 ( setukpairIL
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairIR_type,type,
% 2.77/1.51 setukpairIR: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairIR,definition,
% 2.77/1.51 ( setukpairIR
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(kpairiskpair_type,type,
% 2.77/1.51 kpairiskpair: $o ).
% 2.77/1.51
% 2.77/1.51 thf(kpairiskpair,definition,
% 2.77/1.51 ( kpairiskpair
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(kpair_type,type,
% 2.77/1.51 kpair: $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(kpairp_type,type,
% 2.77/1.51 kpairp: $o ).
% 2.77/1.51
% 2.77/1.51 thf(kpairp,definition,
% 2.77/1.51 ( kpairp
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprod_type,type,
% 2.77/1.51 cartprod: $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(singletonsubset_type,type,
% 2.77/1.51 singletonsubset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(singletonsubset,definition,
% 2.77/1.51 ( singletonsubset
% 2.77/1.51 = ( ! [A: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(singletoninpowerset_type,type,
% 2.77/1.51 singletoninpowerset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(singletoninpowerset,definition,
% 2.77/1.51 ( singletoninpowerset
% 2.77/1.51 = ( ! [A: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(singletoninpowunion_type,type,
% 2.77/1.51 singletoninpowunion: $o ).
% 2.77/1.51
% 2.77/1.51 thf(singletoninpowunion,definition,
% 2.77/1.51 ( singletoninpowunion
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(upairset2E_type,type,
% 2.77/1.51 upairset2E: $o ).
% 2.77/1.51
% 2.77/1.51 thf(upairset2E,definition,
% 2.77/1.51 ( upairset2E
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.77/1.51 => ( ( Xz = Xx )
% 2.77/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(upairsubunion_type,type,
% 2.77/1.51 upairsubunion: $o ).
% 2.77/1.51
% 2.77/1.51 thf(upairsubunion,definition,
% 2.77/1.51 ( upairsubunion
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(upairinpowunion_type,type,
% 2.77/1.51 upairinpowunion: $o ).
% 2.77/1.51
% 2.77/1.51 thf(upairinpowunion,definition,
% 2.77/1.51 ( upairinpowunion
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ubforcartprodlem1_type,type,
% 2.77/1.51 ubforcartprodlem1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ubforcartprodlem1,definition,
% 2.77/1.51 ( ubforcartprodlem1
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ubforcartprodlem2_type,type,
% 2.77/1.51 ubforcartprodlem2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ubforcartprodlem2,definition,
% 2.77/1.51 ( ubforcartprodlem2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ubforcartprodlem3_type,type,
% 2.77/1.51 ubforcartprodlem3: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ubforcartprodlem3,definition,
% 2.77/1.51 ( ubforcartprodlem3
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodpairin_type,type,
% 2.77/1.51 cartprodpairin: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodpairin,definition,
% 2.77/1.51 ( cartprodpairin
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodmempair1_type,type,
% 2.77/1.51 cartprodmempair1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodmempair1,definition,
% 2.77/1.51 ( cartprodmempair1
% 2.77/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.51 => ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ? [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 & ( Xu
% 2.77/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodmempair_type,type,
% 2.77/1.51 cartprodmempair: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodmempair,definition,
% 2.77/1.51 ( cartprodmempair
% 2.77/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.51 => ( iskpair @ Xu ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setunionE2_type,type,
% 2.77/1.51 setunionE2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setunionE2,definition,
% 2.77/1.51 ( setunionE2
% 2.77/1.51 = ( ! [A: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.77/1.51 => ? [X: $i] :
% 2.77/1.51 ( ( in @ X @ A )
% 2.77/1.51 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setunionsingleton1_type,type,
% 2.77/1.51 setunionsingleton1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setunionsingleton1,definition,
% 2.77/1.51 ( setunionsingleton1
% 2.77/1.51 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setunionsingleton2_type,type,
% 2.77/1.51 setunionsingleton2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setunionsingleton2,definition,
% 2.77/1.51 ( setunionsingleton2
% 2.77/1.51 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setunionsingleton_type,type,
% 2.77/1.51 setunionsingleton: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setunionsingleton,definition,
% 2.77/1.51 ( setunionsingleton
% 2.77/1.51 = ( ! [Xx: $i] :
% 2.77/1.51 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.77/1.51 = Xx ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(singleton_type,type,
% 2.77/1.51 singleton: $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(singletonprop_type,type,
% 2.77/1.51 singletonprop: $o ).
% 2.77/1.51
% 2.77/1.51 thf(singletonprop,definition,
% 2.77/1.51 ( singletonprop
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( ( Xphi @ Xx )
% 2.77/1.51 => ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) )
% 2.77/1.51 => ( ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ( Xphi @ Xx ) )
% 2.77/1.51 => ( singleton
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ex1_type,type,
% 2.77/1.51 ex1: $i > ( $i > $o ) > $o ).
% 2.77/1.51
% 2.77/1.51 thf(ex1,definition,
% 2.77/1.51 ( ex1
% 2.77/1.51 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( singleton
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ex1E1_type,type,
% 2.77/1.51 ex1E1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ex1E1,definition,
% 2.77/1.51 ( ex1E1
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ( ex1 @ A
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 => ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ex1I_type,type,
% 2.77/1.51 ex1I: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ex1I,definition,
% 2.77/1.51 ( ex1I
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( Xphi @ Xx )
% 2.77/1.51 => ( ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xy = Xx ) ) )
% 2.77/1.51 => ( ex1 @ A
% 2.77/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ex1I2_type,type,
% 2.77/1.51 ex1I2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ex1I2,definition,
% 2.77/1.51 ( ex1I2
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( ( Xphi @ Xx )
% 2.77/1.51 => ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) )
% 2.77/1.51 => ( ? [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 & ( Xphi @ Xx ) )
% 2.77/1.51 => ( ex1 @ A
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(singletonsuniq_type,type,
% 2.77/1.51 singletonsuniq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(singletonsuniq,definition,
% 2.77/1.51 ( singletonsuniq
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( ( setadjoin @ Xx @ emptyset )
% 2.77/1.51 = ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(atmost1p_type,type,
% 2.77/1.51 atmost1p: $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(atleast2p_type,type,
% 2.77/1.51 atleast2p: $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(atmost2p_type,type,
% 2.77/1.51 atmost2p: $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(upairsetp_type,type,
% 2.77/1.51 upairsetp: $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjL1_type,type,
% 2.77/1.51 setukpairinjL1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjL1,definition,
% 2.77/1.51 ( setukpairinjL1
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.51 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(kfstsingleton_type,type,
% 2.77/1.51 kfstsingleton: $o ).
% 2.77/1.51
% 2.77/1.51 thf(kfstsingleton,definition,
% 2.77/1.51 ( kfstsingleton
% 2.77/1.51 = ( ! [Xu: $i] :
% 2.77/1.51 ( ( iskpair @ Xu )
% 2.77/1.51 => ( singleton
% 2.77/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.77/1.51 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(theprop_type,type,
% 2.77/1.51 theprop: $o ).
% 2.77/1.51
% 2.77/1.51 thf(theprop,definition,
% 2.77/1.51 ( theprop
% 2.77/1.51 = ( ! [X: $i] :
% 2.77/1.51 ( ( singleton @ X )
% 2.77/1.51 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(kfst_type,type,
% 2.77/1.51 kfst: $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(kfstpairEq_type,type,
% 2.77/1.51 kfstpairEq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(kfstpairEq,definition,
% 2.77/1.51 ( kfstpairEq
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.77/1.51 = Xx ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodfstin_type,type,
% 2.77/1.51 cartprodfstin: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodfstin,definition,
% 2.77/1.51 ( cartprodfstin
% 2.77/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.51 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjL2_type,type,
% 2.77/1.51 setukpairinjL2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjL2,definition,
% 2.77/1.51 ( setukpairinjL2
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjL_type,type,
% 2.77/1.51 setukpairinjL: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjL,definition,
% 2.77/1.51 ( setukpairinjL
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.77/1.51 = ( kpair @ Xz @ Xu ) )
% 2.77/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR11_type,type,
% 2.77/1.51 setukpairinjR11: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR11,definition,
% 2.77/1.51 ( setukpairinjR11
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xx = Xy )
% 2.77/1.51 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR12_type,type,
% 2.77/1.51 setukpairinjR12: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR12,definition,
% 2.77/1.51 ( setukpairinjR12
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xx = Xy )
% 2.77/1.51 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.51 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR1_type,type,
% 2.77/1.51 setukpairinjR1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR1,definition,
% 2.77/1.51 ( setukpairinjR1
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.51 => ( ( Xz = Xu )
% 2.77/1.51 => ( Xy = Xu ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(upairequniteq_type,type,
% 2.77/1.51 upairequniteq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(upairequniteq,definition,
% 2.77/1.51 ( upairequniteq
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.51 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.51 = ( setadjoin @ Xz @ emptyset ) )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR2_type,type,
% 2.77/1.51 setukpairinjR2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR2,definition,
% 2.77/1.51 ( setukpairinjR2
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR_type,type,
% 2.77/1.51 setukpairinjR: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setukpairinjR,definition,
% 2.77/1.51 ( setukpairinjR
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.77/1.51 = ( kpair @ Xz @ Xu ) )
% 2.77/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ksndsingleton_type,type,
% 2.77/1.51 ksndsingleton: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ksndsingleton,definition,
% 2.77/1.51 ( ksndsingleton
% 2.77/1.51 = ( ! [Xu: $i] :
% 2.77/1.51 ( ( iskpair @ Xu )
% 2.77/1.51 => ( singleton
% 2.77/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.77/1.51 @ ^ [Xx: $i] :
% 2.77/1.51 ( Xu
% 2.77/1.51 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ksnd_type,type,
% 2.77/1.51 ksnd: $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(ksndpairEq_type,type,
% 2.77/1.51 ksndpairEq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ksndpairEq,definition,
% 2.77/1.51 ( ksndpairEq
% 2.77/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.77/1.51 = Xy ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(kpairsurjEq_type,type,
% 2.77/1.51 kpairsurjEq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(kpairsurjEq,definition,
% 2.77/1.51 ( kpairsurjEq
% 2.77/1.51 = ( ! [Xu: $i] :
% 2.77/1.51 ( ( iskpair @ Xu )
% 2.77/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.77/1.51 = Xu ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodsndin_type,type,
% 2.77/1.51 cartprodsndin: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodsndin,definition,
% 2.77/1.51 ( cartprodsndin
% 2.77/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.51 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodpairmemEL_type,type,
% 2.77/1.51 cartprodpairmemEL: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodpairmemEL,definition,
% 2.77/1.51 ( cartprodpairmemEL
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.77/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodpairmemER_type,type,
% 2.77/1.51 cartprodpairmemER: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodpairmemER,definition,
% 2.77/1.51 ( cartprodpairmemER
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.77/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodmempaircEq_type,type,
% 2.77/1.51 cartprodmempaircEq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodmempaircEq,definition,
% 2.77/1.51 ( cartprodmempaircEq
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( kpair @ Xx @ Xy )
% 2.77/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodfstpairEq_type,type,
% 2.77/1.51 cartprodfstpairEq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodfstpairEq,definition,
% 2.77/1.51 ( cartprodfstpairEq
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.77/1.51 = Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodsndpairEq_type,type,
% 2.77/1.51 cartprodsndpairEq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodsndpairEq,definition,
% 2.77/1.51 ( cartprodsndpairEq
% 2.77/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.77/1.51 = Xy ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodpairsurjEq_type,type,
% 2.77/1.51 cartprodpairsurjEq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(cartprodpairsurjEq,definition,
% 2.77/1.51 ( cartprodpairsurjEq
% 2.77/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.77/1.51 = Xu ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(breln_type,type,
% 2.77/1.51 breln: $i > $i > $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(breln,definition,
% 2.77/1.51 ( breln
% 2.77/1.51 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstr_type,type,
% 2.77/1.51 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrI_type,type,
% 2.77/1.51 dpsetconstrI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrI,definition,
% 2.77/1.51 ( dpsetconstrI
% 2.77/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( Xphi @ Xx @ Xy )
% 2.77/1.51 => ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.51 @ ( dpsetconstr @ A @ B
% 2.77/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrSub_type,type,
% 2.77/1.51 dpsetconstrSub: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrSub,definition,
% 2.77/1.51 ( dpsetconstrSub
% 2.77/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.77/1.51 ( subset
% 2.77/1.51 @ ( dpsetconstr @ A @ B
% 2.77/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.77/1.51 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setOfPairsIsBReln_type,type,
% 2.77/1.51 setOfPairsIsBReln: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setOfPairsIsBReln,definition,
% 2.77/1.51 ( setOfPairsIsBReln
% 2.77/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.77/1.51 ( breln @ A @ B
% 2.77/1.51 @ ( dpsetconstr @ A @ B
% 2.77/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrERa_type,type,
% 2.77/1.51 dpsetconstrERa: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrERa,definition,
% 2.77/1.51 ( dpsetconstrERa
% 2.77/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.51 @ ( dpsetconstr @ A @ B
% 2.77/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrEL1_type,type,
% 2.77/1.51 dpsetconstrEL1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrEL1,definition,
% 2.77/1.51 ( dpsetconstrEL1
% 2.77/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.51 @ ( dpsetconstr @ A @ B
% 2.77/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrEL2_type,type,
% 2.77/1.51 dpsetconstrEL2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrEL2,definition,
% 2.77/1.51 ( dpsetconstrEL2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.51 @ ( dpsetconstr @ A @ B
% 2.77/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrER_type,type,
% 2.77/1.51 dpsetconstrER: $o ).
% 2.77/1.51
% 2.77/1.51 thf(dpsetconstrER,definition,
% 2.77/1.51 ( dpsetconstrER
% 2.77/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.51 @ ( dpsetconstr @ A @ B
% 2.77/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(func_type,type,
% 2.77/1.51 func: $i > $i > $i > $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcSet_type,type,
% 2.77/1.51 funcSet: $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(funcImageSingleton_type,type,
% 2.77/1.51 funcImageSingleton: $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcImageSingleton,definition,
% 2.77/1.51 ( funcImageSingleton
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( singleton
% 2.77/1.51 @ ( dsetconstr @ B
% 2.77/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(apProp_type,type,
% 2.77/1.51 apProp: $o ).
% 2.77/1.51
% 2.77/1.51 thf(apProp,definition,
% 2.77/1.51 ( apProp
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in
% 2.77/1.51 @ ( setunion
% 2.77/1.51 @ ( dsetconstr @ B
% 2.77/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.77/1.51 @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ap_type,type,
% 2.77/1.51 ap: $i > $i > $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(app_type,type,
% 2.77/1.51 app: $o ).
% 2.77/1.51
% 2.77/1.51 thf(app,definition,
% 2.77/1.51 ( app
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(infuncsetfunc_type,type,
% 2.77/1.51 infuncsetfunc: $o ).
% 2.77/1.51
% 2.77/1.51 thf(infuncsetfunc,definition,
% 2.77/1.51 ( infuncsetfunc
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.51 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ap2p_type,type,
% 2.77/1.51 ap2p: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ap2p,definition,
% 2.77/1.51 ( ap2p
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(funcinfuncset_type,type,
% 2.77/1.51 funcinfuncset: $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcinfuncset,definition,
% 2.77/1.51 ( funcinfuncset
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(lamProp_type,type,
% 2.77/1.51 lamProp: $o ).
% 2.77/1.51
% 2.77/1.51 thf(lamProp,definition,
% 2.77/1.51 ( lamProp
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.51 => ( func @ A @ B
% 2.77/1.51 @ ( dpsetconstr @ A @ B
% 2.77/1.51 @ ^ [Xx: $i,Xy: $i] :
% 2.77/1.51 ( ( Xf @ Xx )
% 2.77/1.51 = Xy ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(lam_type,type,
% 2.77/1.51 lam: $i > $i > ( $i > $i ) > $i ).
% 2.77/1.51
% 2.77/1.51 thf(lamp_type,type,
% 2.77/1.51 lamp: $o ).
% 2.77/1.51
% 2.77/1.51 thf(lamp,definition,
% 2.77/1.51 ( lamp
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.51 => ( func @ A @ B
% 2.77/1.51 @ ( lam @ A @ B
% 2.77/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(lam2p_type,type,
% 2.77/1.51 lam2p: $o ).
% 2.77/1.51
% 2.77/1.51 thf(lam2p,definition,
% 2.77/1.51 ( lam2p
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.51 => ( in
% 2.77/1.51 @ ( lam @ A @ B
% 2.77/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.77/1.51 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(brelnall1_type,type,
% 2.77/1.51 brelnall1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(brelnall1,definition,
% 2.77/1.51 ( brelnall1
% 2.77/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.51 ( ( breln @ A @ B @ R )
% 2.77/1.51 => ! [Xphi: $i > $o] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ R )
% 2.77/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(brelnall2_type,type,
% 2.77/1.51 brelnall2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(brelnall2,definition,
% 2.77/1.51 ( brelnall2
% 2.77/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.51 ( ( breln @ A @ B @ R )
% 2.77/1.51 => ! [Xphi: $i > $o] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ R )
% 2.77/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ex1E2_type,type,
% 2.77/1.51 ex1E2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ex1E2,definition,
% 2.77/1.51 ( ex1E2
% 2.77/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.51 ( ( ex1 @ A
% 2.77/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( ( Xphi @ Xx )
% 2.77/1.51 => ( ( Xphi @ Xy )
% 2.77/1.51 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(funcGraphProp1_type,type,
% 2.77/1.51 funcGraphProp1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcGraphProp1,definition,
% 2.77/1.51 ( funcGraphProp1
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(funcGraphProp3_type,type,
% 2.77/1.51 funcGraphProp3: $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcGraphProp3,definition,
% 2.77/1.51 ( funcGraphProp3
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(funcGraphProp2_type,type,
% 2.77/1.51 funcGraphProp2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcGraphProp2,definition,
% 2.77/1.51 ( funcGraphProp2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.77/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(funcextLem_type,type,
% 2.77/1.51 funcextLem: $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcextLem,definition,
% 2.77/1.51 ( funcextLem
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ! [Xg: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xg )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.77/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(funcGraphProp4_type,type,
% 2.77/1.51 funcGraphProp4: $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcGraphProp4,definition,
% 2.77/1.51 ( funcGraphProp4
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.77/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subbreln_type,type,
% 2.77/1.51 subbreln: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subbreln,definition,
% 2.77/1.51 ( subbreln
% 2.77/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.51 ( ( breln @ A @ B @ R )
% 2.77/1.51 => ! [S: $i] :
% 2.77/1.51 ( ( breln @ A @ B @ S )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.77/1.51 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(eqbreln_type,type,
% 2.77/1.51 eqbreln: $o ).
% 2.77/1.51
% 2.77/1.51 thf(eqbreln,definition,
% 2.77/1.51 ( eqbreln
% 2.77/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.51 ( ( breln @ A @ B @ R )
% 2.77/1.51 => ! [S: $i] :
% 2.77/1.51 ( ( breln @ A @ B @ S )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ B )
% 2.77/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.77/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.77/1.51 => ( R = S ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(funcext_type,type,
% 2.77/1.51 funcext: $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcext,definition,
% 2.77/1.51 ( funcext
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ! [Xg: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xg )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.77/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(funcext2_type,type,
% 2.77/1.51 funcext2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(funcext2,definition,
% 2.77/1.51 ( funcext2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.51 => ! [Xg: $i] :
% 2.77/1.51 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.77/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ap2apEq1_type,type,
% 2.77/1.51 ap2apEq1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ap2apEq1,definition,
% 2.77/1.51 ( ap2apEq1
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ap2apEq2_type,type,
% 2.77/1.51 ap2apEq2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ap2apEq2,definition,
% 2.77/1.51 ( ap2apEq2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(beta1_type,type,
% 2.77/1.51 beta1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(beta1,definition,
% 2.77/1.51 ( beta1
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( ap @ A @ B
% 2.77/1.51 @ ( lam @ A @ B
% 2.77/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.77/1.51 @ Xx )
% 2.77/1.51 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(eta1_type,type,
% 2.77/1.51 eta1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(eta1,definition,
% 2.77/1.51 ( eta1
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( func @ A @ B @ Xf )
% 2.77/1.51 => ( ( lam @ A @ B
% 2.77/1.51 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.77/1.51 = Xf ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(lam2lamEq_type,type,
% 2.77/1.51 lam2lamEq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(lam2lamEq,definition,
% 2.77/1.51 ( lam2lamEq
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.51 => ( ( lam @ A @ B
% 2.77/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.77/1.51 = ( lam @ A @ B
% 2.77/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(beta2_type,type,
% 2.77/1.51 beta2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(beta2,definition,
% 2.77/1.51 ( beta2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.51 ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( ap @ A @ B
% 2.77/1.51 @ ( lam @ A @ B
% 2.77/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.77/1.51 @ Xx )
% 2.77/1.51 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(eta2_type,type,
% 2.77/1.51 eta2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(eta2,definition,
% 2.77/1.51 ( eta2
% 2.77/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.51 => ( ( lam @ A @ B
% 2.77/1.51 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.77/1.51 = Xf ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(iffalseProp1_type,type,
% 2.77/1.51 iffalseProp1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(iffalseProp1,definition,
% 2.77/1.51 ( iffalseProp1
% 2.77/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( ~ Xphi
% 2.77/1.51 => ( in @ Xy
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xz: $i] :
% 2.77/1.51 ( ( Xphi
% 2.77/1.51 & ( Xz = Xx ) )
% 2.77/1.51 | ( ~ Xphi
% 2.77/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(iffalseProp2_type,type,
% 2.77/1.51 iffalseProp2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(iffalseProp2,definition,
% 2.77/1.51 ( iffalseProp2
% 2.77/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( ~ Xphi
% 2.77/1.51 => ( ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xz: $i] :
% 2.77/1.51 ( ( Xphi
% 2.77/1.51 & ( Xz = Xx ) )
% 2.77/1.51 | ( ~ Xphi
% 2.77/1.51 & ( Xz = Xy ) ) ) )
% 2.77/1.51 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(iftrueProp1_type,type,
% 2.77/1.51 iftrueProp1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(iftrueProp1,definition,
% 2.77/1.51 ( iftrueProp1
% 2.77/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( Xphi
% 2.77/1.51 => ( in @ Xx
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xz: $i] :
% 2.77/1.51 ( ( Xphi
% 2.77/1.51 & ( Xz = Xx ) )
% 2.77/1.51 | ( ~ Xphi
% 2.77/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(iftrueProp2_type,type,
% 2.77/1.51 iftrueProp2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(iftrueProp2,definition,
% 2.77/1.51 ( iftrueProp2
% 2.77/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( Xphi
% 2.77/1.51 => ( ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xz: $i] :
% 2.77/1.51 ( ( Xphi
% 2.77/1.51 & ( Xz = Xx ) )
% 2.77/1.51 | ( ~ Xphi
% 2.77/1.51 & ( Xz = Xy ) ) ) )
% 2.77/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(ifSingleton_type,type,
% 2.77/1.51 ifSingleton: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ifSingleton,definition,
% 2.77/1.51 ( ifSingleton
% 2.77/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( singleton
% 2.77/1.51 @ ( dsetconstr @ A
% 2.77/1.51 @ ^ [Xz: $i] :
% 2.77/1.51 ( ( Xphi
% 2.77/1.51 & ( Xz = Xx ) )
% 2.77/1.51 | ( ~ Xphi
% 2.77/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(if_type,type,
% 2.77/1.51 if: $i > $o > $i > $i > $i ).
% 2.77/1.51
% 2.77/1.51 thf(ifp_type,type,
% 2.77/1.51 ifp: $o ).
% 2.77/1.51
% 2.77/1.51 thf(ifp,definition,
% 2.77/1.51 ( ifp
% 2.77/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(theeq_type,type,
% 2.77/1.51 theeq: $o ).
% 2.77/1.51
% 2.77/1.51 thf(theeq,definition,
% 2.77/1.51 ( theeq
% 2.77/1.51 = ( ! [X: $i] :
% 2.77/1.51 ( ( singleton @ X )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ X )
% 2.77/1.51 => ( ( setunion @ X )
% 2.77/1.51 = Xx ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(iftrue_type,type,
% 2.77/1.51 iftrue: $o ).
% 2.77/1.51
% 2.77/1.51 thf(iftrue,definition,
% 2.77/1.51 ( iftrue
% 2.77/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( Xphi
% 2.77/1.51 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.77/1.51 = Xx ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(iffalse_type,type,
% 2.77/1.51 iffalse: $o ).
% 2.77/1.51
% 2.77/1.51 thf(iffalse,definition,
% 2.77/1.51 ( iffalse
% 2.77/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( ~ Xphi
% 2.77/1.51 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.77/1.51 = Xy ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(iftrueorfalse_type,type,
% 2.77/1.51 iftrueorfalse: $o ).
% 2.77/1.51
% 2.77/1.51 thf(iftrueorfalse,definition,
% 2.77/1.51 ( iftrueorfalse
% 2.77/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ! [Xy: $i] :
% 2.77/1.51 ( ( in @ Xy @ A )
% 2.77/1.51 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectT_lem_type,type,
% 2.77/1.51 binintersectT_lem: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectT_lem,definition,
% 2.77/1.51 ( binintersectT_lem
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binunionT_lem_type,type,
% 2.77/1.51 binunionT_lem: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binunionT_lem,definition,
% 2.77/1.51 ( binunionT_lem
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powersetT_lem_type,type,
% 2.77/1.51 powersetT_lem: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powersetT_lem,definition,
% 2.77/1.51 ( powersetT_lem
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setminusT_lem_type,type,
% 2.77/1.51 setminusT_lem: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setminusT_lem,definition,
% 2.77/1.51 ( setminusT_lem
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(complementT_lem_type,type,
% 2.77/1.51 complementT_lem: $o ).
% 2.77/1.51
% 2.77/1.51 thf(complementT_lem,definition,
% 2.77/1.51 ( complementT_lem
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(setextT_type,type,
% 2.77/1.51 setextT: $o ).
% 2.77/1.51
% 2.77/1.51 thf(setextT,definition,
% 2.77/1.51 ( setextT
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx @ X )
% 2.77/1.51 => ( in @ Xx @ Y ) ) )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx @ Y )
% 2.77/1.51 => ( in @ Xx @ X ) ) )
% 2.77/1.51 => ( X = Y ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(subsetTI_type,type,
% 2.77/1.51 subsetTI: $o ).
% 2.77/1.51
% 2.77/1.51 thf(subsetTI,definition,
% 2.77/1.51 ( subsetTI
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx @ X )
% 2.77/1.51 => ( in @ Xx @ Y ) ) )
% 2.77/1.51 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powersetTI1_type,type,
% 2.77/1.51 powersetTI1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powersetTI1,definition,
% 2.77/1.51 ( powersetTI1
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ( ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx @ X )
% 2.77/1.51 => ( in @ Xx @ Y ) ) )
% 2.77/1.51 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(powersetTE1_type,type,
% 2.77/1.51 powersetTE1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(powersetTE1,definition,
% 2.77/1.51 ( powersetTE1
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ X @ ( powerset @ Y ) )
% 2.77/1.51 => ( ( in @ Xx @ X )
% 2.77/1.51 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(complementTI1_type,type,
% 2.77/1.51 complementTI1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(complementTI1,definition,
% 2.77/1.51 ( complementTI1
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( in @ Xx @ X )
% 2.77/1.51 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(complementTE1_type,type,
% 2.77/1.51 complementTE1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(complementTE1,definition,
% 2.77/1.51 ( complementTE1
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.77/1.51 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectTELcontra_type,type,
% 2.77/1.51 binintersectTELcontra: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectTELcontra,definition,
% 2.77/1.51 ( binintersectTELcontra
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ~ ( in @ Xx @ X )
% 2.77/1.51 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectTERcontra_type,type,
% 2.77/1.51 binintersectTERcontra: $o ).
% 2.77/1.51
% 2.77/1.51 thf(binintersectTERcontra,definition,
% 2.77/1.51 ( binintersectTERcontra
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ~ ( in @ Xx @ Y )
% 2.77/1.51 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(contrasubsetT_type,type,
% 2.77/1.51 contrasubsetT: $o ).
% 2.77/1.51
% 2.77/1.51 thf(contrasubsetT,definition,
% 2.77/1.51 ( contrasubsetT
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.77/1.51 => ( ( in @ Xx @ Y )
% 2.77/1.51 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(contrasubsetT1_type,type,
% 2.77/1.51 contrasubsetT1: $o ).
% 2.77/1.51
% 2.77/1.51 thf(contrasubsetT1,definition,
% 2.77/1.51 ( contrasubsetT1
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ! [Xx: $i] :
% 2.77/1.51 ( ( in @ Xx @ A )
% 2.77/1.51 => ( ( subset @ X @ Y )
% 2.77/1.51 => ( ~ ( in @ Xx @ Y )
% 2.77/1.51 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(contrasubsetT2_type,type,
% 2.77/1.51 contrasubsetT2: $o ).
% 2.77/1.51
% 2.77/1.51 thf(contrasubsetT2,definition,
% 2.77/1.51 ( contrasubsetT2
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ( ( subset @ X @ Y )
% 2.77/1.51 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(contrasubsetT3_type,type,
% 2.77/1.51 contrasubsetT3: $o ).
% 2.77/1.51
% 2.77/1.51 thf(contrasubsetT3,definition,
% 2.77/1.51 ( contrasubsetT3
% 2.77/1.51 = ( ! [A: $i,X: $i] :
% 2.77/1.51 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.51 => ! [Y: $i] :
% 2.77/1.51 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.51 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.77/1.51 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.77/1.51
% 2.77/1.51 thf(doubleComplementI1,conjecture,
% 2.77/1.51 ( setextAx
% 2.77/1.51 => ( emptysetAx
% 2.77/1.51 => ( setadjoinAx
% 2.77/1.51 => ( powersetAx
% 2.77/1.51 => ( setunionAx
% 2.77/1.51 => ( omega0Ax
% 2.77/1.51 => ( omegaSAx
% 2.77/1.51 => ( omegaIndAx
% 2.77/1.51 => ( replAx
% 2.77/1.51 => ( foundationAx
% 2.77/1.51 => ( wellorderingAx
% 2.77/1.52 => ( descrp
% 2.77/1.52 => ( dsetconstrI
% 2.77/1.52 => ( dsetconstrEL
% 2.77/1.52 => ( dsetconstrER
% 2.77/1.52 => ( exuE1
% 2.77/1.52 => ( prop2setE
% 2.77/1.52 => ( emptysetE
% 2.77/1.52 => ( emptysetimpfalse
% 2.77/1.52 => ( notinemptyset
% 2.77/1.52 => ( exuE3e
% 2.77/1.52 => ( setext
% 2.77/1.52 => ( emptyI
% 2.77/1.52 => ( noeltsimpempty
% 2.77/1.52 => ( setbeta
% 2.77/1.52 => ( nonemptyE1
% 2.77/1.52 => ( nonemptyI
% 2.77/1.52 => ( nonemptyI1
% 2.77/1.52 => ( setadjoinIL
% 2.77/1.52 => ( emptyinunitempty
% 2.77/1.52 => ( setadjoinIR
% 2.77/1.52 => ( setadjoinE
% 2.77/1.52 => ( setadjoinOr
% 2.77/1.52 => ( setoftrueEq
% 2.77/1.52 => ( powersetI
% 2.77/1.52 => ( emptyinPowerset
% 2.77/1.52 => ( emptyInPowerset
% 2.77/1.52 => ( powersetE
% 2.77/1.52 => ( setunionI
% 2.77/1.52 => ( setunionE
% 2.77/1.52 => ( subPowSU
% 2.77/1.52 => ( exuE2
% 2.77/1.52 => ( nonemptyImpWitness
% 2.77/1.52 => ( uniqinunit
% 2.77/1.52 => ( notinsingleton
% 2.77/1.52 => ( eqinunit
% 2.77/1.52 => ( singletonsswitch
% 2.77/1.52 => ( upairsetE
% 2.77/1.52 => ( upairsetIL
% 2.77/1.52 => ( upairsetIR
% 2.77/1.52 => ( emptyE1
% 2.77/1.52 => ( vacuousDall
% 2.77/1.52 => ( quantDeMorgan1
% 2.77/1.52 => ( quantDeMorgan2
% 2.77/1.52 => ( quantDeMorgan3
% 2.77/1.52 => ( quantDeMorgan4
% 2.77/1.52 => ( prop2setI
% 2.77/1.52 => ( prop2set2propI
% 2.77/1.52 => ( notdexE
% 2.77/1.52 => ( notdallE
% 2.77/1.52 => ( exuI1
% 2.77/1.52 => ( exuI3
% 2.77/1.52 => ( exuI2
% 2.77/1.52 => ( inCongP
% 2.77/1.52 => ( in__Cong
% 2.77/1.52 => ( exuE3u
% 2.77/1.52 => ( exu__Cong
% 2.77/1.52 => ( emptyset__Cong
% 2.77/1.52 => ( setadjoin__Cong
% 2.77/1.52 => ( powerset__Cong
% 2.77/1.52 => ( setunion__Cong
% 2.77/1.52 => ( omega__Cong
% 2.77/1.52 => ( exuEu
% 2.77/1.52 => ( descr__Cong
% 2.77/1.52 => ( dsetconstr__Cong
% 2.77/1.52 => ( subsetI1
% 2.77/1.52 => ( eqimpsubset2
% 2.77/1.52 => ( eqimpsubset1
% 2.77/1.52 => ( subsetI2
% 2.77/1.52 => ( emptysetsubset
% 2.77/1.52 => ( subsetE
% 2.77/1.52 => ( subsetE2
% 2.77/1.52 => ( notsubsetI
% 2.77/1.52 => ( notequalI1
% 2.77/1.52 => ( notequalI2
% 2.77/1.52 => ( subsetRefl
% 2.77/1.52 => ( subsetTrans
% 2.77/1.52 => ( setadjoinSub
% 2.77/1.52 => ( setadjoinSub2
% 2.77/1.52 => ( subset2powerset
% 2.77/1.52 => ( setextsub
% 2.77/1.52 => ( subsetemptysetimpeq
% 2.77/1.52 => ( powersetI1
% 2.77/1.52 => ( powersetE1
% 2.77/1.52 => ( inPowerset
% 2.77/1.52 => ( powersetsubset
% 2.77/1.52 => ( sepInPowerset
% 2.77/1.52 => ( sepSubset
% 2.77/1.52 => ( binunionIL
% 2.77/1.52 => ( upairset2IR
% 2.77/1.52 => ( binunionIR
% 2.77/1.52 => ( binunionEcases
% 2.77/1.52 => ( binunionE
% 2.77/1.52 => ( binunionLsub
% 2.77/1.52 => ( binunionRsub
% 2.77/1.52 => ( binintersectI
% 2.77/1.52 => ( binintersectSubset5
% 2.77/1.52 => ( binintersectEL
% 2.77/1.52 => ( binintersectLsub
% 2.77/1.52 => ( binintersectSubset2
% 2.77/1.52 => ( binintersectSubset3
% 2.77/1.52 => ( binintersectER
% 2.77/1.52 => ( disjointsetsI1
% 2.77/1.52 => ( binintersectRsub
% 2.77/1.52 => ( binintersectSubset4
% 2.77/1.52 => ( binintersectSubset1
% 2.77/1.52 => ( bs114d
% 2.77/1.52 => ( setminusI
% 2.77/1.52 => ( setminusEL
% 2.77/1.52 => ( setminusER
% 2.77/1.52 => ( setminusSubset2
% 2.77/1.52 => ( setminusERneg
% 2.77/1.52 => ( setminusELneg
% 2.77/1.52 => ( setminusILneg
% 2.77/1.52 => ( setminusIRneg
% 2.77/1.52 => ( setminusLsub
% 2.77/1.52 => ( setminusSubset1
% 2.77/1.52 => ( symdiffE
% 2.77/1.52 => ( symdiffI1
% 2.77/1.52 => ( symdiffI2
% 2.77/1.52 => ( symdiffIneg1
% 2.77/1.52 => ( symdiffIneg2
% 2.77/1.52 => ( secondinupair
% 2.77/1.52 => ( setukpairIL
% 2.77/1.52 => ( setukpairIR
% 2.77/1.52 => ( kpairiskpair
% 2.77/1.52 => ( kpairp
% 2.77/1.52 => ( singletonsubset
% 2.77/1.52 => ( singletoninpowerset
% 2.77/1.52 => ( singletoninpowunion
% 2.77/1.52 => ( upairset2E
% 2.77/1.52 => ( upairsubunion
% 2.77/1.52 => ( upairinpowunion
% 2.77/1.52 => ( ubforcartprodlem1
% 2.77/1.52 => ( ubforcartprodlem2
% 2.77/1.52 => ( ubforcartprodlem3
% 2.77/1.52 => ( cartprodpairin
% 2.77/1.52 => ( cartprodmempair1
% 2.77/1.52 => ( cartprodmempair
% 2.77/1.52 => ( setunionE2
% 2.77/1.52 => ( setunionsingleton1
% 2.77/1.52 => ( setunionsingleton2
% 2.77/1.52 => ( setunionsingleton
% 2.77/1.52 => ( singletonprop
% 2.77/1.52 => ( ex1E1
% 2.77/1.52 => ( ex1I
% 2.77/1.52 => ( ex1I2
% 2.77/1.52 => ( singletonsuniq
% 2.77/1.52 => ( setukpairinjL1
% 2.77/1.52 => ( kfstsingleton
% 2.77/1.52 => ( theprop
% 2.77/1.52 => ( kfstpairEq
% 2.77/1.52 => ( cartprodfstin
% 2.77/1.52 => ( setukpairinjL2
% 2.77/1.52 => ( setukpairinjL
% 2.77/1.52 => ( setukpairinjR11
% 2.77/1.52 => ( setukpairinjR12
% 2.77/1.52 => ( setukpairinjR1
% 2.77/1.52 => ( upairequniteq
% 2.77/1.52 => ( setukpairinjR2
% 2.77/1.52 => ( setukpairinjR
% 2.77/1.52 => ( ksndsingleton
% 2.77/1.52 => ( ksndpairEq
% 2.77/1.52 => ( kpairsurjEq
% 2.77/1.52 => ( cartprodsndin
% 2.77/1.52 => ( cartprodpairmemEL
% 2.77/1.52 => ( cartprodpairmemER
% 2.77/1.52 => ( cartprodmempaircEq
% 2.77/1.52 => ( cartprodfstpairEq
% 2.77/1.52 => ( cartprodsndpairEq
% 2.77/1.52 => ( cartprodpairsurjEq
% 2.77/1.52 => ( dpsetconstrI
% 2.77/1.52 => ( dpsetconstrSub
% 2.77/1.52 => ( setOfPairsIsBReln
% 2.77/1.52 => ( dpsetconstrERa
% 2.77/1.52 => ( dpsetconstrEL1
% 2.77/1.52 => ( dpsetconstrEL2
% 2.77/1.52 => ( dpsetconstrER
% 2.77/1.52 => ( funcImageSingleton
% 2.77/1.52 => ( apProp
% 2.77/1.52 => ( app
% 2.77/1.52 => ( infuncsetfunc
% 2.77/1.52 => ( ap2p
% 2.77/1.52 => ( funcinfuncset
% 2.77/1.52 => ( lamProp
% 2.77/1.52 => ( lamp
% 2.77/1.52 => ( lam2p
% 2.77/1.52 => ( brelnall1
% 2.77/1.52 => ( brelnall2
% 2.77/1.52 => ( ex1E2
% 2.77/1.52 => ( funcGraphProp1
% 2.77/1.52 => ( funcGraphProp3
% 2.77/1.52 => ( funcGraphProp2
% 2.77/1.52 => ( funcextLem
% 2.77/1.52 => ( funcGraphProp4
% 2.77/1.52 => ( subbreln
% 2.77/1.52 => ( eqbreln
% 2.77/1.52 => ( funcext
% 2.77/1.52 => ( funcext2
% 2.77/1.52 => ( ap2apEq1
% 2.77/1.52 => ( ap2apEq2
% 2.77/1.52 => ( beta1
% 2.77/1.52 => ( eta1
% 2.77/1.52 => ( lam2lamEq
% 2.77/1.52 => ( beta2
% 2.77/1.52 => ( eta2
% 2.77/1.52 => ( iffalseProp1
% 2.77/1.52 => ( iffalseProp2
% 2.77/1.52 => ( iftrueProp1
% 2.77/1.52 => ( iftrueProp2
% 2.77/1.52 => ( ifSingleton
% 2.77/1.52 => ( ifp
% 2.77/1.52 => ( theeq
% 2.77/1.52 => ( iftrue
% 2.77/1.52 => ( iffalse
% 2.77/1.52 => ( iftrueorfalse
% 2.77/1.52 => ( binintersectT_lem
% 2.77/1.52 => ( binunionT_lem
% 2.77/1.52 => ( powersetT_lem
% 2.77/1.52 => ( setminusT_lem
% 2.77/1.52 => ( complementT_lem
% 2.77/1.52 => ( setextT
% 2.77/1.52 => ( subsetTI
% 2.77/1.52 => ( powersetTI1
% 2.77/1.52 => ( powersetTE1
% 2.77/1.52 => ( complementTI1
% 2.77/1.52 => ( complementTE1
% 2.77/1.52 => ( binintersectTELcontra
% 2.77/1.52 => ( binintersectTERcontra
% 2.77/1.52 => ( contrasubsetT
% 2.77/1.52 => ( contrasubsetT1
% 2.77/1.52 => ( contrasubsetT2
% 2.77/1.52 => ( contrasubsetT3
% 2.77/1.52 => ! [A: $i,X: $i] :
% 2.77/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.52 => ! [Xx: $i] :
% 2.77/1.52 ( ( in @ Xx @ A )
% 2.77/1.52 => ( ( in @ Xx @ X )
% 2.77/1.52 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.52 ------------------------
% 2.77/1.54 % (17839)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/183Mi)
% 2.77/1.54 % (17842)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/2Mi)
% 2.77/1.54 % (17843)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_17732 for (2999ds/2Mi)
% 2.77/1.54 % (17840)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_17732 for (2999ds/4Mi)
% 2.77/1.54 % (17844)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_17732 for (2999ds/275Mi)
% 2.77/1.54 % (17841)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_17732 for (2999ds/27Mi)
% 2.77/1.54 % (17845)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/18Mi)
% 2.77/1.55 % (17842)Instruction limit reached!
% 2.77/1.55 % (17842)------------------------------
% 2.77/1.55 % (17842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.55 % (17842)Termination reason: Unknown
% 2.77/1.55 % (17843)Instruction limit reached!
% 2.77/1.55 % (17843)------------------------------
% 2.77/1.55 % (17843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.55 % (17842)Termination phase: shuffling
% 2.77/1.55
% 2.77/1.55 % (17842)Memory used [KB]: 1407
% 2.77/1.55 % (17842)Time elapsed: 0.003 s
% 2.77/1.55 % (17842)Instructions burned: 3 (million)
% 2.77/1.55 % (17842)------------------------------
% 2.77/1.55 % (17842)------------------------------
% 2.77/1.55 % (17843)Termination reason: Unknown
% 2.77/1.55 % (17843)Termination phase: shuffling
% 2.77/1.55
% 2.77/1.55 % (17843)Memory used [KB]: 1407
% 2.77/1.55 % (17843)Time elapsed: 0.003 s
% 2.77/1.55 % (17843)Instructions burned: 3 (million)
% 2.77/1.55 % (17843)------------------------------
% 2.77/1.55 % (17843)------------------------------
% 2.77/1.55 % (17840)Instruction limit reached!
% 2.77/1.55 % (17840)------------------------------
% 2.77/1.55 % (17840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.55 % (17840)Termination reason: Unknown
% 2.77/1.55 % (17840)Termination phase: shuffling
% 2.77/1.55
% 2.77/1.55 % (17840)Memory used [KB]: 1535
% 2.77/1.55 % (17840)Time elapsed: 0.004 s
% 2.77/1.55 % (17840)Instructions burned: 5 (million)
% 2.77/1.55 % (17840)------------------------------
% 2.77/1.55 % (17840)------------------------------
% 2.77/1.55 % (17845)Instruction limit reached!
% 2.77/1.55 % (17845)------------------------------
% 2.77/1.55 % (17845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.55 % (17845)Termination reason: Unknown
% 2.77/1.55 % (17845)Termination phase: shuffling
% 2.77/1.55
% 2.77/1.55 % (17845)Memory used [KB]: 1791
% 2.77/1.55 % (17845)Time elapsed: 0.011 s
% 2.77/1.55 % (17845)Instructions burned: 19 (million)
% 2.77/1.55 % (17845)------------------------------
% 2.77/1.55 % (17845)------------------------------
% 2.77/1.56 % (17841)Instruction limit reached!
% 2.77/1.56 % (17841)------------------------------
% 2.77/1.56 % (17841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.56 % (17841)Termination reason: Unknown
% 2.77/1.56 % (17841)Termination phase: shuffling
% 2.77/1.56
% 2.77/1.56 % (17841)Memory used [KB]: 2046
% 2.77/1.56 % (17841)Time elapsed: 0.015 s
% 2.77/1.56 % (17841)Instructions burned: 29 (million)
% 2.77/1.56 % (17841)------------------------------
% 2.77/1.56 % (17841)------------------------------
% 2.77/1.56 % (17846)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/3Mi)
% 2.77/1.56 % (17847)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_17732 for (2999ds/37Mi)
% 2.77/1.56 % (17848)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_17732 for (2999ds/15Mi)
% 2.77/1.56 % (17846)Instruction limit reached!
% 2.77/1.56 % (17846)------------------------------
% 2.77/1.56 % (17846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.56 % (17846)Termination reason: Unknown
% 2.77/1.56 % (17846)Termination phase: shuffling
% 2.77/1.56
% 2.77/1.56 % (17846)Memory used [KB]: 1407
% 2.77/1.56 % (17846)Time elapsed: 0.003 s
% 2.77/1.56 % (17846)Instructions burned: 3 (million)
% 2.77/1.56 % (17846)------------------------------
% 2.77/1.56 % (17846)------------------------------
% 2.77/1.57 % (17849)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_17732 for (2999ds/3Mi)
% 2.77/1.57 % (17848)Instruction limit reached!
% 2.77/1.57 % (17848)------------------------------
% 2.77/1.57 % (17848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.57 % (17848)Termination reason: Unknown
% 2.77/1.57 % (17848)Termination phase: shuffling
% 2.77/1.57
% 2.77/1.57 % (17848)Memory used [KB]: 1663
% 2.77/1.57 % (17848)Time elapsed: 0.009 s
% 2.77/1.57 % (17848)Instructions burned: 15 (million)
% 2.77/1.57 % (17848)------------------------------
% 2.77/1.57 % (17848)------------------------------
% 2.77/1.57 % (17849)Instruction limit reached!
% 2.77/1.57 % (17849)------------------------------
% 2.77/1.57 % (17849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.57 % (17849)Termination reason: Unknown
% 2.77/1.57 % (17849)Termination phase: shuffling
% 2.77/1.57
% 2.77/1.57 % (17849)Memory used [KB]: 1407
% 2.77/1.57 % (17849)Time elapsed: 0.003 s
% 2.77/1.57 % (17849)Instructions burned: 3 (million)
% 2.77/1.57 % (17849)------------------------------
% 2.77/1.57 % (17849)------------------------------
% 2.77/1.57 % (17850)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_17732 for (2999ds/1041Mi)
% 2.77/1.58 % (17847)Instruction limit reached!
% 2.77/1.58 % (17847)------------------------------
% 2.77/1.58 % (17847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.58 % (17847)Termination reason: Unknown
% 2.77/1.58 % (17847)Termination phase: shuffling
% 2.77/1.58
% 2.77/1.58 % (17847)Memory used [KB]: 2174
% 2.77/1.58 % (17847)Time elapsed: 0.018 s
% 2.77/1.58 % (17847)Instructions burned: 37 (million)
% 2.77/1.58 % (17847)------------------------------
% 2.77/1.58 % (17847)------------------------------
% 2.77/1.58 % (17851)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/7Mi)
% 2.77/1.58 % (17851)Instruction limit reached!
% 2.77/1.58 % (17851)------------------------------
% 2.77/1.58 % (17851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.58 % (17851)Termination reason: Unknown
% 2.77/1.58 % (17851)Termination phase: shuffling
% 2.77/1.58
% 2.77/1.58 % (17851)Memory used [KB]: 1535
% 2.77/1.58 % (17851)Time elapsed: 0.005 s
% 2.77/1.58 % (17851)Instructions burned: 8 (million)
% 2.77/1.58 % (17851)------------------------------
% 2.77/1.58 % (17851)------------------------------
% 2.77/1.59 % (17839)Instruction limit reached!
% 2.77/1.59 % (17839)------------------------------
% 2.77/1.59 % (17839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.59 % (17839)Termination reason: Unknown
% 2.77/1.59 % (17839)Termination phase: Saturation
% 2.77/1.59
% 2.77/1.59 % (17839)Memory used [KB]: 7547
% 2.77/1.59 % (17839)Time elapsed: 0.050 s
% 2.77/1.59 % (17839)Instructions burned: 184 (million)
% 2.77/1.59 % (17839)------------------------------
% 2.77/1.59 % (17839)------------------------------
% 2.77/1.59 % (17853)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_17732 for (2999ds/3Mi)
% 2.77/1.59 % (17852)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/16Mi)
% 2.77/1.59 % (17853)Instruction limit reached!
% 2.77/1.59 % (17853)------------------------------
% 2.77/1.59 % (17853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.77/1.59 % (17853)Termination reason: Unknown
% 2.77/1.59 % (17853)Termination phase: shuffling
% 2.77/1.59
% 2.77/1.59 % (17853)Memory used [KB]: 1407
% 2.77/1.59 % (17853)Time elapsed: 0.003 s
% 2.77/1.59 % (17853)Instructions burned: 3 (million)
% 2.77/1.59 % (17853)------------------------------
% 2.77/1.59 % (17853)------------------------------
% 2.79/1.60 % (17852)Instruction limit reached!
% 2.79/1.60 % (17852)------------------------------
% 2.79/1.60 % (17852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.60 % (17852)Termination reason: Unknown
% 2.79/1.60 % (17852)Termination phase: shuffling
% 2.79/1.60
% 2.79/1.60 % (17852)Memory used [KB]: 1791
% 2.79/1.60 % (17852)Time elapsed: 0.009 s
% 2.79/1.60 % (17852)Instructions burned: 16 (million)
% 2.79/1.60 % (17852)------------------------------
% 2.79/1.60 % (17852)------------------------------
% 2.79/1.60 % (17854)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_17732 for (2999ds/3Mi)
% 2.79/1.60 % (17857)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/3Mi)
% 2.79/1.60 % (17854)Instruction limit reached!
% 2.79/1.60 % (17854)------------------------------
% 2.79/1.60 % (17854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.60 % (17854)Termination reason: Unknown
% 2.79/1.60 % (17854)Termination phase: shuffling
% 2.79/1.60
% 2.79/1.60 % (17854)Memory used [KB]: 1407
% 2.79/1.60 % (17854)Time elapsed: 0.003 s
% 2.79/1.60 % (17854)Instructions burned: 3 (million)
% 2.79/1.60 % (17854)------------------------------
% 2.79/1.60 % (17854)------------------------------
% 2.79/1.60 % (17857)Instruction limit reached!
% 2.79/1.60 % (17857)------------------------------
% 2.79/1.60 % (17857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.60 % (17857)Termination reason: Unknown
% 2.79/1.60 % (17857)Termination phase: shuffling
% 2.79/1.60
% 2.79/1.60 % (17857)Memory used [KB]: 1535
% 2.79/1.60 % (17857)Time elapsed: 0.003 s
% 2.79/1.60 % (17857)Instructions burned: 5 (million)
% 2.79/1.60 % (17857)------------------------------
% 2.79/1.60 % (17857)------------------------------
% 2.79/1.60 % (17855)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/7Mi)
% 2.79/1.60 % (17855)Instruction limit reached!
% 2.79/1.60 % (17855)------------------------------
% 2.79/1.60 % (17855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.60 % (17855)Termination reason: Unknown
% 2.79/1.60 % (17855)Termination phase: shuffling
% 2.79/1.60
% 2.79/1.60 % (17855)Memory used [KB]: 1535
% 2.79/1.60 % (17855)Time elapsed: 0.005 s
% 2.79/1.60 % (17855)Instructions burned: 8 (million)
% 2.79/1.60 % (17855)------------------------------
% 2.79/1.60 % (17855)------------------------------
% 2.79/1.61 % (17858)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/4Mi)
% 2.79/1.61 % (17858)Instruction limit reached!
% 2.79/1.61 % (17858)------------------------------
% 2.79/1.61 % (17858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.61 % (17858)Termination reason: Unknown
% 2.79/1.61 % (17858)Termination phase: shuffling
% 2.79/1.61
% 2.79/1.61 % (17858)Memory used [KB]: 1535
% 2.79/1.61 % (17858)Time elapsed: 0.004 s
% 2.79/1.61 % (17858)Instructions burned: 5 (million)
% 2.79/1.61 % (17858)------------------------------
% 2.79/1.61 % (17858)------------------------------
% 2.79/1.61 % (17861)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/6Mi)
% 2.79/1.61 % (17861)Instruction limit reached!
% 2.79/1.61 % (17861)------------------------------
% 2.79/1.61 % (17861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.61 % (17861)Termination reason: Unknown
% 2.79/1.61 % (17861)Termination phase: shuffling
% 2.79/1.61
% 2.79/1.61 % (17861)Memory used [KB]: 1535
% 2.79/1.61 % (17861)Time elapsed: 0.003 s
% 2.79/1.61 % (17861)Instructions burned: 7 (million)
% 2.79/1.61 % (17861)------------------------------
% 2.79/1.61 % (17861)------------------------------
% 2.79/1.61 % (17859)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/18Mi)
% 2.79/1.61 % (17860)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/710Mi)
% 2.79/1.62 % (17862)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_17732 for (2999ds/902Mi)
% 2.79/1.62 % (17864)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/5Mi)
% 2.79/1.62 % (17859)Instruction limit reached!
% 2.79/1.62 % (17859)------------------------------
% 2.79/1.62 % (17859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.62 % (17859)Termination reason: Unknown
% 2.79/1.62 % (17859)Termination phase: shuffling
% 2.79/1.62
% 2.79/1.62 % (17859)Memory used [KB]: 1791
% 2.79/1.62 % (17859)Time elapsed: 0.010 s
% 2.79/1.62 % (17859)Instructions burned: 19 (million)
% 2.79/1.62 % (17859)------------------------------
% 2.79/1.62 % (17859)------------------------------
% 2.79/1.62 % (17864)Instruction limit reached!
% 2.79/1.62 % (17864)------------------------------
% 2.79/1.62 % (17864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.62 % (17864)Termination reason: Unknown
% 2.79/1.62 % (17864)Termination phase: shuffling
% 2.79/1.62
% 2.79/1.62 % (17864)Memory used [KB]: 1535
% 2.79/1.62 % (17864)Time elapsed: 0.003 s
% 2.79/1.62 % (17864)Instructions burned: 7 (million)
% 2.79/1.62 % (17864)------------------------------
% 2.79/1.62 % (17864)------------------------------
% 2.79/1.63 % (17863)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_17732 for (2999ds/21Mi)
% 2.79/1.63 % (17865)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_17732 for (2999ds/6Mi)
% 2.79/1.63 % (17865)Instruction limit reached!
% 2.79/1.63 % (17865)------------------------------
% 2.79/1.63 % (17865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.63 % (17865)Termination reason: Unknown
% 2.79/1.63 % (17865)Termination phase: shuffling
% 2.79/1.63
% 2.79/1.63 % (17865)Memory used [KB]: 1535
% 2.79/1.63 % (17865)Time elapsed: 0.003 s
% 2.79/1.63 % (17865)Instructions burned: 6 (million)
% 2.79/1.63 % (17865)------------------------------
% 2.79/1.63 % (17865)------------------------------
% 2.79/1.63 % (17863)Instruction limit reached!
% 2.79/1.63 % (17863)------------------------------
% 2.79/1.63 % (17863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.63 % (17863)Termination reason: Unknown
% 2.79/1.63 % (17863)Termination phase: shuffling
% 2.79/1.63
% 2.79/1.63 % (17863)Memory used [KB]: 1791
% 2.79/1.63 % (17863)Time elapsed: 0.011 s
% 2.79/1.63 % (17863)Instructions burned: 21 (million)
% 2.79/1.63 % (17863)------------------------------
% 2.79/1.63 % (17863)------------------------------
% 2.79/1.64 % (17866)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_17732 for (2999ds/377Mi)
% 2.79/1.64 % (17867)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_17732 for (2998ds/779Mi)
% 2.79/1.65 % (17868)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_17732 for (2998ds/19Mi)
% 2.79/1.66 % (17868)Instruction limit reached!
% 2.79/1.66 % (17868)------------------------------
% 2.79/1.66 % (17868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.66 % (17868)Termination reason: Unknown
% 2.79/1.66 % (17868)Termination phase: shuffling
% 2.79/1.66
% 2.79/1.66 % (17868)Memory used [KB]: 1791
% 2.79/1.66 % (17868)Time elapsed: 0.010 s
% 2.79/1.66 % (17868)Instructions burned: 19 (million)
% 2.79/1.66 % (17868)------------------------------
% 2.79/1.66 % (17868)------------------------------
% 2.79/1.67 % (17844)Instruction limit reached!
% 2.79/1.67 % (17844)------------------------------
% 2.79/1.67 % (17844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.67 % (17844)Termination reason: Unknown
% 2.79/1.67 % (17844)Termination phase: Saturation
% 2.79/1.67
% 2.79/1.67 % (17844)Memory used [KB]: 9466
% 2.79/1.67 % (17844)Time elapsed: 0.131 s
% 2.79/1.67 % (17844)Instructions burned: 275 (million)
% 2.79/1.67 % (17844)------------------------------
% 2.79/1.67 % (17844)------------------------------
% 2.79/1.68 % (17869)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_17732 for (2998ds/879Mi)
% 2.79/1.69 % (17870)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_17732 for (2998ds/17Mi)
% 2.79/1.70 % (17870)Instruction limit reached!
% 2.79/1.70 % (17870)------------------------------
% 2.79/1.70 % (17870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.70 % (17870)Termination reason: Unknown
% 2.79/1.70 % (17870)Termination phase: shuffling
% 2.79/1.70
% 2.79/1.70 % (17870)Memory used [KB]: 1791
% 2.79/1.70 % (17870)Time elapsed: 0.010 s
% 2.79/1.70 % (17870)Instructions burned: 19 (million)
% 2.79/1.70 % (17870)------------------------------
% 2.79/1.70 % (17870)------------------------------
% 2.79/1.72 % (17871)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_17732 for (2998ds/3Mi)
% 2.79/1.72 % (17871)Instruction limit reached!
% 2.79/1.72 % (17871)------------------------------
% 2.79/1.72 % (17871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.72 % (17871)Termination reason: Unknown
% 2.79/1.72 % (17871)Termination phase: shuffling
% 2.79/1.72
% 2.79/1.72 % (17871)Memory used [KB]: 1407
% 2.79/1.72 % (17871)Time elapsed: 0.003 s
% 2.79/1.72 % (17871)Instructions burned: 3 (million)
% 2.79/1.72 % (17871)------------------------------
% 2.79/1.72 % (17871)------------------------------
% 2.79/1.73 % (17872)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_17732 for (2998ds/30Mi)
% 2.79/1.75 % (17872)Instruction limit reached!
% 2.79/1.75 % (17872)------------------------------
% 2.79/1.75 % (17872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.75 % (17872)Termination reason: Unknown
% 2.79/1.75 % (17872)Termination phase: shuffling
% 2.79/1.75
% 2.79/1.75 % (17872)Memory used [KB]: 1918
% 2.79/1.75 % (17872)Time elapsed: 0.015 s
% 2.79/1.75 % (17872)Instructions burned: 30 (million)
% 2.79/1.75 % (17872)------------------------------
% 2.79/1.75 % (17872)------------------------------
% 2.79/1.76 % (17873)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_17732 for (2997ds/127Mi)
% 2.79/1.80 % (17866)Instruction limit reached!
% 2.79/1.80 % (17866)------------------------------
% 2.79/1.80 % (17866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.80 % (17866)Termination reason: Unknown
% 2.79/1.80 % (17866)Termination phase: Saturation
% 2.79/1.80
% 2.79/1.80 % (17866)Memory used [KB]: 9338
% 2.79/1.80 % (17866)Time elapsed: 0.162 s
% 2.79/1.80 % (17866)Instructions burned: 377 (million)
% 2.79/1.80 % (17866)------------------------------
% 2.79/1.80 % (17866)------------------------------
% 2.79/1.82 % (17875)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_17732 for (2997ds/100Mi)
% 2.79/1.82 % (17873)Instruction limit reached!
% 2.79/1.82 % (17873)------------------------------
% 2.79/1.82 % (17873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.82 % (17873)Termination reason: Unknown
% 2.79/1.82 % (17873)Termination phase: Property scanning
% 2.79/1.82
% 2.79/1.82 % (17873)Memory used [KB]: 2558
% 2.79/1.82 % (17873)Time elapsed: 0.057 s
% 2.79/1.82 % (17873)Instructions burned: 127 (million)
% 2.79/1.82 % (17873)------------------------------
% 2.79/1.82 % (17873)------------------------------
% 2.79/1.84 % (17876)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_17732 for (2997ds/3Mi)
% 2.79/1.84 % (17876)Instruction limit reached!
% 2.79/1.84 % (17876)------------------------------
% 2.79/1.84 % (17876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.84 % (17876)Termination reason: Unknown
% 2.79/1.84 % (17876)Termination phase: shuffling
% 2.79/1.84
% 2.79/1.84 % (17876)Memory used [KB]: 1407
% 2.79/1.84 % (17876)Time elapsed: 0.003 s
% 2.79/1.84 % (17876)Instructions burned: 3 (million)
% 2.79/1.84 % (17876)------------------------------
% 2.79/1.84 % (17876)------------------------------
% 2.79/1.85 % (17877)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_17732 for (2996ds/20Mi)
% 2.79/1.86 % (17875)Instruction limit reached!
% 2.79/1.86 % (17875)------------------------------
% 2.79/1.86 % (17875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.86 % (17875)Termination reason: Unknown
% 2.79/1.86 % (17875)Termination phase: Function definition elimination
% 2.79/1.86
% 2.79/1.86 % (17875)Memory used [KB]: 2558
% 2.79/1.86 % (17875)Time elapsed: 0.046 s
% 2.79/1.86 % (17875)Instructions burned: 100 (million)
% 2.79/1.86 % (17875)------------------------------
% 2.79/1.86 % (17875)------------------------------
% 2.79/1.86 % (17877)Instruction limit reached!
% 2.79/1.86 % (17877)------------------------------
% 2.79/1.86 % (17877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.86 % (17877)Termination reason: Unknown
% 2.79/1.86 % (17877)Termination phase: shuffling
% 2.79/1.86
% 2.79/1.86 % (17877)Memory used [KB]: 1791
% 2.79/1.86 % (17877)Time elapsed: 0.012 s
% 2.79/1.86 % (17877)Instructions burned: 21 (million)
% 2.79/1.86 % (17877)------------------------------
% 2.79/1.86 % (17877)------------------------------
% 2.79/1.86 % (17862)First to succeed.
% 2.79/1.87 % (17862)Refutation found. Thanks to Tanya!
% 2.79/1.87 % SZS status Theorem for DTF2THF_17732
% 2.79/1.87 % SZS output start Proof for DTF2THF_17732
% 2.79/1.87 thf(func_def_0, type, in: $i > $i > $o).
% 2.79/1.87 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.79/1.87 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.79/1.87 thf(func_def_8, type, powerset: $i > $i).
% 2.79/1.87 thf(func_def_10, type, setunion: $i > $i).
% 2.79/1.87 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.79/1.87 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_26, type, prop2set: $o > $i).
% 2.79/1.87 thf(func_def_36, type, nonempty: $i > $o).
% 2.79/1.87 thf(func_def_69, type, set2prop: $i > $o).
% 2.79/1.87 thf(func_def_88, type, subset: $i > $i > $o).
% 2.79/1.87 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.79/1.87 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.79/1.87 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.79/1.87 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.79/1.87 thf(func_def_135, type, regular: $i > $o).
% 2.79/1.87 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.79/1.87 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.79/1.87 thf(func_def_153, type, iskpair: $i > $o).
% 2.79/1.87 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.79/1.87 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.79/1.87 thf(func_def_177, type, singleton: $i > $o).
% 2.79/1.87 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.79/1.87 thf(func_def_184, type, atmost1p: $i > $o).
% 2.79/1.87 thf(func_def_185, type, atleast2p: $i > $o).
% 2.79/1.87 thf(func_def_186, type, atmost2p: $i > $o).
% 2.79/1.87 thf(func_def_187, type, upairsetp: $i > $o).
% 2.79/1.87 thf(func_def_191, type, kfst: $i > $i).
% 2.79/1.87 thf(func_def_203, type, ksnd: $i > $i).
% 2.79/1.87 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.79/1.87 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.79/1.87 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.79/1.87 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.79/1.87 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.79/1.87 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.79/1.87 thf(func_def_295, type, sP0: $i > $i > $i > $i > $o).
% 2.79/1.87 thf(func_def_297, type, sP2: $i > $i > $o).
% 2.79/1.87 thf(func_def_298, type, sP3: $i > $o).
% 2.79/1.87 thf(func_def_299, type, sP4: $i > $i > $o).
% 2.79/1.87 thf(func_def_300, type, sP5: $i > $i > $o).
% 2.79/1.87 thf(func_def_303, type, sP8: $i > $i > $i > $o).
% 2.79/1.87 thf(func_def_305, type, sP10: $i > $o > $i > $i > $o).
% 2.79/1.87 thf(func_def_309, type, sK14: $i > $i).
% 2.79/1.87 thf(func_def_310, type, sK15: ($i > $i) > $i > $i > $i).
% 2.79/1.87 thf(func_def_317, type, sK22: $i > $i > $o).
% 2.79/1.87 thf(func_def_333, type, sK38: $i > $i).
% 2.79/1.87 thf(func_def_340, type, sK45: $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_344, type, sK49: $i > $o).
% 2.79/1.87 thf(func_def_352, type, sK57: $i > $o).
% 2.79/1.87 thf(func_def_357, type, sK62: $i > $i > $o).
% 2.79/1.87 thf(func_def_389, type, sK94: $i > $i).
% 2.79/1.87 thf(func_def_391, type, sK96: ($i > $i) > $i > $i > $i).
% 2.79/1.87 thf(func_def_402, type, sK107: $i > $i).
% 2.79/1.87 thf(func_def_405, type, sK110: $i > $i).
% 2.79/1.87 thf(func_def_406, type, sK111: ($i > $o) > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_407, type, sK112: ($i > $o) > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_408, type, sK113: $i > $o).
% 2.79/1.87 thf(func_def_409, type, sK114: $i > $o).
% 2.79/1.87 thf(func_def_410, type, sK115: $i > $o).
% 2.79/1.87 thf(func_def_416, type, sK121: $i > $i > $i).
% 2.79/1.87 thf(func_def_428, type, sK133: $i > $i > $i).
% 2.79/1.87 thf(func_def_444, type, sK149: $i > $i > $o).
% 2.79/1.87 thf(func_def_449, type, sK154: $i > $i > $i).
% 2.79/1.87 thf(func_def_463, type, sK168: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_464, type, sK169: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_465, type, sK170: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_466, type, sK171: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_497, type, sK202: $i > $i > $o).
% 2.79/1.87 thf(func_def_509, type, sK214: $i > $i > $i).
% 2.79/1.87 thf(func_def_510, type, sK215: $i > $i > $i).
% 2.79/1.87 thf(func_def_511, type, sK216: $i > $i > $i).
% 2.79/1.87 thf(func_def_512, type, sK217: $i > $i > $i).
% 2.79/1.87 thf(func_def_513, type, sK218: $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_514, type, sK219: $i > $i).
% 2.79/1.87 thf(func_def_515, type, sK220: $i > $i).
% 2.79/1.87 thf(func_def_516, type, sK221: $i > $i).
% 2.79/1.87 thf(func_def_517, type, sK222: $i > $i).
% 2.79/1.87 thf(func_def_518, type, sK223: $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_519, type, sK224: $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_520, type, sK225: $i > $i > $i).
% 2.79/1.87 thf(func_def_521, type, sK226: $i > $i > $i).
% 2.79/1.87 thf(func_def_522, type, sK227: $i > $i > $i).
% 2.79/1.87 thf(func_def_524, type, sK229: $i > $i).
% 2.79/1.87 thf(func_def_525, type, sK230: $i > $i).
% 2.79/1.87 thf(func_def_526, type, sK231: $i > $i).
% 2.79/1.87 thf(func_def_527, type, sK232: $i > $i > $i).
% 2.79/1.87 thf(func_def_529, type, sK234: $i > $i > ($i > $i) > $i).
% 2.79/1.87 thf(func_def_530, type, sK235: $i > $i).
% 2.79/1.87 thf(func_def_556, type, sK261: $i > $i > $i).
% 2.79/1.87 thf(func_def_576, type, sK281: $i > $o).
% 2.79/1.87 thf(func_def_578, type, sK283: ($i > $o) > $i > $i).
% 2.79/1.87 thf(func_def_584, type, sK289: $i > $i).
% 2.79/1.87 thf(func_def_587, type, sK292: $i > $o).
% 2.79/1.87 thf(func_def_590, type, sK295: $i > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_624, type, sK329: $i > $i > $i).
% 2.79/1.87 thf(func_def_637, type, sK342: $i > $o).
% 2.79/1.87 thf(func_def_645, type, sK350: $i > $o).
% 2.79/1.87 thf(func_def_648, type, sK353: $i > $o).
% 2.79/1.87 thf(func_def_651, type, sK356: $i > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_652, type, sK357: $i > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_653, type, sK358: $i > $o).
% 2.79/1.87 thf(func_def_658, type, sK363: $i > $i > $i).
% 2.79/1.87 thf(func_def_667, type, sK372: ($i > $o) > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_668, type, sK373: ($i > $o) > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_672, type, sK377: $i > $o).
% 2.79/1.87 thf(func_def_686, type, sK391: $i > $i > $o).
% 2.79/1.87 thf(func_def_701, type, sK406: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_709, type, sK414: $i > $i > $o).
% 2.79/1.87 thf(func_def_719, type, sK424: ($i > $o) > $i > $i > $i).
% 2.79/1.87 thf(func_def_722, type, sK427: $i > $o).
% 2.79/1.87 thf(func_def_732, type, sK437: $i > $o).
% 2.79/1.87 thf(func_def_751, type, sK456: $i > $i).
% 2.79/1.87 thf(func_def_754, type, sK459: $i > $i > ($i > $i) > $i).
% 2.79/1.87 thf(func_def_765, type, sK470: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_770, type, sK475: $i > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_771, type, sK476: $i > $o).
% 2.79/1.87 thf(func_def_773, type, sK478: $i > $o).
% 2.79/1.87 thf(func_def_781, type, sK486: $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_785, type, sK490: $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_788, type, sK493: $i > $i).
% 2.79/1.87 thf(func_def_793, type, sK498: $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_797, type, sK502: $i > $o).
% 2.79/1.87 thf(func_def_800, type, sK505: $i > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_806, type, sK511: $i > $i > $i).
% 2.79/1.87 thf(func_def_809, type, sK514: $i > $i > $i).
% 2.79/1.87 thf(func_def_810, type, sK515: $i > $i > $i).
% 2.79/1.87 thf(func_def_813, type, sK518: $i > $o).
% 2.79/1.87 thf(func_def_816, type, sK521: $i > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_825, type, sK530: $i > $o).
% 2.79/1.87 thf(func_def_827, type, sK532: ($i > $o) > $i).
% 2.79/1.87 thf(func_def_828, type, sK533: ($i > $o) > $i).
% 2.79/1.87 thf(func_def_831, type, sK536: $i > $o).
% 2.79/1.87 thf(func_def_833, type, sK538: ($i > $o) > $i > $i).
% 2.79/1.87 thf(func_def_851, type, sK556: ($i > $o) > $i > $i).
% 2.79/1.87 thf(func_def_853, type, sK558: $i > $o).
% 2.79/1.87 thf(func_def_855, type, sK560: ($i > $o) > $i).
% 2.79/1.87 thf(func_def_856, type, sK561: $i > $o).
% 2.79/1.87 thf(func_def_857, type, sK562: $i > $i).
% 2.79/1.87 thf(func_def_859, type, sK564: $i > $o).
% 2.79/1.87 thf(func_def_861, type, sK566: ($i > $o) > $i > $i).
% 2.79/1.87 thf(func_def_862, type, sK567: ($i > $o) > $i > $i).
% 2.79/1.87 thf(func_def_865, type, sK570: $o > $i > $i > $i).
% 2.79/1.87 thf(func_def_871, type, sK576: $i > $o).
% 2.79/1.87 thf(func_def_874, type, sK579: $i > $i > $o).
% 2.79/1.87 thf(func_def_885, type, sK590: $i > $o).
% 2.79/1.87 thf(func_def_887, type, sK592: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_894, type, sK599: $i > $i).
% 2.79/1.87 thf(func_def_902, type, sK607: $i > $o).
% 2.79/1.87 thf(func_def_904, type, sK609: $i > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_905, type, sK610: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.79/1.87 thf(func_def_906, type, sK611: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.79/1.87 thf(func_def_909, type, sK614: $i > $o).
% 2.79/1.87 thf(func_def_910, type, sK615: $i > $o).
% 2.79/1.87 thf(func_def_919, type, sK624: $i > $i).
% 2.79/1.87 thf(func_def_922, type, sK627: $i > $i > ($i > $i) > $i).
% 2.79/1.87 thf(func_def_923, type, sK628: $i > $i).
% 2.79/1.87 thf(func_def_932, type, sK637: $i > $i).
% 2.79/1.87 thf(func_def_933, type, sK638: ($i > $i) > $i > $i > $i).
% 2.79/1.87 thf(func_def_951, type, sK656: ($i > $o) > $i).
% 2.79/1.87 thf(func_def_952, type, sK657: $i > $o).
% 2.79/1.87 thf(func_def_960, type, sK665: $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_961, type, sK666: $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_962, type, sK667: ($i > $o) > $i).
% 2.79/1.87 thf(func_def_963, type, sK668: $i > $o).
% 2.79/1.87 thf(func_def_964, type, sK669: $i > $i).
% 2.79/1.87 thf(func_def_967, type, sK672: $i > $o).
% 2.79/1.87 thf(func_def_980, type, sK685: $i > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_981, type, sK686: $i > $o).
% 2.79/1.87 thf(func_def_991, type, sK696: $i > $i > $o).
% 2.79/1.87 thf(func_def_993, type, sK698: $i > $i).
% 2.79/1.87 thf(func_def_994, type, sK699: $i > $i).
% 2.79/1.87 thf(func_def_995, type, sK700: $i > ($i > $i > $o) > $i).
% 2.79/1.87 thf(func_def_996, type, sK701: $i > $i > ($i > $i > $o) > $i).
% 2.79/1.87 thf(func_def_997, type, sK702: $i > ($i > $i > $o) > $i).
% 2.79/1.87 thf(func_def_1003, type, sK708: ($i > $o) > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_1004, type, sK709: ($i > $o) > ($i > $o) > $i).
% 2.79/1.87 thf(func_def_1005, type, sK710: $i > $o).
% 2.79/1.87 thf(func_def_1006, type, sK711: $i > $o).
% 2.79/1.87 thf(func_def_1020, type, sK725: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_1021, type, sK726: $i > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_1026, type, sK731: $i > $o).
% 2.79/1.87 thf(func_def_1039, type, sK744: $i > $i > $i).
% 2.79/1.87 thf(func_def_1050, type, sK755: $i > $o).
% 2.79/1.87 thf(func_def_1057, type, sK762: ($i > $o) > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_1058, type, sK763: ($i > $o) > $i > $i > $i > $i).
% 2.79/1.87 thf(func_def_1062, type, sK767: $i > $o).
% 2.79/1.87 thf(func_def_1068, type, ph773: !>[X0: $tType]:(X0)).
% 2.79/1.87 thf(f5932,plain,(
% 2.79/1.87 $false),
% 2.79/1.87 inference(avatar_sat_refutation,[status(thm)],[f3868,f5923])).
% 2.79/1.87 thf(f5923,plain,(
% 2.79/1.87 ~spl772_11),
% 2.79/1.87 inference(avatar_contradiction_clause,[status(thm)],[f5922])).
% 2.79/1.87 thf(f5922,plain,(
% 2.79/1.87 $false | ~spl772_11),
% 2.79/1.87 inference(trivial_inequality_removal,[status(thm)],[f5913])).
% 2.79/1.87 thf(f5913,plain,(
% 2.79/1.87 ($true != $true) | ~spl772_11),
% 2.79/1.87 inference(superposition,[status(thm)],[f3901,f2790])).
% 2.79/1.87 thf(f2790,plain,(
% 2.79/1.87 ((in @ sK251 @ sK249) = $true)),
% 2.79/1.87 inference(cnf_transformation,[status(thm)],[f1443])).
% 2.79/1.87 thf(f1443,plain,(
% 2.79/1.87 (setunionsingleton1 = $true) & (powersetE = $true) & (dpsetconstrEL2 = $true) & (ex1I = $true) & (brelnall2 = $true) & (dpsetconstrI = $true) & (ex1I2 = $true) & (setukpairIL = $true) & (emptyI = $true) & (subbreln = $true) & (descr__Cong = $true) & (exuI2 = $true) & (cartprodsndpairEq = $true) & (binintersectT_lem = $true) & (notequalI2 = $true) & (($true = (in @ sK249 @ (powerset @ sK250))) & (((in @ sK251 @ sK249) = $true) & ((in @ sK251 @ (setminus @ sK250 @ (setminus @ sK250 @ sK249))) != $true) & ((in @ sK251 @ sK250) = $true))) & (setext = $true) & (exuI3 = $true) & (emptysetAx = $true) & (prop2setE = $true) & (nonemptyE1 = $true) & (subsetRefl = $true) & (binunionEcases = $true) & (dpsetconstrEL1 = $true) & (setukpairinjR2 = $true) & (theprop = $true) & (beta2 = $true) & (setadjoinSub = $true) & (kfstpairEq = $true) & (binintersectTERcontra = $true) & (powersetTI1 = $true) & (singletonsubset = $true) & (setminusERneg = $true) & (binintersectER = $true) & (ifSingleton = $true) & (upairinpowunion = $true) & (secondinupair = $true) & (complementTE1 = $true) & (funcext = $true) & (ap2apEq1 = $true) & (dsetconstrER = $true) & (powersetAx = $true) & (binunionIR = $true) & (setminusLsub = $true) & (sepInPowerset = $true) & (subsetemptysetimpeq = $true) & (cartprodmempaircEq = $true) & (inCongP = $true) & (eta1 = $true) & (cartprodpairmemER = $true) & (cartprodsndin = $true) & (iftrue = $true) & (powersetE1 = $true) & (funcextLem = $true) & (binintersectSubset3 = $true) & (binintersectSubset2 = $true) & (infuncsetfunc = $true) & (emptyInPowerset = $true) & (kfstsingleton = $true) & (contrasubsetT1 = $true) & (setminusILneg = $true) & (subsetTI = $true) & (funcGraphProp4 = $true) & (cartprodmempair = $true) & (setminusIRneg = $true) & (setminusELneg = $true) & (eqbreln = $true) & (subPowSU = $true) & (prop2setI = $true) & (setextT = $true) & (complementTI1 = $true) & (iftrueorfalse = $true) & (upairsubunion = $true) & (iftrueProp1 = $true) & (notequalI1 = $true) & (emptyE1 = $true) & (upairsetE = $true) & (singletoninpowerset = $true) & (iffalseProp2 = $true) & (contrasubsetT3 = $true) & (setadjoinIL = $true) & (noeltsimpempty = $true) & (setadjoinSub2 = $true) & (nonemptyImpWitness = $true) & (setminusT_lem = $true) & (subsetI1 = $true) & (upairset2E = $true) & (setadjoinIR = $true) & (descrp = $true) & (quantDeMorgan2 = $true) & (ex1E1 = $true) & (ifp = $true) & (setukpairinjL2 = $true) & (setbeta = $true) & (setextsub = $true) & (subsetI2 = $true) & (exuEu = $true) & (app = $true) & (symdiffIneg1 = $true) & (ap2p = $true) & (notinemptyset = $true) & (singletoninpowunion = $true) & (funcGraphProp3 = $true) & (eqinunit = $true) & (setukpairinjR1 = $true) & (bs114d = $true) & (upairset2IR = $true) & (setadjoin__Cong = $true) & (dpsetconstrSub = $true) & (wellorderingAx = $true) & (binunionRsub = $true) & (nonemptyI = $true) & (exuE2 = $true) & (ksndpairEq = $true) & (binintersectTELcontra = $true) & (eqimpsubset2 = $true) & (kpairiskpair = $true) & (powersetTE1 = $true) & (singletonsuniq = $true) & (nonemptyI1 = $true) & (funcGraphProp2 = $true) & (singletonprop = $true) & (dpsetconstrER = $true) & (sepSubset = $true) & (cartprodpairsurjEq = $true) & (powerset__Cong = $true) & (binunionLsub = $true) & (notinsingleton = $true) & (exuE3e = $true) & (subset2powerset = $true) & (inPowerset = $true) & (setunionAx = $true) & (kpairsurjEq = $true) & (ap2apEq2 = $true) & (disjointsetsI1 = $true) & (setextAx = $true) & (setOfPairsIsBReln = $true) & (singletonsswitch = $true) & (exuE3u = $true) & (subsetE = $true) & (setunion__Cong = $true) & (symdiffIneg2 = $true) & (powersetI1 = $true) & (iffalseProp1 = $true) & (ex1E2 = $true) & (funcImageSingleton = $true) & (setminusEL = $true) & (setukpairinjR = $true) & (subsetTrans = $true) & (funcinfuncset = $true) & (setoftrueEq = $true) & (setukpairIR = $true) & (ubforcartprodlem3 = $true) & (setunionsingleton2 = $true) & (theeq = $true) & (iffalse = $true) & (binintersectSubset5 = $true) & (dpsetconstrERa = $true) & (ubforcartprodlem2 = $true) & (contrasubsetT = $true) & (setminusER = $true) & (emptyinPowerset = $true) & (cartprodfstin = $true) & (setunionI = $true) & (ubforcartprodlem1 = $true) & (vacuousDall = $true) & (notsubsetI = $true) & (dsetconstrI = $true) & (in__Cong = $true) & (complementT_lem = $true) & (upairsetIR = $true) & (emptyset__Cong = $true) & (upairsetIL = $true) & (binunionIL = $true) & (binintersectLsub = $true) & (kpairp = $true) & (quantDeMorgan4 = $true) & (exu__Cong = $true) & (apProp = $true) & (cartprodpairmemEL = $true) & (prop2set2propI = $true) & (omega__Cong = $true) & (omegaSAx = $true) & (setukpairinjR11 = $true) & (cartprodmempair1 = $true) & (replAx = $true) & (lam2p = $true) & (setunionE = $true) & (binintersectSubset4 = $true) & (notdallE = $true) & (eqimpsubset1 = $true) & (emptysetimpfalse = $true) & (symdiffE = $true) & (foundationAx = $true) & (emptysetsubset = $true) & (setadjoinOr = $true) & (dsetconstr__Cong = $true) & (powersetsubset = $true) & (beta1 = $true) & (setukpairinjL = $true) & (setukpairinjR12 = $true) & (notdexE = $true) & (symdiffI2 = $true) & (dsetconstrEL = $true) & (emptysetE = $true) & (ksndsingleton = $true) & (lam2lamEq = $true) & (omegaIndAx = $true) & (emptyinunitempty = $true) & (setukpairinjL1 = $true) & (setminusSubset1 = $true) & (quantDeMorgan3 = $true) & (setminusI = $true) & (cartprodpairin = $true) & (lamProp = $true) & (setunionE2 = $true) & (exuI1 = $true) & (binintersectRsub = $true) & (lamp = $true) & (symdiffI1 = $true) & (setunionsingleton = $true) & (exuE1 = $true) & (funcGraphProp1 = $true) & (subsetE2 = $true) & (powersetI = $true) & (funcext2 = $true) & (brelnall1 = $true) & (setadjoinE = $true) & (binintersectEL = $true) & (cartprodfstpairEq = $true) & (iftrueProp2 = $true) & (contrasubsetT2 = $true) & (eta2 = $true) & (binintersectI = $true) & (binunionE = $true) & (quantDeMorgan1 = $true) & (omega0Ax = $true) & (uniqinunit = $true) & (powersetT_lem = $true) & (setadjoinAx = $true) & (binunionT_lem = $true) & (setminusSubset2 = $true) & (binintersectSubset1 = $true) & (upairequniteq = $true)),
% 2.79/1.87 inference(skolemisation,[status(esa),new_symbols(skolem,[sK249,sK250,sK251])],[f1059,f1442,f1441])).
% 2.79/1.87 thf(f1441,plain,(
% 2.79/1.87 ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X2 @ (setminus @ X1 @ (setminus @ X1 @ X0))) != $true) & ((in @ X2 @ X1) = $true))) => (($true = (in @ sK249 @ (powerset @ sK250))) & ? [X2 : $i] : (((in @ X2 @ sK249) = $true) & ((in @ X2 @ (setminus @ sK250 @ (setminus @ sK250 @ sK249))) != $true) & ((in @ X2 @ sK250) = $true)))),
% 2.79/1.87 introduced(definition,[],[choice_axiom])).
% 2.79/1.87 thf(f1442,plain,(
% 2.79/1.87 ? [X2 : $i] : (((in @ X2 @ sK249) = $true) & ((in @ X2 @ (setminus @ sK250 @ (setminus @ sK250 @ sK249))) != $true) & ((in @ X2 @ sK250) = $true)) => (((in @ sK251 @ sK249) = $true) & ((in @ sK251 @ (setminus @ sK250 @ (setminus @ sK250 @ sK249))) != $true) & ((in @ sK251 @ sK250) = $true))),
% 2.79/1.87 introduced(definition,[],[choice_axiom])).
% 2.79/1.87 thf(f1059,plain,(
% 2.79/1.87 (setunionsingleton1 = $true) & (powersetE = $true) & (dpsetconstrEL2 = $true) & (ex1I = $true) & (brelnall2 = $true) & (dpsetconstrI = $true) & (ex1I2 = $true) & (setukpairIL = $true) & (emptyI = $true) & (subbreln = $true) & (descr__Cong = $true) & (exuI2 = $true) & (cartprodsndpairEq = $true) & (binintersectT_lem = $true) & (notequalI2 = $true) & ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (((in @ X2 @ X0) = $true) & ((in @ X2 @ (setminus @ X1 @ (setminus @ X1 @ X0))) != $true) & ((in @ X2 @ X1) = $true))) & (setext = $true) & (exuI3 = $true) & (emptysetAx = $true) & (prop2setE = $true) & (nonemptyE1 = $true) & (subsetRefl = $true) & (binunionEcases = $true) & (dpsetconstrEL1 = $true) & (setukpairinjR2 = $true) & (theprop = $true) & (beta2 = $true) & (setadjoinSub = $true) & (kfstpairEq = $true) & (binintersectTERcontra = $true) & (powersetTI1 = $true) & (singletonsubset = $true) & (setminusERneg = $true) & (binintersectER = $true) & (ifSingleton = $true) & (upairinpowunion = $true) & (secondinupair = $true) & (complementTE1 = $true) & (funcext = $true) & (ap2apEq1 = $true) & (dsetconstrER = $true) & (powersetAx = $true) & (binunionIR = $true) & (setminusLsub = $true) & (sepInPowerset = $true) & (subsetemptysetimpeq = $true) & (cartprodmempaircEq = $true) & (inCongP = $true) & (eta1 = $true) & (cartprodpairmemER = $true) & (cartprodsndin = $true) & (iftrue = $true) & (powersetE1 = $true) & (funcextLem = $true) & (binintersectSubset3 = $true) & (binintersectSubset2 = $true) & (infuncsetfunc = $true) & (emptyInPowerset = $true) & (kfstsingleton = $true) & (contrasubsetT1 = $true) & (setminusILneg = $true) & (subsetTI = $true) & (funcGraphProp4 = $true) & (cartprodmempair = $true) & (setminusIRneg = $true) & (setminusELneg = $true) & (eqbreln = $true) & (subPowSU = $true) & (prop2setI = $true) & (setextT = $true) & (complementTI1 = $true) & (iftrueorfalse = $true) & (upairsubunion = $true) & (iftrueProp1 = $true) & (notequalI1 = $true) & (emptyE1 = $true) & (upairsetE = $true) & (singletoninpowerset = $true) & (iffalseProp2 = $true) & (contrasubsetT3 = $true) & (setadjoinIL = $true) & (noeltsimpempty = $true) & (setadjoinSub2 = $true) & (nonemptyImpWitness = $true) & (setminusT_lem = $true) & (subsetI1 = $true) & (upairset2E = $true) & (setadjoinIR = $true) & (descrp = $true) & (quantDeMorgan2 = $true) & (ex1E1 = $true) & (ifp = $true) & (setukpairinjL2 = $true) & (setbeta = $true) & (setextsub = $true) & (subsetI2 = $true) & (exuEu = $true) & (app = $true) & (symdiffIneg1 = $true) & (ap2p = $true) & (notinemptyset = $true) & (singletoninpowunion = $true) & (funcGraphProp3 = $true) & (eqinunit = $true) & (setukpairinjR1 = $true) & (bs114d = $true) & (upairset2IR = $true) & (setadjoin__Cong = $true) & (dpsetconstrSub = $true) & (wellorderingAx = $true) & (binunionRsub = $true) & (nonemptyI = $true) & (exuE2 = $true) & (ksndpairEq = $true) & (binintersectTELcontra = $true) & (eqimpsubset2 = $true) & (kpairiskpair = $true) & (powersetTE1 = $true) & (singletonsuniq = $true) & (nonemptyI1 = $true) & (funcGraphProp2 = $true) & (singletonprop = $true) & (dpsetconstrER = $true) & (sepSubset = $true) & (cartprodpairsurjEq = $true) & (powerset__Cong = $true) & (binunionLsub = $true) & (notinsingleton = $true) & (exuE3e = $true) & (subset2powerset = $true) & (inPowerset = $true) & (setunionAx = $true) & (kpairsurjEq = $true) & (ap2apEq2 = $true) & (disjointsetsI1 = $true) & (setextAx = $true) & (setOfPairsIsBReln = $true) & (singletonsswitch = $true) & (exuE3u = $true) & (subsetE = $true) & (setunion__Cong = $true) & (symdiffIneg2 = $true) & (powersetI1 = $true) & (iffalseProp1 = $true) & (ex1E2 = $true) & (funcImageSingleton = $true) & (setminusEL = $true) & (setukpairinjR = $true) & (subsetTrans = $true) & (funcinfuncset = $true) & (setoftrueEq = $true) & (setukpairIR = $true) & (ubforcartprodlem3 = $true) & (setunionsingleton2 = $true) & (theeq = $true) & (iffalse = $true) & (binintersectSubset5 = $true) & (dpsetconstrERa = $true) & (ubforcartprodlem2 = $true) & (contrasubsetT = $true) & (setminusER = $true) & (emptyinPowerset = $true) & (cartprodfstin = $true) & (setunionI = $true) & (ubforcartprodlem1 = $true) & (vacuousDall = $true) & (notsubsetI = $true) & (dsetconstrI = $true) & (in__Cong = $true) & (complementT_lem = $true) & (upairsetIR = $true) & (emptyset__Cong = $true) & (upairsetIL = $true) & (binunionIL = $true) & (binintersectLsub = $true) & (kpairp = $true) & (quantDeMorgan4 = $true) & (exu__Cong = $true) & (apProp = $true) & (cartprodpairmemEL = $true) & (prop2set2propI = $true) & (omega__Cong = $true) & (omegaSAx = $true) & (setukpairinjR11 = $true) & (cartprodmempair1 = $true) & (replAx = $true) & (lam2p = $true) & (setunionE = $true) & (binintersectSubset4 = $true) & (notdallE = $true) & (eqimpsubset1 = $true) & (emptysetimpfalse = $true) & (symdiffE = $true) & (foundationAx = $true) & (emptysetsubset = $true) & (setadjoinOr = $true) & (dsetconstr__Cong = $true) & (powersetsubset = $true) & (beta1 = $true) & (setukpairinjL = $true) & (setukpairinjR12 = $true) & (notdexE = $true) & (symdiffI2 = $true) & (dsetconstrEL = $true) & (emptysetE = $true) & (ksndsingleton = $true) & (lam2lamEq = $true) & (omegaIndAx = $true) & (emptyinunitempty = $true) & (setukpairinjL1 = $true) & (setminusSubset1 = $true) & (quantDeMorgan3 = $true) & (setminusI = $true) & (cartprodpairin = $true) & (lamProp = $true) & (setunionE2 = $true) & (exuI1 = $true) & (binintersectRsub = $true) & (lamp = $true) & (symdiffI1 = $true) & (setunionsingleton = $true) & (exuE1 = $true) & (funcGraphProp1 = $true) & (subsetE2 = $true) & (powersetI = $true) & (funcext2 = $true) & (brelnall1 = $true) & (setadjoinE = $true) & (binintersectEL = $true) & (cartprodfstpairEq = $true) & (iftrueProp2 = $true) & (contrasubsetT2 = $true) & (eta2 = $true) & (binintersectI = $true) & (binunionE = $true) & (quantDeMorgan1 = $true) & (omega0Ax = $true) & (uniqinunit = $true) & (powersetT_lem = $true) & (setadjoinAx = $true) & (binunionT_lem = $true) & (setminusSubset2 = $true) & (binintersectSubset1 = $true) & (upairequniteq = $true)),
% 2.79/1.87 inference(flattening,[status(thm)],[f1058])).
% 2.79/1.87 thf(f1058,plain,(
% 2.79/1.87 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : ((((in @ X2 @ (setminus @ X1 @ (setminus @ X1 @ X0))) != $true) & ((in @ X2 @ X0) = $true)) & ((in @ X2 @ X1) = $true)) & ((in @ X0 @ (powerset @ X1)) = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.79/1.87 inference(ennf_transformation,[status(thm)],[f732])).
% 2.79/1.87 thf(f732,plain,(
% 2.79/1.87 ~((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) => ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (((in @ X2 @ X1) = $true) => (((in @ X2 @ X0) = $true) => ((in @ X2 @ (setminus @ X1 @ (setminus @ X1 @ X0))) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.79/1.87 inference(fool_elimination,[status(thm)],[f731])).
% 2.79/1.87 thf(f731,plain,(
% 2.79/1.87 ~(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 => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ X1) => ((in @ X2 @ X0) => (in @ X2 @ (setminus @ X1 @ (setminus @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.79/1.87 inference(rectify,[status(thm)],[f249])).
% 2.79/1.87 thf(f249,negated_conjecture,(
% 2.79/1.87 ~(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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ X11) => (in @ X1 @ (setminus @ X3 @ (setminus @ X3 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.79/1.87 inference(negated_conjecture,[status(cth)],[f248])).
% 2.79/1.87 thf(f248,conjecture,(
% 2.79/1.87 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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ X11) => (in @ X1 @ (setminus @ X3 @ (setminus @ X3 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.79/1.87 file('/export/starexec/sandbox2/tmp/tmp.4HrzPZvLLL/DTF2THF_17732.p',doubleComplementI1)).
% 2.79/1.87 thf(f3901,plain,(
% 2.79/1.87 ((in @ sK251 @ sK249) != $true) | ~spl772_11),
% 2.79/1.87 inference(trivial_inequality_removal,[status(thm)],[f3900])).
% 2.79/1.87 thf(f3900,plain,(
% 2.79/1.87 ((in @ sK251 @ sK249) != $true) | ($true != $true) | ~spl772_11),
% 2.79/1.87 inference(forward_demodulation,[status(thm)],[f3893,f2643])).
% 2.79/1.87 thf(f2643,plain,(
% 2.79/1.87 (setminusER = $true)),
% 2.79/1.87 inference(cnf_transformation,[status(thm)],[f1443])).
% 2.79/1.87 thf(f3893,plain,(
% 2.79/1.87 ((in @ sK251 @ sK249) != $true) | (setminusER != $true) | ~spl772_11),
% 2.79/1.87 inference(trivial_inequality_removal,[status(thm)],[f3882])).
% 2.79/1.87 thf(f3882,plain,(
% 2.79/1.87 ((in @ sK251 @ sK249) != $true) | (setminusER != $true) | ($true != $true) | ~spl772_11),
% 2.79/1.87 inference(superposition,[status(thm)],[f2463,f3813])).
% 2.79/1.87 thf(f3813,plain,(
% 2.79/1.87 ((in @ sK251 @ (setminus @ sK250 @ sK249)) = $true) | ~spl772_11),
% 2.79/1.87 inference(avatar_component_clause,[status(thm)],[f3811])).
% 2.79/1.87 thf(f3811,definition,(
% 2.79/1.87 spl772_11 <=> ((in @ sK251 @ (setminus @ sK250 @ sK249)) = $true)),
% 2.79/1.87 introduced(definition,[new_symbols(naming,[spl772_11])],[avatar_definition])).
% 2.79/1.87 thf(f2463,plain,(
% 2.79/1.87 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) != $true) | (setminusER != $true) | ((in @ X1 @ X0) != $true)) )),
% 2.79/1.87 inference(cnf_transformation,[status(thm)],[f1332])).
% 2.79/1.87 thf(f1332,plain,(
% 2.79/1.87 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) != $true) | ((in @ X1 @ (setminus @ X2 @ X0)) != $true)) | (setminusER != $true)) & ((setminusER = $true) | (((in @ sK177 @ sK176) = $true) & ((in @ sK177 @ (setminus @ sK178 @ sK176)) = $true)))),
% 2.79/1.87 inference(skolemisation,[status(esa),new_symbols(skolem,[sK176,sK177,sK178])],[f1330,f1331])).
% 2.79/1.87 thf(f1331,plain,(
% 2.79/1.87 ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) = $true) & ((in @ X4 @ (setminus @ X5 @ X3)) = $true)) => (((in @ sK177 @ sK176) = $true) & ((in @ sK177 @ (setminus @ sK178 @ sK176)) = $true))),
% 2.79/1.87 introduced(definition,[],[choice_axiom])).
% 2.79/1.87 thf(f1330,plain,(
% 2.79/1.87 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) != $true) | ((in @ X1 @ (setminus @ X2 @ X0)) != $true)) | (setminusER != $true)) & ((setminusER = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) = $true) & ((in @ X4 @ (setminus @ X5 @ X3)) = $true)))),
% 2.79/1.87 inference(rectify,[status(thm)],[f1329])).
% 2.79/1.87 thf(f1329,plain,(
% 2.79/1.87 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) != $true) | ((in @ X1 @ (setminus @ X2 @ X0)) != $true)) | (setminusER != $true)) & ((setminusER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) = $true) & ((in @ X1 @ (setminus @ X2 @ X0)) = $true)))),
% 2.79/1.87 inference(nnf_transformation,[status(thm)],[f820])).
% 2.79/1.87 thf(f820,plain,(
% 2.79/1.87 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) != $true) | ((in @ X1 @ (setminus @ X2 @ X0)) != $true)) <=> (setminusER = $true)),
% 2.79/1.87 inference(ennf_transformation,[status(thm)],[f757])).
% 2.79/1.87 thf(f757,plain,(
% 2.79/1.87 (setminusER = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) = $true) => ((in @ X1 @ X0) != $true))),
% 2.79/1.87 inference(flattening,[status(thm)],[f401])).
% 2.79/1.87 thf(f401,plain,(
% 2.79/1.87 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) = $true) => ~((in @ X1 @ X0) = $true)) <=> (setminusER = $true)),
% 2.79/1.87 inference(fool_elimination,[status(thm)],[f400])).
% 2.79/1.87 thf(f400,plain,(
% 2.79/1.87 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (setminus @ X2 @ X0)) => ~(in @ X1 @ X0)) = setminusER)),
% 2.79/1.87 inference(rectify,[status(thm)],[f122])).
% 2.79/1.87 thf(f122,axiom,(
% 2.79/1.87 (! [X4 : $i,X1 : $i,X3 : $i] : ((in @ X1 @ (setminus @ X3 @ X4)) => ~(in @ X1 @ X4)) = setminusER)),
% 2.79/1.87 file('/export/starexec/sandbox2/tmp/tmp.4HrzPZvLLL/DTF2THF_17732.p',setminusER)).
% 2.79/1.87 thf(f3868,plain,(
% 2.79/1.87 spl772_11),
% 2.79/1.87 inference(avatar_split_clause,[status(thm)],[f3805,f3811])).
% 2.79/1.87 thf(f3805,plain,(
% 2.79/1.87 ((in @ sK251 @ (setminus @ sK250 @ sK249)) = $true)),
% 2.79/1.87 inference(trivial_inequality_removal,[status(thm)],[f3804])).
% 2.79/1.87 thf(f3804,plain,(
% 2.79/1.87 ((in @ sK251 @ (setminus @ sK250 @ sK249)) = $true) | ($true != $true)),
% 2.79/1.87 inference(forward_demodulation,[status(thm)],[f3803,f2788])).
% 2.79/1.87 thf(f2788,plain,(
% 2.79/1.87 ((in @ sK251 @ sK250) = $true)),
% 2.79/1.87 inference(cnf_transformation,[status(thm)],[f1443])).
% 2.79/1.87 thf(f3803,plain,(
% 2.79/1.87 ((in @ sK251 @ (setminus @ sK250 @ sK249)) = $true) | ((in @ sK251 @ sK250) != $true)),
% 2.79/1.87 inference(trivial_inequality_removal,[status(thm)],[f3802])).
% 2.79/1.87 thf(f3802,plain,(
% 2.79/1.87 ($true != $true) | ((in @ sK251 @ (setminus @ sK250 @ sK249)) = $true) | ((in @ sK251 @ sK250) != $true)),
% 2.79/1.87 inference(forward_demodulation,[status(thm)],[f3769,f2591])).
% 2.79/1.87 thf(f2591,plain,(
% 2.79/1.87 (setminusI = $true)),
% 2.79/1.87 inference(cnf_transformation,[status(thm)],[f1443])).
% 2.79/1.87 thf(f3769,plain,(
% 2.79/1.87 (setminusI != $true) | ((in @ sK251 @ (setminus @ sK250 @ sK249)) = $true) | ((in @ sK251 @ sK250) != $true)),
% 2.79/1.87 inference(trivial_inequality_removal,[status(thm)],[f3757])).
% 2.79/1.87 thf(f3757,plain,(
% 2.79/1.87 (setminusI != $true) | ((in @ sK251 @ (setminus @ sK250 @ sK249)) = $true) | ((in @ sK251 @ sK250) != $true) | ($true != $true)),
% 2.79/1.87 inference(superposition,[status(thm)],[f2789,f2471])).
% 2.79/1.87 thf(f2471,plain,(
% 2.79/1.87 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) = $true) | ((in @ X1 @ X2) != $true) | ((in @ X1 @ X0) = $true) | (setminusI != $true)) )),
% 2.79/1.87 inference(cnf_transformation,[status(thm)],[f1340])).
% 2.79/1.87 thf(f1340,plain,(
% 2.79/1.87 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true) | ((in @ X1 @ (setminus @ X2 @ X0)) = $true)) | (setminusI != $true)) & ((setminusI = $true) | (((in @ sK183 @ sK182) != $true) & ((in @ sK183 @ sK184) = $true) & ((in @ sK183 @ (setminus @ sK184 @ sK182)) != $true)))),
% 2.79/1.87 inference(skolemisation,[status(esa),new_symbols(skolem,[sK182,sK183,sK184])],[f1338,f1339])).
% 2.79/1.87 thf(f1339,plain,(
% 2.79/1.87 ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) != $true) & ((in @ X4 @ X5) = $true) & ((in @ X4 @ (setminus @ X5 @ X3)) != $true)) => (((in @ sK183 @ sK182) != $true) & ((in @ sK183 @ sK184) = $true) & ((in @ sK183 @ (setminus @ sK184 @ sK182)) != $true))),
% 2.79/1.87 introduced(definition,[],[choice_axiom])).
% 2.79/1.87 thf(f1338,plain,(
% 2.79/1.87 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true) | ((in @ X1 @ (setminus @ X2 @ X0)) = $true)) | (setminusI != $true)) & ((setminusI = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) != $true) & ((in @ X4 @ X5) = $true) & ((in @ X4 @ (setminus @ X5 @ X3)) != $true)))),
% 2.79/1.87 inference(rectify,[status(thm)],[f1337])).
% 2.79/1.87 thf(f1337,plain,(
% 2.79/1.87 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true) | ((in @ X1 @ (setminus @ X2 @ X0)) = $true)) | (setminusI != $true)) & ((setminusI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) != $true) & ((in @ X1 @ X2) = $true) & ((in @ X1 @ (setminus @ X2 @ X0)) != $true)))),
% 2.79/1.87 inference(nnf_transformation,[status(thm)],[f804])).
% 2.79/1.87 thf(f804,plain,(
% 2.79/1.87 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X0) = $true) | ((in @ X1 @ X2) != $true) | ((in @ X1 @ (setminus @ X2 @ X0)) = $true)) <=> (setminusI = $true)),
% 2.79/1.87 inference(flattening,[status(thm)],[f803])).
% 2.79/1.87 thf(f803,plain,(
% 2.79/1.87 (setminusI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X1 @ (setminus @ X2 @ X0)) = $true) | ((in @ X1 @ X0) = $true)) | ((in @ X1 @ X2) != $true))),
% 2.79/1.87 inference(ennf_transformation,[status(thm)],[f754])).
% 2.79/1.87 thf(f754,plain,(
% 2.79/1.87 (setminusI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => (((in @ X1 @ X0) != $true) => ((in @ X1 @ (setminus @ X2 @ X0)) = $true)))),
% 2.79/1.87 inference(flattening,[status(thm)],[f373])).
% 2.79/1.87 thf(f373,plain,(
% 2.79/1.87 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => (~((in @ X1 @ X0) = $true) => ((in @ X1 @ (setminus @ X2 @ X0)) = $true))) <=> (setminusI = $true)),
% 2.79/1.87 inference(fool_elimination,[status(thm)],[f372])).
% 2.79/1.87 thf(f372,plain,(
% 2.79/1.87 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => (~(in @ X1 @ X0) => (in @ X1 @ (setminus @ X2 @ X0)))) = setminusI)),
% 2.79/1.87 inference(rectify,[status(thm)],[f120])).
% 2.79/1.87 thf(f120,axiom,(
% 2.79/1.87 (! [X4 : $i,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ X4) => (in @ X1 @ (setminus @ X3 @ X4)))) = setminusI)),
% 2.79/1.87 file('/export/starexec/sandbox2/tmp/tmp.4HrzPZvLLL/DTF2THF_17732.p',setminusI)).
% 2.79/1.87 thf(f2789,plain,(
% 2.79/1.87 ((in @ sK251 @ (setminus @ sK250 @ (setminus @ sK250 @ sK249))) != $true)),
% 2.79/1.87 inference(cnf_transformation,[status(thm)],[f1443])).
% 2.79/1.87 % SZS output end Proof for DTF2THF_17732
% 2.79/1.87 % (17862)------------------------------
% 2.79/1.87 % (17862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.79/1.87 % (17862)Termination reason: Refutation
% 2.79/1.87
% 2.79/1.87 % (17862)Memory used [KB]: 10874
% 2.79/1.87 % (17862)Time elapsed: 0.249 s
% 2.79/1.87 % (17862)Instructions burned: 564 (million)
% 2.79/1.87 % (17862)------------------------------
% 2.79/1.87 % (17862)------------------------------
% 2.79/1.87 % (17836)Success in time 0.355 s
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