%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU676^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:11 AM UTC 2026
% Result : Theorem 2.70s 1.66s
% Output : Refutation 2.70s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 8
% Syntax : Number of formulae : 45 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 3408 ( 900 equ; 0 cnn)
% Maximal formula atoms : 195 ( 75 avg)
% Number of connectives : 1933 ( 66 ~; 42 |; 611 &; 420 @)
% ( 4 <=>; 790 =>; 0 <=; 0 <~>)
% Maximal formula depth : 206 ( 35 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 358 ( 355 usr; 205 con; 0-4 aty)
% Number of variables : 109 ( 0 ^; 72 !; 37 ?; 109 :)
% 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_243,type,
sP1: $i > $i > $o ).
thf(func_def_244,type,
sP2: $i > $o ).
thf(func_def_245,type,
sP3: $i > $i > $o ).
thf(func_def_246,type,
sP4: $i > $i > $o ).
thf(func_def_249,type,
sP7: $o > $i > $i > $i > $o ).
thf(func_def_258,type,
sK16: $i > $o ).
thf(func_def_260,type,
sK18: ( $i > $o ) > $i ).
thf(func_def_261,type,
sK19: ( $i > $o ) > $i ).
thf(func_def_272,type,
sK30: $i > $o ).
thf(func_def_274,type,
sK32: ( $i > $o ) > $i > $i ).
thf(func_def_275,type,
sK33: ( $i > $o ) > $i > $i ).
thf(func_def_281,type,
sK39: $i > $i > $i ).
thf(func_def_282,type,
sK40: $i > $i > $i ).
thf(func_def_283,type,
sK41: $i > $i > $i ).
thf(func_def_284,type,
sK42: $i > $i > $i ).
thf(func_def_285,type,
sK43: $i > $i > $i ).
thf(func_def_286,type,
sK44: $i > $i > $i > $i ).
thf(func_def_287,type,
sK45: $i > $i ).
thf(func_def_288,type,
sK46: $i > $i ).
thf(func_def_289,type,
sK47: $i > $i ).
thf(func_def_290,type,
sK48: $i > $i ).
thf(func_def_291,type,
sK49: $i > $i > $i ).
thf(func_def_292,type,
sK50: $i > $i > $i > $i ).
thf(func_def_293,type,
sK51: $i > $i > $i > $i ).
thf(func_def_294,type,
sK52: $i > $i > $i ).
thf(func_def_295,type,
sK53: $i > $i > $i ).
thf(func_def_296,type,
sK54: $i > $i ).
thf(func_def_298,type,
sK56: $i > $i ).
thf(func_def_299,type,
sK57: $i > $i ).
thf(func_def_307,type,
sK65: $i > $i > $o ).
thf(func_def_308,type,
sK66: $i > $i ).
thf(func_def_309,type,
sK67: $i > $i ).
thf(func_def_310,type,
sK68: ( $i > $i > $o ) > $i > $i ).
thf(func_def_311,type,
sK69: ( $i > $i > $o ) > $i > $i ).
thf(func_def_312,type,
sK70: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_314,type,
sK72: $i > $o ).
thf(func_def_316,type,
sK74: $i > $o ).
thf(func_def_320,type,
sK78: $i > $i > $i ).
thf(func_def_325,type,
sK83: $i > $o ).
thf(func_def_337,type,
sK95: $i > $i > $i ).
thf(func_def_338,type,
sK96: $i > $o ).
thf(func_def_348,type,
sK106: $i > $o ).
thf(func_def_352,type,
sK110: $i > $i > $i ).
thf(func_def_360,type,
sK118: $i > $o ).
thf(func_def_361,type,
sK119: $i > $o ).
thf(func_def_362,type,
sK120: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_363,type,
sK121: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_364,type,
sK122: $i > $o ).
thf(func_def_367,type,
sK125: $i > ( $i > $o ) > $i ).
thf(func_def_368,type,
sK126: $i > $o ).
thf(func_def_399,type,
sK157: $i > $o ).
thf(func_def_400,type,
sK158: ( $i > $o ) > $i > $i ).
thf(func_def_416,type,
sK174: $i > $o ).
thf(func_def_419,type,
sK177: $i > $i > $o ).
thf(func_def_428,type,
sK186: $i > $i > $i ).
thf(func_def_435,type,
sK193: $i > $i > $i > $i ).
thf(func_def_436,type,
sK194: $i > $i > $i > $i ).
thf(func_def_446,type,
sK204: $i > $i > $i ).
thf(func_def_481,type,
sK239: ( $i > $o ) > $i > $i ).
thf(func_def_483,type,
sK241: $i > $o ).
thf(func_def_486,type,
sK244: $i > $o ).
thf(func_def_488,type,
sK246: ( $i > $o ) > $i > $i ).
thf(func_def_489,type,
sK247: $i > $i > $i ).
thf(func_def_490,type,
sK248: $i > $i > $i ).
thf(func_def_508,type,
sK266: $i > $o ).
thf(func_def_509,type,
sK267: $i > $i ).
thf(func_def_510,type,
sK268: ( $i > $o ) > $i ).
thf(func_def_511,type,
sK269: $i > $o ).
thf(func_def_512,type,
sK270: ( $i > $o ) > $i ).
thf(func_def_515,type,
sK273: $i > $i > $o ).
thf(func_def_525,type,
sK283: $i > $o ).
thf(func_def_526,type,
sK284: $i > $i ).
thf(func_def_527,type,
sK285: ( $i > $o ) > $i ).
thf(func_def_540,type,
sK298: $i > $i > $i ).
thf(func_def_543,type,
sK301: $i > $o ).
thf(func_def_546,type,
sK304: $i > ( $i > $o ) > $i ).
thf(func_def_547,type,
sK305: $i > ( $i > $o ) > $i ).
thf(func_def_557,type,
sK315: $i > $o ).
thf(func_def_564,type,
sK322: $i > $o ).
thf(func_def_565,type,
sK323: $i > $o ).
thf(func_def_566,type,
sK324: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_567,type,
sK325: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_570,type,
sK328: $i > $i ).
thf(func_def_571,type,
sK329: $i > $i ).
thf(func_def_581,type,
sK339: $i > $o ).
thf(func_def_588,type,
sK346: $i > $i > $o ).
thf(func_def_593,type,
sK351: $i > $i > $o ).
thf(func_def_600,type,
sK358: $i > $o ).
thf(func_def_608,type,
sK366: $i > $i ).
thf(func_def_610,type,
sK368: $i > ( $i > $o ) > $i ).
thf(func_def_611,type,
sK369: $i > $o ).
thf(func_def_618,type,
sK376: $i > $i > $o ).
thf(func_def_624,type,
sK382: $i > $i > $i ).
thf(func_def_643,type,
sK401: ( $i > $o ) > $i > $i ).
thf(func_def_645,type,
sK403: $i > $o ).
thf(func_def_650,type,
sK408: $i > $i > $o ).
thf(func_def_653,type,
sK411: $i > $i ).
thf(func_def_662,type,
sK420: $i > $i > $o ).
thf(func_def_670,type,
sK428: $i > $o ).
thf(func_def_685,type,
sK443: $i > $i > $i ).
thf(func_def_694,type,
sK452: $i > ( $i > $o ) > $i ).
thf(func_def_695,type,
sK453: $i > $o ).
thf(func_def_698,type,
sK456: $i > $i ).
thf(func_def_704,type,
sK462: $o > $i > $i > $i ).
thf(func_def_705,type,
sK463: $i > $o ).
thf(func_def_706,type,
sK464: $i > $o ).
thf(func_def_707,type,
sK465: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_708,type,
sK466: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_713,type,
sK471: $i > $i > ( $i > $o ) > $i ).
thf(func_def_714,type,
sK472: $i > $o ).
thf(func_def_748,type,
sK506: $i > $o ).
thf(func_def_756,type,
sK514: $i > $o ).
thf(func_def_758,type,
sK516: $i > ( $i > $o ) > $i ).
thf(func_def_760,type,
sK518: $i > $i ).
thf(func_def_763,type,
sK521: ( $i > $o ) > $i > $i ).
thf(func_def_765,type,
sK523: $i > $o ).
thf(func_def_778,type,
sK536: $i > $i ).
thf(func_def_802,type,
ph560:
!>[X0: $tType] : X0 ).
thf(f2823,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2822]) ).
thf(f2822,plain,
$true != $true,
inference(superposition,[status(thm)],[f2821,f2799]) ).
thf(f2799,plain,
( $true
= ( func @ sK234 @ sK233 @ sK232 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2798]) ).
thf(f2798,plain,
( ( $true != $true )
| ( $true
= ( func @ sK234 @ sK233 @ sK232 ) ) ),
inference(forward_demodulation,[status(thm)],[f2797,f2182]) ).
thf(f2182,plain,
infuncsetfunc = $true,
inference(cnf_transformation,[status(thm)],[f1224]) ).
thf(f1224,plain,
( ( dsetconstr__Cong = $true )
& ( powersetsubset = $true )
& ( setunionI = $true )
& ( in__Cong = $true )
& ( binintersectI = $true )
& ( quantDeMorgan1 = $true )
& ( upairequniteq = $true )
& ( setminusILneg = $true )
& ( disjointsetsI1 = $true )
& ( setunionE2 = $true )
& ( setunionAx = $true )
& ( binunionIL = $true )
& ( upairsetIR = $true )
& ( setukpairinjL = $true )
& ( singletoninpowunion = $true )
& ( powersetI = $true )
& ( subsetE = $true )
& ( emptysetimpfalse = $true )
& ( ex1E1 = $true )
& ( eqinunit = $true )
& ( ksndsingleton = $true )
& ( inPowerset = $true )
& ( subsetE2 = $true )
& ( binintersectSubset3 = $true )
& ( binunionIR = $true )
& ( dpsetconstrI = $true )
& ( dpsetconstrEL1 = $true )
& ( setminusER = $true )
& ( sepInPowerset = $true )
& ( ubforcartprodlem3 = $true )
& ( omegaSAx = $true )
& ( cartprodpairin = $true )
& ( noeltsimpempty = $true )
& ( quantDeMorgan3 = $true )
& ( symdiffIneg1 = $true )
& ( emptysetE = $true )
& ( binunionE = $true )
& ( setunionsingleton = $true )
& ( foundationAx = $true )
& ( ubforcartprodlem2 = $true )
& ( upairinpowunion = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectSubset1 = $true )
& ( app = $true )
& ( emptyinPowerset = $true )
& ( powerset__Cong = $true )
& ( notequalI2 = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ( infuncsetfunc = $true )
& ( setOfPairsIsBReln = $true )
& ( symdiffI2 = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjR2 = $true )
& ( binunionLsub = $true )
& ( singletonsubset = $true )
& ( setbeta = $true )
& ( notequalI1 = $true )
& ( setadjoin__Cong = $true )
& ( setadjoinSub2 = $true )
& ( bs114d = $true )
& ( emptyinunitempty = $true )
& ( upairsetE = $true )
& ( prop2setE = $true )
& ( dsetconstrI = $true )
& ( exuE1 = $true )
& ( prop2setI = $true )
& ( kfstsingleton = $true )
& ( notsubsetI = $true )
& ( eqimpsubset2 = $true )
& ( upairsetIL = $true )
& ( subset2powerset = $true )
& ( ex1I2 = $true )
& ( setadjoinAx = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR = $true )
& ( setoftrueEq = $true )
& ( setukpairinjR12 = $true )
& ( setunionE = $true )
& ( upairset2IR = $true )
& ( $true
!= ( in @ ( ap @ sK234 @ sK233 @ sK232 @ sK235 ) @ sK233 ) )
& ( $true
= ( in @ sK235 @ sK234 ) )
& ( $true
= ( in @ sK232 @ ( funcSet @ sK234 @ sK233 ) ) )
& ( cartprodpairmemEL = $true )
& ( setadjoinOr = $true )
& ( setminusSubset2 = $true )
& ( notdexE = $true )
& ( subsetRefl = $true )
& ( nonemptyImpWitness = $true )
& ( setext = $true )
& ( setadjoinE = $true )
& ( subsetI2 = $true )
& ( exuE3e = $true )
& ( exuEu = $true )
& ( setminusLsub = $true )
& ( theprop = $true )
& ( ksndpairEq = $true )
& ( apProp = $true )
& ( upairset2E = $true )
& ( singletonsuniq = $true )
& ( setunion__Cong = $true )
& ( emptyset__Cong = $true )
& ( sepSubset = $true )
& ( cartprodfstpairEq = $true )
& ( emptyI = $true )
& ( kpairsurjEq = $true )
& ( omegaIndAx = $true )
& ( cartprodmempair = $true )
& ( notinemptyset = $true )
& ( exuE3u = $true )
& ( notdallE = $true )
& ( setukpairinjR1 = $true )
& ( omega__Cong = $true )
& ( cartprodpairsurjEq = $true )
& ( prop2set2propI = $true )
& ( notinsingleton = $true )
& ( cartprodpairmemER = $true )
& ( singletonsswitch = $true )
& ( binintersectEL = $true )
& ( cartprodfstin = $true )
& ( inCongP = $true )
& ( dsetconstrEL = $true )
& ( binintersectLsub = $true )
& ( eqimpsubset1 = $true )
& ( setadjoinIR = $true )
& ( dpsetconstrEL2 = $true )
& ( binunionEcases = $true )
& ( wellorderingAx = $true )
& ( singletoninpowerset = $true )
& ( emptysetsubset = $true )
& ( setminusIRneg = $true )
& ( setminusERneg = $true )
& ( emptysetAx = $true )
& ( binunionRsub = $true )
& ( dpsetconstrERa = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinIL = $true )
& ( singletonprop = $true )
& ( binintersectRsub = $true )
& ( powersetE = $true )
& ( kpairp = $true )
& ( cartprodmempair1 = $true )
& ( binintersectSubset4 = $true )
& ( exuI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectER = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( upairsubunion = $true )
& ( powersetAx = $true )
& ( nonemptyI1 = $true )
& ( setunionsingleton1 = $true )
& ( symdiffE = $true )
& ( kpairiskpair = $true )
& ( setextsub = $true )
& ( setminusEL = $true )
& ( exuI2 = $true )
& ( setukpairIR = $true )
& ( exu__Cong = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjR11 = $true )
& ( nonemptyI = $true )
& ( symdiffIneg2 = $true )
& ( dpsetconstrSub = $true )
& ( setadjoinSub = $true )
& ( quantDeMorgan2 = $true )
& ( setukpairIL = $true )
& ( exuI3 = $true )
& ( secondinupair = $true )
& ( dsetconstrER = $true )
& ( subsetTrans = $true )
& ( subPowSU = $true )
& ( uniqinunit = $true )
& ( subsetI1 = $true )
& ( descrp = $true )
& ( nonemptyE1 = $true )
& ( vacuousDall = $true )
& ( setukpairinjL1 = $true )
& ( setunionsingleton2 = $true )
& ( ex1I = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodsndin = $true )
& ( exuE2 = $true )
& ( kfstpairEq = $true )
& ( emptyE1 = $true )
& ( powersetI1 = $true )
& ( setminusELneg = $true )
& ( omega0Ax = $true )
& ( powersetE1 = $true )
& ( emptyInPowerset = $true )
& ( setminusI = $true )
& ( setukpairinjL2 = $true )
& ( descr__Cong = $true )
& ( setminusSubset1 = $true )
& ( binintersectSubset5 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK232,sK233,sK234,sK235])],[f705,f1223,f1222]) ).
thf(f1222,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ X2 @ X1 @ X0 @ X3 ) @ X1 ) )
& ( ( in @ X3 @ X2 )
= $true ) )
& ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) ) )
=> ( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ sK234 @ sK233 @ sK232 @ X3 ) @ sK233 ) )
& ( $true
= ( in @ X3 @ sK234 ) ) )
& ( $true
= ( in @ sK232 @ ( funcSet @ sK234 @ sK233 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1223,plain,
( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ sK234 @ sK233 @ sK232 @ X3 ) @ sK233 ) )
& ( $true
= ( in @ X3 @ sK234 ) ) )
=> ( ( $true
!= ( in @ ( ap @ sK234 @ sK233 @ sK232 @ sK235 ) @ sK233 ) )
& ( $true
= ( in @ sK235 @ sK234 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f705,plain,
( ( dsetconstr__Cong = $true )
& ( powersetsubset = $true )
& ( setunionI = $true )
& ( in__Cong = $true )
& ( binintersectI = $true )
& ( quantDeMorgan1 = $true )
& ( upairequniteq = $true )
& ( setminusILneg = $true )
& ( disjointsetsI1 = $true )
& ( setunionE2 = $true )
& ( setunionAx = $true )
& ( binunionIL = $true )
& ( upairsetIR = $true )
& ( setukpairinjL = $true )
& ( singletoninpowunion = $true )
& ( powersetI = $true )
& ( subsetE = $true )
& ( emptysetimpfalse = $true )
& ( ex1E1 = $true )
& ( eqinunit = $true )
& ( ksndsingleton = $true )
& ( inPowerset = $true )
& ( subsetE2 = $true )
& ( binintersectSubset3 = $true )
& ( binunionIR = $true )
& ( dpsetconstrI = $true )
& ( dpsetconstrEL1 = $true )
& ( setminusER = $true )
& ( sepInPowerset = $true )
& ( ubforcartprodlem3 = $true )
& ( omegaSAx = $true )
& ( cartprodpairin = $true )
& ( noeltsimpempty = $true )
& ( quantDeMorgan3 = $true )
& ( symdiffIneg1 = $true )
& ( emptysetE = $true )
& ( binunionE = $true )
& ( setunionsingleton = $true )
& ( foundationAx = $true )
& ( ubforcartprodlem2 = $true )
& ( upairinpowunion = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectSubset1 = $true )
& ( app = $true )
& ( emptyinPowerset = $true )
& ( powerset__Cong = $true )
& ( notequalI2 = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ( infuncsetfunc = $true )
& ( setOfPairsIsBReln = $true )
& ( symdiffI2 = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjR2 = $true )
& ( binunionLsub = $true )
& ( singletonsubset = $true )
& ( setbeta = $true )
& ( notequalI1 = $true )
& ( setadjoin__Cong = $true )
& ( setadjoinSub2 = $true )
& ( bs114d = $true )
& ( emptyinunitempty = $true )
& ( upairsetE = $true )
& ( prop2setE = $true )
& ( dsetconstrI = $true )
& ( exuE1 = $true )
& ( prop2setI = $true )
& ( kfstsingleton = $true )
& ( notsubsetI = $true )
& ( eqimpsubset2 = $true )
& ( upairsetIL = $true )
& ( subset2powerset = $true )
& ( ex1I2 = $true )
& ( setadjoinAx = $true )
& ( symdiffI1 = $true )
& ( setukpairinjR = $true )
& ( setoftrueEq = $true )
& ( setukpairinjR12 = $true )
& ( setunionE = $true )
& ( upairset2IR = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ X2 @ X1 @ X0 @ X3 ) @ X1 ) )
& ( ( in @ X3 @ X2 )
= $true ) )
& ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) ) )
& ( cartprodpairmemEL = $true )
& ( setadjoinOr = $true )
& ( setminusSubset2 = $true )
& ( notdexE = $true )
& ( subsetRefl = $true )
& ( nonemptyImpWitness = $true )
& ( setext = $true )
& ( setadjoinE = $true )
& ( subsetI2 = $true )
& ( exuE3e = $true )
& ( exuEu = $true )
& ( setminusLsub = $true )
& ( theprop = $true )
& ( ksndpairEq = $true )
& ( apProp = $true )
& ( upairset2E = $true )
& ( singletonsuniq = $true )
& ( setunion__Cong = $true )
& ( emptyset__Cong = $true )
& ( sepSubset = $true )
& ( cartprodfstpairEq = $true )
& ( emptyI = $true )
& ( kpairsurjEq = $true )
& ( omegaIndAx = $true )
& ( cartprodmempair = $true )
& ( notinemptyset = $true )
& ( exuE3u = $true )
& ( notdallE = $true )
& ( setukpairinjR1 = $true )
& ( omega__Cong = $true )
& ( cartprodpairsurjEq = $true )
& ( prop2set2propI = $true )
& ( notinsingleton = $true )
& ( cartprodpairmemER = $true )
& ( singletonsswitch = $true )
& ( binintersectEL = $true )
& ( cartprodfstin = $true )
& ( inCongP = $true )
& ( dsetconstrEL = $true )
& ( binintersectLsub = $true )
& ( eqimpsubset1 = $true )
& ( setadjoinIR = $true )
& ( dpsetconstrEL2 = $true )
& ( binunionEcases = $true )
& ( wellorderingAx = $true )
& ( singletoninpowerset = $true )
& ( emptysetsubset = $true )
& ( setminusIRneg = $true )
& ( setminusERneg = $true )
& ( emptysetAx = $true )
& ( binunionRsub = $true )
& ( dpsetconstrERa = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinIL = $true )
& ( singletonprop = $true )
& ( binintersectRsub = $true )
& ( powersetE = $true )
& ( kpairp = $true )
& ( cartprodmempair1 = $true )
& ( binintersectSubset4 = $true )
& ( exuI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( binintersectER = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( upairsubunion = $true )
& ( powersetAx = $true )
& ( nonemptyI1 = $true )
& ( setunionsingleton1 = $true )
& ( symdiffE = $true )
& ( kpairiskpair = $true )
& ( setextsub = $true )
& ( setminusEL = $true )
& ( exuI2 = $true )
& ( setukpairIR = $true )
& ( exu__Cong = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjR11 = $true )
& ( nonemptyI = $true )
& ( symdiffIneg2 = $true )
& ( dpsetconstrSub = $true )
& ( setadjoinSub = $true )
& ( quantDeMorgan2 = $true )
& ( setukpairIL = $true )
& ( exuI3 = $true )
& ( secondinupair = $true )
& ( dsetconstrER = $true )
& ( subsetTrans = $true )
& ( subPowSU = $true )
& ( uniqinunit = $true )
& ( subsetI1 = $true )
& ( descrp = $true )
& ( nonemptyE1 = $true )
& ( vacuousDall = $true )
& ( setukpairinjL1 = $true )
& ( setunionsingleton2 = $true )
& ( ex1I = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodsndin = $true )
& ( exuE2 = $true )
& ( kfstpairEq = $true )
& ( emptyE1 = $true )
& ( powersetI1 = $true )
& ( setminusELneg = $true )
& ( omega0Ax = $true )
& ( powersetE1 = $true )
& ( emptyInPowerset = $true )
& ( setminusI = $true )
& ( setukpairinjL2 = $true )
& ( descr__Cong = $true )
& ( setminusSubset1 = $true )
& ( binintersectSubset5 = $true ) ),
inference(flattening,[status(thm)],[f704]) ).
thf(f704,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ X2 @ X1 @ X0 @ X3 ) @ X1 ) )
& ( ( in @ X3 @ X2 )
= $true ) )
& ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) ) )
& ( 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)],[f391]) ).
thf(f391,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ( $true
= ( in @ ( ap @ X2 @ X1 @ X0 @ X3 ) @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f390]) ).
thf(f390,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( funcSet @ X2 @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ ( ap @ X2 @ X1 @ X0 @ X3 ) @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f202]) ).
thf(f202,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
=> ! [X13: $i,X4: $i,X3: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( ap @ X3 @ X4 @ X13 @ X1 ) @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f201]) ).
thf(f201,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
=> ! [X13: $i,X4: $i,X3: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( ap @ X3 @ X4 @ X13 @ X1 ) @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.TbQYBRPofc/DTF2THF_6601.p',ap2p) ).
thf(f2797,plain,
( ( $true
= ( func @ sK234 @ sK233 @ sK232 ) )
| ( infuncsetfunc != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2794]) ).
thf(f2794,plain,
( ( $true
= ( func @ sK234 @ sK233 @ sK232 ) )
| ( $true != $true )
| ( infuncsetfunc != $true ) ),
inference(superposition,[status(thm)],[f2632,f2149]) ).
thf(f2149,plain,
( $true
= ( in @ sK232 @ ( funcSet @ sK234 @ sK233 ) ) ),
inference(cnf_transformation,[status(thm)],[f1224]) ).
thf(f2632,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) )
| ( $true
= ( func @ X1 @ X0 @ X2 ) )
| ( infuncsetfunc != $true ) ),
inference(cnf_transformation,[status(thm)],[f1693]) ).
thf(f1693,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X1 @ X0 @ X2 ) )
| ( $true
!= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) ) )
| ( infuncsetfunc != $true ) )
& ( ( infuncsetfunc = $true )
| ( ( $true
!= ( func @ sK538 @ sK537 @ sK539 ) )
& ( ( in @ sK539 @ ( funcSet @ sK538 @ sK537 ) )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK537,sK538,sK539])],[f1691,f1692]) ).
thf(f1692,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( func @ X4 @ X3 @ X5 )
!= $true )
& ( $true
= ( in @ X5 @ ( funcSet @ X4 @ X3 ) ) ) )
=> ( ( $true
!= ( func @ sK538 @ sK537 @ sK539 ) )
& ( ( in @ sK539 @ ( funcSet @ sK538 @ sK537 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1691,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X1 @ X0 @ X2 ) )
| ( $true
!= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) ) )
| ( infuncsetfunc != $true ) )
& ( ( infuncsetfunc = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( func @ X4 @ X3 @ X5 )
!= $true )
& ( $true
= ( in @ X5 @ ( funcSet @ X4 @ X3 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f1690]) ).
thf(f1690,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X1 @ X0 @ X2 ) )
| ( $true
!= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) ) )
| ( infuncsetfunc != $true ) )
& ( ( infuncsetfunc = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( func @ X1 @ X0 @ X2 ) )
& ( $true
= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f664]) ).
thf(f664,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X1 @ X0 @ X2 ) )
| ( $true
!= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) ) )
<=> ( infuncsetfunc = $true ) ),
inference(ennf_transformation,[status(thm)],[f317]) ).
thf(f317,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) )
=> ( $true
= ( func @ X1 @ X0 @ X2 ) ) )
<=> ( infuncsetfunc = $true ) ),
inference(fool_elimination,[status(thm)],[f316]) ).
thf(f316,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( funcSet @ X1 @ X0 ) )
=> ( func @ X1 @ X0 @ X2 ) )
= infuncsetfunc ),
inference(rectify,[status(thm)],[f200]) ).
thf(f200,axiom,
! [X4: $i,X3: $i,X13: $i] :
( ( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ( func @ X3 @ X4 @ X13 ) )
= infuncsetfunc ),
file('/export/starexec/sandbox/tmp/tmp.TbQYBRPofc/DTF2THF_6601.p',infuncsetfunc) ).
thf(f2821,plain,
( $true
!= ( func @ sK234 @ sK233 @ sK232 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2820]) ).
thf(f2820,plain,
( ( $true != $true )
| ( $true
!= ( func @ sK234 @ sK233 @ sK232 ) ) ),
inference(forward_demodulation,[status(thm)],[f2819,f2188]) ).
thf(f2188,plain,
app = $true,
inference(cnf_transformation,[status(thm)],[f1224]) ).
thf(f2819,plain,
( ( $true
!= ( func @ sK234 @ sK233 @ sK232 ) )
| ( app != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2818]) ).
thf(f2818,plain,
( ( $true
!= ( func @ sK234 @ sK233 @ sK232 ) )
| ( $true != $true )
| ( app != $true ) ),
inference(forward_demodulation,[status(thm)],[f2807,f2150]) ).
thf(f2150,plain,
( $true
= ( in @ sK235 @ sK234 ) ),
inference(cnf_transformation,[status(thm)],[f1224]) ).
thf(f2807,plain,
( ( $true
!= ( in @ sK235 @ sK234 ) )
| ( app != $true )
| ( $true
!= ( func @ sK234 @ sK233 @ sK232 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2802]) ).
thf(f2802,plain,
( ( $true
!= ( func @ sK234 @ sK233 @ sK232 ) )
| ( app != $true )
| ( $true
!= ( in @ sK235 @ sK234 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2151,f2572]) ).
thf(f2572,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ ( ap @ X5 @ X6 @ X4 @ X7 ) @ X6 ) )
| ( $true
!= ( in @ X7 @ X5 ) )
| ( app != $true )
| ( ( func @ X5 @ X6 @ X4 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1622]) ).
thf(f1622,plain,
( ( ( app = $true )
| ( ( $true
!= ( in @ ( ap @ sK497 @ sK498 @ sK496 @ sK499 ) @ sK498 ) )
& ( $true
= ( in @ sK499 @ sK497 ) )
& ( ( func @ sK497 @ sK498 @ sK496 )
= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( $true
= ( in @ ( ap @ X5 @ X6 @ X4 @ X7 ) @ X6 ) )
| ( $true
!= ( in @ X7 @ X5 ) ) )
| ( ( func @ X5 @ X6 @ X4 )
!= $true ) )
| ( app != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK496,sK497,sK498,sK499])],[f1619,f1621,f1620]) ).
thf(f1620,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( ( func @ X1 @ X2 @ X0 )
= $true ) )
=> ( ? [X3: $i] :
( ( ( in @ ( ap @ sK497 @ sK498 @ sK496 @ X3 ) @ sK498 )
!= $true )
& ( ( in @ X3 @ sK497 )
= $true ) )
& ( ( func @ sK497 @ sK498 @ sK496 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1621,plain,
( ? [X3: $i] :
( ( ( in @ ( ap @ sK497 @ sK498 @ sK496 @ X3 ) @ sK498 )
!= $true )
& ( ( in @ X3 @ sK497 )
= $true ) )
=> ( ( $true
!= ( in @ ( ap @ sK497 @ sK498 @ sK496 @ sK499 ) @ sK498 ) )
& ( $true
= ( in @ sK499 @ sK497 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1619,plain,
( ( ( app = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( ( func @ X1 @ X2 @ X0 )
= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ! [X7: $i] :
( ( $true
= ( in @ ( ap @ X5 @ X6 @ X4 @ X7 ) @ X6 ) )
| ( $true
!= ( in @ X7 @ X5 ) ) )
| ( ( func @ X5 @ X6 @ X4 )
!= $true ) )
| ( app != $true ) ) ),
inference(rectify,[status(thm)],[f1618]) ).
thf(f1618,plain,
( ( ( app = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 ) )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( ( func @ X1 @ X2 @ X0 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( $true
= ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 ) )
| ( $true
!= ( in @ X3 @ X1 ) ) )
| ( ( func @ X1 @ X2 @ X0 )
!= $true ) )
| ( app != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f758]) ).
thf(f758,plain,
( ( app = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ! [X3: $i] :
( ( $true
= ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 ) )
| ( $true
!= ( in @ X3 @ X1 ) ) )
| ( ( func @ X1 @ X2 @ X0 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f549]) ).
thf(f549,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X2 @ X0 )
= $true )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( $true
= ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 ) ) ) )
<=> ( app = $true ) ),
inference(fool_elimination,[status(thm)],[f548]) ).
thf(f548,plain,
( app
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X1 @ X2 @ X0 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X0 @ X3 ) @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f199]) ).
thf(f199,axiom,
( app
= ( ! [X13: $i,X3: $i,X4: $i] :
( ( func @ X3 @ X4 @ X13 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( ap @ X3 @ X4 @ X13 @ X1 ) @ X4 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.TbQYBRPofc/DTF2THF_6601.p',app) ).
thf(f2151,plain,
( $true
!= ( in @ ( ap @ sK234 @ sK233 @ sK232 @ sK235 ) @ sK233 ) ),
inference(cnf_transformation,[status(thm)],[f1224]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU676^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n007.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:30:08 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.45 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : SEU676^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.46 % Domain : Set Theory
% 0.19/0.46 % Problem : Functions - Application
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English : (! A:i.! B:i.! f:i.in f (funcSet A B) -> (! x:i.in x A ->
% 0.19/0.46 % in (ap A B f x) B))
% 0.19/0.46
% 0.19/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.46 % Source : [Bro08]
% 0.19/0.46 % Names : ZFC178g [Bro08]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.62 v8.1.0, 0.55 v7.5.0, 0.43 v7.4.0, 0.44 v7.2.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.19/0.46 % Syntax : Number of formulae : 429 ( 200 unt; 228 typ; 200 def)
% 0.19/0.46 % Number of atoms : 1057 ( 293 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 195 ( 5 avg)
% 0.19/0.46 % Number of connectives : 1916 ( 47 ~; 7 |; 36 &;1348 @)
% 0.19/0.46 % ( 14 <=>; 464 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 205 ( 2 avg)
% 0.19/0.46 % Number of types : 2 ( 0 usr)
% 0.19/0.46 % Number of type conns : 115 ( 115 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 231 ( 228 usr; 196 con; 0-4 aty)
% 0.19/0.46 % Number of variables : 623 ( 71 ^; 516 !; 36 ?; 623 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : http://mathgate.info/detsetitem.php?id=228
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 thf(in_type,type,
% 0.19/0.46 in: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu_type,type,
% 0.19/0.46 exu: ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu,definition,
% 0.19/0.46 ( exu
% 0.19/0.46 = ( ^ [Xphi: $i > $o] :
% 0.19/0.46 ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextAx_type,type,
% 0.19/0.46 setextAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setextAx,definition,
% 0.19/0.46 ( setextAx
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 <=> ( in @ Xx @ B ) )
% 0.19/0.46 => ( A = B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset_type,type,
% 0.19/0.46 emptyset: $i ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetAx_type,type,
% 0.19/0.46 emptysetAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetAx,definition,
% 0.19/0.46 ( emptysetAx
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin_type,type,
% 0.19/0.46 setadjoin: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinAx_type,type,
% 0.19/0.46 setadjoinAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinAx,definition,
% 0.19/0.46 ( setadjoinAx
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 <=> ( ( Xy = Xx )
% 0.19/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powerset_type,type,
% 0.19/0.46 powerset: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(powersetAx_type,type,
% 0.19/0.46 powersetAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetAx,definition,
% 0.19/0.46 ( powersetAx
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 <=> ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunion_type,type,
% 0.19/0.46 setunion: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setunionAx_type,type,
% 0.19/0.46 setunionAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionAx,definition,
% 0.19/0.46 ( setunionAx
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 <=> ? [B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omega_type,type,
% 0.19/0.46 omega: $i ).
% 0.19/0.46
% 0.19/0.46 thf(omega0Ax_type,type,
% 0.19/0.46 omega0Ax: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omega0Ax,definition,
% 0.19/0.46 ( omega0Ax
% 0.19/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omegaSAx_type,type,
% 0.19/0.46 omegaSAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omegaSAx,definition,
% 0.19/0.46 ( omegaSAx
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ omega )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omegaIndAx_type,type,
% 0.19/0.46 omegaIndAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omegaIndAx,definition,
% 0.19/0.46 ( omegaIndAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( ( in @ emptyset @ A )
% 0.19/0.46 & ! [Xx: $i] :
% 0.19/0.46 ( ( ( in @ Xx @ omega )
% 0.19/0.46 & ( in @ Xx @ A ) )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ omega )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(replAx_type,type,
% 0.19/0.46 replAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(replAx,definition,
% 0.19/0.46 ( replAx
% 0.19/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.46 => ? [B: $i] :
% 0.19/0.46 ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 <=> ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(foundationAx_type,type,
% 0.19/0.46 foundationAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(foundationAx,definition,
% 0.19/0.46 ( foundationAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.46 => ? [B: $i] :
% 0.19/0.46 ( ( in @ B @ A )
% 0.19/0.46 & ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(wellorderingAx_type,type,
% 0.19/0.46 wellorderingAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(wellorderingAx,definition,
% 0.19/0.46 ( wellorderingAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ? [B: $i] :
% 0.19/0.46 ( ! [C: $i] :
% 0.19/0.46 ( ( in @ C @ B )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ A ) ) )
% 0.19/0.46 & ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ( in @ Xx @ A )
% 0.19/0.46 & ( in @ Xy @ A ) )
% 0.19/0.46 => ( ! [C: $i] :
% 0.19/0.46 ( ( in @ C @ B )
% 0.19/0.46 => ( ( in @ Xx @ C )
% 0.19/0.46 <=> ( in @ Xy @ C ) ) )
% 0.19/0.46 => ( Xx = Xy ) ) )
% 0.19/0.46 & ! [C: $i,D: $i] :
% 0.19/0.46 ( ( ( in @ C @ B )
% 0.19/0.46 & ( in @ D @ B ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ D ) )
% 0.19/0.46 | ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ D )
% 0.19/0.46 => ( in @ Xx @ C ) ) ) )
% 0.19/0.46 & ! [C: $i] :
% 0.19/0.46 ( ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.46 => ? [D: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ D @ B )
% 0.19/0.46 & ( in @ Xx @ C )
% 0.19/0.46 & ~ ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ D )
% 0.19/0.46 & ( in @ Xy @ C ) )
% 0.19/0.46 & ! [E: $i] :
% 0.19/0.46 ( ( in @ E @ B )
% 0.19/0.46 => ( ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ E )
% 0.19/0.46 => ( in @ Xy @ D ) )
% 0.19/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(descr_type,type,
% 0.19/0.46 descr: ( $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(descrp_type,type,
% 0.19/0.46 descrp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(descrp,definition,
% 0.19/0.46 ( descrp
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 @ ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr_type,type,
% 0.19/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrI_type,type,
% 0.19/0.46 dsetconstrI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrI,definition,
% 0.19/0.46 ( dsetconstrI
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrEL_type,type,
% 0.19/0.46 dsetconstrEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrEL,definition,
% 0.19/0.46 ( dsetconstrEL
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrER_type,type,
% 0.19/0.46 dsetconstrER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrER,definition,
% 0.19/0.46 ( dsetconstrER
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE1_type,type,
% 0.19/0.46 exuE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE1,definition,
% 0.19/0.46 ( exuE1
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(prop2set_type,type,
% 0.19/0.46 prop2set: $o > $i ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setE_type,type,
% 0.19/0.46 prop2setE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setE,definition,
% 0.19/0.46 ( prop2setE
% 0.19/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetE_type,type,
% 0.19/0.46 emptysetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetE,definition,
% 0.19/0.46 ( emptysetE
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ emptyset )
% 0.19/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetimpfalse_type,type,
% 0.19/0.46 emptysetimpfalse: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetimpfalse,definition,
% 0.19/0.46 ( emptysetimpfalse
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ emptyset )
% 0.19/0.46 => $false ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notinemptyset_type,type,
% 0.19/0.46 notinemptyset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notinemptyset,definition,
% 0.19/0.46 ( notinemptyset
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3e_type,type,
% 0.19/0.46 exuE3e: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3e,definition,
% 0.19/0.46 ( exuE3e
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setext_type,type,
% 0.19/0.46 setext: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setext,definition,
% 0.19/0.46 ( setext
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 => ( A = B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyI_type,type,
% 0.19/0.46 emptyI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyI,definition,
% 0.19/0.46 ( emptyI
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ A )
% 0.19/0.46 => ( A = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(noeltsimpempty_type,type,
% 0.19/0.46 noeltsimpempty: $o ).
% 0.19/0.46
% 0.19/0.46 thf(noeltsimpempty,definition,
% 0.19/0.46 ( noeltsimpempty
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ A )
% 0.19/0.46 => ( A = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setbeta_type,type,
% 0.19/0.46 setbeta: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setbeta,definition,
% 0.19/0.46 ( setbeta
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonempty_type,type,
% 0.19/0.46 nonempty: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonempty,definition,
% 0.19/0.46 ( nonempty
% 0.19/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyE1_type,type,
% 0.19/0.46 nonemptyE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyE1,definition,
% 0.19/0.46 ( nonemptyE1
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( nonempty @ A )
% 0.19/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI_type,type,
% 0.19/0.46 nonemptyI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI,definition,
% 0.19/0.46 ( nonemptyI
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( nonempty
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI1_type,type,
% 0.19/0.46 nonemptyI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI1,definition,
% 0.19/0.46 ( nonemptyI1
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIL_type,type,
% 0.19/0.46 setadjoinIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIL,definition,
% 0.19/0.46 ( setadjoinIL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinunitempty_type,type,
% 0.19/0.46 emptyinunitempty: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinunitempty,definition,
% 0.19/0.46 ( emptyinunitempty
% 0.19/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIR_type,type,
% 0.19/0.46 setadjoinIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIR,definition,
% 0.19/0.46 ( setadjoinIR
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinE_type,type,
% 0.19/0.46 setadjoinE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinE,definition,
% 0.19/0.46 ( setadjoinE
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 => ! [Xphi: $o] :
% 0.19/0.46 ( ( ( Xy = Xx )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => ( ( ( in @ Xy @ A )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinOr_type,type,
% 0.19/0.46 setadjoinOr: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinOr,definition,
% 0.19/0.46 ( setadjoinOr
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 => ( ( Xy = Xx )
% 0.19/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setoftrueEq_type,type,
% 0.19/0.46 setoftrueEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setoftrueEq,definition,
% 0.19/0.46 ( setoftrueEq
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : $true )
% 0.19/0.46 = A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI_type,type,
% 0.19/0.46 powersetI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI,definition,
% 0.19/0.46 ( powersetI
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinPowerset_type,type,
% 0.19/0.46 emptyinPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinPowerset,definition,
% 0.19/0.46 ( emptyinPowerset
% 0.19/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyInPowerset_type,type,
% 0.19/0.46 emptyInPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyInPowerset,definition,
% 0.19/0.46 ( emptyInPowerset
% 0.19/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE_type,type,
% 0.19/0.46 powersetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE,definition,
% 0.19/0.46 ( powersetE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionI_type,type,
% 0.19/0.46 setunionI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionI,definition,
% 0.19/0.46 ( setunionI
% 0.19/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( ( in @ B @ A )
% 0.19/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE_type,type,
% 0.19/0.46 setunionE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE,definition,
% 0.19/0.46 ( setunionE
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 => ! [Xphi: $o] :
% 0.19/0.46 ( ! [B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( ( in @ B @ A )
% 0.19/0.46 => Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subPowSU_type,type,
% 0.19/0.46 subPowSU: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subPowSU,definition,
% 0.19/0.46 ( subPowSU
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE2_type,type,
% 0.19/0.46 exuE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE2,definition,
% 0.19/0.46 ( exuE2
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyImpWitness_type,type,
% 0.19/0.46 nonemptyImpWitness: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyImpWitness,definition,
% 0.19/0.46 ( nonemptyImpWitness
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( nonempty @ A )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & $true ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(uniqinunit_type,type,
% 0.19/0.46 uniqinunit: $o ).
% 0.19/0.46
% 0.19/0.46 thf(uniqinunit,definition,
% 0.19/0.46 ( uniqinunit
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notinsingleton_type,type,
% 0.19/0.46 notinsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notinsingleton,definition,
% 0.19/0.46 ( notinsingleton
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx != Xy )
% 0.19/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqinunit_type,type,
% 0.19/0.46 eqinunit: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqinunit,definition,
% 0.19/0.46 ( eqinunit
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsswitch_type,type,
% 0.19/0.46 singletonsswitch: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsswitch,definition,
% 0.19/0.46 ( singletonsswitch
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetE_type,type,
% 0.19/0.46 upairsetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetE,definition,
% 0.19/0.46 ( upairsetE
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.46 => ( ( Xz = Xx )
% 0.19/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIL_type,type,
% 0.19/0.46 upairsetIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIL,definition,
% 0.19/0.46 ( upairsetIL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIR_type,type,
% 0.19/0.46 upairsetIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIR,definition,
% 0.19/0.46 ( upairsetIR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyE1_type,type,
% 0.19/0.46 emptyE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyE1,definition,
% 0.19/0.46 ( emptyE1
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ( ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = emptyset )
% 0.19/0.46 => $false ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(vacuousDall_type,type,
% 0.19/0.46 vacuousDall: $o ).
% 0.19/0.46
% 0.19/0.46 thf(vacuousDall,definition,
% 0.19/0.46 ( vacuousDall
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ emptyset )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan1_type,type,
% 0.19/0.46 quantDeMorgan1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan1,definition,
% 0.19/0.46 ( quantDeMorgan1
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ~ ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan2_type,type,
% 0.19/0.46 quantDeMorgan2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan2,definition,
% 0.19/0.46 ( quantDeMorgan2
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ~ ( Xphi @ Xx ) )
% 0.19/0.46 => ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan3_type,type,
% 0.19/0.46 quantDeMorgan3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan3,definition,
% 0.19/0.46 ( quantDeMorgan3
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan4_type,type,
% 0.19/0.46 quantDeMorgan4: $o ).
% 0.19/0.46
% 0.19/0.46 thf(quantDeMorgan4,definition,
% 0.19/0.46 ( quantDeMorgan4
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ~ ( Xphi @ Xx ) )
% 0.19/0.46 => ~ ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setI_type,type,
% 0.19/0.46 prop2setI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setI,definition,
% 0.19/0.46 ( prop2setI
% 0.19/0.46 = ( ! [Xphi: $o] :
% 0.19/0.46 ( Xphi
% 0.19/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(set2prop_type,type,
% 0.19/0.46 set2prop: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2set2propI_type,type,
% 0.19/0.46 prop2set2propI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2set2propI,definition,
% 0.19/0.46 ( prop2set2propI
% 0.19/0.46 = ( ! [Xphi: $o] :
% 0.19/0.46 ( Xphi
% 0.19/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notdexE_type,type,
% 0.19/0.46 notdexE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notdexE,definition,
% 0.19/0.46 ( notdexE
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notdallE_type,type,
% 0.19/0.46 notdallE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notdallE,definition,
% 0.19/0.46 ( notdallE
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ~ ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuI1_type,type,
% 0.19/0.46 exuI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuI1,definition,
% 0.19/0.46 ( exuI1
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuI3_type,type,
% 0.19/0.46 exuI3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuI3,definition,
% 0.19/0.46 ( exuI3
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuI2_type,type,
% 0.19/0.46 exuI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuI2,definition,
% 0.19/0.46 ( exuI2
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ? [Xx: $i] :
% 0.19/0.46 ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 <=> ( Xy = Xx ) )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(inCongP_type,type,
% 0.19/0.46 inCongP: $o ).
% 0.19/0.46
% 0.19/0.46 thf(inCongP,definition,
% 0.19/0.46 ( inCongP
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(in__Cong_type,type,
% 0.19/0.46 in__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(in__Cong,definition,
% 0.19/0.46 ( in__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3u_type,type,
% 0.19/0.46 exuE3u: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3u,definition,
% 0.19/0.46 ( exuE3u
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exu__Cong_type,type,
% 0.19/0.46 exu__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu__Cong,definition,
% 0.19/0.46 ( exu__Cong
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 <=> ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset__Cong_type,type,
% 0.19/0.46 emptyset__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset__Cong,definition,
% 0.19/0.46 ( emptyset__Cong
% 0.19/0.46 = ( emptyset = emptyset ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin__Cong_type,type,
% 0.19/0.46 setadjoin__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin__Cong,definition,
% 0.19/0.46 ( setadjoin__Cong
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ! [Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( Xz = Xu )
% 0.19/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powerset__Cong_type,type,
% 0.19/0.46 powerset__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powerset__Cong,definition,
% 0.19/0.46 ( powerset__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( powerset @ A )
% 0.19/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunion__Cong_type,type,
% 0.19/0.46 setunion__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunion__Cong,definition,
% 0.19/0.46 ( setunion__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( setunion @ A )
% 0.19/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omega__Cong_type,type,
% 0.19/0.46 omega__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omega__Cong,definition,
% 0.19/0.46 ( omega__Cong
% 0.19/0.46 = ( omega = omega ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuEu_type,type,
% 0.19/0.46 exuEu: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuEu,definition,
% 0.19/0.46 ( exuEu
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(descr__Cong_type,type,
% 0.19/0.46 descr__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(descr__Cong,definition,
% 0.19/0.46 ( descr__Cong
% 0.19/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.46 => ( ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr__Cong_type,type,
% 0.19/0.46 dsetconstr__Cong: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr__Cong,definition,
% 0.19/0.46 ( dsetconstr__Cong
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( Xx = Xy )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.46 => ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 = ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subset_type,type,
% 0.19/0.46 subset: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(disjoint_type,type,
% 0.19/0.46 disjoint: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setsmeet_type,type,
% 0.19/0.46 setsmeet: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI1_type,type,
% 0.19/0.46 subsetI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI1,definition,
% 0.19/0.46 ( subsetI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset2_type,type,
% 0.19/0.46 eqimpsubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset2,definition,
% 0.19/0.46 ( eqimpsubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset1_type,type,
% 0.19/0.46 eqimpsubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqimpsubset1,definition,
% 0.19/0.46 ( eqimpsubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI2_type,type,
% 0.19/0.46 subsetI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetI2,definition,
% 0.19/0.46 ( subsetI2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetsubset_type,type,
% 0.19/0.46 emptysetsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetsubset,definition,
% 0.19/0.46 ( emptysetsubset
% 0.19/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE_type,type,
% 0.19/0.46 subsetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE,definition,
% 0.19/0.46 ( subsetE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE2_type,type,
% 0.19/0.46 subsetE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetE2,definition,
% 0.19/0.46 ( subsetE2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notsubsetI_type,type,
% 0.19/0.46 notsubsetI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notsubsetI,definition,
% 0.19/0.46 ( notsubsetI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI1_type,type,
% 0.19/0.46 notequalI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI1,definition,
% 0.19/0.46 ( notequalI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ~ ( subset @ A @ B )
% 0.19/0.46 => ( A != B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI2_type,type,
% 0.19/0.46 notequalI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notequalI2,definition,
% 0.19/0.46 ( notequalI2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ( A != B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetRefl_type,type,
% 0.19/0.46 subsetRefl: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetRefl,definition,
% 0.19/0.46 ( subsetRefl
% 0.19/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTrans_type,type,
% 0.19/0.46 subsetTrans: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetTrans,definition,
% 0.19/0.46 ( subsetTrans
% 0.19/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( subset @ B @ C )
% 0.19/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub_type,type,
% 0.19/0.46 setadjoinSub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub,definition,
% 0.19/0.46 ( setadjoinSub
% 0.19/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub2_type,type,
% 0.19/0.46 setadjoinSub2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinSub2,definition,
% 0.19/0.46 ( setadjoinSub2
% 0.19/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subset2powerset_type,type,
% 0.19/0.46 subset2powerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subset2powerset,definition,
% 0.19/0.46 ( subset2powerset
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextsub_type,type,
% 0.19/0.46 setextsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setextsub,definition,
% 0.19/0.46 ( setextsub
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( subset @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subsetemptysetimpeq_type,type,
% 0.19/0.46 subsetemptysetimpeq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subsetemptysetimpeq,definition,
% 0.19/0.46 ( subsetemptysetimpeq
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( subset @ A @ emptyset )
% 0.19/0.46 => ( A = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI1_type,type,
% 0.19/0.46 powersetI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI1,definition,
% 0.19/0.46 ( powersetI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ B @ A )
% 0.19/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE1_type,type,
% 0.19/0.46 powersetE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE1,definition,
% 0.19/0.46 ( powersetE1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(inPowerset_type,type,
% 0.19/0.46 inPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(inPowerset,definition,
% 0.19/0.46 ( inPowerset
% 0.19/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetsubset_type,type,
% 0.19/0.46 powersetsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetsubset,definition,
% 0.19/0.46 ( powersetsubset
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sepInPowerset_type,type,
% 0.19/0.46 sepInPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(sepInPowerset,definition,
% 0.19/0.46 ( sepInPowerset
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( in
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sepSubset_type,type,
% 0.19/0.46 sepSubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(sepSubset,definition,
% 0.19/0.46 ( sepSubset
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( subset
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunion_type,type,
% 0.19/0.46 binunion: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIL_type,type,
% 0.19/0.46 binunionIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIL,definition,
% 0.19/0.46 ( binunionIL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2IR_type,type,
% 0.19/0.46 upairset2IR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2IR,definition,
% 0.19/0.46 ( upairset2IR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIR_type,type,
% 0.19/0.46 binunionIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionIR,definition,
% 0.19/0.46 ( binunionIR
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionEcases_type,type,
% 0.19/0.46 binunionEcases: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionEcases,definition,
% 0.19/0.46 ( binunionEcases
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.19/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.46 => ( ( ( in @ Xx @ A )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => ( ( ( in @ Xx @ B )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionE_type,type,
% 0.19/0.46 binunionE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionE,definition,
% 0.19/0.46 ( binunionE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionLsub_type,type,
% 0.19/0.46 binunionLsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionLsub,definition,
% 0.19/0.46 ( binunionLsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binunionRsub_type,type,
% 0.19/0.46 binunionRsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binunionRsub,definition,
% 0.19/0.46 ( binunionRsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersect_type,type,
% 0.19/0.46 binintersect: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectI_type,type,
% 0.19/0.46 binintersectI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectI,definition,
% 0.19/0.46 ( binintersectI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset5_type,type,
% 0.19/0.46 binintersectSubset5: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset5,definition,
% 0.19/0.46 ( binintersectSubset5
% 0.19/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.46 ( ( subset @ C @ A )
% 0.19/0.46 => ( ( subset @ C @ B )
% 0.19/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectEL_type,type,
% 0.19/0.46 binintersectEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectEL,definition,
% 0.19/0.46 ( binintersectEL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectLsub_type,type,
% 0.19/0.46 binintersectLsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectLsub,definition,
% 0.19/0.46 ( binintersectLsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset2_type,type,
% 0.19/0.46 binintersectSubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset2,definition,
% 0.19/0.46 ( binintersectSubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( binintersect @ A @ B )
% 0.19/0.46 = A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset3_type,type,
% 0.19/0.46 binintersectSubset3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset3,definition,
% 0.19/0.46 ( binintersectSubset3
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( ( binintersect @ A @ B )
% 0.19/0.46 = B )
% 0.19/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectER_type,type,
% 0.19/0.46 binintersectER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectER,definition,
% 0.19/0.46 ( binintersectER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(disjointsetsI1_type,type,
% 0.19/0.46 disjointsetsI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(disjointsetsI1,definition,
% 0.19/0.46 ( disjointsetsI1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( in @ Xx @ B ) )
% 0.19/0.46 => ( ( binintersect @ A @ B )
% 0.19/0.46 = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectRsub_type,type,
% 0.19/0.46 binintersectRsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectRsub,definition,
% 0.19/0.46 ( binintersectRsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset4_type,type,
% 0.19/0.46 binintersectSubset4: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset4,definition,
% 0.19/0.46 ( binintersectSubset4
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ B @ A )
% 0.19/0.46 => ( ( binintersect @ A @ B )
% 0.19/0.46 = B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset1_type,type,
% 0.19/0.46 binintersectSubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(binintersectSubset1,definition,
% 0.19/0.46 ( binintersectSubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( ( binintersect @ A @ B )
% 0.19/0.46 = A )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(bs114d_type,type,
% 0.19/0.46 bs114d: $o ).
% 0.19/0.46
% 0.19/0.46 thf(bs114d,definition,
% 0.19/0.46 ( bs114d
% 0.19/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.19/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(regular_type,type,
% 0.19/0.46 regular: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminus_type,type,
% 0.19/0.46 setminus: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setminusI_type,type,
% 0.19/0.46 setminusI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusI,definition,
% 0.19/0.46 ( setminusI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusEL_type,type,
% 0.19/0.46 setminusEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusEL,definition,
% 0.19/0.46 ( setminusEL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusER_type,type,
% 0.19/0.46 setminusER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusER,definition,
% 0.19/0.46 ( setminusER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset2_type,type,
% 0.19/0.46 setminusSubset2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset2,definition,
% 0.19/0.46 ( setminusSubset2
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( subset @ A @ B )
% 0.19/0.46 => ( ( setminus @ A @ B )
% 0.19/0.46 = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusERneg_type,type,
% 0.19/0.46 setminusERneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusERneg,definition,
% 0.19/0.46 ( setminusERneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusELneg_type,type,
% 0.19/0.46 setminusELneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusELneg,definition,
% 0.19/0.46 ( setminusELneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusILneg_type,type,
% 0.19/0.46 setminusILneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusILneg,definition,
% 0.19/0.46 ( setminusILneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusIRneg_type,type,
% 0.19/0.46 setminusIRneg: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusIRneg,definition,
% 0.19/0.46 ( setminusIRneg
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusLsub_type,type,
% 0.19/0.46 setminusLsub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusLsub,definition,
% 0.19/0.46 ( setminusLsub
% 0.19/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset1_type,type,
% 0.19/0.46 setminusSubset1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setminusSubset1,definition,
% 0.19/0.46 ( setminusSubset1
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( ( setminus @ A @ B )
% 0.19/0.46 = emptyset )
% 0.19/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiff_type,type,
% 0.19/0.46 symdiff: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffE_type,type,
% 0.19/0.46 symdiffE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffE,definition,
% 0.19/0.46 ( symdiffE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.19/0.46 => ! [Xphi: $o] :
% 0.19/0.46 ( ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => Xphi ) )
% 0.19/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI1_type,type,
% 0.19/0.46 symdiffI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI1,definition,
% 0.19/0.46 ( symdiffI1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI2_type,type,
% 0.19/0.46 symdiffI2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffI2,definition,
% 0.19/0.46 ( symdiffI2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg1_type,type,
% 0.19/0.46 symdiffIneg1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg1,definition,
% 0.19/0.46 ( symdiffIneg1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg2_type,type,
% 0.19/0.46 symdiffIneg2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(symdiffIneg2,definition,
% 0.19/0.46 ( symdiffIneg2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ~ ( in @ Xx @ A )
% 0.19/0.46 => ( ~ ( in @ Xx @ B )
% 0.19/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(iskpair_type,type,
% 0.19/0.46 iskpair: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(secondinupair_type,type,
% 0.19/0.46 secondinupair: $o ).
% 0.19/0.46
% 0.19/0.46 thf(secondinupair,definition,
% 0.19/0.46 ( secondinupair
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIL_type,type,
% 0.19/0.46 setukpairIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIL,definition,
% 0.19/0.46 ( setukpairIL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIR_type,type,
% 0.19/0.46 setukpairIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairIR,definition,
% 0.19/0.46 ( setukpairIR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kpairiskpair_type,type,
% 0.19/0.46 kpairiskpair: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kpairiskpair,definition,
% 0.19/0.46 ( kpairiskpair
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kpair_type,type,
% 0.19/0.46 kpair: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(kpairp_type,type,
% 0.19/0.46 kpairp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kpairp,definition,
% 0.19/0.46 ( kpairp
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprod_type,type,
% 0.19/0.46 cartprod: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsubset_type,type,
% 0.19/0.46 singletonsubset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsubset,definition,
% 0.19/0.46 ( singletonsubset
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowerset_type,type,
% 0.19/0.46 singletoninpowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowerset,definition,
% 0.19/0.46 ( singletoninpowerset
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowunion_type,type,
% 0.19/0.46 singletoninpowunion: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletoninpowunion,definition,
% 0.19/0.46 ( singletoninpowunion
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2E_type,type,
% 0.19/0.46 upairset2E: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairset2E,definition,
% 0.19/0.46 ( upairset2E
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.46 => ( ( Xz = Xx )
% 0.19/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsubunion_type,type,
% 0.19/0.46 upairsubunion: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsubunion,definition,
% 0.19/0.46 ( upairsubunion
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairinpowunion_type,type,
% 0.19/0.46 upairinpowunion: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairinpowunion,definition,
% 0.19/0.46 ( upairinpowunion
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem1_type,type,
% 0.19/0.46 ubforcartprodlem1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem1,definition,
% 0.19/0.46 ( ubforcartprodlem1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem2_type,type,
% 0.19/0.46 ubforcartprodlem2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem2,definition,
% 0.19/0.46 ( ubforcartprodlem2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem3_type,type,
% 0.19/0.46 ubforcartprodlem3: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ubforcartprodlem3,definition,
% 0.19/0.46 ( ubforcartprodlem3
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairin_type,type,
% 0.19/0.46 cartprodpairin: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairin,definition,
% 0.19/0.46 ( cartprodpairin
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair1_type,type,
% 0.19/0.46 cartprodmempair1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair1,definition,
% 0.19/0.46 ( cartprodmempair1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 & ( Xu
% 0.19/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair_type,type,
% 0.19/0.46 cartprodmempair: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempair,definition,
% 0.19/0.46 ( cartprodmempair
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE2_type,type,
% 0.19/0.46 setunionE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE2,definition,
% 0.19/0.46 ( setunionE2
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 => ? [X: $i] :
% 0.19/0.46 ( ( in @ X @ A )
% 0.19/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton1_type,type,
% 0.19/0.46 setunionsingleton1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton1,definition,
% 0.19/0.46 ( setunionsingleton1
% 0.19/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton2_type,type,
% 0.19/0.46 setunionsingleton2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton2,definition,
% 0.19/0.46 ( setunionsingleton2
% 0.19/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton_type,type,
% 0.19/0.46 setunionsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionsingleton,definition,
% 0.19/0.46 ( setunionsingleton
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.19/0.46 = Xx ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singleton_type,type,
% 0.19/0.46 singleton: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(singleton,definition,
% 0.19/0.46 ( singleton
% 0.19/0.46 = ( ^ [A: $i] :
% 0.19/0.46 ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( A
% 0.19/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletonprop_type,type,
% 0.19/0.46 singletonprop: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonprop,definition,
% 0.19/0.46 ( singletonprop
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) )
% 0.19/0.46 => ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1_type,type,
% 0.19/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1,definition,
% 0.19/0.46 ( ex1
% 0.19/0.46 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( singleton
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E1_type,type,
% 0.19/0.46 ex1E1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1E1,definition,
% 0.19/0.46 ( ex1E1
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ( ex1 @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I_type,type,
% 0.19/0.46 ex1I: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I,definition,
% 0.19/0.46 ( ex1I
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xy = Xx ) ) )
% 0.19/0.46 => ( ex1 @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I2_type,type,
% 0.19/0.46 ex1I2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ex1I2,definition,
% 0.19/0.46 ( ex1I2
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) )
% 0.19/0.46 => ( ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & ( Xphi @ Xx ) )
% 0.19/0.46 => ( ex1 @ A
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsuniq_type,type,
% 0.19/0.46 singletonsuniq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsuniq,definition,
% 0.19/0.46 ( singletonsuniq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.19/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(atmost1p_type,type,
% 0.19/0.46 atmost1p: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(atleast2p_type,type,
% 0.19/0.46 atleast2p: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(atmost2p_type,type,
% 0.19/0.46 atmost2p: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetp_type,type,
% 0.19/0.46 upairsetp: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL1_type,type,
% 0.19/0.46 setukpairinjL1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL1,definition,
% 0.19/0.46 ( setukpairinjL1
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kfstsingleton_type,type,
% 0.19/0.46 kfstsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kfstsingleton,definition,
% 0.19/0.46 ( kfstsingleton
% 0.19/0.46 = ( ! [Xu: $i] :
% 0.19/0.46 ( ( iskpair @ Xu )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(theprop_type,type,
% 0.19/0.46 theprop: $o ).
% 0.19/0.46
% 0.19/0.46 thf(theprop,definition,
% 0.19/0.46 ( theprop
% 0.19/0.46 = ( ! [X: $i] :
% 0.19/0.46 ( ( singleton @ X )
% 0.19/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kfst_type,type,
% 0.19/0.46 kfst: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(kfstpairEq_type,type,
% 0.19/0.46 kfstpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kfstpairEq,definition,
% 0.19/0.46 ( kfstpairEq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xx ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstin_type,type,
% 0.19/0.46 cartprodfstin: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstin,definition,
% 0.19/0.46 ( cartprodfstin
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL2_type,type,
% 0.19/0.46 setukpairinjL2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL2,definition,
% 0.19/0.46 ( setukpairinjL2
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL_type,type,
% 0.19/0.46 setukpairinjL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjL,definition,
% 0.19/0.46 ( setukpairinjL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.46 = ( kpair @ Xz @ Xu ) )
% 0.19/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR11_type,type,
% 0.19/0.46 setukpairinjR11: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR11,definition,
% 0.19/0.46 ( setukpairinjR11
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR12_type,type,
% 0.19/0.46 setukpairinjR12: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR12,definition,
% 0.19/0.46 ( setukpairinjR12
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR1_type,type,
% 0.19/0.46 setukpairinjR1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR1,definition,
% 0.19/0.46 ( setukpairinjR1
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( ( Xz = Xu )
% 0.19/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairequniteq_type,type,
% 0.19/0.46 upairequniteq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairequniteq,definition,
% 0.19/0.46 ( upairequniteq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR2_type,type,
% 0.19/0.46 setukpairinjR2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR2,definition,
% 0.19/0.46 ( setukpairinjR2
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR_type,type,
% 0.19/0.46 setukpairinjR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setukpairinjR,definition,
% 0.19/0.46 ( setukpairinjR
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.46 = ( kpair @ Xz @ Xu ) )
% 0.19/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ksndsingleton_type,type,
% 0.19/0.46 ksndsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ksndsingleton,definition,
% 0.19/0.46 ( ksndsingleton
% 0.19/0.46 = ( ! [Xu: $i] :
% 0.19/0.46 ( ( iskpair @ Xu )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.46 @ ^ [Xx: $i] :
% 0.19/0.46 ( Xu
% 0.19/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(ksnd_type,type,
% 0.19/0.46 ksnd: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(ksndpairEq_type,type,
% 0.19/0.46 ksndpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(ksndpairEq,definition,
% 0.19/0.46 ( ksndpairEq
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xy ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(kpairsurjEq_type,type,
% 0.19/0.46 kpairsurjEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(kpairsurjEq,definition,
% 0.19/0.46 ( kpairsurjEq
% 0.19/0.46 = ( ! [Xu: $i] :
% 0.19/0.46 ( ( iskpair @ Xu )
% 0.19/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.46 = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndin_type,type,
% 0.19/0.46 cartprodsndin: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndin,definition,
% 0.19/0.46 ( cartprodsndin
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemEL_type,type,
% 0.19/0.46 cartprodpairmemEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemEL,definition,
% 0.19/0.46 ( cartprodpairmemEL
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemER_type,type,
% 0.19/0.46 cartprodpairmemER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairmemER,definition,
% 0.19/0.46 ( cartprodpairmemER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempaircEq_type,type,
% 0.19/0.46 cartprodmempaircEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodmempaircEq,definition,
% 0.19/0.46 ( cartprodmempaircEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( kpair @ Xx @ Xy )
% 0.19/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstpairEq_type,type,
% 0.19/0.46 cartprodfstpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodfstpairEq,definition,
% 0.19/0.46 ( cartprodfstpairEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndpairEq_type,type,
% 0.19/0.46 cartprodsndpairEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodsndpairEq,definition,
% 0.19/0.46 ( cartprodsndpairEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.46 = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairsurjEq_type,type,
% 0.19/0.46 cartprodpairsurjEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(cartprodpairsurjEq,definition,
% 0.19/0.46 ( cartprodpairsurjEq
% 0.19/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.46 = Xu ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(breln_type,type,
% 0.19/0.46 breln: $i > $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(breln,definition,
% 0.19/0.46 ( breln
% 0.19/0.46 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstr_type,type,
% 0.19/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrI_type,type,
% 0.19/0.46 dpsetconstrI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrI,definition,
% 0.19/0.46 ( dpsetconstrI
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.19/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrSub_type,type,
% 0.19/0.46 dpsetconstrSub: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrSub,definition,
% 0.19/0.46 ( dpsetconstrSub
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.46 ( subset
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.19/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setOfPairsIsBReln_type,type,
% 0.19/0.46 setOfPairsIsBReln: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setOfPairsIsBReln,definition,
% 0.19/0.46 ( setOfPairsIsBReln
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.46 ( breln @ A @ B
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrERa_type,type,
% 0.19/0.46 dpsetconstrERa: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrERa,definition,
% 0.19/0.46 ( dpsetconstrERa
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ B )
% 0.19/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL1_type,type,
% 0.19/0.46 dpsetconstrEL1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL1,definition,
% 0.19/0.46 ( dpsetconstrEL1
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL2_type,type,
% 0.19/0.46 dpsetconstrEL2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrEL2,definition,
% 0.19/0.46 ( dpsetconstrEL2
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrER_type,type,
% 0.19/0.46 dpsetconstrER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dpsetconstrER,definition,
% 0.19/0.46 ( dpsetconstrER
% 0.19/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.46 @ ( dpsetconstr @ A @ B
% 0.19/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(func_type,type,
% 0.19/0.46 func: $i > $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(func,definition,
% 0.19/0.46 ( func
% 0.19/0.46 = ( ^ [A: $i,B: $i,R: $i] :
% 0.19/0.46 ( ( breln @ A @ B @ R )
% 0.19/0.46 & ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ex1 @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcSet_type,type,
% 0.19/0.46 funcSet: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(funcSet,definition,
% 0.19/0.46 ( funcSet
% 0.19/0.46 = ( ^ [A: $i,B: $i] :
% 0.19/0.46 ( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
% 0.19/0.46 @ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(funcImageSingleton_type,type,
% 0.19/0.46 funcImageSingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(funcImageSingleton,definition,
% 0.19/0.46 ( funcImageSingleton
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( singleton
% 0.19/0.46 @ ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(apProp_type,type,
% 0.19/0.46 apProp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(apProp,definition,
% 0.19/0.46 ( apProp
% 0.19/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.46 ( ( func @ A @ B @ Xf )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in
% 0.19/0.46 @ ( setunion
% 0.19/0.46 @ ( dsetconstr @ B
% 0.19/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.19/0.46 @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap_type,type,
% 0.30/0.46 ap: $i > $i > $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(ap,definition,
% 0.30/0.46 ( ap
% 0.30/0.46 = ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
% 0.30/0.46 ( setunion
% 0.30/0.46 @ ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(app_type,type,
% 0.30/0.46 app: $o ).
% 0.30/0.46
% 0.30/0.46 thf(app,definition,
% 0.30/0.46 ( app
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( func @ A @ B @ Xf )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(infuncsetfunc_type,type,
% 0.30/0.46 infuncsetfunc: $o ).
% 0.30/0.46
% 0.30/0.46 thf(infuncsetfunc,definition,
% 0.30/0.46 ( infuncsetfunc
% 0.30/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ap2p,conjecture,
% 0.30/0.46 ( setextAx
% 0.30/0.46 => ( emptysetAx
% 0.30/0.46 => ( setadjoinAx
% 0.30/0.46 => ( powersetAx
% 0.30/0.46 => ( setunionAx
% 0.30/0.46 => ( omega0Ax
% 0.30/0.46 => ( omegaSAx
% 0.30/0.46 => ( omegaIndAx
% 0.30/0.46 => ( replAx
% 0.30/0.46 => ( foundationAx
% 0.30/0.46 => ( wellorderingAx
% 0.30/0.46 => ( descrp
% 0.30/0.46 => ( dsetconstrI
% 0.30/0.46 => ( dsetconstrEL
% 0.30/0.46 => ( dsetconstrER
% 0.30/0.46 => ( exuE1
% 0.30/0.46 => ( prop2setE
% 0.30/0.46 => ( emptysetE
% 0.30/0.46 => ( emptysetimpfalse
% 0.30/0.46 => ( notinemptyset
% 0.30/0.46 => ( exuE3e
% 0.30/0.46 => ( setext
% 0.30/0.46 => ( emptyI
% 0.30/0.46 => ( noeltsimpempty
% 0.30/0.46 => ( setbeta
% 0.30/0.46 => ( nonemptyE1
% 0.30/0.46 => ( nonemptyI
% 0.30/0.46 => ( nonemptyI1
% 0.30/0.46 => ( setadjoinIL
% 0.30/0.46 => ( emptyinunitempty
% 0.30/0.46 => ( setadjoinIR
% 0.30/0.46 => ( setadjoinE
% 0.30/0.46 => ( setadjoinOr
% 0.30/0.46 => ( setoftrueEq
% 0.30/0.46 => ( powersetI
% 0.30/0.46 => ( emptyinPowerset
% 0.30/0.46 => ( emptyInPowerset
% 0.30/0.46 => ( powersetE
% 0.30/0.46 => ( setunionI
% 0.30/0.46 => ( setunionE
% 0.30/0.46 => ( subPowSU
% 0.30/0.46 => ( exuE2
% 0.30/0.46 => ( nonemptyImpWitness
% 0.30/0.46 => ( uniqinunit
% 0.30/0.46 => ( notinsingleton
% 0.30/0.46 => ( eqinunit
% 0.30/0.46 => ( singletonsswitch
% 0.30/0.46 => ( upairsetE
% 0.30/0.46 => ( upairsetIL
% 0.30/0.46 => ( upairsetIR
% 0.30/0.46 => ( emptyE1
% 0.30/0.46 => ( vacuousDall
% 0.30/0.46 => ( quantDeMorgan1
% 0.30/0.46 => ( quantDeMorgan2
% 0.30/0.46 => ( quantDeMorgan3
% 0.30/0.46 => ( quantDeMorgan4
% 0.30/0.46 => ( prop2setI
% 0.30/0.46 => ( prop2set2propI
% 0.30/0.46 => ( notdexE
% 0.30/0.46 => ( notdallE
% 0.30/0.46 => ( exuI1
% 0.30/0.46 => ( exuI3
% 0.30/0.46 => ( exuI2
% 0.30/0.46 => ( inCongP
% 0.30/0.46 => ( in__Cong
% 0.30/0.46 => ( exuE3u
% 0.30/0.46 => ( exu__Cong
% 0.30/0.46 => ( emptyset__Cong
% 0.30/0.46 => ( setadjoin__Cong
% 0.30/0.46 => ( powerset__Cong
% 0.30/0.46 => ( setunion__Cong
% 0.30/0.46 => ( omega__Cong
% 0.30/0.46 => ( exuEu
% 0.30/0.46 => ( descr__Cong
% 0.30/0.46 => ( dsetconstr__Cong
% 0.30/0.46 => ( subsetI1
% 0.30/0.46 => ( eqimpsubset2
% 0.30/0.46 => ( eqimpsubset1
% 0.30/0.46 => ( subsetI2
% 0.30/0.46 => ( emptysetsubset
% 0.30/0.46 => ( subsetE
% 0.30/0.46 => ( subsetE2
% 0.30/0.46 => ( notsubsetI
% 0.30/0.46 => ( notequalI1
% 0.30/0.46 => ( notequalI2
% 0.30/0.46 => ( subsetRefl
% 0.30/0.46 => ( subsetTrans
% 0.30/0.46 => ( setadjoinSub
% 0.30/0.46 => ( setadjoinSub2
% 0.30/0.46 => ( subset2powerset
% 0.30/0.46 => ( setextsub
% 0.30/0.46 => ( subsetemptysetimpeq
% 0.30/0.46 => ( powersetI1
% 0.30/0.46 => ( powersetE1
% 0.30/0.46 => ( inPowerset
% 0.30/0.46 => ( powersetsubset
% 0.30/0.46 => ( sepInPowerset
% 0.30/0.46 => ( sepSubset
% 0.30/0.46 => ( binunionIL
% 0.30/0.46 => ( upairset2IR
% 0.30/0.46 => ( binunionIR
% 0.30/0.46 => ( binunionEcases
% 0.30/0.46 => ( binunionE
% 0.30/0.46 => ( binunionLsub
% 0.30/0.46 => ( binunionRsub
% 0.30/0.46 => ( binintersectI
% 0.30/0.46 => ( binintersectSubset5
% 0.30/0.46 => ( binintersectEL
% 0.30/0.46 => ( binintersectLsub
% 0.30/0.46 => ( binintersectSubset2
% 0.30/0.46 => ( binintersectSubset3
% 0.30/0.46 => ( binintersectER
% 0.30/0.46 => ( disjointsetsI1
% 0.30/0.46 => ( binintersectRsub
% 0.30/0.46 => ( binintersectSubset4
% 0.30/0.46 => ( binintersectSubset1
% 0.30/0.46 => ( bs114d
% 0.30/0.46 => ( setminusI
% 0.30/0.46 => ( setminusEL
% 0.30/0.46 => ( setminusER
% 0.30/0.46 => ( setminusSubset2
% 0.30/0.46 => ( setminusERneg
% 0.30/0.46 => ( setminusELneg
% 0.30/0.46 => ( setminusILneg
% 0.30/0.46 => ( setminusIRneg
% 0.30/0.46 => ( setminusLsub
% 0.30/0.46 => ( setminusSubset1
% 0.30/0.46 => ( symdiffE
% 0.30/0.46 => ( symdiffI1
% 0.30/0.46 => ( symdiffI2
% 0.30/0.46 => ( symdiffIneg1
% 0.30/0.46 => ( symdiffIneg2
% 0.30/0.46 => ( secondinupair
% 0.30/0.46 => ( setukpairIL
% 0.30/0.46 => ( setukpairIR
% 0.30/0.46 => ( kpairiskpair
% 0.30/0.46 => ( kpairp
% 0.30/0.46 => ( singletonsubset
% 0.30/0.46 => ( singletoninpowerset
% 0.30/0.46 => ( singletoninpowunion
% 0.30/0.46 => ( upairset2E
% 0.30/0.46 => ( upairsubunion
% 0.30/0.46 => ( upairinpowunion
% 0.30/0.46 => ( ubforcartprodlem1
% 0.30/0.46 => ( ubforcartprodlem2
% 0.30/0.46 => ( ubforcartprodlem3
% 0.30/0.46 => ( cartprodpairin
% 0.30/0.46 => ( cartprodmempair1
% 0.30/0.46 => ( cartprodmempair
% 0.30/0.46 => ( setunionE2
% 0.30/0.46 => ( setunionsingleton1
% 0.30/0.46 => ( setunionsingleton2
% 0.30/0.46 => ( setunionsingleton
% 0.30/0.46 => ( singletonprop
% 0.30/0.46 => ( ex1E1
% 0.30/0.46 => ( ex1I
% 0.30/0.46 => ( ex1I2
% 0.30/0.46 => ( singletonsuniq
% 0.30/0.46 => ( setukpairinjL1
% 0.30/0.46 => ( kfstsingleton
% 0.30/0.46 => ( theprop
% 0.30/0.46 => ( kfstpairEq
% 0.30/0.46 => ( cartprodfstin
% 0.30/0.46 => ( setukpairinjL2
% 0.30/0.46 => ( setukpairinjL
% 0.30/0.46 => ( setukpairinjR11
% 0.30/0.46 => ( setukpairinjR12
% 0.30/0.46 => ( setukpairinjR1
% 0.30/0.46 => ( upairequniteq
% 0.30/0.46 => ( setukpairinjR2
% 0.30/0.46 => ( setukpairinjR
% 0.30/0.46 => ( ksndsingleton
% 0.30/0.46 => ( ksndpairEq
% 0.30/0.46 => ( kpairsurjEq
% 0.30/0.46 => ( cartprodsndin
% 0.30/0.46 => ( cartprodpairmemEL
% 0.30/0.46 => ( cartprodpairmemER
% 0.30/0.46 => ( cartprodmempaircEq
% 0.30/0.46 => ( cartprodfstpairEq
% 0.30/0.46 => ( cartprodsndpairEq
% 0.30/0.46 => ( cartprodpairsurjEq
% 0.30/0.46 => ( dpsetconstrI
% 0.30/0.46 => ( dpsetconstrSub
% 0.30/0.46 => ( setOfPairsIsBReln
% 0.30/0.46 => ( dpsetconstrERa
% 0.30/0.46 => ( dpsetconstrEL1
% 0.30/0.46 => ( dpsetconstrEL2
% 0.30/0.46 => ( dpsetconstrER
% 0.30/0.46 => ( funcImageSingleton
% 0.30/0.46 => ( apProp
% 0.30/0.46 => ( app
% 0.30/0.46 => ( infuncsetfunc
% 0.30/0.46 => ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.47/1.46 ---- Embedded in THF ---
% 2.47/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.47/1.46 %%% Generated on Tue Feb 24 12:30:09 EST 2026
% 2.47/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.47/1.46 %%% Logic specification used:
% 2.47/1.46 %%% thf(spec, logic, $$dhol).
% 2.47/1.46
% 2.47/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.TbQYBRPofc/DTF2DTF6601.p
% See solution above
% 2.47/1.46 ------------------------
% 2.67/1.50 ---- Cleaned THF ---
% 2.67/1.50 thf(in_type,type,
% 2.67/1.50 in: $i > $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(exu_type,type,
% 2.67/1.50 exu: ( $i > $o ) > $o ).
% 2.67/1.50
% 2.67/1.50 thf(exu,definition,
% 2.67/1.50 ( exu
% 2.67/1.50 = ( ^ [Xphi: $i > $o] :
% 2.67/1.50 ? [Xx: $i] :
% 2.67/1.50 ( ( Xphi @ Xx )
% 2.67/1.50 & ! [Xy: $i] :
% 2.67/1.50 ( ( Xphi @ Xy )
% 2.67/1.50 => ( Xx = Xy ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setextAx_type,type,
% 2.67/1.50 setextAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setextAx,definition,
% 2.67/1.50 ( setextAx
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 <=> ( in @ Xx @ B ) )
% 2.67/1.50 => ( A = B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptyset_type,type,
% 2.67/1.50 emptyset: $i ).
% 2.67/1.50
% 2.67/1.50 thf(emptysetAx_type,type,
% 2.67/1.50 emptysetAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptysetAx,definition,
% 2.67/1.50 ( emptysetAx
% 2.67/1.50 = ( ! [Xx: $i] :
% 2.67/1.50 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoin_type,type,
% 2.67/1.50 setadjoin: $i > $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinAx_type,type,
% 2.67/1.50 setadjoinAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinAx,definition,
% 2.67/1.50 ( setadjoinAx
% 2.67/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.67/1.50 <=> ( ( Xy = Xx )
% 2.67/1.50 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(powerset_type,type,
% 2.67/1.50 powerset: $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(powersetAx_type,type,
% 2.67/1.50 powersetAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(powersetAx,definition,
% 2.67/1.50 ( powersetAx
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( in @ B @ ( powerset @ A ) )
% 2.67/1.50 <=> ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setunion_type,type,
% 2.67/1.50 setunion: $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(setunionAx_type,type,
% 2.67/1.50 setunionAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setunionAx,definition,
% 2.67/1.50 ( setunionAx
% 2.67/1.50 = ( ! [A: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( setunion @ A ) )
% 2.67/1.50 <=> ? [B: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 & ( in @ B @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(omega_type,type,
% 2.67/1.50 omega: $i ).
% 2.67/1.50
% 2.67/1.50 thf(omega0Ax_type,type,
% 2.67/1.50 omega0Ax: $o ).
% 2.67/1.50
% 2.67/1.50 thf(omega0Ax,definition,
% 2.67/1.50 ( omega0Ax
% 2.67/1.50 = ( in @ emptyset @ omega ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(omegaSAx_type,type,
% 2.67/1.50 omegaSAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(omegaSAx,definition,
% 2.67/1.50 ( omegaSAx
% 2.67/1.50 = ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ omega )
% 2.67/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(omegaIndAx_type,type,
% 2.67/1.50 omegaIndAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(omegaIndAx,definition,
% 2.67/1.50 ( omegaIndAx
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ( ( ( in @ emptyset @ A )
% 2.67/1.50 & ! [Xx: $i] :
% 2.67/1.50 ( ( ( in @ Xx @ omega )
% 2.67/1.50 & ( in @ Xx @ A ) )
% 2.67/1.50 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.67/1.50 => ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ omega )
% 2.67/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(replAx_type,type,
% 2.67/1.50 replAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(replAx,definition,
% 2.67/1.50 ( replAx
% 2.67/1.50 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( exu
% 2.67/1.50 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.67/1.50 => ? [B: $i] :
% 2.67/1.50 ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 <=> ? [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ A )
% 2.67/1.50 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(foundationAx_type,type,
% 2.67/1.50 foundationAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(foundationAx,definition,
% 2.67/1.50 ( foundationAx
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.67/1.50 => ? [B: $i] :
% 2.67/1.50 ( ( in @ B @ A )
% 2.67/1.50 & ~ ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(wellorderingAx_type,type,
% 2.67/1.50 wellorderingAx: $o ).
% 2.67/1.50
% 2.67/1.50 thf(wellorderingAx,definition,
% 2.67/1.50 ( wellorderingAx
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ? [B: $i] :
% 2.67/1.50 ( ! [C: $i] :
% 2.67/1.50 ( ( in @ C @ B )
% 2.67/1.50 => ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ C )
% 2.67/1.50 => ( in @ Xx @ A ) ) )
% 2.67/1.50 & ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( ( in @ Xx @ A )
% 2.67/1.50 & ( in @ Xy @ A ) )
% 2.67/1.50 => ( ! [C: $i] :
% 2.67/1.50 ( ( in @ C @ B )
% 2.67/1.50 => ( ( in @ Xx @ C )
% 2.67/1.50 <=> ( in @ Xy @ C ) ) )
% 2.67/1.50 => ( Xx = Xy ) ) )
% 2.67/1.50 & ! [C: $i,D: $i] :
% 2.67/1.50 ( ( ( in @ C @ B )
% 2.67/1.50 & ( in @ D @ B ) )
% 2.67/1.50 => ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ C )
% 2.67/1.50 => ( in @ Xx @ D ) )
% 2.67/1.50 | ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ D )
% 2.67/1.50 => ( in @ Xx @ C ) ) ) )
% 2.67/1.50 & ! [C: $i] :
% 2.67/1.50 ( ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ C )
% 2.67/1.50 => ( in @ Xx @ A ) )
% 2.67/1.50 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.67/1.50 => ? [D: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ D @ B )
% 2.67/1.50 & ( in @ Xx @ C )
% 2.67/1.50 & ~ ? [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ D )
% 2.67/1.50 & ( in @ Xy @ C ) )
% 2.67/1.50 & ! [E: $i] :
% 2.67/1.50 ( ( in @ E @ B )
% 2.67/1.50 => ( ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ E )
% 2.67/1.50 => ( in @ Xy @ D ) )
% 2.67/1.50 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(descr_type,type,
% 2.67/1.50 descr: ( $i > $o ) > $i ).
% 2.67/1.50
% 2.67/1.50 thf(descrp_type,type,
% 2.67/1.50 descrp: $o ).
% 2.67/1.50
% 2.67/1.50 thf(descrp,definition,
% 2.67/1.50 ( descrp
% 2.67/1.50 = ( ! [Xphi: $i > $o] :
% 2.67/1.50 ( ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 => ( Xphi
% 2.67/1.50 @ ( descr
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dsetconstr_type,type,
% 2.67/1.50 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.67/1.50
% 2.67/1.50 thf(dsetconstrI_type,type,
% 2.67/1.50 dsetconstrI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dsetconstrI,definition,
% 2.67/1.50 ( dsetconstrI
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ( Xphi @ Xx )
% 2.67/1.50 => ( in @ Xx
% 2.67/1.50 @ ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dsetconstrEL_type,type,
% 2.67/1.50 dsetconstrEL: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dsetconstrEL,definition,
% 2.67/1.50 ( dsetconstrEL
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx
% 2.67/1.50 @ ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.67/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dsetconstrER_type,type,
% 2.67/1.50 dsetconstrER: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dsetconstrER,definition,
% 2.67/1.50 ( dsetconstrER
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx
% 2.67/1.50 @ ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.67/1.50 => ( Xphi @ Xx ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(exuE1_type,type,
% 2.67/1.50 exuE1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(exuE1,definition,
% 2.67/1.50 ( exuE1
% 2.67/1.50 = ( ! [Xphi: $i > $o] :
% 2.67/1.50 ( ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 => ? [Xx: $i] :
% 2.67/1.50 ( ( Xphi @ Xx )
% 2.67/1.50 & ! [Xy: $i] :
% 2.67/1.50 ( ( Xphi @ Xy )
% 2.67/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(prop2set_type,type,
% 2.67/1.50 prop2set: $o > $i ).
% 2.67/1.50
% 2.67/1.50 thf(prop2setE_type,type,
% 2.67/1.50 prop2setE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(prop2setE,definition,
% 2.67/1.50 ( prop2setE
% 2.67/1.50 = ( ! [Xphi: $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.67/1.50 => Xphi ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptysetE_type,type,
% 2.67/1.50 emptysetE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptysetE,definition,
% 2.67/1.50 ( emptysetE
% 2.67/1.50 = ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ emptyset )
% 2.67/1.50 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptysetimpfalse_type,type,
% 2.67/1.50 emptysetimpfalse: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptysetimpfalse,definition,
% 2.67/1.50 ( emptysetimpfalse
% 2.67/1.50 = ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ emptyset )
% 2.67/1.50 => $false ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(notinemptyset_type,type,
% 2.67/1.50 notinemptyset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(notinemptyset,definition,
% 2.67/1.50 ( notinemptyset
% 2.67/1.50 = ( ! [Xx: $i] :
% 2.67/1.50 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(exuE3e_type,type,
% 2.67/1.50 exuE3e: $o ).
% 2.67/1.50
% 2.67/1.50 thf(exuE3e,definition,
% 2.67/1.50 ( exuE3e
% 2.67/1.50 = ( ! [Xphi: $i > $o] :
% 2.67/1.50 ( ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setext_type,type,
% 2.67/1.50 setext: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setext,definition,
% 2.67/1.50 ( setext
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ Xx @ B ) )
% 2.67/1.50 => ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 => ( in @ Xx @ A ) )
% 2.67/1.50 => ( A = B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptyI_type,type,
% 2.67/1.50 emptyI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptyI,definition,
% 2.67/1.50 ( emptyI
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ~ ( in @ Xx @ A )
% 2.67/1.50 => ( A = emptyset ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(noeltsimpempty_type,type,
% 2.67/1.50 noeltsimpempty: $o ).
% 2.67/1.50
% 2.67/1.50 thf(noeltsimpempty,definition,
% 2.67/1.50 ( noeltsimpempty
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ~ ( in @ Xx @ A )
% 2.67/1.50 => ( A = emptyset ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setbeta_type,type,
% 2.67/1.50 setbeta: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setbeta,definition,
% 2.67/1.50 ( setbeta
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ( in @ Xx
% 2.67/1.50 @ ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.67/1.50 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(nonempty_type,type,
% 2.67/1.50 nonempty: $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(nonempty,definition,
% 2.67/1.50 ( nonempty
% 2.67/1.50 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(nonemptyE1_type,type,
% 2.67/1.50 nonemptyE1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(nonemptyE1,definition,
% 2.67/1.50 ( nonemptyE1
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ( ( nonempty @ A )
% 2.67/1.50 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(nonemptyI_type,type,
% 2.67/1.50 nonemptyI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(nonemptyI,definition,
% 2.67/1.50 ( nonemptyI
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ( Xphi @ Xx )
% 2.67/1.50 => ( nonempty
% 2.67/1.50 @ ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(nonemptyI1_type,type,
% 2.67/1.50 nonemptyI1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(nonemptyI1,definition,
% 2.67/1.50 ( nonemptyI1
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.67/1.50 => ( nonempty @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinIL_type,type,
% 2.67/1.50 setadjoinIL: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinIL,definition,
% 2.67/1.50 ( setadjoinIL
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptyinunitempty_type,type,
% 2.67/1.50 emptyinunitempty: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptyinunitempty,definition,
% 2.67/1.50 ( emptyinunitempty
% 2.67/1.50 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinIR_type,type,
% 2.67/1.50 setadjoinIR: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinIR,definition,
% 2.67/1.50 ( setadjoinIR
% 2.67/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ A )
% 2.67/1.50 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinE_type,type,
% 2.67/1.50 setadjoinE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinE,definition,
% 2.67/1.50 ( setadjoinE
% 2.67/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.67/1.50 => ! [Xphi: $o] :
% 2.67/1.50 ( ( ( Xy = Xx )
% 2.67/1.50 => Xphi )
% 2.67/1.50 => ( ( ( in @ Xy @ A )
% 2.67/1.50 => Xphi )
% 2.67/1.50 => Xphi ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinOr_type,type,
% 2.67/1.50 setadjoinOr: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinOr,definition,
% 2.67/1.50 ( setadjoinOr
% 2.67/1.50 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.67/1.50 => ( ( Xy = Xx )
% 2.67/1.50 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setoftrueEq_type,type,
% 2.67/1.50 setoftrueEq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setoftrueEq,definition,
% 2.67/1.50 ( setoftrueEq
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ( ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xx: $i] : $true )
% 2.67/1.50 = A ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(powersetI_type,type,
% 2.67/1.50 powersetI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(powersetI,definition,
% 2.67/1.50 ( powersetI
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 => ( in @ Xx @ A ) )
% 2.67/1.50 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptyinPowerset_type,type,
% 2.67/1.50 emptyinPowerset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptyinPowerset,definition,
% 2.67/1.50 ( emptyinPowerset
% 2.67/1.50 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptyInPowerset_type,type,
% 2.67/1.50 emptyInPowerset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptyInPowerset,definition,
% 2.67/1.50 ( emptyInPowerset
% 2.67/1.50 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(powersetE_type,type,
% 2.67/1.50 powersetE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(powersetE,definition,
% 2.67/1.50 ( powersetE
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ B @ ( powerset @ A ) )
% 2.67/1.50 => ( ( in @ Xx @ B )
% 2.67/1.50 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setunionI_type,type,
% 2.67/1.50 setunionI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setunionI,definition,
% 2.67/1.50 ( setunionI
% 2.67/1.50 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 => ( ( in @ B @ A )
% 2.67/1.50 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setunionE_type,type,
% 2.67/1.50 setunionE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setunionE,definition,
% 2.67/1.50 ( setunionE
% 2.67/1.50 = ( ! [A: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( setunion @ A ) )
% 2.67/1.50 => ! [Xphi: $o] :
% 2.67/1.50 ( ! [B: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 => ( ( in @ B @ A )
% 2.67/1.50 => Xphi ) )
% 2.67/1.50 => Xphi ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(subPowSU_type,type,
% 2.67/1.50 subPowSU: $o ).
% 2.67/1.50
% 2.67/1.50 thf(subPowSU,definition,
% 2.67/1.50 ( subPowSU
% 2.67/1.50 = ( ! [A: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(exuE2_type,type,
% 2.67/1.50 exuE2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(exuE2,definition,
% 2.67/1.50 ( exuE2
% 2.67/1.50 = ( ! [Xphi: $i > $o] :
% 2.67/1.50 ( ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 => ? [Xx: $i] :
% 2.67/1.50 ! [Xy: $i] :
% 2.67/1.50 ( ( Xphi @ Xy )
% 2.67/1.50 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(nonemptyImpWitness_type,type,
% 2.67/1.50 nonemptyImpWitness: $o ).
% 2.67/1.50
% 2.67/1.50 thf(nonemptyImpWitness,definition,
% 2.67/1.50 ( nonemptyImpWitness
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ( ( nonempty @ A )
% 2.67/1.50 => ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & $true ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(uniqinunit_type,type,
% 2.67/1.50 uniqinunit: $o ).
% 2.67/1.50
% 2.67/1.50 thf(uniqinunit,definition,
% 2.67/1.50 ( uniqinunit
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.67/1.50 => ( Xx = Xy ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(notinsingleton_type,type,
% 2.67/1.50 notinsingleton: $o ).
% 2.67/1.50
% 2.67/1.50 thf(notinsingleton,definition,
% 2.67/1.50 ( notinsingleton
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xx != Xy )
% 2.67/1.50 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(eqinunit_type,type,
% 2.67/1.50 eqinunit: $o ).
% 2.67/1.50
% 2.67/1.50 thf(eqinunit,definition,
% 2.67/1.50 ( eqinunit
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xx = Xy )
% 2.67/1.50 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(singletonsswitch_type,type,
% 2.67/1.50 singletonsswitch: $o ).
% 2.67/1.50
% 2.67/1.50 thf(singletonsswitch,definition,
% 2.67/1.50 ( singletonsswitch
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.67/1.50 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(upairsetE_type,type,
% 2.67/1.50 upairsetE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(upairsetE,definition,
% 2.67/1.50 ( upairsetE
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.67/1.50 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.67/1.50 => ( ( Xz = Xx )
% 2.67/1.50 | ( Xz = Xy ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(upairsetIL_type,type,
% 2.67/1.50 upairsetIL: $o ).
% 2.67/1.50
% 2.67/1.50 thf(upairsetIL,definition,
% 2.67/1.50 ( upairsetIL
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(upairsetIR_type,type,
% 2.67/1.50 upairsetIR: $o ).
% 2.67/1.50
% 2.67/1.50 thf(upairsetIR,definition,
% 2.67/1.50 ( upairsetIR
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptyE1_type,type,
% 2.67/1.50 emptyE1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptyE1,definition,
% 2.67/1.50 ( emptyE1
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ( Xphi @ Xx ) )
% 2.67/1.50 => ( ( ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 = emptyset )
% 2.67/1.50 => $false ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(vacuousDall_type,type,
% 2.67/1.50 vacuousDall: $o ).
% 2.67/1.50
% 2.67/1.50 thf(vacuousDall,definition,
% 2.67/1.50 ( vacuousDall
% 2.67/1.50 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ emptyset )
% 2.67/1.50 => ( Xphi @ Xx ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(quantDeMorgan1_type,type,
% 2.67/1.50 quantDeMorgan1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(quantDeMorgan1,definition,
% 2.67/1.50 ( quantDeMorgan1
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ~ ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( Xphi @ Xx ) )
% 2.67/1.50 => ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(quantDeMorgan2_type,type,
% 2.67/1.50 quantDeMorgan2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(quantDeMorgan2,definition,
% 2.67/1.50 ( quantDeMorgan2
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ~ ( Xphi @ Xx ) )
% 2.67/1.50 => ~ ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(quantDeMorgan3_type,type,
% 2.67/1.50 quantDeMorgan3: $o ).
% 2.67/1.50
% 2.67/1.50 thf(quantDeMorgan3,definition,
% 2.67/1.50 ( quantDeMorgan3
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ~ ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ( Xphi @ Xx ) )
% 2.67/1.50 => ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(quantDeMorgan4_type,type,
% 2.67/1.50 quantDeMorgan4: $o ).
% 2.67/1.50
% 2.67/1.50 thf(quantDeMorgan4,definition,
% 2.67/1.50 ( quantDeMorgan4
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ~ ( Xphi @ Xx ) )
% 2.67/1.50 => ~ ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(prop2setI_type,type,
% 2.67/1.50 prop2setI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(prop2setI,definition,
% 2.67/1.50 ( prop2setI
% 2.67/1.50 = ( ! [Xphi: $o] :
% 2.67/1.50 ( Xphi
% 2.67/1.50 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(set2prop_type,type,
% 2.67/1.50 set2prop: $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(prop2set2propI_type,type,
% 2.67/1.50 prop2set2propI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(prop2set2propI,definition,
% 2.67/1.50 ( prop2set2propI
% 2.67/1.50 = ( ! [Xphi: $o] :
% 2.67/1.50 ( Xphi
% 2.67/1.50 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(notdexE_type,type,
% 2.67/1.50 notdexE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(notdexE,definition,
% 2.67/1.50 ( notdexE
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ~ ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ( Xphi @ Xx ) )
% 2.67/1.50 => ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(notdallE_type,type,
% 2.67/1.50 notdallE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(notdallE,definition,
% 2.67/1.50 ( notdallE
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ~ ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( Xphi @ Xx ) )
% 2.67/1.50 => ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(exuI1_type,type,
% 2.67/1.50 exuI1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(exuI1,definition,
% 2.67/1.50 ( exuI1
% 2.67/1.50 = ( ! [Xphi: $i > $o] :
% 2.67/1.50 ( ? [Xx: $i] :
% 2.67/1.50 ( ( Xphi @ Xx )
% 2.67/1.50 & ! [Xy: $i] :
% 2.67/1.50 ( ( Xphi @ Xy )
% 2.67/1.50 => ( Xx = Xy ) ) )
% 2.67/1.50 => ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(exuI3_type,type,
% 2.67/1.50 exuI3: $o ).
% 2.67/1.50
% 2.67/1.50 thf(exuI3,definition,
% 2.67/1.50 ( exuI3
% 2.67/1.50 = ( ! [Xphi: $i > $o] :
% 2.67/1.50 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.67/1.50 => ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xphi @ Xx )
% 2.67/1.50 => ( ( Xphi @ Xy )
% 2.67/1.50 => ( Xx = Xy ) ) )
% 2.67/1.50 => ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(exuI2_type,type,
% 2.67/1.50 exuI2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(exuI2,definition,
% 2.67/1.50 ( exuI2
% 2.67/1.50 = ( ! [Xphi: $i > $o] :
% 2.67/1.50 ( ? [Xx: $i] :
% 2.67/1.50 ! [Xy: $i] :
% 2.67/1.50 ( ( Xphi @ Xy )
% 2.67/1.50 <=> ( Xy = Xx ) )
% 2.67/1.50 => ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(inCongP_type,type,
% 2.67/1.50 inCongP: $o ).
% 2.67/1.50
% 2.67/1.50 thf(inCongP,definition,
% 2.67/1.50 ( inCongP
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( A = B )
% 2.67/1.50 => ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xx = Xy )
% 2.67/1.50 => ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(in__Cong_type,type,
% 2.67/1.50 in__Cong: $o ).
% 2.67/1.50
% 2.67/1.50 thf(in__Cong,definition,
% 2.67/1.50 ( in__Cong
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( A = B )
% 2.67/1.50 => ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xx = Xy )
% 2.67/1.50 => ( ( in @ Xx @ A )
% 2.67/1.50 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(exuE3u_type,type,
% 2.67/1.50 exuE3u: $o ).
% 2.67/1.50
% 2.67/1.50 thf(exuE3u,definition,
% 2.67/1.50 ( exuE3u
% 2.67/1.50 = ( ! [Xphi: $i > $o] :
% 2.67/1.50 ( ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 => ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xphi @ Xx )
% 2.67/1.50 => ( ( Xphi @ Xy )
% 2.67/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(exu__Cong_type,type,
% 2.67/1.50 exu__Cong: $o ).
% 2.67/1.50
% 2.67/1.50 thf(exu__Cong,definition,
% 2.67/1.50 ( exu__Cong
% 2.67/1.50 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.67/1.50 ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xx = Xy )
% 2.67/1.50 => ( ( Xphi @ Xx )
% 2.67/1.50 <=> ( Xpsi @ Xy ) ) )
% 2.67/1.50 => ( ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 <=> ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptyset__Cong_type,type,
% 2.67/1.50 emptyset__Cong: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptyset__Cong,definition,
% 2.67/1.50 ( emptyset__Cong
% 2.67/1.50 = ( emptyset = emptyset ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoin__Cong_type,type,
% 2.67/1.50 setadjoin__Cong: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoin__Cong,definition,
% 2.67/1.50 ( setadjoin__Cong
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xx = Xy )
% 2.67/1.50 => ! [Xz: $i,Xu: $i] :
% 2.67/1.50 ( ( Xz = Xu )
% 2.67/1.50 => ( ( setadjoin @ Xx @ Xz )
% 2.67/1.50 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(powerset__Cong_type,type,
% 2.67/1.50 powerset__Cong: $o ).
% 2.67/1.50
% 2.67/1.50 thf(powerset__Cong,definition,
% 2.67/1.50 ( powerset__Cong
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( A = B )
% 2.67/1.50 => ( ( powerset @ A )
% 2.67/1.50 = ( powerset @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setunion__Cong_type,type,
% 2.67/1.50 setunion__Cong: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setunion__Cong,definition,
% 2.67/1.50 ( setunion__Cong
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( A = B )
% 2.67/1.50 => ( ( setunion @ A )
% 2.67/1.50 = ( setunion @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(omega__Cong_type,type,
% 2.67/1.50 omega__Cong: $o ).
% 2.67/1.50
% 2.67/1.50 thf(omega__Cong,definition,
% 2.67/1.50 ( omega__Cong
% 2.67/1.50 = ( omega = omega ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(exuEu_type,type,
% 2.67/1.50 exuEu: $o ).
% 2.67/1.50
% 2.67/1.50 thf(exuEu,definition,
% 2.67/1.50 ( exuEu
% 2.67/1.50 = ( ! [Xphi: $i > $o] :
% 2.67/1.50 ( ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 => ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xphi @ Xx )
% 2.67/1.50 => ( ( Xphi @ Xy )
% 2.67/1.50 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(descr__Cong_type,type,
% 2.67/1.50 descr__Cong: $o ).
% 2.67/1.50
% 2.67/1.50 thf(descr__Cong,definition,
% 2.67/1.50 ( descr__Cong
% 2.67/1.50 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.67/1.50 ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xx = Xy )
% 2.67/1.50 => ( ( Xphi @ Xx )
% 2.67/1.50 <=> ( Xpsi @ Xy ) ) )
% 2.67/1.50 => ( ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 => ( ( exu
% 2.67/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.67/1.50 => ( ( descr
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 = ( descr
% 2.67/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dsetconstr__Cong_type,type,
% 2.67/1.50 dsetconstr__Cong: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dsetconstr__Cong,definition,
% 2.67/1.50 ( dsetconstr__Cong
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( A = B )
% 2.67/1.50 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( ( Xx = Xy )
% 2.67/1.50 => ( ( Xphi @ Xx )
% 2.67/1.50 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.67/1.50 => ( ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 = ( dsetconstr @ B
% 2.67/1.50 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(subset_type,type,
% 2.67/1.50 subset: $i > $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(disjoint_type,type,
% 2.67/1.50 disjoint: $i > $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(setsmeet_type,type,
% 2.67/1.50 setsmeet: $i > $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(subsetI1_type,type,
% 2.67/1.50 subsetI1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(subsetI1,definition,
% 2.67/1.50 ( subsetI1
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ Xx @ B ) )
% 2.67/1.50 => ( subset @ A @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(eqimpsubset2_type,type,
% 2.67/1.50 eqimpsubset2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(eqimpsubset2,definition,
% 2.67/1.50 ( eqimpsubset2
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( A = B )
% 2.67/1.50 => ( subset @ B @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(eqimpsubset1_type,type,
% 2.67/1.50 eqimpsubset1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(eqimpsubset1,definition,
% 2.67/1.50 ( eqimpsubset1
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( A = B )
% 2.67/1.50 => ( subset @ A @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(subsetI2_type,type,
% 2.67/1.50 subsetI2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(subsetI2,definition,
% 2.67/1.50 ( subsetI2
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ Xx @ B ) )
% 2.67/1.50 => ( subset @ A @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(emptysetsubset_type,type,
% 2.67/1.50 emptysetsubset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(emptysetsubset,definition,
% 2.67/1.50 ( emptysetsubset
% 2.67/1.50 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(subsetE_type,type,
% 2.67/1.50 subsetE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(subsetE,definition,
% 2.67/1.50 ( subsetE
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( subset @ A @ B )
% 2.67/1.50 => ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(subsetE2_type,type,
% 2.67/1.50 subsetE2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(subsetE2,definition,
% 2.67/1.50 ( subsetE2
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( subset @ A @ B )
% 2.67/1.50 => ( ~ ( in @ Xx @ B )
% 2.67/1.50 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(notsubsetI_type,type,
% 2.67/1.50 notsubsetI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(notsubsetI,definition,
% 2.67/1.50 ( notsubsetI
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ~ ( in @ Xx @ B )
% 2.67/1.50 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(notequalI1_type,type,
% 2.67/1.50 notequalI1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(notequalI1,definition,
% 2.67/1.50 ( notequalI1
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ~ ( subset @ A @ B )
% 2.67/1.50 => ( A != B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(notequalI2_type,type,
% 2.67/1.50 notequalI2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(notequalI2,definition,
% 2.67/1.50 ( notequalI2
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ~ ( in @ Xx @ B )
% 2.67/1.50 => ( A != B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(subsetRefl_type,type,
% 2.67/1.50 subsetRefl: $o ).
% 2.67/1.50
% 2.67/1.50 thf(subsetRefl,definition,
% 2.67/1.50 ( subsetRefl
% 2.67/1.50 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(subsetTrans_type,type,
% 2.67/1.50 subsetTrans: $o ).
% 2.67/1.50
% 2.67/1.50 thf(subsetTrans,definition,
% 2.67/1.50 ( subsetTrans
% 2.67/1.50 = ( ! [A: $i,B: $i,C: $i] :
% 2.67/1.50 ( ( subset @ A @ B )
% 2.67/1.50 => ( ( subset @ B @ C )
% 2.67/1.50 => ( subset @ A @ C ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinSub_type,type,
% 2.67/1.50 setadjoinSub: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinSub,definition,
% 2.67/1.50 ( setadjoinSub
% 2.67/1.50 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinSub2_type,type,
% 2.67/1.50 setadjoinSub2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setadjoinSub2,definition,
% 2.67/1.50 ( setadjoinSub2
% 2.67/1.50 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.67/1.50 ( ( subset @ A @ B )
% 2.67/1.50 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(subset2powerset_type,type,
% 2.67/1.50 subset2powerset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(subset2powerset,definition,
% 2.67/1.50 ( subset2powerset
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( subset @ A @ B )
% 2.67/1.50 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setextsub_type,type,
% 2.67/1.50 setextsub: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setextsub,definition,
% 2.67/1.50 ( setextsub
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( subset @ A @ B )
% 2.67/1.50 => ( ( subset @ B @ A )
% 2.67/1.50 => ( A = B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(subsetemptysetimpeq_type,type,
% 2.67/1.50 subsetemptysetimpeq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(subsetemptysetimpeq,definition,
% 2.67/1.50 ( subsetemptysetimpeq
% 2.67/1.50 = ( ! [A: $i] :
% 2.67/1.50 ( ( subset @ A @ emptyset )
% 2.67/1.50 => ( A = emptyset ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(powersetI1_type,type,
% 2.67/1.50 powersetI1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(powersetI1,definition,
% 2.67/1.50 ( powersetI1
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( subset @ B @ A )
% 2.67/1.50 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(powersetE1_type,type,
% 2.67/1.50 powersetE1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(powersetE1,definition,
% 2.67/1.50 ( powersetE1
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( in @ B @ ( powerset @ A ) )
% 2.67/1.50 => ( subset @ B @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(inPowerset_type,type,
% 2.67/1.50 inPowerset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(inPowerset,definition,
% 2.67/1.50 ( inPowerset
% 2.67/1.50 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(powersetsubset_type,type,
% 2.67/1.50 powersetsubset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(powersetsubset,definition,
% 2.67/1.50 ( powersetsubset
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( subset @ A @ B )
% 2.67/1.50 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(sepInPowerset_type,type,
% 2.67/1.50 sepInPowerset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(sepInPowerset,definition,
% 2.67/1.50 ( sepInPowerset
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( in
% 2.67/1.50 @ ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 @ ( powerset @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(sepSubset_type,type,
% 2.67/1.50 sepSubset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(sepSubset,definition,
% 2.67/1.50 ( sepSubset
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( subset
% 2.67/1.50 @ ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 @ A ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binunion_type,type,
% 2.67/1.50 binunion: $i > $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(binunionIL_type,type,
% 2.67/1.50 binunionIL: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binunionIL,definition,
% 2.67/1.50 ( binunionIL
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(upairset2IR_type,type,
% 2.67/1.50 upairset2IR: $o ).
% 2.67/1.50
% 2.67/1.50 thf(upairset2IR,definition,
% 2.67/1.50 ( upairset2IR
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binunionIR_type,type,
% 2.67/1.50 binunionIR: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binunionIR,definition,
% 2.67/1.50 ( binunionIR
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binunionEcases_type,type,
% 2.67/1.50 binunionEcases: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binunionEcases,definition,
% 2.67/1.50 ( binunionEcases
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.67/1.50 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.67/1.50 => ( ( ( in @ Xx @ A )
% 2.67/1.50 => Xphi )
% 2.67/1.50 => ( ( ( in @ Xx @ B )
% 2.67/1.50 => Xphi )
% 2.67/1.50 => Xphi ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binunionE_type,type,
% 2.67/1.50 binunionE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binunionE,definition,
% 2.67/1.50 ( binunionE
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.67/1.50 => ( ( in @ Xx @ A )
% 2.67/1.50 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binunionLsub_type,type,
% 2.67/1.50 binunionLsub: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binunionLsub,definition,
% 2.67/1.50 ( binunionLsub
% 2.67/1.50 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binunionRsub_type,type,
% 2.67/1.50 binunionRsub: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binunionRsub,definition,
% 2.67/1.50 ( binunionRsub
% 2.67/1.50 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersect_type,type,
% 2.67/1.50 binintersect: $i > $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectI_type,type,
% 2.67/1.50 binintersectI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectI,definition,
% 2.67/1.50 ( binintersectI
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ( in @ Xx @ B )
% 2.67/1.50 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset5_type,type,
% 2.67/1.50 binintersectSubset5: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset5,definition,
% 2.67/1.50 ( binintersectSubset5
% 2.67/1.50 = ( ! [A: $i,B: $i,C: $i] :
% 2.67/1.50 ( ( subset @ C @ A )
% 2.67/1.50 => ( ( subset @ C @ B )
% 2.67/1.50 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectEL_type,type,
% 2.67/1.50 binintersectEL: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectEL,definition,
% 2.67/1.50 ( binintersectEL
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.67/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectLsub_type,type,
% 2.67/1.50 binintersectLsub: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectLsub,definition,
% 2.67/1.50 ( binintersectLsub
% 2.67/1.50 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset2_type,type,
% 2.67/1.50 binintersectSubset2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset2,definition,
% 2.67/1.50 ( binintersectSubset2
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( subset @ A @ B )
% 2.67/1.50 => ( ( binintersect @ A @ B )
% 2.67/1.50 = A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset3_type,type,
% 2.67/1.50 binintersectSubset3: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset3,definition,
% 2.67/1.50 ( binintersectSubset3
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( ( binintersect @ A @ B )
% 2.67/1.50 = B )
% 2.67/1.50 => ( subset @ B @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectER_type,type,
% 2.67/1.50 binintersectER: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectER,definition,
% 2.67/1.50 ( binintersectER
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.67/1.50 => ( in @ Xx @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(disjointsetsI1_type,type,
% 2.67/1.50 disjointsetsI1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(disjointsetsI1,definition,
% 2.67/1.50 ( disjointsetsI1
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ~ ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ( in @ Xx @ B ) )
% 2.67/1.50 => ( ( binintersect @ A @ B )
% 2.67/1.50 = emptyset ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectRsub_type,type,
% 2.67/1.50 binintersectRsub: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectRsub,definition,
% 2.67/1.50 ( binintersectRsub
% 2.67/1.50 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset4_type,type,
% 2.67/1.50 binintersectSubset4: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset4,definition,
% 2.67/1.50 ( binintersectSubset4
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( subset @ B @ A )
% 2.67/1.50 => ( ( binintersect @ A @ B )
% 2.67/1.50 = B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset1_type,type,
% 2.67/1.50 binintersectSubset1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(binintersectSubset1,definition,
% 2.67/1.50 ( binintersectSubset1
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( ( binintersect @ A @ B )
% 2.67/1.50 = A )
% 2.67/1.50 => ( subset @ A @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(bs114d_type,type,
% 2.67/1.50 bs114d: $o ).
% 2.67/1.50
% 2.67/1.50 thf(bs114d,definition,
% 2.67/1.50 ( bs114d
% 2.67/1.50 = ( ! [A: $i,B: $i,C: $i] :
% 2.67/1.50 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.67/1.50 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(regular_type,type,
% 2.67/1.50 regular: $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminus_type,type,
% 2.67/1.50 setminus: $i > $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(setminusI_type,type,
% 2.67/1.50 setminusI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusI,definition,
% 2.67/1.50 ( setminusI
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ~ ( in @ Xx @ B )
% 2.67/1.50 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setminusEL_type,type,
% 2.67/1.50 setminusEL: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusEL,definition,
% 2.67/1.50 ( setminusEL
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.67/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setminusER_type,type,
% 2.67/1.50 setminusER: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusER,definition,
% 2.67/1.50 ( setminusER
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.67/1.50 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setminusSubset2_type,type,
% 2.67/1.50 setminusSubset2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusSubset2,definition,
% 2.67/1.50 ( setminusSubset2
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( subset @ A @ B )
% 2.67/1.50 => ( ( setminus @ A @ B )
% 2.67/1.50 = emptyset ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setminusERneg_type,type,
% 2.67/1.50 setminusERneg: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusERneg,definition,
% 2.67/1.50 ( setminusERneg
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.67/1.50 => ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setminusELneg_type,type,
% 2.67/1.50 setminusELneg: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusELneg,definition,
% 2.67/1.50 ( setminusELneg
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.67/1.50 => ( ~ ( in @ Xx @ B )
% 2.67/1.50 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setminusILneg_type,type,
% 2.67/1.50 setminusILneg: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusILneg,definition,
% 2.67/1.50 ( setminusILneg
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ~ ( in @ Xx @ A )
% 2.67/1.50 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setminusIRneg_type,type,
% 2.67/1.50 setminusIRneg: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusIRneg,definition,
% 2.67/1.50 ( setminusIRneg
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ B )
% 2.67/1.50 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setminusLsub_type,type,
% 2.67/1.50 setminusLsub: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusLsub,definition,
% 2.67/1.50 ( setminusLsub
% 2.67/1.50 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setminusSubset1_type,type,
% 2.67/1.50 setminusSubset1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setminusSubset1,definition,
% 2.67/1.50 ( setminusSubset1
% 2.67/1.50 = ( ! [A: $i,B: $i] :
% 2.67/1.50 ( ( ( setminus @ A @ B )
% 2.67/1.50 = emptyset )
% 2.67/1.50 => ( subset @ A @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(symdiff_type,type,
% 2.67/1.50 symdiff: $i > $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffE_type,type,
% 2.67/1.50 symdiffE: $o ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffE,definition,
% 2.67/1.50 ( symdiffE
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.67/1.50 => ! [Xphi: $o] :
% 2.67/1.50 ( ( ( in @ Xx @ A )
% 2.67/1.50 => ( ~ ( in @ Xx @ B )
% 2.67/1.50 => Xphi ) )
% 2.67/1.50 => ( ( ~ ( in @ Xx @ A )
% 2.67/1.50 => ( ( in @ Xx @ B )
% 2.67/1.50 => Xphi ) )
% 2.67/1.50 => Xphi ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffI1_type,type,
% 2.67/1.50 symdiffI1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffI1,definition,
% 2.67/1.50 ( symdiffI1
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ~ ( in @ Xx @ B )
% 2.67/1.50 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffI2_type,type,
% 2.67/1.50 symdiffI2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffI2,definition,
% 2.67/1.50 ( symdiffI2
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ~ ( in @ Xx @ A )
% 2.67/1.50 => ( ( in @ Xx @ B )
% 2.67/1.50 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffIneg1_type,type,
% 2.67/1.50 symdiffIneg1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffIneg1,definition,
% 2.67/1.50 ( symdiffIneg1
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ( in @ Xx @ B )
% 2.67/1.50 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffIneg2_type,type,
% 2.67/1.50 symdiffIneg2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(symdiffIneg2,definition,
% 2.67/1.50 ( symdiffIneg2
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ~ ( in @ Xx @ A )
% 2.67/1.50 => ( ~ ( in @ Xx @ B )
% 2.67/1.50 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(iskpair_type,type,
% 2.67/1.50 iskpair: $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(secondinupair_type,type,
% 2.67/1.50 secondinupair: $o ).
% 2.67/1.50
% 2.67/1.50 thf(secondinupair,definition,
% 2.67/1.50 ( secondinupair
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairIL_type,type,
% 2.67/1.50 setukpairIL: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairIL,definition,
% 2.67/1.50 ( setukpairIL
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairIR_type,type,
% 2.67/1.50 setukpairIR: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairIR,definition,
% 2.67/1.50 ( setukpairIR
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(kpairiskpair_type,type,
% 2.67/1.50 kpairiskpair: $o ).
% 2.67/1.50
% 2.67/1.50 thf(kpairiskpair,definition,
% 2.67/1.50 ( kpairiskpair
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(kpair_type,type,
% 2.67/1.50 kpair: $i > $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(kpairp_type,type,
% 2.67/1.50 kpairp: $o ).
% 2.67/1.50
% 2.67/1.50 thf(kpairp,definition,
% 2.67/1.50 ( kpairp
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprod_type,type,
% 2.67/1.50 cartprod: $i > $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(singletonsubset_type,type,
% 2.67/1.50 singletonsubset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(singletonsubset,definition,
% 2.67/1.50 ( singletonsubset
% 2.67/1.50 = ( ! [A: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(singletoninpowerset_type,type,
% 2.67/1.50 singletoninpowerset: $o ).
% 2.67/1.50
% 2.67/1.50 thf(singletoninpowerset,definition,
% 2.67/1.50 ( singletoninpowerset
% 2.67/1.50 = ( ! [A: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(singletoninpowunion_type,type,
% 2.67/1.50 singletoninpowunion: $o ).
% 2.67/1.50
% 2.67/1.50 thf(singletoninpowunion,definition,
% 2.67/1.50 ( singletoninpowunion
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(upairset2E_type,type,
% 2.67/1.50 upairset2E: $o ).
% 2.67/1.50
% 2.67/1.50 thf(upairset2E,definition,
% 2.67/1.50 ( upairset2E
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.67/1.50 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.67/1.50 => ( ( Xz = Xx )
% 2.67/1.50 | ( Xz = Xy ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(upairsubunion_type,type,
% 2.67/1.50 upairsubunion: $o ).
% 2.67/1.50
% 2.67/1.50 thf(upairsubunion,definition,
% 2.67/1.50 ( upairsubunion
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(upairinpowunion_type,type,
% 2.67/1.50 upairinpowunion: $o ).
% 2.67/1.50
% 2.67/1.50 thf(upairinpowunion,definition,
% 2.67/1.50 ( upairinpowunion
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ubforcartprodlem1_type,type,
% 2.67/1.50 ubforcartprodlem1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(ubforcartprodlem1,definition,
% 2.67/1.50 ( ubforcartprodlem1
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ubforcartprodlem2_type,type,
% 2.67/1.50 ubforcartprodlem2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(ubforcartprodlem2,definition,
% 2.67/1.50 ( ubforcartprodlem2
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ubforcartprodlem3_type,type,
% 2.67/1.50 ubforcartprodlem3: $o ).
% 2.67/1.50
% 2.67/1.50 thf(ubforcartprodlem3,definition,
% 2.67/1.50 ( ubforcartprodlem3
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodpairin_type,type,
% 2.67/1.50 cartprodpairin: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodpairin,definition,
% 2.67/1.50 ( cartprodpairin
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodmempair1_type,type,
% 2.67/1.50 cartprodmempair1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodmempair1,definition,
% 2.67/1.50 ( cartprodmempair1
% 2.67/1.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.67/1.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.67/1.50 => ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ? [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 & ( Xu
% 2.67/1.50 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodmempair_type,type,
% 2.67/1.50 cartprodmempair: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodmempair,definition,
% 2.67/1.50 ( cartprodmempair
% 2.67/1.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.67/1.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.67/1.50 => ( iskpair @ Xu ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setunionE2_type,type,
% 2.67/1.50 setunionE2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setunionE2,definition,
% 2.67/1.50 ( setunionE2
% 2.67/1.50 = ( ! [A: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ ( setunion @ A ) )
% 2.67/1.50 => ? [X: $i] :
% 2.67/1.50 ( ( in @ X @ A )
% 2.67/1.50 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setunionsingleton1_type,type,
% 2.67/1.50 setunionsingleton1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setunionsingleton1,definition,
% 2.67/1.50 ( setunionsingleton1
% 2.67/1.50 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setunionsingleton2_type,type,
% 2.67/1.50 setunionsingleton2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setunionsingleton2,definition,
% 2.67/1.50 ( setunionsingleton2
% 2.67/1.50 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setunionsingleton_type,type,
% 2.67/1.50 setunionsingleton: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setunionsingleton,definition,
% 2.67/1.50 ( setunionsingleton
% 2.67/1.50 = ( ! [Xx: $i] :
% 2.67/1.50 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.67/1.50 = Xx ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(singleton_type,type,
% 2.67/1.50 singleton: $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(singleton,definition,
% 2.67/1.50 ( singleton
% 2.67/1.50 = ( ^ [A: $i] :
% 2.67/1.50 ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ( A
% 2.67/1.50 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(singletonprop_type,type,
% 2.67/1.50 singletonprop: $o ).
% 2.67/1.50
% 2.67/1.50 thf(singletonprop,definition,
% 2.67/1.50 ( singletonprop
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ A )
% 2.67/1.50 => ( ( Xphi @ Xx )
% 2.67/1.50 => ( ( Xphi @ Xy )
% 2.67/1.50 => ( Xx = Xy ) ) ) ) )
% 2.67/1.50 => ( ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ( Xphi @ Xx ) )
% 2.67/1.50 => ( singleton
% 2.67/1.50 @ ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ex1_type,type,
% 2.67/1.50 ex1: $i > ( $i > $o ) > $o ).
% 2.67/1.50
% 2.67/1.50 thf(ex1,definition,
% 2.67/1.50 ( ex1
% 2.67/1.50 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( singleton
% 2.67/1.50 @ ( dsetconstr @ A
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ex1E1_type,type,
% 2.67/1.50 ex1E1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(ex1E1,definition,
% 2.67/1.50 ( ex1E1
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ( ex1 @ A
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.67/1.50 => ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ex1I_type,type,
% 2.67/1.50 ex1I: $o ).
% 2.67/1.50
% 2.67/1.50 thf(ex1I,definition,
% 2.67/1.50 ( ex1I
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ( Xphi @ Xx )
% 2.67/1.50 => ( ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ A )
% 2.67/1.50 => ( ( Xphi @ Xy )
% 2.67/1.50 => ( Xy = Xx ) ) )
% 2.67/1.50 => ( ex1 @ A
% 2.67/1.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ex1I2_type,type,
% 2.67/1.50 ex1I2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(ex1I2,definition,
% 2.67/1.50 ( ex1I2
% 2.67/1.50 = ( ! [A: $i,Xphi: $i > $o] :
% 2.67/1.50 ( ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ A )
% 2.67/1.50 => ( ( Xphi @ Xx )
% 2.67/1.50 => ( ( Xphi @ Xy )
% 2.67/1.50 => ( Xx = Xy ) ) ) ) )
% 2.67/1.50 => ( ? [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 & ( Xphi @ Xx ) )
% 2.67/1.50 => ( ex1 @ A
% 2.67/1.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(singletonsuniq_type,type,
% 2.67/1.50 singletonsuniq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(singletonsuniq,definition,
% 2.67/1.50 ( singletonsuniq
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( ( setadjoin @ Xx @ emptyset )
% 2.67/1.50 = ( setadjoin @ Xy @ emptyset ) )
% 2.67/1.50 => ( Xx = Xy ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(atmost1p_type,type,
% 2.67/1.50 atmost1p: $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(atleast2p_type,type,
% 2.67/1.50 atleast2p: $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(atmost2p_type,type,
% 2.67/1.50 atmost2p: $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(upairsetp_type,type,
% 2.67/1.50 upairsetp: $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjL1_type,type,
% 2.67/1.50 setukpairinjL1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjL1,definition,
% 2.67/1.50 ( setukpairinjL1
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.67/1.50 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.67/1.50 => ( Xx = Xz ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(kfstsingleton_type,type,
% 2.67/1.50 kfstsingleton: $o ).
% 2.67/1.50
% 2.67/1.50 thf(kfstsingleton,definition,
% 2.67/1.50 ( kfstsingleton
% 2.67/1.50 = ( ! [Xu: $i] :
% 2.67/1.50 ( ( iskpair @ Xu )
% 2.67/1.50 => ( singleton
% 2.67/1.50 @ ( dsetconstr @ ( setunion @ Xu )
% 2.67/1.50 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(theprop_type,type,
% 2.67/1.50 theprop: $o ).
% 2.67/1.50
% 2.67/1.50 thf(theprop,definition,
% 2.67/1.50 ( theprop
% 2.67/1.50 = ( ! [X: $i] :
% 2.67/1.50 ( ( singleton @ X )
% 2.67/1.50 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(kfst_type,type,
% 2.67/1.50 kfst: $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(kfstpairEq_type,type,
% 2.67/1.50 kfstpairEq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(kfstpairEq,definition,
% 2.67/1.50 ( kfstpairEq
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.67/1.50 = Xx ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodfstin_type,type,
% 2.67/1.50 cartprodfstin: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodfstin,definition,
% 2.67/1.50 ( cartprodfstin
% 2.67/1.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.67/1.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.67/1.50 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjL2_type,type,
% 2.67/1.50 setukpairinjL2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjL2,definition,
% 2.67/1.50 ( setukpairinjL2
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.67/1.50 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.67/1.50 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.67/1.50 => ( Xx = Xz ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjL_type,type,
% 2.67/1.50 setukpairinjL: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjL,definition,
% 2.67/1.50 ( setukpairinjL
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.67/1.50 ( ( ( kpair @ Xx @ Xy )
% 2.67/1.50 = ( kpair @ Xz @ Xu ) )
% 2.67/1.50 => ( Xx = Xz ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR11_type,type,
% 2.67/1.50 setukpairinjR11: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR11,definition,
% 2.67/1.50 ( setukpairinjR11
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xx = Xy )
% 2.67/1.50 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.67/1.50 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR12_type,type,
% 2.67/1.50 setukpairinjR12: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR12,definition,
% 2.67/1.50 ( setukpairinjR12
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( Xx = Xy )
% 2.67/1.50 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.67/1.50 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR1_type,type,
% 2.67/1.50 setukpairinjR1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR1,definition,
% 2.67/1.50 ( setukpairinjR1
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.67/1.50 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.67/1.50 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.67/1.50 => ( ( Xz = Xu )
% 2.67/1.50 => ( Xy = Xu ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(upairequniteq_type,type,
% 2.67/1.50 upairequniteq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(upairequniteq,definition,
% 2.67/1.50 ( upairequniteq
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.67/1.50 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.67/1.50 = ( setadjoin @ Xz @ emptyset ) )
% 2.67/1.50 => ( Xx = Xy ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR2_type,type,
% 2.67/1.50 setukpairinjR2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR2,definition,
% 2.67/1.50 ( setukpairinjR2
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.67/1.50 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.67/1.50 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.67/1.50 => ( Xy = Xu ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR_type,type,
% 2.67/1.50 setukpairinjR: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setukpairinjR,definition,
% 2.67/1.50 ( setukpairinjR
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.67/1.50 ( ( ( kpair @ Xx @ Xy )
% 2.67/1.50 = ( kpair @ Xz @ Xu ) )
% 2.67/1.50 => ( Xy = Xu ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ksndsingleton_type,type,
% 2.67/1.50 ksndsingleton: $o ).
% 2.67/1.50
% 2.67/1.50 thf(ksndsingleton,definition,
% 2.67/1.50 ( ksndsingleton
% 2.67/1.50 = ( ! [Xu: $i] :
% 2.67/1.50 ( ( iskpair @ Xu )
% 2.67/1.50 => ( singleton
% 2.67/1.50 @ ( dsetconstr @ ( setunion @ Xu )
% 2.67/1.50 @ ^ [Xx: $i] :
% 2.67/1.50 ( Xu
% 2.67/1.50 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ksnd_type,type,
% 2.67/1.50 ksnd: $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(ksndpairEq_type,type,
% 2.67/1.50 ksndpairEq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(ksndpairEq,definition,
% 2.67/1.50 ( ksndpairEq
% 2.67/1.50 = ( ! [Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.67/1.50 = Xy ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(kpairsurjEq_type,type,
% 2.67/1.50 kpairsurjEq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(kpairsurjEq,definition,
% 2.67/1.50 ( kpairsurjEq
% 2.67/1.50 = ( ! [Xu: $i] :
% 2.67/1.50 ( ( iskpair @ Xu )
% 2.67/1.50 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.67/1.50 = Xu ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodsndin_type,type,
% 2.67/1.50 cartprodsndin: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodsndin,definition,
% 2.67/1.50 ( cartprodsndin
% 2.67/1.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.67/1.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.67/1.50 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodpairmemEL_type,type,
% 2.67/1.50 cartprodpairmemEL: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodpairmemEL,definition,
% 2.67/1.50 ( cartprodpairmemEL
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.67/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodpairmemER_type,type,
% 2.67/1.50 cartprodpairmemER: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodpairmemER,definition,
% 2.67/1.50 ( cartprodpairmemER
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.67/1.50 => ( in @ Xy @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodmempaircEq_type,type,
% 2.67/1.50 cartprodmempaircEq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodmempaircEq,definition,
% 2.67/1.50 ( cartprodmempaircEq
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( ( kpair @ Xx @ Xy )
% 2.67/1.50 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodfstpairEq_type,type,
% 2.67/1.50 cartprodfstpairEq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodfstpairEq,definition,
% 2.67/1.50 ( cartprodfstpairEq
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.67/1.50 = Xx ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodsndpairEq_type,type,
% 2.67/1.50 cartprodsndpairEq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodsndpairEq,definition,
% 2.67/1.50 ( cartprodsndpairEq
% 2.67/1.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.67/1.50 = Xy ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodpairsurjEq_type,type,
% 2.67/1.50 cartprodpairsurjEq: $o ).
% 2.67/1.50
% 2.67/1.50 thf(cartprodpairsurjEq,definition,
% 2.67/1.50 ( cartprodpairsurjEq
% 2.67/1.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.67/1.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.67/1.50 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.67/1.50 = Xu ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(breln_type,type,
% 2.67/1.50 breln: $i > $i > $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(breln,definition,
% 2.67/1.50 ( breln
% 2.67/1.50 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstr_type,type,
% 2.67/1.50 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrI_type,type,
% 2.67/1.50 dpsetconstrI: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrI,definition,
% 2.67/1.50 ( dpsetconstrI
% 2.67/1.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( ( Xphi @ Xx @ Xy )
% 2.67/1.50 => ( in @ ( kpair @ Xx @ Xy )
% 2.67/1.50 @ ( dpsetconstr @ A @ B
% 2.67/1.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrSub_type,type,
% 2.67/1.50 dpsetconstrSub: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrSub,definition,
% 2.67/1.50 ( dpsetconstrSub
% 2.67/1.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.67/1.50 ( subset
% 2.67/1.50 @ ( dpsetconstr @ A @ B
% 2.67/1.50 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.67/1.50 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(setOfPairsIsBReln_type,type,
% 2.67/1.50 setOfPairsIsBReln: $o ).
% 2.67/1.50
% 2.67/1.50 thf(setOfPairsIsBReln,definition,
% 2.67/1.50 ( setOfPairsIsBReln
% 2.67/1.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.67/1.50 ( breln @ A @ B
% 2.67/1.50 @ ( dpsetconstr @ A @ B
% 2.67/1.50 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrERa_type,type,
% 2.67/1.50 dpsetconstrERa: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrERa,definition,
% 2.67/1.50 ( dpsetconstrERa
% 2.67/1.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ! [Xy: $i] :
% 2.67/1.50 ( ( in @ Xy @ B )
% 2.67/1.50 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.67/1.50 @ ( dpsetconstr @ A @ B
% 2.67/1.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.67/1.50 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrEL1_type,type,
% 2.67/1.50 dpsetconstrEL1: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrEL1,definition,
% 2.67/1.50 ( dpsetconstrEL1
% 2.67/1.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ ( kpair @ Xx @ Xy )
% 2.67/1.50 @ ( dpsetconstr @ A @ B
% 2.67/1.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.67/1.50 => ( in @ Xx @ A ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrEL2_type,type,
% 2.67/1.50 dpsetconstrEL2: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrEL2,definition,
% 2.67/1.50 ( dpsetconstrEL2
% 2.67/1.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ ( kpair @ Xx @ Xy )
% 2.67/1.50 @ ( dpsetconstr @ A @ B
% 2.67/1.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.67/1.50 => ( in @ Xy @ B ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrER_type,type,
% 2.67/1.50 dpsetconstrER: $o ).
% 2.67/1.50
% 2.67/1.50 thf(dpsetconstrER,definition,
% 2.67/1.50 ( dpsetconstrER
% 2.67/1.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.67/1.50 ( ( in @ ( kpair @ Xx @ Xy )
% 2.67/1.50 @ ( dpsetconstr @ A @ B
% 2.67/1.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.67/1.50 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(func_type,type,
% 2.67/1.50 func: $i > $i > $i > $o ).
% 2.67/1.50
% 2.67/1.50 thf(func,definition,
% 2.67/1.50 ( func
% 2.67/1.50 = ( ^ [A: $i,B: $i,R: $i] :
% 2.67/1.50 ( ( breln @ A @ B @ R )
% 2.67/1.50 & ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( ex1 @ B
% 2.67/1.50 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(funcSet_type,type,
% 2.67/1.50 funcSet: $i > $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(funcSet,definition,
% 2.67/1.50 ( funcSet
% 2.67/1.50 = ( ^ [A: $i,B: $i] :
% 2.67/1.50 ( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
% 2.67/1.50 @ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(funcImageSingleton_type,type,
% 2.67/1.50 funcImageSingleton: $o ).
% 2.67/1.50
% 2.67/1.50 thf(funcImageSingleton,definition,
% 2.67/1.50 ( funcImageSingleton
% 2.67/1.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.67/1.50 ( ( func @ A @ B @ Xf )
% 2.67/1.50 => ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( singleton
% 2.67/1.50 @ ( dsetconstr @ B
% 2.67/1.50 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(apProp_type,type,
% 2.67/1.50 apProp: $o ).
% 2.67/1.50
% 2.67/1.50 thf(apProp,definition,
% 2.67/1.50 ( apProp
% 2.67/1.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.67/1.50 ( ( func @ A @ B @ Xf )
% 2.67/1.50 => ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in
% 2.67/1.50 @ ( setunion
% 2.67/1.50 @ ( dsetconstr @ B
% 2.67/1.50 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.67/1.50 @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ap_type,type,
% 2.67/1.50 ap: $i > $i > $i > $i > $i ).
% 2.67/1.50
% 2.67/1.50 thf(ap,definition,
% 2.67/1.50 ( ap
% 2.67/1.50 = ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
% 2.67/1.50 ( setunion
% 2.67/1.50 @ ( dsetconstr @ B
% 2.67/1.50 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(app_type,type,
% 2.67/1.50 app: $o ).
% 2.67/1.50
% 2.67/1.50 thf(app,definition,
% 2.67/1.50 ( app
% 2.67/1.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.67/1.50 ( ( func @ A @ B @ Xf )
% 2.67/1.50 => ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(infuncsetfunc_type,type,
% 2.67/1.50 infuncsetfunc: $o ).
% 2.67/1.50
% 2.67/1.50 thf(infuncsetfunc,definition,
% 2.67/1.50 ( infuncsetfunc
% 2.67/1.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.67/1.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.67/1.50 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.67/1.50
% 2.67/1.50 thf(ap2p,conjecture,
% 2.67/1.50 ( setextAx
% 2.67/1.50 => ( emptysetAx
% 2.67/1.50 => ( setadjoinAx
% 2.67/1.50 => ( powersetAx
% 2.67/1.50 => ( setunionAx
% 2.67/1.50 => ( omega0Ax
% 2.67/1.50 => ( omegaSAx
% 2.67/1.50 => ( omegaIndAx
% 2.67/1.50 => ( replAx
% 2.67/1.50 => ( foundationAx
% 2.67/1.50 => ( wellorderingAx
% 2.67/1.50 => ( descrp
% 2.67/1.50 => ( dsetconstrI
% 2.67/1.50 => ( dsetconstrEL
% 2.67/1.50 => ( dsetconstrER
% 2.67/1.50 => ( exuE1
% 2.67/1.50 => ( prop2setE
% 2.67/1.50 => ( emptysetE
% 2.67/1.50 => ( emptysetimpfalse
% 2.67/1.50 => ( notinemptyset
% 2.67/1.50 => ( exuE3e
% 2.67/1.50 => ( setext
% 2.67/1.50 => ( emptyI
% 2.67/1.50 => ( noeltsimpempty
% 2.67/1.50 => ( setbeta
% 2.67/1.50 => ( nonemptyE1
% 2.67/1.50 => ( nonemptyI
% 2.67/1.50 => ( nonemptyI1
% 2.67/1.50 => ( setadjoinIL
% 2.67/1.50 => ( emptyinunitempty
% 2.67/1.50 => ( setadjoinIR
% 2.67/1.50 => ( setadjoinE
% 2.67/1.50 => ( setadjoinOr
% 2.67/1.50 => ( setoftrueEq
% 2.67/1.50 => ( powersetI
% 2.67/1.50 => ( emptyinPowerset
% 2.67/1.50 => ( emptyInPowerset
% 2.67/1.50 => ( powersetE
% 2.67/1.50 => ( setunionI
% 2.67/1.50 => ( setunionE
% 2.67/1.50 => ( subPowSU
% 2.67/1.50 => ( exuE2
% 2.67/1.50 => ( nonemptyImpWitness
% 2.67/1.50 => ( uniqinunit
% 2.67/1.50 => ( notinsingleton
% 2.67/1.50 => ( eqinunit
% 2.67/1.50 => ( singletonsswitch
% 2.67/1.50 => ( upairsetE
% 2.67/1.50 => ( upairsetIL
% 2.67/1.50 => ( upairsetIR
% 2.67/1.50 => ( emptyE1
% 2.67/1.50 => ( vacuousDall
% 2.67/1.50 => ( quantDeMorgan1
% 2.67/1.50 => ( quantDeMorgan2
% 2.67/1.50 => ( quantDeMorgan3
% 2.67/1.50 => ( quantDeMorgan4
% 2.67/1.50 => ( prop2setI
% 2.67/1.50 => ( prop2set2propI
% 2.67/1.50 => ( notdexE
% 2.67/1.50 => ( notdallE
% 2.67/1.50 => ( exuI1
% 2.67/1.50 => ( exuI3
% 2.67/1.50 => ( exuI2
% 2.67/1.50 => ( inCongP
% 2.67/1.50 => ( in__Cong
% 2.67/1.50 => ( exuE3u
% 2.67/1.50 => ( exu__Cong
% 2.67/1.50 => ( emptyset__Cong
% 2.67/1.50 => ( setadjoin__Cong
% 2.67/1.50 => ( powerset__Cong
% 2.67/1.50 => ( setunion__Cong
% 2.67/1.50 => ( omega__Cong
% 2.67/1.50 => ( exuEu
% 2.67/1.50 => ( descr__Cong
% 2.67/1.50 => ( dsetconstr__Cong
% 2.67/1.50 => ( subsetI1
% 2.67/1.50 => ( eqimpsubset2
% 2.67/1.50 => ( eqimpsubset1
% 2.67/1.50 => ( subsetI2
% 2.67/1.50 => ( emptysetsubset
% 2.67/1.50 => ( subsetE
% 2.67/1.50 => ( subsetE2
% 2.67/1.50 => ( notsubsetI
% 2.67/1.50 => ( notequalI1
% 2.67/1.50 => ( notequalI2
% 2.67/1.50 => ( subsetRefl
% 2.67/1.50 => ( subsetTrans
% 2.67/1.50 => ( setadjoinSub
% 2.67/1.50 => ( setadjoinSub2
% 2.67/1.50 => ( subset2powerset
% 2.67/1.50 => ( setextsub
% 2.67/1.50 => ( subsetemptysetimpeq
% 2.67/1.50 => ( powersetI1
% 2.67/1.50 => ( powersetE1
% 2.67/1.50 => ( inPowerset
% 2.67/1.50 => ( powersetsubset
% 2.67/1.50 => ( sepInPowerset
% 2.67/1.50 => ( sepSubset
% 2.67/1.50 => ( binunionIL
% 2.67/1.50 => ( upairset2IR
% 2.67/1.50 => ( binunionIR
% 2.67/1.50 => ( binunionEcases
% 2.67/1.50 => ( binunionE
% 2.67/1.50 => ( binunionLsub
% 2.67/1.50 => ( binunionRsub
% 2.67/1.50 => ( binintersectI
% 2.67/1.50 => ( binintersectSubset5
% 2.67/1.50 => ( binintersectEL
% 2.67/1.50 => ( binintersectLsub
% 2.67/1.50 => ( binintersectSubset2
% 2.67/1.50 => ( binintersectSubset3
% 2.67/1.50 => ( binintersectER
% 2.67/1.50 => ( disjointsetsI1
% 2.67/1.50 => ( binintersectRsub
% 2.67/1.50 => ( binintersectSubset4
% 2.67/1.50 => ( binintersectSubset1
% 2.67/1.50 => ( bs114d
% 2.67/1.50 => ( setminusI
% 2.67/1.50 => ( setminusEL
% 2.67/1.50 => ( setminusER
% 2.67/1.50 => ( setminusSubset2
% 2.67/1.50 => ( setminusERneg
% 2.67/1.50 => ( setminusELneg
% 2.67/1.50 => ( setminusILneg
% 2.67/1.50 => ( setminusIRneg
% 2.67/1.50 => ( setminusLsub
% 2.67/1.50 => ( setminusSubset1
% 2.67/1.50 => ( symdiffE
% 2.67/1.50 => ( symdiffI1
% 2.67/1.50 => ( symdiffI2
% 2.67/1.50 => ( symdiffIneg1
% 2.67/1.50 => ( symdiffIneg2
% 2.67/1.50 => ( secondinupair
% 2.67/1.50 => ( setukpairIL
% 2.67/1.50 => ( setukpairIR
% 2.67/1.50 => ( kpairiskpair
% 2.67/1.50 => ( kpairp
% 2.67/1.50 => ( singletonsubset
% 2.67/1.50 => ( singletoninpowerset
% 2.67/1.50 => ( singletoninpowunion
% 2.67/1.50 => ( upairset2E
% 2.67/1.50 => ( upairsubunion
% 2.67/1.50 => ( upairinpowunion
% 2.67/1.50 => ( ubforcartprodlem1
% 2.67/1.50 => ( ubforcartprodlem2
% 2.67/1.50 => ( ubforcartprodlem3
% 2.67/1.50 => ( cartprodpairin
% 2.67/1.50 => ( cartprodmempair1
% 2.67/1.50 => ( cartprodmempair
% 2.67/1.50 => ( setunionE2
% 2.67/1.50 => ( setunionsingleton1
% 2.67/1.50 => ( setunionsingleton2
% 2.67/1.50 => ( setunionsingleton
% 2.67/1.50 => ( singletonprop
% 2.67/1.50 => ( ex1E1
% 2.67/1.50 => ( ex1I
% 2.67/1.50 => ( ex1I2
% 2.67/1.50 => ( singletonsuniq
% 2.67/1.50 => ( setukpairinjL1
% 2.67/1.50 => ( kfstsingleton
% 2.67/1.50 => ( theprop
% 2.67/1.50 => ( kfstpairEq
% 2.67/1.50 => ( cartprodfstin
% 2.67/1.50 => ( setukpairinjL2
% 2.67/1.50 => ( setukpairinjL
% 2.67/1.50 => ( setukpairinjR11
% 2.67/1.50 => ( setukpairinjR12
% 2.67/1.50 => ( setukpairinjR1
% 2.67/1.50 => ( upairequniteq
% 2.67/1.50 => ( setukpairinjR2
% 2.67/1.50 => ( setukpairinjR
% 2.67/1.50 => ( ksndsingleton
% 2.67/1.50 => ( ksndpairEq
% 2.67/1.50 => ( kpairsurjEq
% 2.67/1.50 => ( cartprodsndin
% 2.67/1.50 => ( cartprodpairmemEL
% 2.67/1.50 => ( cartprodpairmemER
% 2.67/1.50 => ( cartprodmempaircEq
% 2.67/1.50 => ( cartprodfstpairEq
% 2.67/1.50 => ( cartprodsndpairEq
% 2.67/1.50 => ( cartprodpairsurjEq
% 2.67/1.50 => ( dpsetconstrI
% 2.67/1.50 => ( dpsetconstrSub
% 2.67/1.50 => ( setOfPairsIsBReln
% 2.67/1.50 => ( dpsetconstrERa
% 2.67/1.50 => ( dpsetconstrEL1
% 2.67/1.50 => ( dpsetconstrEL2
% 2.67/1.50 => ( dpsetconstrER
% 2.67/1.50 => ( funcImageSingleton
% 2.67/1.50 => ( apProp
% 2.67/1.50 => ( app
% 2.67/1.50 => ( infuncsetfunc
% 2.67/1.50 => ! [A: $i,B: $i,Xf: $i] :
% 2.67/1.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.67/1.50 => ! [Xx: $i] :
% 2.67/1.50 ( ( in @ Xx @ A )
% 2.67/1.50 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.67/1.50 ------------------------
% 2.67/1.52 % (6826)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/183Mi)
% 2.67/1.53 % (6828)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_6601 for (2999ds/27Mi)
% 2.67/1.53 % (6830)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_6601 for (2999ds/2Mi)
% 2.67/1.53 % (6831)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_6601 for (2999ds/275Mi)
% 2.67/1.53 % (6832)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/18Mi)
% 2.67/1.53 % (6830)Instruction limit reached!
% 2.67/1.53 % (6830)------------------------------
% 2.67/1.53 % (6830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.53 % (6830)Termination reason: Unknown
% 2.67/1.53 % (6830)Termination phase: shuffling
% 2.67/1.53
% 2.67/1.53 % (6830)Memory used [KB]: 1407
% 2.67/1.53 % (6830)Time elapsed: 0.003 s
% 2.67/1.53 % (6830)Instructions burned: 2 (million)
% 2.67/1.53 % (6830)------------------------------
% 2.67/1.53 % (6830)------------------------------
% 2.67/1.54 % (6827)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_6601 for (2999ds/4Mi)
% 2.67/1.54 % (6832)Instruction limit reached!
% 2.67/1.54 % (6832)------------------------------
% 2.67/1.54 % (6832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.54 % (6832)Termination reason: Unknown
% 2.67/1.54 % (6832)Termination phase: shuffling
% 2.67/1.54
% 2.67/1.54 % (6832)Memory used [KB]: 1663
% 2.67/1.54 % (6832)Time elapsed: 0.012 s
% 2.67/1.54 % (6832)Instructions burned: 18 (million)
% 2.67/1.54 % (6832)------------------------------
% 2.67/1.54 % (6832)------------------------------
% 2.67/1.54 % (6827)Instruction limit reached!
% 2.67/1.54 % (6827)------------------------------
% 2.67/1.54 % (6827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.54 % (6827)Termination reason: Unknown
% 2.67/1.54 % (6827)Termination phase: shuffling
% 2.67/1.54
% 2.67/1.54 % (6828)Instruction limit reached!
% 2.67/1.54 % (6828)------------------------------
% 2.67/1.54 % (6828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.54 % (6827)Memory used [KB]: 1407
% 2.67/1.54 % (6828)Termination reason: Unknown
% 2.67/1.54 % (6827)Time elapsed: 0.005 s
% 2.67/1.54 % (6828)Termination phase: shuffling
% 2.67/1.54
% 2.67/1.54 % (6827)Instructions burned: 4 (million)
% 2.67/1.54 % (6828)Memory used [KB]: 1918
% 2.67/1.54 % (6828)Time elapsed: 0.014 s
% 2.67/1.54 % (6827)------------------------------
% 2.67/1.54 % (6827)------------------------------
% 2.67/1.54 % (6828)Instructions burned: 27 (million)
% 2.67/1.54 % (6828)------------------------------
% 2.67/1.54 % (6828)------------------------------
% 2.67/1.54 % (6829)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/2Mi)
% 2.67/1.54 % (6829)Instruction limit reached!
% 2.67/1.54 % (6829)------------------------------
% 2.67/1.54 % (6829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.54 % (6829)Termination reason: Unknown
% 2.67/1.54 % (6829)Termination phase: shuffling
% 2.67/1.54
% 2.67/1.54 % (6829)Memory used [KB]: 1279
% 2.67/1.54 % (6829)Time elapsed: 0.004 s
% 2.67/1.54 % (6829)Instructions burned: 2 (million)
% 2.67/1.54 % (6829)------------------------------
% 2.67/1.54 % (6829)------------------------------
% 2.67/1.55 % (6837)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/3Mi)
% 2.67/1.55 % (6837)Instruction limit reached!
% 2.67/1.55 % (6837)------------------------------
% 2.67/1.55 % (6837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.55 % (6837)Termination reason: Unknown
% 2.67/1.55 % (6837)Termination phase: shuffling
% 2.67/1.55
% 2.67/1.55 % (6837)Memory used [KB]: 1407
% 2.67/1.55 % (6837)Time elapsed: 0.004 s
% 2.67/1.55 % (6837)Instructions burned: 4 (million)
% 2.67/1.55 % (6837)------------------------------
% 2.67/1.55 % (6837)------------------------------
% 2.67/1.56 % (6841)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_6601 for (2999ds/3Mi)
% 2.67/1.56 % (6839)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_6601 for (2999ds/37Mi)
% 2.67/1.56 % (6841)Instruction limit reached!
% 2.67/1.56 % (6841)------------------------------
% 2.67/1.56 % (6841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.56 % (6841)Termination reason: Unknown
% 2.67/1.56 % (6841)Termination phase: shuffling
% 2.67/1.56
% 2.67/1.56 % (6841)Memory used [KB]: 1407
% 2.67/1.56 % (6841)Time elapsed: 0.003 s
% 2.67/1.56 % (6841)Instructions burned: 4 (million)
% 2.67/1.56 % (6841)------------------------------
% 2.67/1.56 % (6841)------------------------------
% 2.67/1.56 % (6844)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/7Mi)
% 2.67/1.56 % (6840)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_6601 for (2999ds/15Mi)
% 2.67/1.56 % (6844)Instruction limit reached!
% 2.67/1.56 % (6844)------------------------------
% 2.67/1.56 % (6844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.56 % (6844)Termination reason: Unknown
% 2.67/1.56 % (6844)Termination phase: shuffling
% 2.67/1.56
% 2.67/1.56 % (6844)Memory used [KB]: 1535
% 2.67/1.56 % (6844)Time elapsed: 0.004 s
% 2.67/1.56 % (6844)Instructions burned: 9 (million)
% 2.67/1.56 % (6844)------------------------------
% 2.67/1.56 % (6844)------------------------------
% 2.67/1.56 % (6842)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_6601 for (2999ds/1041Mi)
% 2.67/1.57 % (6846)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/16Mi)
% 2.67/1.57 % (6840)Instruction limit reached!
% 2.67/1.57 % (6840)------------------------------
% 2.67/1.57 % (6840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.57 % (6840)Termination reason: Unknown
% 2.67/1.57 % (6840)Termination phase: shuffling
% 2.67/1.57
% 2.67/1.57 % (6840)Memory used [KB]: 1663
% 2.67/1.57 % (6840)Time elapsed: 0.008 s
% 2.67/1.57 % (6840)Instructions burned: 17 (million)
% 2.67/1.57 % (6840)------------------------------
% 2.67/1.57 % (6840)------------------------------
% 2.67/1.57 % (6839)Instruction limit reached!
% 2.67/1.57 % (6839)------------------------------
% 2.67/1.57 % (6839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.57 % (6839)Termination reason: Unknown
% 2.67/1.57 % (6839)Termination phase: Property scanning
% 2.67/1.57
% 2.67/1.57 % (6839)Memory used [KB]: 2046
% 2.67/1.57 % (6839)Time elapsed: 0.013 s
% 2.67/1.57 % (6839)Instructions burned: 39 (million)
% 2.67/1.57 % (6839)------------------------------
% 2.67/1.57 % (6839)------------------------------
% 2.67/1.57 % (6847)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_6601 for (2999ds/3Mi)
% 2.67/1.57 % (6846)Instruction limit reached!
% 2.67/1.57 % (6846)------------------------------
% 2.67/1.57 % (6846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.57 % (6846)Termination reason: Unknown
% 2.67/1.57 % (6846)Termination phase: shuffling
% 2.67/1.57
% 2.67/1.57 % (6846)Memory used [KB]: 1791
% 2.67/1.57 % (6846)Time elapsed: 0.006 s
% 2.67/1.57 % (6846)Instructions burned: 18 (million)
% 2.67/1.57 % (6846)------------------------------
% 2.67/1.57 % (6846)------------------------------
% 2.67/1.57 % (6847)Instruction limit reached!
% 2.67/1.57 % (6847)------------------------------
% 2.67/1.57 % (6847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.57 % (6847)Termination reason: Unknown
% 2.67/1.57 % (6847)Termination phase: shuffling
% 2.67/1.57
% 2.67/1.57 % (6847)Memory used [KB]: 1407
% 2.67/1.57 % (6847)Time elapsed: 0.002 s
% 2.67/1.57 % (6847)Instructions burned: 3 (million)
% 2.67/1.57 % (6847)------------------------------
% 2.67/1.57 % (6847)------------------------------
% 2.67/1.57 % (6826)Instruction limit reached!
% 2.67/1.57 % (6826)------------------------------
% 2.67/1.57 % (6826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.57 % (6826)Termination reason: Unknown
% 2.67/1.57 % (6826)Termination phase: Saturation
% 2.67/1.57
% 2.67/1.57 % (6826)Memory used [KB]: 7547
% 2.67/1.57 % (6826)Time elapsed: 0.051 s
% 2.67/1.57 % (6826)Instructions burned: 186 (million)
% 2.67/1.57 % (6826)------------------------------
% 2.67/1.57 % (6826)------------------------------
% 2.67/1.58 % (6848)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_6601 for (2999ds/3Mi)
% 2.67/1.58 % (6849)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/7Mi)
% 2.67/1.58 % (6848)Instruction limit reached!
% 2.67/1.58 % (6848)------------------------------
% 2.67/1.58 % (6848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.58 % (6848)Termination reason: Unknown
% 2.67/1.58 % (6848)Termination phase: shuffling
% 2.67/1.58
% 2.67/1.58 % (6848)Memory used [KB]: 1407
% 2.67/1.58 % (6848)Time elapsed: 0.002 s
% 2.67/1.58 % (6848)Instructions burned: 4 (million)
% 2.67/1.58 % (6848)------------------------------
% 2.67/1.58 % (6848)------------------------------
% 2.67/1.58 % (6849)Instruction limit reached!
% 2.67/1.58 % (6849)------------------------------
% 2.67/1.58 % (6849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.58 % (6849)Termination reason: Unknown
% 2.67/1.58 % (6849)Termination phase: shuffling
% 2.67/1.58
% 2.67/1.58 % (6849)Memory used [KB]: 1407
% 2.67/1.58 % (6849)Time elapsed: 0.003 s
% 2.67/1.58 % (6849)Instructions burned: 7 (million)
% 2.67/1.58 % (6849)------------------------------
% 2.67/1.58 % (6849)------------------------------
% 2.67/1.58 % (6850)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/3Mi)
% 2.67/1.58 % (6851)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/4Mi)
% 2.67/1.58 % (6850)Instruction limit reached!
% 2.67/1.58 % (6850)------------------------------
% 2.67/1.58 % (6850)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.58 % (6850)Termination reason: Unknown
% 2.67/1.58 % (6850)Termination phase: shuffling
% 2.67/1.58
% 2.67/1.58 % (6850)Memory used [KB]: 1407
% 2.67/1.58 % (6850)Time elapsed: 0.002 s
% 2.67/1.58 % (6850)Instructions burned: 3 (million)
% 2.67/1.58 % (6850)------------------------------
% 2.67/1.58 % (6850)------------------------------
% 2.67/1.58 % (6851)Instruction limit reached!
% 2.67/1.58 % (6851)------------------------------
% 2.67/1.58 % (6851)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.58 % (6851)Termination reason: Unknown
% 2.67/1.58 % (6851)Termination phase: shuffling
% 2.67/1.58
% 2.67/1.58 % (6851)Memory used [KB]: 1535
% 2.67/1.58 % (6851)Time elapsed: 0.003 s
% 2.67/1.58 % (6851)Instructions burned: 7 (million)
% 2.67/1.58 % (6851)------------------------------
% 2.67/1.58 % (6851)------------------------------
% 2.67/1.59 % (6852)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/18Mi)
% 2.67/1.59 % (6853)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/710Mi)
% 2.67/1.59 % (6854)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/6Mi)
% 2.67/1.59 % (6855)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_6601 for (2999ds/902Mi)
% 2.67/1.59 % (6854)Instruction limit reached!
% 2.67/1.59 % (6854)------------------------------
% 2.67/1.59 % (6854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.67/1.59 % (6854)Termination reason: Unknown
% 2.67/1.59 % (6854)Termination phase: shuffling
% 2.67/1.59
% 2.67/1.59 % (6854)Memory used [KB]: 1535
% 2.67/1.59 % (6854)Time elapsed: 0.004 s
% 2.67/1.59 % (6854)Instructions burned: 7 (million)
% 2.67/1.59 % (6854)------------------------------
% 2.67/1.59 % (6854)------------------------------
% 2.67/1.59 % (6856)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_6601 for (2999ds/21Mi)
% 2.70/1.60 % (6852)Instruction limit reached!
% 2.70/1.60 % (6852)------------------------------
% 2.70/1.60 % (6852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (6852)Termination reason: Unknown
% 2.70/1.60 % (6852)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (6852)Memory used [KB]: 1791
% 2.70/1.60 % (6852)Time elapsed: 0.011 s
% 2.70/1.60 % (6852)Instructions burned: 19 (million)
% 2.70/1.60 % (6852)------------------------------
% 2.70/1.60 % (6852)------------------------------
% 2.70/1.60 % (6856)Instruction limit reached!
% 2.70/1.60 % (6856)------------------------------
% 2.70/1.60 % (6856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (6856)Termination reason: Unknown
% 2.70/1.60 % (6856)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (6856)Memory used [KB]: 1791
% 2.70/1.60 % (6856)Time elapsed: 0.007 s
% 2.70/1.60 % (6856)Instructions burned: 21 (million)
% 2.70/1.60 % (6856)------------------------------
% 2.70/1.60 % (6856)------------------------------
% 2.70/1.60 % (6857)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/5Mi)
% 2.70/1.60 % (6857)Instruction limit reached!
% 2.70/1.60 % (6857)------------------------------
% 2.70/1.60 % (6857)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.60 % (6857)Termination reason: Unknown
% 2.70/1.60 % (6857)Termination phase: shuffling
% 2.70/1.60
% 2.70/1.60 % (6857)Memory used [KB]: 1535
% 2.70/1.60 % (6857)Time elapsed: 0.004 s
% 2.70/1.60 % (6857)Instructions burned: 8 (million)
% 2.70/1.60 % (6857)------------------------------
% 2.70/1.60 % (6857)------------------------------
% 2.70/1.61 % (6859)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_6601 for (2999ds/377Mi)
% 2.70/1.61 % (6860)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_6601 for (2999ds/779Mi)
% 2.70/1.62 % (6858)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_6601 for (2999ds/6Mi)
% 2.70/1.62 % (6858)Instruction limit reached!
% 2.70/1.62 % (6858)------------------------------
% 2.70/1.62 % (6858)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.62 % (6858)Termination reason: Unknown
% 2.70/1.62 % (6858)Termination phase: shuffling
% 2.70/1.62
% 2.70/1.62 % (6858)Memory used [KB]: 1407
% 2.70/1.62 % (6858)Time elapsed: 0.006 s
% 2.70/1.62 % (6858)Instructions burned: 7 (million)
% 2.70/1.62 % (6858)------------------------------
% 2.70/1.62 % (6858)------------------------------
% 2.70/1.63 % (6831)Instruction limit reached!
% 2.70/1.63 % (6831)------------------------------
% 2.70/1.63 % (6831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.63 % (6831)Termination reason: Unknown
% 2.70/1.63 % (6831)Termination phase: Saturation
% 2.70/1.63
% 2.70/1.63 % (6831)Memory used [KB]: 8443
% 2.70/1.63 % (6831)Time elapsed: 0.100 s
% 2.70/1.63 % (6831)Instructions burned: 276 (million)
% 2.70/1.63 % (6831)------------------------------
% 2.70/1.63 % (6831)------------------------------
% 2.70/1.63 % (6862)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_6601 for (2999ds/19Mi)
% 2.70/1.64 % (6863)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_6601 for (2998ds/879Mi)
% 2.70/1.64 % (6862)Instruction limit reached!
% 2.70/1.64 % (6862)------------------------------
% 2.70/1.64 % (6862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.64 % (6862)Termination reason: Unknown
% 2.70/1.64 % (6862)Termination phase: shuffling
% 2.70/1.64
% 2.70/1.64 % (6862)Memory used [KB]: 1791
% 2.70/1.64 % (6862)Time elapsed: 0.010 s
% 2.70/1.64 % (6862)Instructions burned: 21 (million)
% 2.70/1.64 % (6862)------------------------------
% 2.70/1.64 % (6862)------------------------------
% 2.70/1.65 % (6864)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_6601 for (2998ds/17Mi)
% 2.70/1.66 % (6855)First to succeed.
% 2.70/1.66 % (6864)Instruction limit reached!
% 2.70/1.66 % (6864)------------------------------
% 2.70/1.66 % (6864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.66 % (6864)Termination reason: Unknown
% 2.70/1.66 % (6864)Termination phase: shuffling
% 2.70/1.66
% 2.70/1.66 % (6864)Memory used [KB]: 1791
% 2.70/1.66 % (6864)Time elapsed: 0.007 s
% 2.70/1.66 % (6864)Instructions burned: 18 (million)
% 2.70/1.66 % (6864)------------------------------
% 2.70/1.66 % (6864)------------------------------
% 2.70/1.66 % (6855)Refutation found. Thanks to Tanya!
% 2.70/1.66 % SZS status Theorem for DTF2THF_6601
% 2.70/1.66 % SZS output start Proof for DTF2THF_6601
% 2.70/1.66 thf(func_def_0, type, in: $i > $i > $o).
% 2.70/1.66 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.70/1.66 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.70/1.66 thf(func_def_8, type, powerset: $i > $i).
% 2.70/1.66 thf(func_def_10, type, setunion: $i > $i).
% 2.70/1.66 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.70/1.66 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_26, type, prop2set: $o > $i).
% 2.70/1.66 thf(func_def_36, type, nonempty: $i > $o).
% 2.70/1.66 thf(func_def_69, type, set2prop: $i > $o).
% 2.70/1.66 thf(func_def_88, type, subset: $i > $i > $o).
% 2.70/1.66 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.70/1.66 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.70/1.66 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.70/1.66 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.70/1.66 thf(func_def_135, type, regular: $i > $o).
% 2.70/1.66 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.70/1.66 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.70/1.66 thf(func_def_153, type, iskpair: $i > $o).
% 2.70/1.66 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.70/1.66 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.70/1.66 thf(func_def_177, type, singleton: $i > $o).
% 2.70/1.66 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.70/1.66 thf(func_def_184, type, atmost1p: $i > $o).
% 2.70/1.66 thf(func_def_185, type, atleast2p: $i > $o).
% 2.70/1.66 thf(func_def_186, type, atmost2p: $i > $o).
% 2.70/1.66 thf(func_def_187, type, upairsetp: $i > $o).
% 2.70/1.66 thf(func_def_191, type, kfst: $i > $i).
% 2.70/1.66 thf(func_def_203, type, ksnd: $i > $i).
% 2.70/1.66 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.70/1.66 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.70/1.66 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.70/1.66 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.70/1.66 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.70/1.66 thf(func_def_243, type, sP1: $i > $i > $o).
% 2.70/1.66 thf(func_def_244, type, sP2: $i > $o).
% 2.70/1.66 thf(func_def_245, type, sP3: $i > $i > $o).
% 2.70/1.66 thf(func_def_246, type, sP4: $i > $i > $o).
% 2.70/1.66 thf(func_def_249, type, sP7: $o > $i > $i > $i > $o).
% 2.70/1.66 thf(func_def_258, type, sK16: $i > $o).
% 2.70/1.66 thf(func_def_260, type, sK18: ($i > $o) > $i).
% 2.70/1.66 thf(func_def_261, type, sK19: ($i > $o) > $i).
% 2.70/1.66 thf(func_def_272, type, sK30: $i > $o).
% 2.70/1.66 thf(func_def_274, type, sK32: ($i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_275, type, sK33: ($i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_281, type, sK39: $i > $i > $i).
% 2.70/1.66 thf(func_def_282, type, sK40: $i > $i > $i).
% 2.70/1.66 thf(func_def_283, type, sK41: $i > $i > $i).
% 2.70/1.66 thf(func_def_284, type, sK42: $i > $i > $i).
% 2.70/1.66 thf(func_def_285, type, sK43: $i > $i > $i).
% 2.70/1.66 thf(func_def_286, type, sK44: $i > $i > $i > $i).
% 2.70/1.66 thf(func_def_287, type, sK45: $i > $i).
% 2.70/1.66 thf(func_def_288, type, sK46: $i > $i).
% 2.70/1.66 thf(func_def_289, type, sK47: $i > $i).
% 2.70/1.66 thf(func_def_290, type, sK48: $i > $i).
% 2.70/1.66 thf(func_def_291, type, sK49: $i > $i > $i).
% 2.70/1.66 thf(func_def_292, type, sK50: $i > $i > $i > $i).
% 2.70/1.66 thf(func_def_293, type, sK51: $i > $i > $i > $i).
% 2.70/1.66 thf(func_def_294, type, sK52: $i > $i > $i).
% 2.70/1.66 thf(func_def_295, type, sK53: $i > $i > $i).
% 2.70/1.66 thf(func_def_296, type, sK54: $i > $i).
% 2.70/1.66 thf(func_def_298, type, sK56: $i > $i).
% 2.70/1.66 thf(func_def_299, type, sK57: $i > $i).
% 2.70/1.66 thf(func_def_307, type, sK65: $i > $i > $o).
% 2.70/1.66 thf(func_def_308, type, sK66: $i > $i).
% 2.70/1.66 thf(func_def_309, type, sK67: $i > $i).
% 2.70/1.66 thf(func_def_310, type, sK68: ($i > $i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_311, type, sK69: ($i > $i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_312, type, sK70: $i > ($i > $i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_314, type, sK72: $i > $o).
% 2.70/1.66 thf(func_def_316, type, sK74: $i > $o).
% 2.70/1.66 thf(func_def_320, type, sK78: $i > $i > $i).
% 2.70/1.66 thf(func_def_325, type, sK83: $i > $o).
% 2.70/1.66 thf(func_def_337, type, sK95: $i > $i > $i).
% 2.70/1.66 thf(func_def_338, type, sK96: $i > $o).
% 2.70/1.66 thf(func_def_348, type, sK106: $i > $o).
% 2.70/1.66 thf(func_def_352, type, sK110: $i > $i > $i).
% 2.70/1.66 thf(func_def_360, type, sK118: $i > $o).
% 2.70/1.66 thf(func_def_361, type, sK119: $i > $o).
% 2.70/1.66 thf(func_def_362, type, sK120: ($i > $o) > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_363, type, sK121: ($i > $o) > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_364, type, sK122: $i > $o).
% 2.70/1.66 thf(func_def_367, type, sK125: $i > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_368, type, sK126: $i > $o).
% 2.70/1.66 thf(func_def_399, type, sK157: $i > $o).
% 2.70/1.66 thf(func_def_400, type, sK158: ($i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_416, type, sK174: $i > $o).
% 2.70/1.66 thf(func_def_419, type, sK177: $i > $i > $o).
% 2.70/1.66 thf(func_def_428, type, sK186: $i > $i > $i).
% 2.70/1.66 thf(func_def_435, type, sK193: $i > $i > $i > $i).
% 2.70/1.66 thf(func_def_436, type, sK194: $i > $i > $i > $i).
% 2.70/1.66 thf(func_def_446, type, sK204: $i > $i > $i).
% 2.70/1.66 thf(func_def_481, type, sK239: ($i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_483, type, sK241: $i > $o).
% 2.70/1.66 thf(func_def_486, type, sK244: $i > $o).
% 2.70/1.66 thf(func_def_488, type, sK246: ($i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_489, type, sK247: $i > $i > $i).
% 2.70/1.66 thf(func_def_490, type, sK248: $i > $i > $i).
% 2.70/1.66 thf(func_def_508, type, sK266: $i > $o).
% 2.70/1.66 thf(func_def_509, type, sK267: $i > $i).
% 2.70/1.66 thf(func_def_510, type, sK268: ($i > $o) > $i).
% 2.70/1.66 thf(func_def_511, type, sK269: $i > $o).
% 2.70/1.66 thf(func_def_512, type, sK270: ($i > $o) > $i).
% 2.70/1.66 thf(func_def_515, type, sK273: $i > $i > $o).
% 2.70/1.66 thf(func_def_525, type, sK283: $i > $o).
% 2.70/1.66 thf(func_def_526, type, sK284: $i > $i).
% 2.70/1.66 thf(func_def_527, type, sK285: ($i > $o) > $i).
% 2.70/1.66 thf(func_def_540, type, sK298: $i > $i > $i).
% 2.70/1.66 thf(func_def_543, type, sK301: $i > $o).
% 2.70/1.66 thf(func_def_546, type, sK304: $i > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_547, type, sK305: $i > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_557, type, sK315: $i > $o).
% 2.70/1.66 thf(func_def_564, type, sK322: $i > $o).
% 2.70/1.66 thf(func_def_565, type, sK323: $i > $o).
% 2.70/1.66 thf(func_def_566, type, sK324: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.70/1.66 thf(func_def_567, type, sK325: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.70/1.66 thf(func_def_570, type, sK328: $i > $i).
% 2.70/1.66 thf(func_def_571, type, sK329: $i > $i).
% 2.70/1.66 thf(func_def_581, type, sK339: $i > $o).
% 2.70/1.66 thf(func_def_588, type, sK346: $i > $i > $o).
% 2.70/1.66 thf(func_def_593, type, sK351: $i > $i > $o).
% 2.70/1.66 thf(func_def_600, type, sK358: $i > $o).
% 2.70/1.66 thf(func_def_608, type, sK366: $i > $i).
% 2.70/1.66 thf(func_def_610, type, sK368: $i > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_611, type, sK369: $i > $o).
% 2.70/1.66 thf(func_def_618, type, sK376: $i > $i > $o).
% 2.70/1.66 thf(func_def_624, type, sK382: $i > $i > $i).
% 2.70/1.66 thf(func_def_643, type, sK401: ($i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_645, type, sK403: $i > $o).
% 2.70/1.66 thf(func_def_650, type, sK408: $i > $i > $o).
% 2.70/1.66 thf(func_def_653, type, sK411: $i > $i).
% 2.70/1.66 thf(func_def_662, type, sK420: $i > $i > $o).
% 2.70/1.66 thf(func_def_670, type, sK428: $i > $o).
% 2.70/1.66 thf(func_def_685, type, sK443: $i > $i > $i).
% 2.70/1.66 thf(func_def_694, type, sK452: $i > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_695, type, sK453: $i > $o).
% 2.70/1.66 thf(func_def_698, type, sK456: $i > $i).
% 2.70/1.66 thf(func_def_704, type, sK462: $o > $i > $i > $i).
% 2.70/1.66 thf(func_def_705, type, sK463: $i > $o).
% 2.70/1.66 thf(func_def_706, type, sK464: $i > $o).
% 2.70/1.66 thf(func_def_707, type, sK465: ($i > $o) > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_708, type, sK466: ($i > $o) > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_713, type, sK471: $i > $i > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_714, type, sK472: $i > $o).
% 2.70/1.66 thf(func_def_748, type, sK506: $i > $o).
% 2.70/1.66 thf(func_def_756, type, sK514: $i > $o).
% 2.70/1.66 thf(func_def_758, type, sK516: $i > ($i > $o) > $i).
% 2.70/1.66 thf(func_def_760, type, sK518: $i > $i).
% 2.70/1.66 thf(func_def_763, type, sK521: ($i > $o) > $i > $i).
% 2.70/1.66 thf(func_def_765, type, sK523: $i > $o).
% 2.70/1.66 thf(func_def_778, type, sK536: $i > $i).
% 2.70/1.66 thf(func_def_802, type, ph560: !>[X0: $tType]:(X0)).
% 2.70/1.66 thf(f2823,plain,(
% 2.70/1.66 $false),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f2822])).
% 2.70/1.66 thf(f2822,plain,(
% 2.70/1.66 ($true != $true)),
% 2.70/1.66 inference(superposition,[status(thm)],[f2821,f2799])).
% 2.70/1.66 thf(f2799,plain,(
% 2.70/1.66 ($true = (func @ sK234 @ sK233 @ sK232))),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f2798])).
% 2.70/1.66 thf(f2798,plain,(
% 2.70/1.66 ($true != $true) | ($true = (func @ sK234 @ sK233 @ sK232))),
% 2.70/1.66 inference(forward_demodulation,[status(thm)],[f2797,f2182])).
% 2.70/1.66 thf(f2182,plain,(
% 2.70/1.66 (infuncsetfunc = $true)),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f1224])).
% 2.70/1.66 thf(f1224,plain,(
% 2.70/1.66 (dsetconstr__Cong = $true) & (powersetsubset = $true) & (setunionI = $true) & (in__Cong = $true) & (binintersectI = $true) & (quantDeMorgan1 = $true) & (upairequniteq = $true) & (setminusILneg = $true) & (disjointsetsI1 = $true) & (setunionE2 = $true) & (setunionAx = $true) & (binunionIL = $true) & (upairsetIR = $true) & (setukpairinjL = $true) & (singletoninpowunion = $true) & (powersetI = $true) & (subsetE = $true) & (emptysetimpfalse = $true) & (ex1E1 = $true) & (eqinunit = $true) & (ksndsingleton = $true) & (inPowerset = $true) & (subsetE2 = $true) & (binintersectSubset3 = $true) & (binunionIR = $true) & (dpsetconstrI = $true) & (dpsetconstrEL1 = $true) & (setminusER = $true) & (sepInPowerset = $true) & (ubforcartprodlem3 = $true) & (omegaSAx = $true) & (cartprodpairin = $true) & (noeltsimpempty = $true) & (quantDeMorgan3 = $true) & (symdiffIneg1 = $true) & (emptysetE = $true) & (binunionE = $true) & (setunionsingleton = $true) & (foundationAx = $true) & (ubforcartprodlem2 = $true) & (upairinpowunion = $true) & (quantDeMorgan4 = $true) & (binintersectSubset1 = $true) & (app = $true) & (emptyinPowerset = $true) & (powerset__Cong = $true) & (notequalI2 = $true) & (setextAx = $true) & (replAx = $true) & (infuncsetfunc = $true) & (setOfPairsIsBReln = $true) & (symdiffI2 = $true) & (ubforcartprodlem1 = $true) & (setukpairinjR2 = $true) & (binunionLsub = $true) & (singletonsubset = $true) & (setbeta = $true) & (notequalI1 = $true) & (setadjoin__Cong = $true) & (setadjoinSub2 = $true) & (bs114d = $true) & (emptyinunitempty = $true) & (upairsetE = $true) & (prop2setE = $true) & (dsetconstrI = $true) & (exuE1 = $true) & (prop2setI = $true) & (kfstsingleton = $true) & (notsubsetI = $true) & (eqimpsubset2 = $true) & (upairsetIL = $true) & (subset2powerset = $true) & (ex1I2 = $true) & (setadjoinAx = $true) & (symdiffI1 = $true) & (setukpairinjR = $true) & (setoftrueEq = $true) & (setukpairinjR12 = $true) & (setunionE = $true) & (upairset2IR = $true) & ((($true != (in @ (ap @ sK234 @ sK233 @ sK232 @ sK235) @ sK233)) & ($true = (in @ sK235 @ sK234))) & ($true = (in @ sK232 @ (funcSet @ sK234 @ sK233)))) & (cartprodpairmemEL = $true) & (setadjoinOr = $true) & (setminusSubset2 = $true) & (notdexE = $true) & (subsetRefl = $true) & (nonemptyImpWitness = $true) & (setext = $true) & (setadjoinE = $true) & (subsetI2 = $true) & (exuE3e = $true) & (exuEu = $true) & (setminusLsub = $true) & (theprop = $true) & (ksndpairEq = $true) & (apProp = $true) & (upairset2E = $true) & (singletonsuniq = $true) & (setunion__Cong = $true) & (emptyset__Cong = $true) & (sepSubset = $true) & (cartprodfstpairEq = $true) & (emptyI = $true) & (kpairsurjEq = $true) & (omegaIndAx = $true) & (cartprodmempair = $true) & (notinemptyset = $true) & (exuE3u = $true) & (notdallE = $true) & (setukpairinjR1 = $true) & (omega__Cong = $true) & (cartprodpairsurjEq = $true) & (prop2set2propI = $true) & (notinsingleton = $true) & (cartprodpairmemER = $true) & (singletonsswitch = $true) & (binintersectEL = $true) & (cartprodfstin = $true) & (inCongP = $true) & (dsetconstrEL = $true) & (binintersectLsub = $true) & (eqimpsubset1 = $true) & (setadjoinIR = $true) & (dpsetconstrEL2 = $true) & (binunionEcases = $true) & (wellorderingAx = $true) & (singletoninpowerset = $true) & (emptysetsubset = $true) & (setminusIRneg = $true) & (setminusERneg = $true) & (emptysetAx = $true) & (binunionRsub = $true) & (dpsetconstrERa = $true) & (binintersectSubset2 = $true) & (setadjoinIL = $true) & (singletonprop = $true) & (binintersectRsub = $true) & (powersetE = $true) & (kpairp = $true) & (cartprodmempair1 = $true) & (binintersectSubset4 = $true) & (exuI1 = $true) & (subsetemptysetimpeq = $true) & (binintersectER = $true) & (funcImageSingleton = $true) & (dpsetconstrER = $true) & (upairsubunion = $true) & (powersetAx = $true) & (nonemptyI1 = $true) & (setunionsingleton1 = $true) & (symdiffE = $true) & (kpairiskpair = $true) & (setextsub = $true) & (setminusEL = $true) & (exuI2 = $true) & (setukpairIR = $true) & (exu__Cong = $true) & (cartprodmempaircEq = $true) & (setukpairinjR11 = $true) & (nonemptyI = $true) & (symdiffIneg2 = $true) & (dpsetconstrSub = $true) & (setadjoinSub = $true) & (quantDeMorgan2 = $true) & (setukpairIL = $true) & (exuI3 = $true) & (secondinupair = $true) & (dsetconstrER = $true) & (subsetTrans = $true) & (subPowSU = $true) & (uniqinunit = $true) & (subsetI1 = $true) & (descrp = $true) & (nonemptyE1 = $true) & (vacuousDall = $true) & (setukpairinjL1 = $true) & (setunionsingleton2 = $true) & (ex1I = $true) & (cartprodsndpairEq = $true) & (cartprodsndin = $true) & (exuE2 = $true) & (kfstpairEq = $true) & (emptyE1 = $true) & (powersetI1 = $true) & (setminusELneg = $true) & (omega0Ax = $true) & (powersetE1 = $true) & (emptyInPowerset = $true) & (setminusI = $true) & (setukpairinjL2 = $true) & (descr__Cong = $true) & (setminusSubset1 = $true) & (binintersectSubset5 = $true)),
% 2.70/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK232,sK233,sK234,sK235])],[f705,f1223,f1222])).
% 2.70/1.66 thf(f1222,plain,(
% 2.70/1.66 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true != (in @ (ap @ X2 @ X1 @ X0 @ X3) @ X1)) & ((in @ X3 @ X2) = $true)) & ($true = (in @ X0 @ (funcSet @ X2 @ X1)))) => (? [X3 : $i] : (($true != (in @ (ap @ sK234 @ sK233 @ sK232 @ X3) @ sK233)) & ($true = (in @ X3 @ sK234))) & ($true = (in @ sK232 @ (funcSet @ sK234 @ sK233))))),
% 2.70/1.66 introduced(definition,[],[choice_axiom])).
% 2.70/1.66 thf(f1223,plain,(
% 2.70/1.66 ? [X3 : $i] : (($true != (in @ (ap @ sK234 @ sK233 @ sK232 @ X3) @ sK233)) & ($true = (in @ X3 @ sK234))) => (($true != (in @ (ap @ sK234 @ sK233 @ sK232 @ sK235) @ sK233)) & ($true = (in @ sK235 @ sK234)))),
% 2.70/1.66 introduced(definition,[],[choice_axiom])).
% 2.70/1.66 thf(f705,plain,(
% 2.70/1.66 (dsetconstr__Cong = $true) & (powersetsubset = $true) & (setunionI = $true) & (in__Cong = $true) & (binintersectI = $true) & (quantDeMorgan1 = $true) & (upairequniteq = $true) & (setminusILneg = $true) & (disjointsetsI1 = $true) & (setunionE2 = $true) & (setunionAx = $true) & (binunionIL = $true) & (upairsetIR = $true) & (setukpairinjL = $true) & (singletoninpowunion = $true) & (powersetI = $true) & (subsetE = $true) & (emptysetimpfalse = $true) & (ex1E1 = $true) & (eqinunit = $true) & (ksndsingleton = $true) & (inPowerset = $true) & (subsetE2 = $true) & (binintersectSubset3 = $true) & (binunionIR = $true) & (dpsetconstrI = $true) & (dpsetconstrEL1 = $true) & (setminusER = $true) & (sepInPowerset = $true) & (ubforcartprodlem3 = $true) & (omegaSAx = $true) & (cartprodpairin = $true) & (noeltsimpempty = $true) & (quantDeMorgan3 = $true) & (symdiffIneg1 = $true) & (emptysetE = $true) & (binunionE = $true) & (setunionsingleton = $true) & (foundationAx = $true) & (ubforcartprodlem2 = $true) & (upairinpowunion = $true) & (quantDeMorgan4 = $true) & (binintersectSubset1 = $true) & (app = $true) & (emptyinPowerset = $true) & (powerset__Cong = $true) & (notequalI2 = $true) & (setextAx = $true) & (replAx = $true) & (infuncsetfunc = $true) & (setOfPairsIsBReln = $true) & (symdiffI2 = $true) & (ubforcartprodlem1 = $true) & (setukpairinjR2 = $true) & (binunionLsub = $true) & (singletonsubset = $true) & (setbeta = $true) & (notequalI1 = $true) & (setadjoin__Cong = $true) & (setadjoinSub2 = $true) & (bs114d = $true) & (emptyinunitempty = $true) & (upairsetE = $true) & (prop2setE = $true) & (dsetconstrI = $true) & (exuE1 = $true) & (prop2setI = $true) & (kfstsingleton = $true) & (notsubsetI = $true) & (eqimpsubset2 = $true) & (upairsetIL = $true) & (subset2powerset = $true) & (ex1I2 = $true) & (setadjoinAx = $true) & (symdiffI1 = $true) & (setukpairinjR = $true) & (setoftrueEq = $true) & (setukpairinjR12 = $true) & (setunionE = $true) & (upairset2IR = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true != (in @ (ap @ X2 @ X1 @ X0 @ X3) @ X1)) & ((in @ X3 @ X2) = $true)) & ($true = (in @ X0 @ (funcSet @ X2 @ X1)))) & (cartprodpairmemEL = $true) & (setadjoinOr = $true) & (setminusSubset2 = $true) & (notdexE = $true) & (subsetRefl = $true) & (nonemptyImpWitness = $true) & (setext = $true) & (setadjoinE = $true) & (subsetI2 = $true) & (exuE3e = $true) & (exuEu = $true) & (setminusLsub = $true) & (theprop = $true) & (ksndpairEq = $true) & (apProp = $true) & (upairset2E = $true) & (singletonsuniq = $true) & (setunion__Cong = $true) & (emptyset__Cong = $true) & (sepSubset = $true) & (cartprodfstpairEq = $true) & (emptyI = $true) & (kpairsurjEq = $true) & (omegaIndAx = $true) & (cartprodmempair = $true) & (notinemptyset = $true) & (exuE3u = $true) & (notdallE = $true) & (setukpairinjR1 = $true) & (omega__Cong = $true) & (cartprodpairsurjEq = $true) & (prop2set2propI = $true) & (notinsingleton = $true) & (cartprodpairmemER = $true) & (singletonsswitch = $true) & (binintersectEL = $true) & (cartprodfstin = $true) & (inCongP = $true) & (dsetconstrEL = $true) & (binintersectLsub = $true) & (eqimpsubset1 = $true) & (setadjoinIR = $true) & (dpsetconstrEL2 = $true) & (binunionEcases = $true) & (wellorderingAx = $true) & (singletoninpowerset = $true) & (emptysetsubset = $true) & (setminusIRneg = $true) & (setminusERneg = $true) & (emptysetAx = $true) & (binunionRsub = $true) & (dpsetconstrERa = $true) & (binintersectSubset2 = $true) & (setadjoinIL = $true) & (singletonprop = $true) & (binintersectRsub = $true) & (powersetE = $true) & (kpairp = $true) & (cartprodmempair1 = $true) & (binintersectSubset4 = $true) & (exuI1 = $true) & (subsetemptysetimpeq = $true) & (binintersectER = $true) & (funcImageSingleton = $true) & (dpsetconstrER = $true) & (upairsubunion = $true) & (powersetAx = $true) & (nonemptyI1 = $true) & (setunionsingleton1 = $true) & (symdiffE = $true) & (kpairiskpair = $true) & (setextsub = $true) & (setminusEL = $true) & (exuI2 = $true) & (setukpairIR = $true) & (exu__Cong = $true) & (cartprodmempaircEq = $true) & (setukpairinjR11 = $true) & (nonemptyI = $true) & (symdiffIneg2 = $true) & (dpsetconstrSub = $true) & (setadjoinSub = $true) & (quantDeMorgan2 = $true) & (setukpairIL = $true) & (exuI3 = $true) & (secondinupair = $true) & (dsetconstrER = $true) & (subsetTrans = $true) & (subPowSU = $true) & (uniqinunit = $true) & (subsetI1 = $true) & (descrp = $true) & (nonemptyE1 = $true) & (vacuousDall = $true) & (setukpairinjL1 = $true) & (setunionsingleton2 = $true) & (ex1I = $true) & (cartprodsndpairEq = $true) & (cartprodsndin = $true) & (exuE2 = $true) & (kfstpairEq = $true) & (emptyE1 = $true) & (powersetI1 = $true) & (setminusELneg = $true) & (omega0Ax = $true) & (powersetE1 = $true) & (emptyInPowerset = $true) & (setminusI = $true) & (setukpairinjL2 = $true) & (descr__Cong = $true) & (setminusSubset1 = $true) & (binintersectSubset5 = $true)),
% 2.70/1.66 inference(flattening,[status(thm)],[f704])).
% 2.70/1.66 thf(f704,plain,(
% 2.70/1.66 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true != (in @ (ap @ X2 @ X1 @ X0 @ X3) @ X1)) & ((in @ X3 @ X2) = $true)) & ($true = (in @ X0 @ (funcSet @ X2 @ X1)))) & (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)),
% 2.70/1.66 inference(ennf_transformation,[status(thm)],[f391])).
% 2.70/1.66 thf(f391,plain,(
% 2.70/1.66 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X2 @ X1))) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => ($true = (in @ (ap @ X2 @ X1 @ X0 @ X3) @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.70/1.66 inference(fool_elimination,[status(thm)],[f390])).
% 2.70/1.66 thf(f390,plain,(
% 2.70/1.66 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (funcSet @ X2 @ X1)) => ! [X3 : $i] : ((in @ X3 @ X2) => (in @ (ap @ X2 @ X1 @ X0 @ X3) @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.70/1.66 inference(rectify,[status(thm)],[f202])).
% 2.70/1.66 thf(f202,negated_conjecture,(
% 2.70/1.66 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => ! [X13 : $i,X4 : $i,X3 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (ap @ X3 @ X4 @ X13 @ X1) @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.70/1.66 inference(negated_conjecture,[status(cth)],[f201])).
% 2.70/1.66 thf(f201,conjecture,(
% 2.70/1.66 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => ! [X13 : $i,X4 : $i,X3 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (ap @ X3 @ X4 @ X13 @ X1) @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.70/1.66 file('/export/starexec/sandbox/tmp/tmp.TbQYBRPofc/DTF2THF_6601.p',ap2p)).
% 2.70/1.66 thf(f2797,plain,(
% 2.70/1.66 ($true = (func @ sK234 @ sK233 @ sK232)) | (infuncsetfunc != $true)),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f2794])).
% 2.70/1.66 thf(f2794,plain,(
% 2.70/1.66 ($true = (func @ sK234 @ sK233 @ sK232)) | ($true != $true) | (infuncsetfunc != $true)),
% 2.70/1.66 inference(superposition,[status(thm)],[f2632,f2149])).
% 2.70/1.66 thf(f2149,plain,(
% 2.70/1.66 ($true = (in @ sK232 @ (funcSet @ sK234 @ sK233)))),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f1224])).
% 2.70/1.66 thf(f2632,plain,(
% 2.70/1.66 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X2 @ (funcSet @ X1 @ X0))) | ($true = (func @ X1 @ X0 @ X2)) | (infuncsetfunc != $true)) )),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f1693])).
% 2.70/1.66 thf(f1693,plain,(
% 2.70/1.66 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X1 @ X0 @ X2)) | ($true != (in @ X2 @ (funcSet @ X1 @ X0)))) | (infuncsetfunc != $true)) & ((infuncsetfunc = $true) | (($true != (func @ sK538 @ sK537 @ sK539)) & ((in @ sK539 @ (funcSet @ sK538 @ sK537)) = $true)))),
% 2.70/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK537,sK538,sK539])],[f1691,f1692])).
% 2.70/1.66 thf(f1692,plain,(
% 2.70/1.66 ? [X3 : $i,X4 : $i,X5 : $i] : (((func @ X4 @ X3 @ X5) != $true) & ($true = (in @ X5 @ (funcSet @ X4 @ X3)))) => (($true != (func @ sK538 @ sK537 @ sK539)) & ((in @ sK539 @ (funcSet @ sK538 @ sK537)) = $true))),
% 2.70/1.66 introduced(definition,[],[choice_axiom])).
% 2.70/1.66 thf(f1691,plain,(
% 2.70/1.66 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X1 @ X0 @ X2)) | ($true != (in @ X2 @ (funcSet @ X1 @ X0)))) | (infuncsetfunc != $true)) & ((infuncsetfunc = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((func @ X4 @ X3 @ X5) != $true) & ($true = (in @ X5 @ (funcSet @ X4 @ X3)))))),
% 2.70/1.66 inference(rectify,[status(thm)],[f1690])).
% 2.70/1.66 thf(f1690,plain,(
% 2.70/1.66 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X1 @ X0 @ X2)) | ($true != (in @ X2 @ (funcSet @ X1 @ X0)))) | (infuncsetfunc != $true)) & ((infuncsetfunc = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (func @ X1 @ X0 @ X2)) & ($true = (in @ X2 @ (funcSet @ X1 @ X0)))))),
% 2.70/1.66 inference(nnf_transformation,[status(thm)],[f664])).
% 2.70/1.66 thf(f664,plain,(
% 2.70/1.66 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X1 @ X0 @ X2)) | ($true != (in @ X2 @ (funcSet @ X1 @ X0)))) <=> (infuncsetfunc = $true)),
% 2.70/1.66 inference(ennf_transformation,[status(thm)],[f317])).
% 2.70/1.66 thf(f317,plain,(
% 2.70/1.66 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X1 @ X0))) => ($true = (func @ X1 @ X0 @ X2))) <=> (infuncsetfunc = $true)),
% 2.70/1.66 inference(fool_elimination,[status(thm)],[f316])).
% 2.70/1.66 thf(f316,plain,(
% 2.70/1.66 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (funcSet @ X1 @ X0)) => (func @ X1 @ X0 @ X2)) = infuncsetfunc)),
% 2.70/1.66 inference(rectify,[status(thm)],[f200])).
% 2.70/1.66 thf(f200,axiom,(
% 2.70/1.66 (! [X4 : $i,X3 : $i,X13 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => (func @ X3 @ X4 @ X13)) = infuncsetfunc)),
% 2.70/1.66 file('/export/starexec/sandbox/tmp/tmp.TbQYBRPofc/DTF2THF_6601.p',infuncsetfunc)).
% 2.70/1.66 thf(f2821,plain,(
% 2.70/1.66 ($true != (func @ sK234 @ sK233 @ sK232))),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f2820])).
% 2.70/1.66 thf(f2820,plain,(
% 2.70/1.66 ($true != $true) | ($true != (func @ sK234 @ sK233 @ sK232))),
% 2.70/1.66 inference(forward_demodulation,[status(thm)],[f2819,f2188])).
% 2.70/1.66 thf(f2188,plain,(
% 2.70/1.66 (app = $true)),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f1224])).
% 2.70/1.66 thf(f2819,plain,(
% 2.70/1.66 ($true != (func @ sK234 @ sK233 @ sK232)) | (app != $true)),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f2818])).
% 2.70/1.66 thf(f2818,plain,(
% 2.70/1.66 ($true != (func @ sK234 @ sK233 @ sK232)) | ($true != $true) | (app != $true)),
% 2.70/1.66 inference(forward_demodulation,[status(thm)],[f2807,f2150])).
% 2.70/1.66 thf(f2150,plain,(
% 2.70/1.66 ($true = (in @ sK235 @ sK234))),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f1224])).
% 2.70/1.66 thf(f2807,plain,(
% 2.70/1.66 ($true != (in @ sK235 @ sK234)) | (app != $true) | ($true != (func @ sK234 @ sK233 @ sK232))),
% 2.70/1.66 inference(trivial_inequality_removal,[status(thm)],[f2802])).
% 2.70/1.66 thf(f2802,plain,(
% 2.70/1.66 ($true != (func @ sK234 @ sK233 @ sK232)) | (app != $true) | ($true != (in @ sK235 @ sK234)) | ($true != $true)),
% 2.70/1.66 inference(superposition,[status(thm)],[f2151,f2572])).
% 2.70/1.66 thf(f2572,plain,(
% 2.70/1.66 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (($true = (in @ (ap @ X5 @ X6 @ X4 @ X7) @ X6)) | ($true != (in @ X7 @ X5)) | (app != $true) | ((func @ X5 @ X6 @ X4) != $true)) )),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f1622])).
% 2.70/1.66 thf(f1622,plain,(
% 2.70/1.66 ((app = $true) | ((($true != (in @ (ap @ sK497 @ sK498 @ sK496 @ sK499) @ sK498)) & ($true = (in @ sK499 @ sK497))) & ((func @ sK497 @ sK498 @ sK496) = $true))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (($true = (in @ (ap @ X5 @ X6 @ X4 @ X7) @ X6)) | ($true != (in @ X7 @ X5))) | ((func @ X5 @ X6 @ X4) != $true)) | (app != $true))),
% 2.70/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK496,sK497,sK498,sK499])],[f1619,f1621,f1620])).
% 2.70/1.66 thf(f1620,plain,(
% 2.70/1.66 ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true != (in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2)) & ($true = (in @ X3 @ X1))) & ((func @ X1 @ X2 @ X0) = $true)) => (? [X3 : $i] : (((in @ (ap @ sK497 @ sK498 @ sK496 @ X3) @ sK498) != $true) & ((in @ X3 @ sK497) = $true)) & ((func @ sK497 @ sK498 @ sK496) = $true))),
% 2.70/1.66 introduced(definition,[],[choice_axiom])).
% 2.70/1.66 thf(f1621,plain,(
% 2.70/1.66 ? [X3 : $i] : (((in @ (ap @ sK497 @ sK498 @ sK496 @ X3) @ sK498) != $true) & ((in @ X3 @ sK497) = $true)) => (($true != (in @ (ap @ sK497 @ sK498 @ sK496 @ sK499) @ sK498)) & ($true = (in @ sK499 @ sK497)))),
% 2.70/1.66 introduced(definition,[],[choice_axiom])).
% 2.70/1.66 thf(f1619,plain,(
% 2.70/1.66 ((app = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true != (in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2)) & ($true = (in @ X3 @ X1))) & ((func @ X1 @ X2 @ X0) = $true))) & (! [X4 : $i,X5 : $i,X6 : $i] : (! [X7 : $i] : (($true = (in @ (ap @ X5 @ X6 @ X4 @ X7) @ X6)) | ($true != (in @ X7 @ X5))) | ((func @ X5 @ X6 @ X4) != $true)) | (app != $true))),
% 2.70/1.66 inference(rectify,[status(thm)],[f1618])).
% 2.70/1.66 thf(f1618,plain,(
% 2.70/1.66 ((app = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (? [X3 : $i] : (($true != (in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2)) & ($true = (in @ X3 @ X1))) & ((func @ X1 @ X2 @ X0) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (($true = (in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2)) | ($true != (in @ X3 @ X1))) | ((func @ X1 @ X2 @ X0) != $true)) | (app != $true))),
% 2.70/1.66 inference(nnf_transformation,[status(thm)],[f758])).
% 2.70/1.66 thf(f758,plain,(
% 2.70/1.66 (app = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (! [X3 : $i] : (($true = (in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2)) | ($true != (in @ X3 @ X1))) | ((func @ X1 @ X2 @ X0) != $true))),
% 2.70/1.66 inference(ennf_transformation,[status(thm)],[f549])).
% 2.70/1.66 thf(f549,plain,(
% 2.70/1.66 ! [X0 : $i,X1 : $i,X2 : $i] : (((func @ X1 @ X2 @ X0) = $true) => ! [X3 : $i] : (($true = (in @ X3 @ X1)) => ($true = (in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2)))) <=> (app = $true)),
% 2.70/1.66 inference(fool_elimination,[status(thm)],[f548])).
% 2.70/1.66 thf(f548,plain,(
% 2.70/1.66 (app = ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X1 @ X2 @ X0) => ! [X3 : $i] : ((in @ X3 @ X1) => (in @ (ap @ X1 @ X2 @ X0 @ X3) @ X2))))),
% 2.70/1.66 inference(rectify,[status(thm)],[f199])).
% 2.70/1.66 thf(f199,axiom,(
% 2.70/1.66 (app = ! [X13 : $i,X3 : $i,X4 : $i] : ((func @ X3 @ X4 @ X13) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (ap @ X3 @ X4 @ X13 @ X1) @ X4))))),
% 2.70/1.66 file('/export/starexec/sandbox/tmp/tmp.TbQYBRPofc/DTF2THF_6601.p',app)).
% 2.70/1.66 thf(f2151,plain,(
% 2.70/1.66 ($true != (in @ (ap @ sK234 @ sK233 @ sK232 @ sK235) @ sK233))),
% 2.70/1.66 inference(cnf_transformation,[status(thm)],[f1224])).
% 2.70/1.66 % SZS output end Proof for DTF2THF_6601
% 2.70/1.66 % (6855)------------------------------
% 2.70/1.66 % (6855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.70/1.66 % (6855)Termination reason: Refutation
% 2.70/1.66
% 2.70/1.66 % (6855)Memory used [KB]: 8571
% 2.70/1.66 % (6855)Time elapsed: 0.070 s
% 2.70/1.66 % (6855)Instructions burned: 220 (million)
% 2.70/1.66 % (6855)------------------------------
% 2.70/1.66 % (6855)------------------------------
% 2.70/1.66 % (6822)Success in time 0.142 s
%------------------------------------------------------------------------------