%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU712^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n001.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 09:36:35 AM UTC 2026
% Result : Theorem 2.68s 1.77s
% Output : Refutation 2.68s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 8
% Syntax : Number of formulae : 54 ( 16 unt; 0 typ; 0 def)
% Number of atoms : 3916 (1053 equ; 0 cnn)
% Maximal formula atoms : 230 ( 72 avg)
% Number of connectives : 2088 ( 72 ~; 48 |; 713 &; 320 @)
% ( 6 <=>; 929 =>; 0 <=; 0 <~>)
% Maximal formula depth : 236 ( 34 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 422 ( 419 usr; 238 con; 0-4 aty)
% Number of variables : 80 ( 0 ^; 56 !; 24 ?; 80 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_283,type,
sP3: $i > $o > $i > $i > $o ).
thf(func_def_284,type,
sP4: $i > $i > $o ).
thf(func_def_285,type,
sP5: $i > $o ).
thf(func_def_286,type,
sP6: $i > $i > $o ).
thf(func_def_287,type,
sP7: $i > $i > $o ).
thf(func_def_289,type,
sP9: $i > $i > $i > $i > $o ).
thf(func_def_295,type,
sK15: $i > $i > ( $i > $i ) > $i ).
thf(func_def_296,type,
sK16: $i > $i ).
thf(func_def_303,type,
sK23: $i > $i > $i ).
thf(func_def_310,type,
sK30: $i > $o ).
thf(func_def_311,type,
sK31: ( $i > $o ) > $i ).
thf(func_def_313,type,
sK33: $i > $i ).
thf(func_def_315,type,
sK35: $i > ( $i > $i ) > $i > $i ).
thf(func_def_322,type,
sK42: $i > $o ).
thf(func_def_340,type,
sK60: $i > $o ).
thf(func_def_341,type,
sK61: $i > $i ).
thf(func_def_342,type,
sK62: ( $i > $o ) > $i ).
thf(func_def_350,type,
sK70: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_351,type,
sK71: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_352,type,
sK72: $i > $o ).
thf(func_def_353,type,
sK73: $i > $o ).
thf(func_def_358,type,
sK78: $i > $i ).
thf(func_def_360,type,
sK80: $i > $i ).
thf(func_def_361,type,
sK81: $i > $i ).
thf(func_def_368,type,
sK88: $i > $o ).
thf(func_def_370,type,
sK90: $i > ( $i > $o ) > $i ).
thf(func_def_378,type,
sK98: $i > $o ).
thf(func_def_413,type,
sK133: $o > $i > $i > $i ).
thf(func_def_414,type,
sK134: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_415,type,
sK135: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_416,type,
sK136: $i > $o ).
thf(func_def_417,type,
sK137: $i > $o ).
thf(func_def_420,type,
sK140: $i > $i > $o ).
thf(func_def_424,type,
sK144: $i > $o ).
thf(func_def_426,type,
sK146: ( $i > $o ) > $i > $i ).
thf(func_def_427,type,
sK147: ( $i > $o ) > $i > $i ).
thf(func_def_431,type,
sK151: $i > $o ).
thf(func_def_440,type,
sK160: $i > $i > $i > $i > $i ).
thf(func_def_441,type,
sK161: $i > $o ).
thf(func_def_452,type,
sK172: $i > $i ).
thf(func_def_482,type,
sK202: $i > $i > $o ).
thf(func_def_483,type,
sK203: $i > $i ).
thf(func_def_484,type,
sK204: $i > $i ).
thf(func_def_485,type,
sK205: ( $i > $i > $o ) > $i > $i ).
thf(func_def_486,type,
sK206: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_487,type,
sK207: ( $i > $i > $o ) > $i > $i ).
thf(func_def_510,type,
sK230: $i > $i > $i ).
thf(func_def_526,type,
sK246: $i > $o ).
thf(func_def_528,type,
sK248: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_529,type,
sK249: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_530,type,
sK250: $i > $i > $i > $i > $i ).
thf(func_def_536,type,
sK256: $i > $i ).
thf(func_def_539,type,
sK259: $i > ( $i > $i ) > $i > $i ).
thf(func_def_546,type,
sK266: $i > $i > $i > $i ).
thf(func_def_547,type,
sK267: $i > $i > $i > $i ).
thf(func_def_551,type,
sK271: $i > $o ).
thf(func_def_553,type,
sK273: $i > ( $i > $o ) > $i ).
thf(func_def_554,type,
sK274: $i > $o ).
thf(func_def_557,type,
sK277: $i > $i > $i > $i > $i ).
thf(func_def_564,type,
sK284: $i > $i > $i ).
thf(func_def_565,type,
sK285: $i > $i > $i ).
thf(func_def_566,type,
sK286: $i > $i > $i ).
thf(func_def_567,type,
sK287: $i > $i > $i ).
thf(func_def_568,type,
sK288: $i > $i > $i > $i ).
thf(func_def_569,type,
sK289: $i > $i ).
thf(func_def_570,type,
sK290: $i > $i ).
thf(func_def_571,type,
sK291: $i > $i ).
thf(func_def_572,type,
sK292: $i > $i ).
thf(func_def_573,type,
sK293: $i > $i > $i > $i ).
thf(func_def_574,type,
sK294: $i > $i > $i > $i ).
thf(func_def_575,type,
sK295: $i > $i > $i ).
thf(func_def_576,type,
sK296: $i > $i > $i ).
thf(func_def_577,type,
sK297: $i > $i > $i ).
thf(func_def_579,type,
sK299: $i > $i ).
thf(func_def_580,type,
sK300: $i > $i ).
thf(func_def_581,type,
sK301: $i > $i ).
thf(func_def_582,type,
sK302: $i > $i > $i ).
thf(func_def_586,type,
sK306: $i > $o ).
thf(func_def_599,type,
sK319: ( $i > $o ) > $i > $i ).
thf(func_def_601,type,
sK321: $i > $o ).
thf(func_def_610,type,
sK330: $i > $o ).
thf(func_def_626,type,
sK346: $i > $o ).
thf(func_def_629,type,
sK349: $i > ( $i > $o ) > $i ).
thf(func_def_632,type,
sK352: $i > $i > $o ).
thf(func_def_643,type,
sK363: $i > $i > $i > $i > $i ).
thf(func_def_644,type,
sK364: $i > $i > $i > $i > $i ).
thf(func_def_649,type,
sK369: $i > $o ).
thf(func_def_651,type,
sK371: ( $i > $o ) > $i ).
thf(func_def_652,type,
sK372: ( $i > $o ) > $i ).
thf(func_def_659,type,
sK379: $i > $i ).
thf(func_def_661,type,
sK381: $i > $o ).
thf(func_def_669,type,
sK389: $i > $i > $o ).
thf(func_def_670,type,
sK390: $i > $o ).
thf(func_def_673,type,
sK393: $i > $i > ( $i > $o ) > $i ).
thf(func_def_684,type,
sK404: $i > $o ).
thf(func_def_690,type,
sK410: $i > $i > $o ).
thf(func_def_693,type,
sK413: $i > $o ).
thf(func_def_695,type,
sK415: $i > ( $i > $o ) > $i ).
thf(func_def_701,type,
sK421: $i > $i > $o ).
thf(func_def_706,type,
sK426: $i > $o ).
thf(func_def_709,type,
sK429: $i > ( $i > $o ) > $i ).
thf(func_def_710,type,
sK430: $i > ( $i > $o ) > $i ).
thf(func_def_719,type,
sK439: $i > $i ).
thf(func_def_722,type,
sK442: $i > ( $i > $i ) > $i > $i ).
thf(func_def_724,type,
sK444: $i > $o ).
thf(func_def_726,type,
sK446: ( $i > $o ) > $i > $i ).
thf(func_def_733,type,
sK453: $i > $i > $i ).
thf(func_def_734,type,
sK454: $i > $i > $i ).
thf(func_def_757,type,
sK477: $i > ( $i > $o ) > $i ).
thf(func_def_758,type,
sK478: $i > $o ).
thf(func_def_761,type,
sK481: ( $i > $o ) > $i > $i ).
thf(func_def_763,type,
sK483: $i > $o ).
thf(func_def_774,type,
sK494: $i > $i > $i ).
thf(func_def_784,type,
sK504: $i > $i > $i ).
thf(func_def_800,type,
sK520: $i > $i > $o ).
thf(func_def_806,type,
sK526: $i > $i > $i ).
thf(func_def_820,type,
sK540: $i > $i > $o ).
thf(func_def_826,type,
sK546: $i > $o ).
thf(func_def_835,type,
sK555: $i > $o ).
thf(func_def_838,type,
sK558: $i > $i ).
thf(func_def_850,type,
sK570: $i > $i ).
thf(func_def_859,type,
sK579: $i > $i > $i > $i > $i ).
thf(func_def_860,type,
sK580: $i > $i > $i > $i > $i ).
thf(func_def_861,type,
sK581: $i > $i > $i > $i > $i ).
thf(func_def_862,type,
sK582: $i > $i > $i > $i > $i ).
thf(func_def_876,type,
sK596: $i > $i > $i ).
thf(func_def_878,type,
sK598: $i > $o ).
thf(func_def_906,type,
sK626: $i > $o ).
thf(func_def_907,type,
sK627: ( $i > $o ) > $i ).
thf(func_def_908,type,
sK628: $i > $o ).
thf(func_def_909,type,
sK629: $i > $i ).
thf(func_def_910,type,
sK630: $i > ( $i > $i ) > $i > $i ).
thf(func_def_912,type,
sK632: $i > $i ).
thf(func_def_914,type,
sK634: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_915,type,
sK635: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_919,type,
sK639: $i > $o ).
thf(func_def_924,type,
sK644: $i > $o ).
thf(func_def_925,type,
sK645: $i > $o ).
thf(func_def_926,type,
sK646: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_927,type,
sK647: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_930,type,
sK650: $i > $i > $i ).
thf(func_def_945,type,
sK665: $i > ( $i > $i ) > $i > $i ).
thf(func_def_947,type,
sK667: $i > $i ).
thf(func_def_954,type,
sK674: $i > $i > $i ).
thf(func_def_958,type,
sK678: ( $i > $o ) > $i > $i ).
thf(func_def_960,type,
sK680: $i > $o ).
thf(func_def_999,type,
ph719:
!>[X0: $tType] : X0 ).
thf(f4368,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f4361]) ).
thf(f4361,plain,
$true != $true,
inference(superposition,[status(thm)],[f3488,f3426]) ).
thf(f3426,plain,
( ( subset @ sK698 @ sK699 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3425]) ).
thf(f3425,plain,
( ( ( subset @ sK698 @ sK699 )
= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3415,f3064]) ).
thf(f3064,plain,
powersetE1 = $true,
inference(cnf_transformation,[status(thm)],[f2063]) ).
thf(f2063,plain,
( ( binintersectER = $true )
& ( dpsetconstrEL1 = $true )
& ( upairsetIL = $true )
& ( in__Cong = $true )
& ( subbreln = $true )
& ( dpsetconstrEL2 = $true )
& ( inCongP = $true )
& ( omega__Cong = $true )
& ( powersetI1 = $true )
& ( binunionLsub = $true )
& ( upairsetIR = $true )
& ( exu__Cong = $true )
& ( funcinfuncset = $true )
& ( setbeta = $true )
& ( notdallE = $true )
& ( dsetconstr__Cong = $true )
& ( setminusLsub = $true )
& ( subsetRefl = $true )
& ( beta1 = $true )
& ( funcext2 = $true )
& ( setukpairinjL2 = $true )
& ( wellorderingAx = $true )
& ( apProp = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset4 = $true )
& ( setukpairinjR1 = $true )
& ( ap2p = $true )
& ( setextAx = $true )
& ( ex1I2 = $true )
& ( lam2lamEq = $true )
& ( dsetconstrEL = $true )
& ( binintersectI = $true )
& ( funcGraphProp1 = $true )
& ( setunionsingleton2 = $true )
& ( dsetconstrER = $true )
& ( ksndpairEq = $true )
& ( cartprodpairin = $true )
& ( upairinpowunion = $true )
& ( brelnall2 = $true )
& ( setukpairIL = $true )
& ( cartprodpairmemEL = $true )
& ( ap2apEq1 = $true )
& ( setminusI = $true )
& ( singletonsuniq = $true )
& ( binunionIR = $true )
& ( emptyinPowerset = $true )
& ( cartprodfstin = $true )
& ( setadjoinIR = $true )
& ( iffalseProp1 = $true )
& ( ubforcartprodlem3 = $true )
& ( prop2setI = $true )
& ( setOfPairsIsBReln = $true )
& ( funcImageSingleton = $true )
& ( notequalI1 = $true )
& ( cartprodmempair = $true )
& ( iffalseProp2 = $true )
& ( ubforcartprodlem2 = $true )
& ( binintersectSubset1 = $true )
& ( setunionE = $true )
& ( app = $true )
& ( ex1I = $true )
& ( eta1 = $true )
& ( setminusER = $true )
& ( setext = $true )
& ( upairsetE = $true )
& ( cartprodsndin = $true )
& ( binunionIL = $true )
& ( setunionI = $true )
& ( setunionE2 = $true )
& ( notinemptyset = $true )
& ( binunionE = $true )
& ( omegaIndAx = $true )
& ( upairset2IR = $true )
& ( subsetI2 = $true )
& ( subPowSU = $true )
& ( symdiffI2 = $true )
& ( powersetI = $true )
& ( infuncsetfunc = $true )
& ( quantDeMorgan4 = $true )
& ( setukpairinjR2 = $true )
& ( dpsetconstrI = $true )
& ( eqimpsubset2 = $true )
& ( eqimpsubset1 = $true )
& ( exuE3e = $true )
& ( disjointsetsI1 = $true )
& ( funcext = $true )
& ( exuE1 = $true )
& ( setminusERneg = $true )
& ( kfstsingleton = $true )
& ( setadjoin__Cong = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyI = $true )
& ( symdiffIneg2 = $true )
& ( setunionsingleton1 = $true )
& ( setukpairIR = $true )
& ( nonemptyImpWitness = $true )
& ( singletoninpowerset = $true )
& ( ( in @ sK698 @ ( powerset @ sK699 ) )
= $true )
& ( ( in @ ( powerset @ sK698 ) @ ( powerset @ ( powerset @ sK699 ) ) )
!= $true )
& ( emptyset__Cong = $true )
& ( replAx = $true )
& ( setadjoinE = $true )
& ( binunionT_lem = $true )
& ( ap2apEq2 = $true )
& ( sepSubset = $true )
& ( setminusEL = $true )
& ( secondinupair = $true )
& ( subsetTrans = $true )
& ( funcGraphProp4 = $true )
& ( singletonsubset = $true )
& ( emptyInPowerset = $true )
& ( subsetI1 = $true )
& ( uniqinunit = $true )
& ( symdiffE = $true )
& ( dpsetconstrSub = $true )
& ( iftrueProp2 = $true )
& ( theeq = $true )
& ( dsetconstrI = $true )
& ( binintersectT_lem = $true )
& ( lam2p = $true )
& ( setadjoinSub2 = $true )
& ( iffalse = $true )
& ( iftrueProp1 = $true )
& ( powersetsubset = $true )
& ( binintersectLsub = $true )
& ( singletonsswitch = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( binunionRsub = $true )
& ( exuE3u = $true )
& ( emptysetAx = $true )
& ( exuEu = $true )
& ( upairsubunion = $true )
& ( cartprodmempaircEq = $true )
& ( vacuousDall = $true )
& ( notequalI2 = $true )
& ( eqinunit = $true )
& ( kpairsurjEq = $true )
& ( prop2set2propI = $true )
& ( eta2 = $true )
& ( binintersectSubset3 = $true )
& ( binunionEcases = $true )
& ( exuI2 = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodfstpairEq = $true )
& ( inPowerset = $true )
& ( noeltsimpempty = $true )
& ( subsetE2 = $true )
& ( emptysetimpfalse = $true )
& ( foundationAx = $true )
& ( cartprodmempair1 = $true )
& ( prop2setE = $true )
& ( kfstpairEq = $true )
& ( notsubsetI = $true )
& ( powersetE1 = $true )
& ( setadjoinSub = $true )
& ( dpsetconstrER = $true )
& ( emptyinunitempty = $true )
& ( symdiffIneg1 = $true )
& ( quantDeMorgan1 = $true )
& ( dpsetconstrERa = $true )
& ( iftrue = $true )
& ( ksndsingleton = $true )
& ( emptyI = $true )
& ( ifSingleton = $true )
& ( emptyE1 = $true )
& ( subsetemptysetimpeq = $true )
& ( funcGraphProp2 = $true )
& ( setukpairinjL = $true )
& ( binintersectSubset2 = $true )
& ( emptysetE = $true )
& ( setukpairinjR11 = $true )
& ( exuI1 = $true )
& ( iftrueorfalse = $true )
& ( lamProp = $true )
& ( setminusILneg = $true )
& ( descr__Cong = $true )
& ( nonemptyI1 = $true )
& ( descrp = $true )
& ( kpairiskpair = $true )
& ( singletonprop = $true )
& ( theprop = $true )
& ( upairset2E = $true )
& ( setunion__Cong = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjR = $true )
& ( subsetE = $true )
& ( eqbreln = $true )
& ( bs114d = $true )
& ( ifp = $true )
& ( setminusELneg = $true )
& ( setadjoinAx = $true )
& ( exuI3 = $true )
& ( symdiffI1 = $true )
& ( powerset__Cong = $true )
& ( funcGraphProp3 = $true )
& ( kpairp = $true )
& ( powersetE = $true )
& ( setextsub = $true )
& ( lamp = $true )
& ( notdexE = $true )
& ( subset2powerset = $true )
& ( emptysetsubset = $true )
& ( setminusSubset1 = $true )
& ( omegaSAx = $true )
& ( setukpairinjR12 = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset5 = $true )
& ( powersetAx = $true )
& ( setadjoinIL = $true )
& ( cartprodsndpairEq = $true )
& ( notinsingleton = $true )
& ( binintersectEL = $true )
& ( nonemptyE1 = $true )
& ( setminusSubset2 = $true )
& ( quantDeMorgan3 = $true )
& ( brelnall1 = $true )
& ( upairequniteq = $true )
& ( beta2 = $true )
& ( exuE2 = $true )
& ( setukpairinjL1 = $true )
& ( setadjoinOr = $true )
& ( cartprodpairmemER = $true )
& ( binintersectRsub = $true )
& ( funcextLem = $true )
& ( setminusIRneg = $true )
& ( setunionsingleton = $true )
& ( ex1E1 = $true )
& ( ex1E2 = $true )
& ( singletoninpowunion = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK698,sK699])],[f856,f2062]) ).
thf(f2062,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ( $true
!= ( in @ ( powerset @ X0 ) @ ( powerset @ ( powerset @ X1 ) ) ) ) )
=> ( ( ( in @ sK698 @ ( powerset @ sK699 ) )
= $true )
& ( ( in @ ( powerset @ sK698 ) @ ( powerset @ ( powerset @ sK699 ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f856,plain,
( ( binintersectER = $true )
& ( dpsetconstrEL1 = $true )
& ( upairsetIL = $true )
& ( in__Cong = $true )
& ( subbreln = $true )
& ( dpsetconstrEL2 = $true )
& ( inCongP = $true )
& ( omega__Cong = $true )
& ( powersetI1 = $true )
& ( binunionLsub = $true )
& ( upairsetIR = $true )
& ( exu__Cong = $true )
& ( funcinfuncset = $true )
& ( setbeta = $true )
& ( notdallE = $true )
& ( dsetconstr__Cong = $true )
& ( setminusLsub = $true )
& ( subsetRefl = $true )
& ( beta1 = $true )
& ( funcext2 = $true )
& ( setukpairinjL2 = $true )
& ( wellorderingAx = $true )
& ( apProp = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset4 = $true )
& ( setukpairinjR1 = $true )
& ( ap2p = $true )
& ( setextAx = $true )
& ( ex1I2 = $true )
& ( lam2lamEq = $true )
& ( dsetconstrEL = $true )
& ( binintersectI = $true )
& ( funcGraphProp1 = $true )
& ( setunionsingleton2 = $true )
& ( dsetconstrER = $true )
& ( ksndpairEq = $true )
& ( cartprodpairin = $true )
& ( upairinpowunion = $true )
& ( brelnall2 = $true )
& ( setukpairIL = $true )
& ( cartprodpairmemEL = $true )
& ( ap2apEq1 = $true )
& ( setminusI = $true )
& ( singletonsuniq = $true )
& ( binunionIR = $true )
& ( emptyinPowerset = $true )
& ( cartprodfstin = $true )
& ( setadjoinIR = $true )
& ( iffalseProp1 = $true )
& ( ubforcartprodlem3 = $true )
& ( prop2setI = $true )
& ( setOfPairsIsBReln = $true )
& ( funcImageSingleton = $true )
& ( notequalI1 = $true )
& ( cartprodmempair = $true )
& ( iffalseProp2 = $true )
& ( ubforcartprodlem2 = $true )
& ( binintersectSubset1 = $true )
& ( setunionE = $true )
& ( app = $true )
& ( ex1I = $true )
& ( eta1 = $true )
& ( setminusER = $true )
& ( setext = $true )
& ( upairsetE = $true )
& ( cartprodsndin = $true )
& ( binunionIL = $true )
& ( setunionI = $true )
& ( setunionE2 = $true )
& ( notinemptyset = $true )
& ( binunionE = $true )
& ( omegaIndAx = $true )
& ( upairset2IR = $true )
& ( subsetI2 = $true )
& ( subPowSU = $true )
& ( symdiffI2 = $true )
& ( powersetI = $true )
& ( infuncsetfunc = $true )
& ( quantDeMorgan4 = $true )
& ( setukpairinjR2 = $true )
& ( dpsetconstrI = $true )
& ( eqimpsubset2 = $true )
& ( eqimpsubset1 = $true )
& ( exuE3e = $true )
& ( disjointsetsI1 = $true )
& ( funcext = $true )
& ( exuE1 = $true )
& ( setminusERneg = $true )
& ( kfstsingleton = $true )
& ( setadjoin__Cong = $true )
& ( quantDeMorgan2 = $true )
& ( nonemptyI = $true )
& ( symdiffIneg2 = $true )
& ( setunionsingleton1 = $true )
& ( setukpairIR = $true )
& ( nonemptyImpWitness = $true )
& ( singletoninpowerset = $true )
& ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ( $true
!= ( in @ ( powerset @ X0 ) @ ( powerset @ ( powerset @ X1 ) ) ) ) )
& ( emptyset__Cong = $true )
& ( replAx = $true )
& ( setadjoinE = $true )
& ( binunionT_lem = $true )
& ( ap2apEq2 = $true )
& ( sepSubset = $true )
& ( setminusEL = $true )
& ( secondinupair = $true )
& ( subsetTrans = $true )
& ( funcGraphProp4 = $true )
& ( singletonsubset = $true )
& ( emptyInPowerset = $true )
& ( subsetI1 = $true )
& ( uniqinunit = $true )
& ( symdiffE = $true )
& ( dpsetconstrSub = $true )
& ( iftrueProp2 = $true )
& ( theeq = $true )
& ( dsetconstrI = $true )
& ( binintersectT_lem = $true )
& ( lam2p = $true )
& ( setadjoinSub2 = $true )
& ( iffalse = $true )
& ( iftrueProp1 = $true )
& ( powersetsubset = $true )
& ( binintersectLsub = $true )
& ( singletonsswitch = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( binunionRsub = $true )
& ( exuE3u = $true )
& ( emptysetAx = $true )
& ( exuEu = $true )
& ( upairsubunion = $true )
& ( cartprodmempaircEq = $true )
& ( vacuousDall = $true )
& ( notequalI2 = $true )
& ( eqinunit = $true )
& ( kpairsurjEq = $true )
& ( prop2set2propI = $true )
& ( eta2 = $true )
& ( binintersectSubset3 = $true )
& ( binunionEcases = $true )
& ( exuI2 = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodfstpairEq = $true )
& ( inPowerset = $true )
& ( noeltsimpempty = $true )
& ( subsetE2 = $true )
& ( emptysetimpfalse = $true )
& ( foundationAx = $true )
& ( cartprodmempair1 = $true )
& ( prop2setE = $true )
& ( kfstpairEq = $true )
& ( notsubsetI = $true )
& ( powersetE1 = $true )
& ( setadjoinSub = $true )
& ( dpsetconstrER = $true )
& ( emptyinunitempty = $true )
& ( symdiffIneg1 = $true )
& ( quantDeMorgan1 = $true )
& ( dpsetconstrERa = $true )
& ( iftrue = $true )
& ( ksndsingleton = $true )
& ( emptyI = $true )
& ( ifSingleton = $true )
& ( emptyE1 = $true )
& ( subsetemptysetimpeq = $true )
& ( funcGraphProp2 = $true )
& ( setukpairinjL = $true )
& ( binintersectSubset2 = $true )
& ( emptysetE = $true )
& ( setukpairinjR11 = $true )
& ( exuI1 = $true )
& ( iftrueorfalse = $true )
& ( lamProp = $true )
& ( setminusILneg = $true )
& ( descr__Cong = $true )
& ( nonemptyI1 = $true )
& ( descrp = $true )
& ( kpairiskpair = $true )
& ( singletonprop = $true )
& ( theprop = $true )
& ( upairset2E = $true )
& ( setunion__Cong = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjR = $true )
& ( subsetE = $true )
& ( eqbreln = $true )
& ( bs114d = $true )
& ( ifp = $true )
& ( setminusELneg = $true )
& ( setadjoinAx = $true )
& ( exuI3 = $true )
& ( symdiffI1 = $true )
& ( powerset__Cong = $true )
& ( funcGraphProp3 = $true )
& ( kpairp = $true )
& ( powersetE = $true )
& ( setextsub = $true )
& ( lamp = $true )
& ( notdexE = $true )
& ( subset2powerset = $true )
& ( emptysetsubset = $true )
& ( setminusSubset1 = $true )
& ( omegaSAx = $true )
& ( setukpairinjR12 = $true )
& ( sepInPowerset = $true )
& ( binintersectSubset5 = $true )
& ( powersetAx = $true )
& ( setadjoinIL = $true )
& ( cartprodsndpairEq = $true )
& ( notinsingleton = $true )
& ( binintersectEL = $true )
& ( nonemptyE1 = $true )
& ( setminusSubset2 = $true )
& ( quantDeMorgan3 = $true )
& ( brelnall1 = $true )
& ( upairequniteq = $true )
& ( beta2 = $true )
& ( exuE2 = $true )
& ( setukpairinjL1 = $true )
& ( setadjoinOr = $true )
& ( cartprodpairmemER = $true )
& ( binintersectRsub = $true )
& ( funcextLem = $true )
& ( setminusIRneg = $true )
& ( setunionsingleton = $true )
& ( ex1E1 = $true )
& ( ex1E2 = $true )
& ( singletoninpowunion = $true ) ),
inference(flattening,[status(thm)],[f855]) ).
thf(f855,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ( $true
!= ( in @ ( powerset @ X0 ) @ ( powerset @ ( powerset @ X1 ) ) ) ) )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f476]) ).
thf(f476,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ( $true
= ( in @ ( powerset @ X0 ) @ ( powerset @ ( powerset @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f475]) ).
thf(f475,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ( in @ ( powerset @ X0 ) @ ( powerset @ ( powerset @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f234]) ).
thf(f234,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ( in @ ( powerset @ X11 ) @ ( powerset @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f233]) ).
thf(f233,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ( in @ ( powerset @ X11 ) @ ( powerset @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.PfJgBtH2Qy/DTF2THF_5724.p',powersetT_lem) ).
thf(f3415,plain,
( ( ( subset @ sK698 @ sK699 )
= $true )
| ( powersetE1 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3411]) ).
thf(f3411,plain,
( ( ( subset @ sK698 @ sK699 )
= $true )
| ( powersetE1 != $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f2285,f3121]) ).
thf(f3121,plain,
( ( in @ sK698 @ ( powerset @ sK699 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f2063]) ).
thf(f2285,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( powersetE1 != $true )
| ( ( subset @ X1 @ X0 )
= $true ) ),
inference(cnf_transformation,[status(thm)],[f1240]) ).
thf(f1240,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( ( subset @ X1 @ X0 )
= $true ) )
| ( powersetE1 != $true ) )
& ( ( powersetE1 = $true )
| ( ( ( in @ sK165 @ ( powerset @ sK164 ) )
= $true )
& ( ( subset @ sK165 @ sK164 )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK164,sK165])],[f1238,f1239]) ).
thf(f1239,plain,
( ? [X2: $i,X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ X2 ) ) )
& ( ( subset @ X3 @ X2 )
!= $true ) )
=> ( ( ( in @ sK165 @ ( powerset @ sK164 ) )
= $true )
& ( ( subset @ sK165 @ sK164 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1238,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( ( subset @ X1 @ X0 )
= $true ) )
| ( powersetE1 != $true ) )
& ( ( powersetE1 = $true )
| ? [X2: $i,X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ X2 ) ) )
& ( ( subset @ X3 @ X2 )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1237]) ).
thf(f1237,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( ( subset @ X1 @ X0 )
= $true ) )
| ( powersetE1 != $true ) )
& ( ( powersetE1 = $true )
| ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
& ( ( subset @ X1 @ X0 )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f871]) ).
thf(f871,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( ( subset @ X1 @ X0 )
= $true ) )
<=> ( powersetE1 = $true ) ),
inference(ennf_transformation,[status(thm)],[f432]) ).
thf(f432,plain,
( ( powersetE1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ( ( subset @ X1 @ X0 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f431]) ).
thf(f431,plain,
( powersetE1
= ( ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ( subset @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f96]) ).
thf(f96,axiom,
( powersetE1
= ( ! [X3: $i,X4: $i] :
( ( in @ X4 @ ( powerset @ X3 ) )
=> ( subset @ X4 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.PfJgBtH2Qy/DTF2THF_5724.p',powersetE1) ).
thf(f3488,plain,
( ( subset @ sK698 @ sK699 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3487]) ).
thf(f3487,plain,
( ( ( subset @ sK698 @ sK699 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3483,f3095]) ).
thf(f3095,plain,
powersetsubset = $true,
inference(cnf_transformation,[status(thm)],[f2063]) ).
thf(f3483,plain,
( ( powersetsubset != $true )
| ( ( subset @ sK698 @ sK699 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3475]) ).
thf(f3475,plain,
( ( $true != $true )
| ( powersetsubset != $true )
| ( ( subset @ sK698 @ sK699 )
!= $true ) ),
inference(superposition,[status(thm)],[f3457,f2127]) ).
thf(f2127,plain,
! [X2: $i,X3: $i] :
( ( ( subset @ ( powerset @ X2 ) @ ( powerset @ X3 ) )
= $true )
| ( ( subset @ X2 @ X3 )
!= $true )
| ( powersetsubset != $true ) ),
inference(cnf_transformation,[status(thm)],[f1060]) ).
thf(f1060,plain,
( ( ( powersetsubset = $true )
| ( ( ( subset @ sK44 @ sK45 )
= $true )
& ( ( subset @ ( powerset @ sK44 ) @ ( powerset @ sK45 ) )
!= $true ) ) )
& ( ! [X2: $i,X3: $i] :
( ( ( subset @ X2 @ X3 )
!= $true )
| ( ( subset @ ( powerset @ X2 ) @ ( powerset @ X3 ) )
= $true ) )
| ( powersetsubset != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK44,sK45])],[f1058,f1059]) ).
thf(f1059,plain,
( ? [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
& ( ( subset @ ( powerset @ X0 ) @ ( powerset @ X1 ) )
!= $true ) )
=> ( ( ( subset @ sK44 @ sK45 )
= $true )
& ( ( subset @ ( powerset @ sK44 ) @ ( powerset @ sK45 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1058,plain,
( ( ( powersetsubset = $true )
| ? [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
& ( ( subset @ ( powerset @ X0 ) @ ( powerset @ X1 ) )
!= $true ) ) )
& ( ! [X2: $i,X3: $i] :
( ( ( subset @ X2 @ X3 )
!= $true )
| ( ( subset @ ( powerset @ X2 ) @ ( powerset @ X3 ) )
= $true ) )
| ( powersetsubset != $true ) ) ),
inference(rectify,[status(thm)],[f1057]) ).
thf(f1057,plain,
( ( ( powersetsubset = $true )
| ? [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
& ( ( subset @ ( powerset @ X0 ) @ ( powerset @ X1 ) )
!= $true ) ) )
& ( ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
| ( ( subset @ ( powerset @ X0 ) @ ( powerset @ X1 ) )
= $true ) )
| ( powersetsubset != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f860]) ).
thf(f860,plain,
( ( powersetsubset = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
| ( ( subset @ ( powerset @ X0 ) @ ( powerset @ X1 ) )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f494]) ).
thf(f494,plain,
( ( powersetsubset = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
=> ( ( subset @ ( powerset @ X0 ) @ ( powerset @ X1 ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f493]) ).
thf(f493,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
=> ( subset @ ( powerset @ X0 ) @ ( powerset @ X1 ) ) )
= powersetsubset ),
inference(rectify,[status(thm)],[f98]) ).
thf(f98,axiom,
! [X3: $i,X4: $i] :
( ( ( subset @ X3 @ X4 )
=> ( subset @ ( powerset @ X3 ) @ ( powerset @ X4 ) ) )
= powersetsubset ),
file('/export/starexec/sandbox2/tmp/tmp.PfJgBtH2Qy/DTF2THF_5724.p',powersetsubset) ).
thf(f3457,plain,
( ( subset @ ( powerset @ sK698 ) @ ( powerset @ sK699 ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3456]) ).
thf(f3456,plain,
( ( $true != $true )
| ( ( subset @ ( powerset @ sK698 ) @ ( powerset @ sK699 ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f3447,f3017]) ).
thf(f3017,plain,
subset2powerset = $true,
inference(cnf_transformation,[status(thm)],[f2063]) ).
thf(f3447,plain,
( ( ( subset @ ( powerset @ sK698 ) @ ( powerset @ sK699 ) )
!= $true )
| ( subset2powerset != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3439]) ).
thf(f3439,plain,
( ( $true != $true )
| ( subset2powerset != $true )
| ( ( subset @ ( powerset @ sK698 ) @ ( powerset @ sK699 ) )
!= $true ) ),
inference(superposition,[status(thm)],[f3120,f2565]) ).
thf(f2565,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true )
| ( subset2powerset != $true ) ),
inference(cnf_transformation,[status(thm)],[f1552]) ).
thf(f1552,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
| ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ( ( ( subset @ sK374 @ sK373 )
= $true )
& ( ( in @ sK374 @ ( powerset @ sK373 ) )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK373,sK374])],[f1550,f1551]) ).
thf(f1551,plain,
( ? [X2: $i,X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
& ( $true
!= ( in @ X3 @ ( powerset @ X2 ) ) ) )
=> ( ( ( subset @ sK374 @ sK373 )
= $true )
& ( ( in @ sK374 @ ( powerset @ sK373 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1550,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
| ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X2: $i,X3: $i] :
( ( ( subset @ X3 @ X2 )
= $true )
& ( $true
!= ( in @ X3 @ ( powerset @ X2 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f1549]) ).
thf(f1549,plain,
( ( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
| ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
& ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f977]) ).
thf(f977,plain,
( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
!= $true )
| ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
<=> ( subset2powerset = $true ) ),
inference(ennf_transformation,[status(thm)],[f354]) ).
thf(f354,plain,
( ! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
= $true )
=> ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
<=> ( subset2powerset = $true ) ),
inference(fool_elimination,[status(thm)],[f353]) ).
thf(f353,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X1 @ X0 )
=> ( in @ X1 @ ( powerset @ X0 ) ) )
= subset2powerset ),
inference(rectify,[status(thm)],[f92]) ).
thf(f92,axiom,
! [X4: $i,X3: $i] :
( ( ( subset @ X3 @ X4 )
=> ( in @ X3 @ ( powerset @ X4 ) ) )
= subset2powerset ),
file('/export/starexec/sandbox2/tmp/tmp.PfJgBtH2Qy/DTF2THF_5724.p',subset2powerset) ).
thf(f3120,plain,
( ( in @ ( powerset @ sK698 ) @ ( powerset @ ( powerset @ sK699 ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f2063]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : SEU712^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.14 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.35 % Computer : n001.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Tue Feb 24 12:32:35 EST 2026
% 0.15/0.36 % CPUTime :
% 0.15/0.36 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.26/0.49 ---- Original DTF file ---
% 0.26/0.49 thf(spec,logic,$$dhol).
% 0.26/0.49 %------------------------------------------------------------------------------
% 0.26/0.49 % File : SEU712^1 : TPTP v9.2.1. Released v3.7.0.
% 0.26/0.49 % Domain : Set Theory
% 0.26/0.49 % Problem : Typed Set Theory - Types of Set Operators
% 0.26/0.49 % Version : Especial.
% 0.26/0.49 % English : (! A:i.! X:i.in X (powerset A) ->
% 0.26/0.49 % in (powerset X) (powerset (powerset A)))
% 0.26/0.49
% 0.26/0.49 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.26/0.49 % Source : [Bro08]
% 0.26/0.49 % Names : ZFC214g [Bro08]
% 0.26/0.49
% 0.26/0.49 % Status : Theorem
% 0.26/0.49 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 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, 0.80 v5.2.0, 1.00 v3.7.0
% 0.26/0.49 % Syntax : Number of formulae : 499 ( 232 unt; 266 typ; 232 def)
% 0.26/0.49 % Number of atoms : 1298 ( 356 equ; 0 cnn)
% 0.26/0.49 % Maximal formula atoms : 230 ( 5 avg)
% 0.26/0.49 % Number of connectives : 2517 ( 55 ~; 12 |; 44 &;1781 @)
% 0.26/0.49 % ( 14 <=>; 611 =>; 0 <=; 0 <~>)
% 0.26/0.49 % Maximal formula depth : 235 ( 2 avg)
% 0.26/0.49 % Number of types : 2 ( 0 usr)
% 0.26/0.49 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.26/0.49 % Number of symbols : 269 ( 266 usr; 232 con; 0-4 aty)
% 0.26/0.49 % Number of variables : 780 ( 74 ^; 671 !; 35 ?; 780 :)
% 0.26/0.49 % SPC : TH0_THM_EQU_NAR
% 0.26/0.49
% 0.26/0.49 % Comments : http://mathgate.info/detsetitem.php?id=269
% 0.26/0.49 %------------------------------------------------------------------------------
% 0.26/0.49 thf(in_type,type,
% 0.26/0.49 in: $i > $i > $o ).
% 0.26/0.49
% 0.26/0.49 thf(exu_type,type,
% 0.26/0.49 exu: ( $i > $o ) > $o ).
% 0.26/0.49
% 0.26/0.49 thf(exu,definition,
% 0.26/0.49 ( exu
% 0.26/0.49 = ( ^ [Xphi: $i > $o] :
% 0.26/0.49 ? [Xx: $i] :
% 0.26/0.49 ( ( Xphi @ Xx )
% 0.26/0.49 & ! [Xy: $i] :
% 0.26/0.49 ( ( Xphi @ Xy )
% 0.26/0.49 => ( Xx = Xy ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setextAx_type,type,
% 0.26/0.49 setextAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setextAx,definition,
% 0.26/0.49 ( setextAx
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 <=> ( in @ Xx @ B ) )
% 0.26/0.49 => ( A = B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptyset_type,type,
% 0.26/0.49 emptyset: $i ).
% 0.26/0.49
% 0.26/0.49 thf(emptysetAx_type,type,
% 0.26/0.49 emptysetAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptysetAx,definition,
% 0.26/0.49 ( emptysetAx
% 0.26/0.49 = ( ! [Xx: $i] :
% 0.26/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoin_type,type,
% 0.26/0.49 setadjoin: $i > $i > $i ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinAx_type,type,
% 0.26/0.49 setadjoinAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinAx,definition,
% 0.26/0.49 ( setadjoinAx
% 0.26/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.26/0.49 <=> ( ( Xy = Xx )
% 0.26/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(powerset_type,type,
% 0.26/0.49 powerset: $i > $i ).
% 0.26/0.49
% 0.26/0.49 thf(powersetAx_type,type,
% 0.26/0.49 powersetAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(powersetAx,definition,
% 0.26/0.49 ( powersetAx
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.26/0.49 <=> ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setunion_type,type,
% 0.26/0.49 setunion: $i > $i ).
% 0.26/0.49
% 0.26/0.49 thf(setunionAx_type,type,
% 0.26/0.49 setunionAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setunionAx,definition,
% 0.26/0.49 ( setunionAx
% 0.26/0.49 = ( ! [A: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.26/0.49 <=> ? [B: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 & ( in @ B @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(omega_type,type,
% 0.26/0.49 omega: $i ).
% 0.26/0.49
% 0.26/0.49 thf(omega0Ax_type,type,
% 0.26/0.49 omega0Ax: $o ).
% 0.26/0.49
% 0.26/0.49 thf(omega0Ax,definition,
% 0.26/0.49 ( omega0Ax
% 0.26/0.49 = ( in @ emptyset @ omega ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(omegaSAx_type,type,
% 0.26/0.49 omegaSAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(omegaSAx,definition,
% 0.26/0.49 ( omegaSAx
% 0.26/0.49 = ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ omega )
% 0.26/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(omegaIndAx_type,type,
% 0.26/0.49 omegaIndAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(omegaIndAx,definition,
% 0.26/0.49 ( omegaIndAx
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ( ( ( in @ emptyset @ A )
% 0.26/0.49 & ! [Xx: $i] :
% 0.26/0.49 ( ( ( in @ Xx @ omega )
% 0.26/0.49 & ( in @ Xx @ A ) )
% 0.26/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.26/0.49 => ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ omega )
% 0.26/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(replAx_type,type,
% 0.26/0.49 replAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(replAx,definition,
% 0.26/0.49 ( replAx
% 0.26/0.49 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( exu
% 0.26/0.49 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.26/0.49 => ? [B: $i] :
% 0.26/0.49 ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 <=> ? [Xy: $i] :
% 0.26/0.49 ( ( in @ Xy @ A )
% 0.26/0.49 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(foundationAx_type,type,
% 0.26/0.49 foundationAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(foundationAx,definition,
% 0.26/0.49 ( foundationAx
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.26/0.49 => ? [B: $i] :
% 0.26/0.49 ( ( in @ B @ A )
% 0.26/0.49 & ~ ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(wellorderingAx_type,type,
% 0.26/0.49 wellorderingAx: $o ).
% 0.26/0.49
% 0.26/0.49 thf(wellorderingAx,definition,
% 0.26/0.49 ( wellorderingAx
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ? [B: $i] :
% 0.26/0.49 ( ! [C: $i] :
% 0.26/0.49 ( ( in @ C @ B )
% 0.26/0.49 => ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ C )
% 0.26/0.49 => ( in @ Xx @ A ) ) )
% 0.26/0.49 & ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( ( in @ Xx @ A )
% 0.26/0.49 & ( in @ Xy @ A ) )
% 0.26/0.49 => ( ! [C: $i] :
% 0.26/0.49 ( ( in @ C @ B )
% 0.26/0.49 => ( ( in @ Xx @ C )
% 0.26/0.49 <=> ( in @ Xy @ C ) ) )
% 0.26/0.49 => ( Xx = Xy ) ) )
% 0.26/0.49 & ! [C: $i,D: $i] :
% 0.26/0.49 ( ( ( in @ C @ B )
% 0.26/0.49 & ( in @ D @ B ) )
% 0.26/0.49 => ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ C )
% 0.26/0.49 => ( in @ Xx @ D ) )
% 0.26/0.49 | ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ D )
% 0.26/0.49 => ( in @ Xx @ C ) ) ) )
% 0.26/0.49 & ! [C: $i] :
% 0.26/0.49 ( ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ C )
% 0.26/0.49 => ( in @ Xx @ A ) )
% 0.26/0.49 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.26/0.49 => ? [D: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ D @ B )
% 0.26/0.49 & ( in @ Xx @ C )
% 0.26/0.49 & ~ ? [Xy: $i] :
% 0.26/0.49 ( ( in @ Xy @ D )
% 0.26/0.49 & ( in @ Xy @ C ) )
% 0.26/0.49 & ! [E: $i] :
% 0.26/0.49 ( ( in @ E @ B )
% 0.26/0.49 => ( ! [Xy: $i] :
% 0.26/0.49 ( ( in @ Xy @ E )
% 0.26/0.49 => ( in @ Xy @ D ) )
% 0.26/0.49 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(descr_type,type,
% 0.26/0.49 descr: ( $i > $o ) > $i ).
% 0.26/0.49
% 0.26/0.49 thf(descrp_type,type,
% 0.26/0.49 descrp: $o ).
% 0.26/0.49
% 0.26/0.49 thf(descrp,definition,
% 0.26/0.49 ( descrp
% 0.26/0.49 = ( ! [Xphi: $i > $o] :
% 0.26/0.49 ( ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 => ( Xphi
% 0.26/0.49 @ ( descr
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(dsetconstr_type,type,
% 0.26/0.49 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.26/0.49
% 0.26/0.49 thf(dsetconstrI_type,type,
% 0.26/0.49 dsetconstrI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(dsetconstrI,definition,
% 0.26/0.49 ( dsetconstrI
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( ( Xphi @ Xx )
% 0.26/0.49 => ( in @ Xx
% 0.26/0.49 @ ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(dsetconstrEL_type,type,
% 0.26/0.49 dsetconstrEL: $o ).
% 0.26/0.49
% 0.26/0.49 thf(dsetconstrEL,definition,
% 0.26/0.49 ( dsetconstrEL
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx
% 0.26/0.49 @ ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.26/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(dsetconstrER_type,type,
% 0.26/0.49 dsetconstrER: $o ).
% 0.26/0.49
% 0.26/0.49 thf(dsetconstrER,definition,
% 0.26/0.49 ( dsetconstrER
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx
% 0.26/0.49 @ ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.26/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(exuE1_type,type,
% 0.26/0.49 exuE1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(exuE1,definition,
% 0.26/0.49 ( exuE1
% 0.26/0.49 = ( ! [Xphi: $i > $o] :
% 0.26/0.49 ( ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 => ? [Xx: $i] :
% 0.26/0.49 ( ( Xphi @ Xx )
% 0.26/0.49 & ! [Xy: $i] :
% 0.26/0.49 ( ( Xphi @ Xy )
% 0.26/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(prop2set_type,type,
% 0.26/0.49 prop2set: $o > $i ).
% 0.26/0.49
% 0.26/0.49 thf(prop2setE_type,type,
% 0.26/0.49 prop2setE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(prop2setE,definition,
% 0.26/0.49 ( prop2setE
% 0.26/0.49 = ( ! [Xphi: $o,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.26/0.49 => Xphi ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptysetE_type,type,
% 0.26/0.49 emptysetE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptysetE,definition,
% 0.26/0.49 ( emptysetE
% 0.26/0.49 = ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ emptyset )
% 0.26/0.49 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptysetimpfalse_type,type,
% 0.26/0.49 emptysetimpfalse: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptysetimpfalse,definition,
% 0.26/0.49 ( emptysetimpfalse
% 0.26/0.49 = ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ emptyset )
% 0.26/0.49 => $false ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(notinemptyset_type,type,
% 0.26/0.49 notinemptyset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(notinemptyset,definition,
% 0.26/0.49 ( notinemptyset
% 0.26/0.49 = ( ! [Xx: $i] :
% 0.26/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(exuE3e_type,type,
% 0.26/0.49 exuE3e: $o ).
% 0.26/0.49
% 0.26/0.49 thf(exuE3e,definition,
% 0.26/0.49 ( exuE3e
% 0.26/0.49 = ( ! [Xphi: $i > $o] :
% 0.26/0.49 ( ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setext_type,type,
% 0.26/0.49 setext: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setext,definition,
% 0.26/0.49 ( setext
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ Xx @ B ) )
% 0.26/0.49 => ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 => ( in @ Xx @ A ) )
% 0.26/0.49 => ( A = B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptyI_type,type,
% 0.26/0.49 emptyI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptyI,definition,
% 0.26/0.49 ( emptyI
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ~ ( in @ Xx @ A )
% 0.26/0.49 => ( A = emptyset ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(noeltsimpempty_type,type,
% 0.26/0.49 noeltsimpempty: $o ).
% 0.26/0.49
% 0.26/0.49 thf(noeltsimpempty,definition,
% 0.26/0.49 ( noeltsimpempty
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ~ ( in @ Xx @ A )
% 0.26/0.49 => ( A = emptyset ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setbeta_type,type,
% 0.26/0.49 setbeta: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setbeta,definition,
% 0.26/0.49 ( setbeta
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( ( in @ Xx
% 0.26/0.49 @ ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.26/0.49 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(nonempty_type,type,
% 0.26/0.49 nonempty: $i > $o ).
% 0.26/0.49
% 0.26/0.49 thf(nonempty,definition,
% 0.26/0.49 ( nonempty
% 0.26/0.49 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(nonemptyE1_type,type,
% 0.26/0.49 nonemptyE1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(nonemptyE1,definition,
% 0.26/0.49 ( nonemptyE1
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ( ( nonempty @ A )
% 0.26/0.49 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(nonemptyI_type,type,
% 0.26/0.49 nonemptyI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(nonemptyI,definition,
% 0.26/0.49 ( nonemptyI
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( ( Xphi @ Xx )
% 0.26/0.49 => ( nonempty
% 0.26/0.49 @ ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(nonemptyI1_type,type,
% 0.26/0.49 nonemptyI1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(nonemptyI1,definition,
% 0.26/0.49 ( nonemptyI1
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.26/0.49 => ( nonempty @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinIL_type,type,
% 0.26/0.49 setadjoinIL: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinIL,definition,
% 0.26/0.49 ( setadjoinIL
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptyinunitempty_type,type,
% 0.26/0.49 emptyinunitempty: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptyinunitempty,definition,
% 0.26/0.49 ( emptyinunitempty
% 0.26/0.49 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinIR_type,type,
% 0.26/0.49 setadjoinIR: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinIR,definition,
% 0.26/0.49 ( setadjoinIR
% 0.26/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.49 ( ( in @ Xy @ A )
% 0.26/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinE_type,type,
% 0.26/0.49 setadjoinE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinE,definition,
% 0.26/0.49 ( setadjoinE
% 0.26/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.26/0.49 => ! [Xphi: $o] :
% 0.26/0.49 ( ( ( Xy = Xx )
% 0.26/0.49 => Xphi )
% 0.26/0.49 => ( ( ( in @ Xy @ A )
% 0.26/0.49 => Xphi )
% 0.26/0.49 => Xphi ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinOr_type,type,
% 0.26/0.49 setadjoinOr: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinOr,definition,
% 0.26/0.49 ( setadjoinOr
% 0.26/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.26/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.26/0.49 => ( ( Xy = Xx )
% 0.26/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setoftrueEq_type,type,
% 0.26/0.49 setoftrueEq: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setoftrueEq,definition,
% 0.26/0.49 ( setoftrueEq
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ( ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xx: $i] : $true )
% 0.26/0.49 = A ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(powersetI_type,type,
% 0.26/0.49 powersetI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(powersetI,definition,
% 0.26/0.49 ( powersetI
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 => ( in @ Xx @ A ) )
% 0.26/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptyinPowerset_type,type,
% 0.26/0.49 emptyinPowerset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptyinPowerset,definition,
% 0.26/0.49 ( emptyinPowerset
% 0.26/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptyInPowerset_type,type,
% 0.26/0.49 emptyInPowerset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptyInPowerset,definition,
% 0.26/0.49 ( emptyInPowerset
% 0.26/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(powersetE_type,type,
% 0.26/0.49 powersetE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(powersetE,definition,
% 0.26/0.49 ( powersetE
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.26/0.49 => ( ( in @ Xx @ B )
% 0.26/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setunionI_type,type,
% 0.26/0.49 setunionI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setunionI,definition,
% 0.26/0.49 ( setunionI
% 0.26/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 => ( ( in @ B @ A )
% 0.26/0.49 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setunionE_type,type,
% 0.26/0.49 setunionE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setunionE,definition,
% 0.26/0.49 ( setunionE
% 0.26/0.49 = ( ! [A: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.26/0.49 => ! [Xphi: $o] :
% 0.26/0.49 ( ! [B: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 => ( ( in @ B @ A )
% 0.26/0.49 => Xphi ) )
% 0.26/0.49 => Xphi ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(subPowSU_type,type,
% 0.26/0.49 subPowSU: $o ).
% 0.26/0.49
% 0.26/0.49 thf(subPowSU,definition,
% 0.26/0.49 ( subPowSU
% 0.26/0.49 = ( ! [A: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(exuE2_type,type,
% 0.26/0.49 exuE2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(exuE2,definition,
% 0.26/0.49 ( exuE2
% 0.26/0.49 = ( ! [Xphi: $i > $o] :
% 0.26/0.49 ( ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 => ? [Xx: $i] :
% 0.26/0.49 ! [Xy: $i] :
% 0.26/0.49 ( ( Xphi @ Xy )
% 0.26/0.49 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(nonemptyImpWitness_type,type,
% 0.26/0.49 nonemptyImpWitness: $o ).
% 0.26/0.49
% 0.26/0.49 thf(nonemptyImpWitness,definition,
% 0.26/0.49 ( nonemptyImpWitness
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ( ( nonempty @ A )
% 0.26/0.49 => ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 & $true ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(uniqinunit_type,type,
% 0.26/0.49 uniqinunit: $o ).
% 0.26/0.49
% 0.26/0.49 thf(uniqinunit,definition,
% 0.26/0.49 ( uniqinunit
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(notinsingleton_type,type,
% 0.26/0.49 notinsingleton: $o ).
% 0.26/0.49
% 0.26/0.49 thf(notinsingleton,definition,
% 0.26/0.49 ( notinsingleton
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xx != Xy )
% 0.26/0.49 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(eqinunit_type,type,
% 0.26/0.49 eqinunit: $o ).
% 0.26/0.49
% 0.26/0.49 thf(eqinunit,definition,
% 0.26/0.49 ( eqinunit
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xx = Xy )
% 0.26/0.49 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(singletonsswitch_type,type,
% 0.26/0.49 singletonsswitch: $o ).
% 0.26/0.49
% 0.26/0.49 thf(singletonsswitch,definition,
% 0.26/0.49 ( singletonsswitch
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(upairsetE_type,type,
% 0.26/0.49 upairsetE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(upairsetE,definition,
% 0.26/0.49 ( upairsetE
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.26/0.49 => ( ( Xz = Xx )
% 0.26/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(upairsetIL_type,type,
% 0.26/0.49 upairsetIL: $o ).
% 0.26/0.49
% 0.26/0.49 thf(upairsetIL,definition,
% 0.26/0.49 ( upairsetIL
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(upairsetIR_type,type,
% 0.26/0.49 upairsetIR: $o ).
% 0.26/0.49
% 0.26/0.49 thf(upairsetIR,definition,
% 0.26/0.49 ( upairsetIR
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptyE1_type,type,
% 0.26/0.49 emptyE1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptyE1,definition,
% 0.26/0.49 ( emptyE1
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.49 ( ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 & ( Xphi @ Xx ) )
% 0.26/0.49 => ( ( ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 = emptyset )
% 0.26/0.49 => $false ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(vacuousDall_type,type,
% 0.26/0.49 vacuousDall: $o ).
% 0.26/0.49
% 0.26/0.49 thf(vacuousDall,definition,
% 0.26/0.49 ( vacuousDall
% 0.26/0.49 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ emptyset )
% 0.26/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(quantDeMorgan1_type,type,
% 0.26/0.49 quantDeMorgan1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(quantDeMorgan1,definition,
% 0.26/0.49 ( quantDeMorgan1
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.49 ( ~ ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( Xphi @ Xx ) )
% 0.26/0.49 => ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(quantDeMorgan2_type,type,
% 0.26/0.49 quantDeMorgan2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(quantDeMorgan2,definition,
% 0.26/0.49 ( quantDeMorgan2
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ~ ( Xphi @ Xx ) )
% 0.26/0.49 => ~ ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(quantDeMorgan3_type,type,
% 0.26/0.49 quantDeMorgan3: $o ).
% 0.26/0.49
% 0.26/0.49 thf(quantDeMorgan3,definition,
% 0.26/0.49 ( quantDeMorgan3
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.49 ( ~ ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 & ( Xphi @ Xx ) )
% 0.26/0.49 => ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(quantDeMorgan4_type,type,
% 0.26/0.49 quantDeMorgan4: $o ).
% 0.26/0.49
% 0.26/0.49 thf(quantDeMorgan4,definition,
% 0.26/0.49 ( quantDeMorgan4
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.49 ( ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 & ~ ( Xphi @ Xx ) )
% 0.26/0.49 => ~ ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(prop2setI_type,type,
% 0.26/0.49 prop2setI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(prop2setI,definition,
% 0.26/0.49 ( prop2setI
% 0.26/0.49 = ( ! [Xphi: $o] :
% 0.26/0.49 ( Xphi
% 0.26/0.49 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(set2prop_type,type,
% 0.26/0.49 set2prop: $i > $o ).
% 0.26/0.49
% 0.26/0.49 thf(prop2set2propI_type,type,
% 0.26/0.49 prop2set2propI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(prop2set2propI,definition,
% 0.26/0.49 ( prop2set2propI
% 0.26/0.49 = ( ! [Xphi: $o] :
% 0.26/0.49 ( Xphi
% 0.26/0.49 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(notdexE_type,type,
% 0.26/0.49 notdexE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(notdexE,definition,
% 0.26/0.49 ( notdexE
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.49 ( ~ ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 & ( Xphi @ Xx ) )
% 0.26/0.49 => ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(notdallE_type,type,
% 0.26/0.49 notdallE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(notdallE,definition,
% 0.26/0.49 ( notdallE
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.49 ( ~ ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( Xphi @ Xx ) )
% 0.26/0.49 => ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(exuI1_type,type,
% 0.26/0.49 exuI1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(exuI1,definition,
% 0.26/0.49 ( exuI1
% 0.26/0.49 = ( ! [Xphi: $i > $o] :
% 0.26/0.49 ( ? [Xx: $i] :
% 0.26/0.49 ( ( Xphi @ Xx )
% 0.26/0.49 & ! [Xy: $i] :
% 0.26/0.49 ( ( Xphi @ Xy )
% 0.26/0.49 => ( Xx = Xy ) ) )
% 0.26/0.49 => ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(exuI3_type,type,
% 0.26/0.49 exuI3: $o ).
% 0.26/0.49
% 0.26/0.49 thf(exuI3,definition,
% 0.26/0.49 ( exuI3
% 0.26/0.49 = ( ! [Xphi: $i > $o] :
% 0.26/0.49 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.26/0.49 => ( ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xphi @ Xx )
% 0.26/0.49 => ( ( Xphi @ Xy )
% 0.26/0.49 => ( Xx = Xy ) ) )
% 0.26/0.49 => ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(exuI2_type,type,
% 0.26/0.49 exuI2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(exuI2,definition,
% 0.26/0.49 ( exuI2
% 0.26/0.49 = ( ! [Xphi: $i > $o] :
% 0.26/0.49 ( ? [Xx: $i] :
% 0.26/0.49 ! [Xy: $i] :
% 0.26/0.49 ( ( Xphi @ Xy )
% 0.26/0.49 <=> ( Xy = Xx ) )
% 0.26/0.49 => ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(inCongP_type,type,
% 0.26/0.49 inCongP: $o ).
% 0.26/0.49
% 0.26/0.49 thf(inCongP,definition,
% 0.26/0.49 ( inCongP
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( A = B )
% 0.26/0.49 => ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xx = Xy )
% 0.26/0.49 => ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(in__Cong_type,type,
% 0.26/0.49 in__Cong: $o ).
% 0.26/0.49
% 0.26/0.49 thf(in__Cong,definition,
% 0.26/0.49 ( in__Cong
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( A = B )
% 0.26/0.49 => ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xx = Xy )
% 0.26/0.49 => ( ( in @ Xx @ A )
% 0.26/0.49 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(exuE3u_type,type,
% 0.26/0.49 exuE3u: $o ).
% 0.26/0.49
% 0.26/0.49 thf(exuE3u,definition,
% 0.26/0.49 ( exuE3u
% 0.26/0.49 = ( ! [Xphi: $i > $o] :
% 0.26/0.49 ( ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 => ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xphi @ Xx )
% 0.26/0.49 => ( ( Xphi @ Xy )
% 0.26/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(exu__Cong_type,type,
% 0.26/0.49 exu__Cong: $o ).
% 0.26/0.49
% 0.26/0.49 thf(exu__Cong,definition,
% 0.26/0.49 ( exu__Cong
% 0.26/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.26/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xx = Xy )
% 0.26/0.49 => ( ( Xphi @ Xx )
% 0.26/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.26/0.49 => ( ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 <=> ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptyset__Cong_type,type,
% 0.26/0.49 emptyset__Cong: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptyset__Cong,definition,
% 0.26/0.49 ( emptyset__Cong
% 0.26/0.49 = ( emptyset = emptyset ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoin__Cong_type,type,
% 0.26/0.49 setadjoin__Cong: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoin__Cong,definition,
% 0.26/0.49 ( setadjoin__Cong
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xx = Xy )
% 0.26/0.49 => ! [Xz: $i,Xu: $i] :
% 0.26/0.49 ( ( Xz = Xu )
% 0.26/0.49 => ( ( setadjoin @ Xx @ Xz )
% 0.26/0.49 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(powerset__Cong_type,type,
% 0.26/0.49 powerset__Cong: $o ).
% 0.26/0.49
% 0.26/0.49 thf(powerset__Cong,definition,
% 0.26/0.49 ( powerset__Cong
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( A = B )
% 0.26/0.49 => ( ( powerset @ A )
% 0.26/0.49 = ( powerset @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setunion__Cong_type,type,
% 0.26/0.49 setunion__Cong: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setunion__Cong,definition,
% 0.26/0.49 ( setunion__Cong
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( A = B )
% 0.26/0.49 => ( ( setunion @ A )
% 0.26/0.49 = ( setunion @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(omega__Cong_type,type,
% 0.26/0.49 omega__Cong: $o ).
% 0.26/0.49
% 0.26/0.49 thf(omega__Cong,definition,
% 0.26/0.49 ( omega__Cong
% 0.26/0.49 = ( omega = omega ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(exuEu_type,type,
% 0.26/0.49 exuEu: $o ).
% 0.26/0.49
% 0.26/0.49 thf(exuEu,definition,
% 0.26/0.49 ( exuEu
% 0.26/0.49 = ( ! [Xphi: $i > $o] :
% 0.26/0.49 ( ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 => ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xphi @ Xx )
% 0.26/0.49 => ( ( Xphi @ Xy )
% 0.26/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(descr__Cong_type,type,
% 0.26/0.49 descr__Cong: $o ).
% 0.26/0.49
% 0.26/0.49 thf(descr__Cong,definition,
% 0.26/0.49 ( descr__Cong
% 0.26/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.26/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.26/0.49 ( ( Xx = Xy )
% 0.26/0.49 => ( ( Xphi @ Xx )
% 0.26/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.26/0.49 => ( ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 => ( ( exu
% 0.26/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.26/0.49 => ( ( descr
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 = ( descr
% 0.26/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(dsetconstr__Cong_type,type,
% 0.26/0.49 dsetconstr__Cong: $o ).
% 0.26/0.49
% 0.26/0.49 thf(dsetconstr__Cong,definition,
% 0.26/0.49 ( dsetconstr__Cong
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( A = B )
% 0.26/0.49 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ! [Xy: $i] :
% 0.26/0.49 ( ( in @ Xy @ B )
% 0.26/0.49 => ( ( Xx = Xy )
% 0.26/0.49 => ( ( Xphi @ Xx )
% 0.26/0.49 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.26/0.49 => ( ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 = ( dsetconstr @ B
% 0.26/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(subset_type,type,
% 0.26/0.49 subset: $i > $i > $o ).
% 0.26/0.49
% 0.26/0.49 thf(disjoint_type,type,
% 0.26/0.49 disjoint: $i > $i > $o ).
% 0.26/0.49
% 0.26/0.49 thf(setsmeet_type,type,
% 0.26/0.49 setsmeet: $i > $i > $o ).
% 0.26/0.49
% 0.26/0.49 thf(subsetI1_type,type,
% 0.26/0.49 subsetI1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(subsetI1,definition,
% 0.26/0.49 ( subsetI1
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ Xx @ B ) )
% 0.26/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(eqimpsubset2_type,type,
% 0.26/0.49 eqimpsubset2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(eqimpsubset2,definition,
% 0.26/0.49 ( eqimpsubset2
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( A = B )
% 0.26/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(eqimpsubset1_type,type,
% 0.26/0.49 eqimpsubset1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(eqimpsubset1,definition,
% 0.26/0.49 ( eqimpsubset1
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( A = B )
% 0.26/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(subsetI2_type,type,
% 0.26/0.49 subsetI2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(subsetI2,definition,
% 0.26/0.49 ( subsetI2
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ! [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ Xx @ B ) )
% 0.26/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(emptysetsubset_type,type,
% 0.26/0.49 emptysetsubset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(emptysetsubset,definition,
% 0.26/0.49 ( emptysetsubset
% 0.26/0.49 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(subsetE_type,type,
% 0.26/0.49 subsetE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(subsetE,definition,
% 0.26/0.49 ( subsetE
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( subset @ A @ B )
% 0.26/0.49 => ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(subsetE2_type,type,
% 0.26/0.49 subsetE2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(subsetE2,definition,
% 0.26/0.49 ( subsetE2
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( subset @ A @ B )
% 0.26/0.49 => ( ~ ( in @ Xx @ B )
% 0.26/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(notsubsetI_type,type,
% 0.26/0.49 notsubsetI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(notsubsetI,definition,
% 0.26/0.49 ( notsubsetI
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( ~ ( in @ Xx @ B )
% 0.26/0.49 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(notequalI1_type,type,
% 0.26/0.49 notequalI1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(notequalI1,definition,
% 0.26/0.49 ( notequalI1
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ~ ( subset @ A @ B )
% 0.26/0.49 => ( A != B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(notequalI2_type,type,
% 0.26/0.49 notequalI2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(notequalI2,definition,
% 0.26/0.49 ( notequalI2
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( ~ ( in @ Xx @ B )
% 0.26/0.49 => ( A != B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(subsetRefl_type,type,
% 0.26/0.49 subsetRefl: $o ).
% 0.26/0.49
% 0.26/0.49 thf(subsetRefl,definition,
% 0.26/0.49 ( subsetRefl
% 0.26/0.49 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(subsetTrans_type,type,
% 0.26/0.49 subsetTrans: $o ).
% 0.26/0.49
% 0.26/0.49 thf(subsetTrans,definition,
% 0.26/0.49 ( subsetTrans
% 0.26/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.26/0.49 ( ( subset @ A @ B )
% 0.26/0.49 => ( ( subset @ B @ C )
% 0.26/0.49 => ( subset @ A @ C ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinSub_type,type,
% 0.26/0.49 setadjoinSub: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinSub,definition,
% 0.26/0.49 ( setadjoinSub
% 0.26/0.49 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinSub2_type,type,
% 0.26/0.49 setadjoinSub2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setadjoinSub2,definition,
% 0.26/0.49 ( setadjoinSub2
% 0.26/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.26/0.49 ( ( subset @ A @ B )
% 0.26/0.49 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(subset2powerset_type,type,
% 0.26/0.49 subset2powerset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(subset2powerset,definition,
% 0.26/0.49 ( subset2powerset
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( subset @ A @ B )
% 0.26/0.49 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setextsub_type,type,
% 0.26/0.49 setextsub: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setextsub,definition,
% 0.26/0.49 ( setextsub
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( subset @ A @ B )
% 0.26/0.49 => ( ( subset @ B @ A )
% 0.26/0.49 => ( A = B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(subsetemptysetimpeq_type,type,
% 0.26/0.49 subsetemptysetimpeq: $o ).
% 0.26/0.49
% 0.26/0.49 thf(subsetemptysetimpeq,definition,
% 0.26/0.49 ( subsetemptysetimpeq
% 0.26/0.49 = ( ! [A: $i] :
% 0.26/0.49 ( ( subset @ A @ emptyset )
% 0.26/0.49 => ( A = emptyset ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(powersetI1_type,type,
% 0.26/0.49 powersetI1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(powersetI1,definition,
% 0.26/0.49 ( powersetI1
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( subset @ B @ A )
% 0.26/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(powersetE1_type,type,
% 0.26/0.49 powersetE1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(powersetE1,definition,
% 0.26/0.49 ( powersetE1
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.26/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(inPowerset_type,type,
% 0.26/0.49 inPowerset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(inPowerset,definition,
% 0.26/0.49 ( inPowerset
% 0.26/0.49 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(powersetsubset_type,type,
% 0.26/0.49 powersetsubset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(powersetsubset,definition,
% 0.26/0.49 ( powersetsubset
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( subset @ A @ B )
% 0.26/0.49 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(sepInPowerset_type,type,
% 0.26/0.49 sepInPowerset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(sepInPowerset,definition,
% 0.26/0.49 ( sepInPowerset
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.49 ( in
% 0.26/0.49 @ ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 @ ( powerset @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(sepSubset_type,type,
% 0.26/0.49 sepSubset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(sepSubset,definition,
% 0.26/0.49 ( sepSubset
% 0.26/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.49 ( subset
% 0.26/0.49 @ ( dsetconstr @ A
% 0.26/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.49 @ A ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binunion_type,type,
% 0.26/0.49 binunion: $i > $i > $i ).
% 0.26/0.49
% 0.26/0.49 thf(binunionIL_type,type,
% 0.26/0.49 binunionIL: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binunionIL,definition,
% 0.26/0.49 ( binunionIL
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(upairset2IR_type,type,
% 0.26/0.49 upairset2IR: $o ).
% 0.26/0.49
% 0.26/0.49 thf(upairset2IR,definition,
% 0.26/0.49 ( upairset2IR
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binunionIR_type,type,
% 0.26/0.49 binunionIR: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binunionIR,definition,
% 0.26/0.49 ( binunionIR
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binunionEcases_type,type,
% 0.26/0.49 binunionEcases: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binunionEcases,definition,
% 0.26/0.49 ( binunionEcases
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.26/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.26/0.49 => ( ( ( in @ Xx @ A )
% 0.26/0.49 => Xphi )
% 0.26/0.49 => ( ( ( in @ Xx @ B )
% 0.26/0.49 => Xphi )
% 0.26/0.49 => Xphi ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binunionE_type,type,
% 0.26/0.49 binunionE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binunionE,definition,
% 0.26/0.49 ( binunionE
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.26/0.49 => ( ( in @ Xx @ A )
% 0.26/0.49 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binunionLsub_type,type,
% 0.26/0.49 binunionLsub: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binunionLsub,definition,
% 0.26/0.49 ( binunionLsub
% 0.26/0.49 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binunionRsub_type,type,
% 0.26/0.49 binunionRsub: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binunionRsub,definition,
% 0.26/0.49 ( binunionRsub
% 0.26/0.49 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersect_type,type,
% 0.26/0.49 binintersect: $i > $i > $i ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectI_type,type,
% 0.26/0.49 binintersectI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectI,definition,
% 0.26/0.49 ( binintersectI
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( ( in @ Xx @ B )
% 0.26/0.49 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset5_type,type,
% 0.26/0.49 binintersectSubset5: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset5,definition,
% 0.26/0.49 ( binintersectSubset5
% 0.26/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.26/0.49 ( ( subset @ C @ A )
% 0.26/0.49 => ( ( subset @ C @ B )
% 0.26/0.49 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectEL_type,type,
% 0.26/0.49 binintersectEL: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectEL,definition,
% 0.26/0.49 ( binintersectEL
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.26/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectLsub_type,type,
% 0.26/0.49 binintersectLsub: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectLsub,definition,
% 0.26/0.49 ( binintersectLsub
% 0.26/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset2_type,type,
% 0.26/0.49 binintersectSubset2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset2,definition,
% 0.26/0.49 ( binintersectSubset2
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( subset @ A @ B )
% 0.26/0.49 => ( ( binintersect @ A @ B )
% 0.26/0.49 = A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset3_type,type,
% 0.26/0.49 binintersectSubset3: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset3,definition,
% 0.26/0.49 ( binintersectSubset3
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( ( binintersect @ A @ B )
% 0.26/0.49 = B )
% 0.26/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectER_type,type,
% 0.26/0.49 binintersectER: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectER,definition,
% 0.26/0.49 ( binintersectER
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.26/0.49 => ( in @ Xx @ B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(disjointsetsI1_type,type,
% 0.26/0.49 disjointsetsI1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(disjointsetsI1,definition,
% 0.26/0.49 ( disjointsetsI1
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ~ ? [Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 & ( in @ Xx @ B ) )
% 0.26/0.49 => ( ( binintersect @ A @ B )
% 0.26/0.49 = emptyset ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectRsub_type,type,
% 0.26/0.49 binintersectRsub: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectRsub,definition,
% 0.26/0.49 ( binintersectRsub
% 0.26/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset4_type,type,
% 0.26/0.49 binintersectSubset4: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset4,definition,
% 0.26/0.49 ( binintersectSubset4
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( subset @ B @ A )
% 0.26/0.49 => ( ( binintersect @ A @ B )
% 0.26/0.49 = B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset1_type,type,
% 0.26/0.49 binintersectSubset1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(binintersectSubset1,definition,
% 0.26/0.49 ( binintersectSubset1
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( ( binintersect @ A @ B )
% 0.26/0.49 = A )
% 0.26/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(bs114d_type,type,
% 0.26/0.49 bs114d: $o ).
% 0.26/0.49
% 0.26/0.49 thf(bs114d,definition,
% 0.26/0.49 ( bs114d
% 0.26/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.26/0.49 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.26/0.49 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(regular_type,type,
% 0.26/0.49 regular: $i > $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminus_type,type,
% 0.26/0.49 setminus: $i > $i > $i ).
% 0.26/0.49
% 0.26/0.49 thf(setminusI_type,type,
% 0.26/0.49 setminusI: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusI,definition,
% 0.26/0.49 ( setminusI
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( ~ ( in @ Xx @ B )
% 0.26/0.49 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setminusEL_type,type,
% 0.26/0.49 setminusEL: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusEL,definition,
% 0.26/0.49 ( setminusEL
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setminusER_type,type,
% 0.26/0.49 setminusER: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusER,definition,
% 0.26/0.49 ( setminusER
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.49 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setminusSubset2_type,type,
% 0.26/0.49 setminusSubset2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusSubset2,definition,
% 0.26/0.49 ( setminusSubset2
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( subset @ A @ B )
% 0.26/0.49 => ( ( setminus @ A @ B )
% 0.26/0.49 = emptyset ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setminusERneg_type,type,
% 0.26/0.49 setminusERneg: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusERneg,definition,
% 0.26/0.49 ( setminusERneg
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.49 => ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setminusELneg_type,type,
% 0.26/0.49 setminusELneg: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusELneg,definition,
% 0.26/0.49 ( setminusELneg
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.26/0.49 => ( ~ ( in @ Xx @ B )
% 0.26/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setminusILneg_type,type,
% 0.26/0.49 setminusILneg: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusILneg,definition,
% 0.26/0.49 ( setminusILneg
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ~ ( in @ Xx @ A )
% 0.26/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setminusIRneg_type,type,
% 0.26/0.49 setminusIRneg: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusIRneg,definition,
% 0.26/0.49 ( setminusIRneg
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ B )
% 0.26/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setminusLsub_type,type,
% 0.26/0.49 setminusLsub: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusLsub,definition,
% 0.26/0.49 ( setminusLsub
% 0.26/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setminusSubset1_type,type,
% 0.26/0.49 setminusSubset1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setminusSubset1,definition,
% 0.26/0.49 ( setminusSubset1
% 0.26/0.49 = ( ! [A: $i,B: $i] :
% 0.26/0.49 ( ( ( setminus @ A @ B )
% 0.26/0.49 = emptyset )
% 0.26/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(symdiff_type,type,
% 0.26/0.49 symdiff: $i > $i > $i ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffE_type,type,
% 0.26/0.49 symdiffE: $o ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffE,definition,
% 0.26/0.49 ( symdiffE
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.26/0.49 => ! [Xphi: $o] :
% 0.26/0.49 ( ( ( in @ Xx @ A )
% 0.26/0.49 => ( ~ ( in @ Xx @ B )
% 0.26/0.49 => Xphi ) )
% 0.26/0.49 => ( ( ~ ( in @ Xx @ A )
% 0.26/0.49 => ( ( in @ Xx @ B )
% 0.26/0.49 => Xphi ) )
% 0.26/0.49 => Xphi ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffI1_type,type,
% 0.26/0.49 symdiffI1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffI1,definition,
% 0.26/0.49 ( symdiffI1
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( ~ ( in @ Xx @ B )
% 0.26/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffI2_type,type,
% 0.26/0.49 symdiffI2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffI2,definition,
% 0.26/0.49 ( symdiffI2
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ~ ( in @ Xx @ A )
% 0.26/0.49 => ( ( in @ Xx @ B )
% 0.26/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffIneg1_type,type,
% 0.26/0.49 symdiffIneg1: $o ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffIneg1,definition,
% 0.26/0.49 ( symdiffIneg1
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( ( in @ Xx @ B )
% 0.26/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffIneg2_type,type,
% 0.26/0.49 symdiffIneg2: $o ).
% 0.26/0.49
% 0.26/0.49 thf(symdiffIneg2,definition,
% 0.26/0.49 ( symdiffIneg2
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ~ ( in @ Xx @ A )
% 0.26/0.49 => ( ~ ( in @ Xx @ B )
% 0.26/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(iskpair_type,type,
% 0.26/0.49 iskpair: $i > $o ).
% 0.26/0.49
% 0.26/0.49 thf(secondinupair_type,type,
% 0.26/0.49 secondinupair: $o ).
% 0.26/0.49
% 0.26/0.49 thf(secondinupair,definition,
% 0.26/0.49 ( secondinupair
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setukpairIL_type,type,
% 0.26/0.49 setukpairIL: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setukpairIL,definition,
% 0.26/0.49 ( setukpairIL
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(setukpairIR_type,type,
% 0.26/0.49 setukpairIR: $o ).
% 0.26/0.49
% 0.26/0.49 thf(setukpairIR,definition,
% 0.26/0.49 ( setukpairIR
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(kpairiskpair_type,type,
% 0.26/0.49 kpairiskpair: $o ).
% 0.26/0.49
% 0.26/0.49 thf(kpairiskpair,definition,
% 0.26/0.49 ( kpairiskpair
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(kpair_type,type,
% 0.26/0.49 kpair: $i > $i > $i ).
% 0.26/0.49
% 0.26/0.49 thf(kpairp_type,type,
% 0.26/0.49 kpairp: $o ).
% 0.26/0.49
% 0.26/0.49 thf(kpairp,definition,
% 0.26/0.49 ( kpairp
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(cartprod_type,type,
% 0.26/0.49 cartprod: $i > $i > $i ).
% 0.26/0.49
% 0.26/0.49 thf(singletonsubset_type,type,
% 0.26/0.49 singletonsubset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(singletonsubset,definition,
% 0.26/0.49 ( singletonsubset
% 0.26/0.49 = ( ! [A: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(singletoninpowerset_type,type,
% 0.26/0.49 singletoninpowerset: $o ).
% 0.26/0.49
% 0.26/0.49 thf(singletoninpowerset,definition,
% 0.26/0.49 ( singletoninpowerset
% 0.26/0.49 = ( ! [A: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(singletoninpowunion_type,type,
% 0.26/0.49 singletoninpowunion: $o ).
% 0.26/0.49
% 0.26/0.49 thf(singletoninpowunion,definition,
% 0.26/0.49 ( singletoninpowunion
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(upairset2E_type,type,
% 0.26/0.49 upairset2E: $o ).
% 0.26/0.49
% 0.26/0.49 thf(upairset2E,definition,
% 0.26/0.49 ( upairset2E
% 0.26/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.26/0.49 => ( ( Xz = Xx )
% 0.26/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.26/0.49
% 0.26/0.49 thf(upairsubunion_type,type,
% 0.26/0.49 upairsubunion: $o ).
% 0.26/0.49
% 0.26/0.49 thf(upairsubunion,definition,
% 0.26/0.49 ( upairsubunion
% 0.26/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.49 ( ( in @ Xx @ A )
% 0.26/0.49 => ! [Xy: $i] :
% 0.26/0.49 ( ( in @ Xy @ B )
% 0.26/0.50 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(upairinpowunion_type,type,
% 0.26/0.50 upairinpowunion: $o ).
% 0.26/0.50
% 0.26/0.50 thf(upairinpowunion,definition,
% 0.26/0.50 ( upairinpowunion
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ubforcartprodlem1_type,type,
% 0.26/0.50 ubforcartprodlem1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ubforcartprodlem1,definition,
% 0.26/0.50 ( ubforcartprodlem1
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ubforcartprodlem2_type,type,
% 0.26/0.50 ubforcartprodlem2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ubforcartprodlem2,definition,
% 0.26/0.50 ( ubforcartprodlem2
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ubforcartprodlem3_type,type,
% 0.26/0.50 ubforcartprodlem3: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ubforcartprodlem3,definition,
% 0.26/0.50 ( ubforcartprodlem3
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodpairin_type,type,
% 0.26/0.50 cartprodpairin: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodpairin,definition,
% 0.26/0.50 ( cartprodpairin
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodmempair1_type,type,
% 0.26/0.50 cartprodmempair1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodmempair1,definition,
% 0.26/0.50 ( cartprodmempair1
% 0.26/0.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.50 => ? [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 & ? [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 & ( Xu
% 0.26/0.50 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodmempair_type,type,
% 0.26/0.50 cartprodmempair: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodmempair,definition,
% 0.26/0.50 ( cartprodmempair
% 0.26/0.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.50 => ( iskpair @ Xu ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setunionE2_type,type,
% 0.26/0.50 setunionE2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setunionE2,definition,
% 0.26/0.50 ( setunionE2
% 0.26/0.50 = ( ! [A: $i,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ ( setunion @ A ) )
% 0.26/0.50 => ? [X: $i] :
% 0.26/0.50 ( ( in @ X @ A )
% 0.26/0.50 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setunionsingleton1_type,type,
% 0.26/0.50 setunionsingleton1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setunionsingleton1,definition,
% 0.26/0.50 ( setunionsingleton1
% 0.26/0.50 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setunionsingleton2_type,type,
% 0.26/0.50 setunionsingleton2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setunionsingleton2,definition,
% 0.26/0.50 ( setunionsingleton2
% 0.26/0.50 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setunionsingleton_type,type,
% 0.26/0.50 setunionsingleton: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setunionsingleton,definition,
% 0.26/0.50 ( setunionsingleton
% 0.26/0.50 = ( ! [Xx: $i] :
% 0.26/0.50 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.26/0.50 = Xx ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(singleton_type,type,
% 0.26/0.50 singleton: $i > $o ).
% 0.26/0.50
% 0.26/0.50 thf(singletonprop_type,type,
% 0.26/0.50 singletonprop: $o ).
% 0.26/0.50
% 0.26/0.50 thf(singletonprop,definition,
% 0.26/0.50 ( singletonprop
% 0.26/0.50 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( ( Xphi @ Xx )
% 0.26/0.50 => ( ( Xphi @ Xy )
% 0.26/0.50 => ( Xx = Xy ) ) ) ) )
% 0.26/0.50 => ( ? [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 & ( Xphi @ Xx ) )
% 0.26/0.50 => ( singleton
% 0.26/0.50 @ ( dsetconstr @ A
% 0.26/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ex1_type,type,
% 0.26/0.50 ex1: $i > ( $i > $o ) > $o ).
% 0.26/0.50
% 0.26/0.50 thf(ex1,definition,
% 0.26/0.50 ( ex1
% 0.26/0.50 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.26/0.50 ( singleton
% 0.26/0.50 @ ( dsetconstr @ A
% 0.26/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ex1E1_type,type,
% 0.26/0.50 ex1E1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ex1E1,definition,
% 0.26/0.50 ( ex1E1
% 0.26/0.50 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.50 ( ( ex1 @ A
% 0.26/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.50 => ? [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ex1I_type,type,
% 0.26/0.50 ex1I: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ex1I,definition,
% 0.26/0.50 ( ex1I
% 0.26/0.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( ( Xphi @ Xx )
% 0.26/0.50 => ( ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( ( Xphi @ Xy )
% 0.26/0.50 => ( Xy = Xx ) ) )
% 0.26/0.50 => ( ex1 @ A
% 0.26/0.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ex1I2_type,type,
% 0.26/0.50 ex1I2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ex1I2,definition,
% 0.26/0.50 ( ex1I2
% 0.26/0.50 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( ( Xphi @ Xx )
% 0.26/0.50 => ( ( Xphi @ Xy )
% 0.26/0.50 => ( Xx = Xy ) ) ) ) )
% 0.26/0.50 => ( ? [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 & ( Xphi @ Xx ) )
% 0.26/0.50 => ( ex1 @ A
% 0.26/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(singletonsuniq_type,type,
% 0.26/0.50 singletonsuniq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(singletonsuniq,definition,
% 0.26/0.50 ( singletonsuniq
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( ( setadjoin @ Xx @ emptyset )
% 0.26/0.50 = ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.50 => ( Xx = Xy ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(atmost1p_type,type,
% 0.26/0.50 atmost1p: $i > $o ).
% 0.26/0.50
% 0.26/0.50 thf(atleast2p_type,type,
% 0.26/0.50 atleast2p: $i > $o ).
% 0.26/0.50
% 0.26/0.50 thf(atmost2p_type,type,
% 0.26/0.50 atmost2p: $i > $o ).
% 0.26/0.50
% 0.26/0.50 thf(upairsetp_type,type,
% 0.26/0.50 upairsetp: $i > $o ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjL1_type,type,
% 0.26/0.50 setukpairinjL1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjL1,definition,
% 0.26/0.50 ( setukpairinjL1
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.50 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.50 => ( Xx = Xz ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(kfstsingleton_type,type,
% 0.26/0.50 kfstsingleton: $o ).
% 0.26/0.50
% 0.26/0.50 thf(kfstsingleton,definition,
% 0.26/0.50 ( kfstsingleton
% 0.26/0.50 = ( ! [Xu: $i] :
% 0.26/0.50 ( ( iskpair @ Xu )
% 0.26/0.50 => ( singleton
% 0.26/0.50 @ ( dsetconstr @ ( setunion @ Xu )
% 0.26/0.50 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(theprop_type,type,
% 0.26/0.50 theprop: $o ).
% 0.26/0.50
% 0.26/0.50 thf(theprop,definition,
% 0.26/0.50 ( theprop
% 0.26/0.50 = ( ! [X: $i] :
% 0.26/0.50 ( ( singleton @ X )
% 0.26/0.50 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(kfst_type,type,
% 0.26/0.50 kfst: $i > $i ).
% 0.26/0.50
% 0.26/0.50 thf(kfstpairEq_type,type,
% 0.26/0.50 kfstpairEq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(kfstpairEq,definition,
% 0.26/0.50 ( kfstpairEq
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.26/0.50 = Xx ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodfstin_type,type,
% 0.26/0.50 cartprodfstin: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodfstin,definition,
% 0.26/0.50 ( cartprodfstin
% 0.26/0.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.50 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjL2_type,type,
% 0.26/0.50 setukpairinjL2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjL2,definition,
% 0.26/0.50 ( setukpairinjL2
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.50 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.50 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.50 => ( Xx = Xz ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjL_type,type,
% 0.26/0.50 setukpairinjL: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjL,definition,
% 0.26/0.50 ( setukpairinjL
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.50 ( ( ( kpair @ Xx @ Xy )
% 0.26/0.50 = ( kpair @ Xz @ Xu ) )
% 0.26/0.50 => ( Xx = Xz ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR11_type,type,
% 0.26/0.50 setukpairinjR11: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR11,definition,
% 0.26/0.50 ( setukpairinjR11
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( Xx = Xy )
% 0.26/0.50 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.50 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR12_type,type,
% 0.26/0.50 setukpairinjR12: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR12,definition,
% 0.26/0.50 ( setukpairinjR12
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( Xx = Xy )
% 0.26/0.50 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.50 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR1_type,type,
% 0.26/0.50 setukpairinjR1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR1,definition,
% 0.26/0.50 ( setukpairinjR1
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.50 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.50 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.50 => ( ( Xz = Xu )
% 0.26/0.50 => ( Xy = Xu ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(upairequniteq_type,type,
% 0.26/0.50 upairequniteq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(upairequniteq,definition,
% 0.26/0.50 ( upairequniteq
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.26/0.50 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.26/0.50 = ( setadjoin @ Xz @ emptyset ) )
% 0.26/0.50 => ( Xx = Xy ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR2_type,type,
% 0.26/0.50 setukpairinjR2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR2,definition,
% 0.26/0.50 ( setukpairinjR2
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.50 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.26/0.50 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.26/0.50 => ( Xy = Xu ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR_type,type,
% 0.26/0.50 setukpairinjR: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setukpairinjR,definition,
% 0.26/0.50 ( setukpairinjR
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.26/0.50 ( ( ( kpair @ Xx @ Xy )
% 0.26/0.50 = ( kpair @ Xz @ Xu ) )
% 0.26/0.50 => ( Xy = Xu ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ksndsingleton_type,type,
% 0.26/0.50 ksndsingleton: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ksndsingleton,definition,
% 0.26/0.50 ( ksndsingleton
% 0.26/0.50 = ( ! [Xu: $i] :
% 0.26/0.50 ( ( iskpair @ Xu )
% 0.26/0.50 => ( singleton
% 0.26/0.50 @ ( dsetconstr @ ( setunion @ Xu )
% 0.26/0.50 @ ^ [Xx: $i] :
% 0.26/0.50 ( Xu
% 0.26/0.50 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ksnd_type,type,
% 0.26/0.50 ksnd: $i > $i ).
% 0.26/0.50
% 0.26/0.50 thf(ksndpairEq_type,type,
% 0.26/0.50 ksndpairEq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ksndpairEq,definition,
% 0.26/0.50 ( ksndpairEq
% 0.26/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.26/0.50 = Xy ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(kpairsurjEq_type,type,
% 0.26/0.50 kpairsurjEq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(kpairsurjEq,definition,
% 0.26/0.50 ( kpairsurjEq
% 0.26/0.50 = ( ! [Xu: $i] :
% 0.26/0.50 ( ( iskpair @ Xu )
% 0.26/0.50 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.26/0.50 = Xu ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodsndin_type,type,
% 0.26/0.50 cartprodsndin: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodsndin,definition,
% 0.26/0.50 ( cartprodsndin
% 0.26/0.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.50 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodpairmemEL_type,type,
% 0.26/0.50 cartprodpairmemEL: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodpairmemEL,definition,
% 0.26/0.50 ( cartprodpairmemEL
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.26/0.50 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodpairmemER_type,type,
% 0.26/0.50 cartprodpairmemER: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodpairmemER,definition,
% 0.26/0.50 ( cartprodpairmemER
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.26/0.50 => ( in @ Xy @ B ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodmempaircEq_type,type,
% 0.26/0.50 cartprodmempaircEq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodmempaircEq,definition,
% 0.26/0.50 ( cartprodmempaircEq
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( kpair @ Xx @ Xy )
% 0.26/0.50 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodfstpairEq_type,type,
% 0.26/0.50 cartprodfstpairEq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodfstpairEq,definition,
% 0.26/0.50 ( cartprodfstpairEq
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.26/0.50 = Xx ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodsndpairEq_type,type,
% 0.26/0.50 cartprodsndpairEq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodsndpairEq,definition,
% 0.26/0.50 ( cartprodsndpairEq
% 0.26/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.26/0.50 = Xy ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodpairsurjEq_type,type,
% 0.26/0.50 cartprodpairsurjEq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(cartprodpairsurjEq,definition,
% 0.26/0.50 ( cartprodpairsurjEq
% 0.26/0.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.26/0.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.26/0.50 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.26/0.50 = Xu ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(breln_type,type,
% 0.26/0.50 breln: $i > $i > $i > $o ).
% 0.26/0.50
% 0.26/0.50 thf(breln,definition,
% 0.26/0.50 ( breln
% 0.26/0.50 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstr_type,type,
% 0.26/0.50 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrI_type,type,
% 0.26/0.50 dpsetconstrI: $o ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrI,definition,
% 0.26/0.50 ( dpsetconstrI
% 0.26/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( Xphi @ Xx @ Xy )
% 0.26/0.50 => ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.50 @ ( dpsetconstr @ A @ B
% 0.26/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrSub_type,type,
% 0.26/0.50 dpsetconstrSub: $o ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrSub,definition,
% 0.26/0.50 ( dpsetconstrSub
% 0.26/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.26/0.50 ( subset
% 0.26/0.50 @ ( dpsetconstr @ A @ B
% 0.26/0.50 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.26/0.50 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(setOfPairsIsBReln_type,type,
% 0.26/0.50 setOfPairsIsBReln: $o ).
% 0.26/0.50
% 0.26/0.50 thf(setOfPairsIsBReln,definition,
% 0.26/0.50 ( setOfPairsIsBReln
% 0.26/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.26/0.50 ( breln @ A @ B
% 0.26/0.50 @ ( dpsetconstr @ A @ B
% 0.26/0.50 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrERa_type,type,
% 0.26/0.50 dpsetconstrERa: $o ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrERa,definition,
% 0.26/0.50 ( dpsetconstrERa
% 0.26/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.50 @ ( dpsetconstr @ A @ B
% 0.26/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.50 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrEL1_type,type,
% 0.26/0.50 dpsetconstrEL1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrEL1,definition,
% 0.26/0.50 ( dpsetconstrEL1
% 0.26/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.50 @ ( dpsetconstr @ A @ B
% 0.26/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.50 => ( in @ Xx @ A ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrEL2_type,type,
% 0.26/0.50 dpsetconstrEL2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrEL2,definition,
% 0.26/0.50 ( dpsetconstrEL2
% 0.26/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.50 @ ( dpsetconstr @ A @ B
% 0.26/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.50 => ( in @ Xy @ B ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrER_type,type,
% 0.26/0.50 dpsetconstrER: $o ).
% 0.26/0.50
% 0.26/0.50 thf(dpsetconstrER,definition,
% 0.26/0.50 ( dpsetconstrER
% 0.26/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( in @ ( kpair @ Xx @ Xy )
% 0.26/0.50 @ ( dpsetconstr @ A @ B
% 0.26/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.26/0.50 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(func_type,type,
% 0.26/0.50 func: $i > $i > $i > $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcSet_type,type,
% 0.26/0.50 funcSet: $i > $i > $i ).
% 0.26/0.50
% 0.26/0.50 thf(funcImageSingleton_type,type,
% 0.26/0.50 funcImageSingleton: $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcImageSingleton,definition,
% 0.26/0.50 ( funcImageSingleton
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( singleton
% 0.26/0.50 @ ( dsetconstr @ B
% 0.26/0.50 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(apProp_type,type,
% 0.26/0.50 apProp: $o ).
% 0.26/0.50
% 0.26/0.50 thf(apProp,definition,
% 0.26/0.50 ( apProp
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in
% 0.26/0.50 @ ( setunion
% 0.26/0.50 @ ( dsetconstr @ B
% 0.26/0.50 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.26/0.50 @ B ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ap_type,type,
% 0.26/0.50 ap: $i > $i > $i > $i > $i ).
% 0.26/0.50
% 0.26/0.50 thf(app_type,type,
% 0.26/0.50 app: $o ).
% 0.26/0.50
% 0.26/0.50 thf(app,definition,
% 0.26/0.50 ( app
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(infuncsetfunc_type,type,
% 0.26/0.50 infuncsetfunc: $o ).
% 0.26/0.50
% 0.26/0.50 thf(infuncsetfunc,definition,
% 0.26/0.50 ( infuncsetfunc
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.50 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ap2p_type,type,
% 0.26/0.50 ap2p: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ap2p,definition,
% 0.26/0.50 ( ap2p
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(funcinfuncset_type,type,
% 0.26/0.50 funcinfuncset: $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcinfuncset,definition,
% 0.26/0.50 ( funcinfuncset
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(lamProp_type,type,
% 0.26/0.50 lamProp: $o ).
% 0.26/0.50
% 0.26/0.50 thf(lamProp,definition,
% 0.26/0.50 ( lamProp
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.50 => ( func @ A @ B
% 0.26/0.50 @ ( dpsetconstr @ A @ B
% 0.26/0.50 @ ^ [Xx: $i,Xy: $i] :
% 0.26/0.50 ( ( Xf @ Xx )
% 0.26/0.50 = Xy ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(lam_type,type,
% 0.26/0.50 lam: $i > $i > ( $i > $i ) > $i ).
% 0.26/0.50
% 0.26/0.50 thf(lamp_type,type,
% 0.26/0.50 lamp: $o ).
% 0.26/0.50
% 0.26/0.50 thf(lamp,definition,
% 0.26/0.50 ( lamp
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.50 => ( func @ A @ B
% 0.26/0.50 @ ( lam @ A @ B
% 0.26/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(lam2p_type,type,
% 0.26/0.50 lam2p: $o ).
% 0.26/0.50
% 0.26/0.50 thf(lam2p,definition,
% 0.26/0.50 ( lam2p
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.50 => ( in
% 0.26/0.50 @ ( lam @ A @ B
% 0.26/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.26/0.50 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(brelnall1_type,type,
% 0.26/0.50 brelnall1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(brelnall1,definition,
% 0.26/0.50 ( brelnall1
% 0.26/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.50 ( ( breln @ A @ B @ R )
% 0.26/0.50 => ! [Xphi: $i > $o] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.50 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ R )
% 0.26/0.50 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(brelnall2_type,type,
% 0.26/0.50 brelnall2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(brelnall2,definition,
% 0.26/0.50 ( brelnall2
% 0.26/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.50 ( ( breln @ A @ B @ R )
% 0.26/0.50 => ! [Xphi: $i > $o] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.50 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ R )
% 0.26/0.50 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ex1E2_type,type,
% 0.26/0.50 ex1E2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ex1E2,definition,
% 0.26/0.50 ( ex1E2
% 0.26/0.50 = ( ! [A: $i,Xphi: $i > $o] :
% 0.26/0.50 ( ( ex1 @ A
% 0.26/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( ( Xphi @ Xx )
% 0.26/0.50 => ( ( Xphi @ Xy )
% 0.26/0.50 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(funcGraphProp1_type,type,
% 0.26/0.50 funcGraphProp1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcGraphProp1,definition,
% 0.26/0.50 ( funcGraphProp1
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(funcGraphProp3_type,type,
% 0.26/0.50 funcGraphProp3: $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcGraphProp3,definition,
% 0.26/0.50 ( funcGraphProp3
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(funcGraphProp2_type,type,
% 0.26/0.50 funcGraphProp2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcGraphProp2,definition,
% 0.26/0.50 ( funcGraphProp2
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.26/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.50 = Xy ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(funcextLem_type,type,
% 0.26/0.50 funcextLem: $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcextLem,definition,
% 0.26/0.50 ( funcextLem
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ! [Xg: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xg )
% 0.26/0.50 => ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.50 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.26/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(funcGraphProp4_type,type,
% 0.26/0.50 funcGraphProp4: $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcGraphProp4,definition,
% 0.26/0.50 ( funcGraphProp4
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.26/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.50 = Xy ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(subbreln_type,type,
% 0.26/0.50 subbreln: $o ).
% 0.26/0.50
% 0.26/0.50 thf(subbreln,definition,
% 0.26/0.50 ( subbreln
% 0.26/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.50 ( ( breln @ A @ B @ R )
% 0.26/0.50 => ! [S: $i] :
% 0.26/0.50 ( ( breln @ A @ B @ S )
% 0.26/0.50 => ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.26/0.50 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(eqbreln_type,type,
% 0.26/0.50 eqbreln: $o ).
% 0.26/0.50
% 0.26/0.50 thf(eqbreln,definition,
% 0.26/0.50 ( eqbreln
% 0.26/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.26/0.50 ( ( breln @ A @ B @ R )
% 0.26/0.50 => ! [S: $i] :
% 0.26/0.50 ( ( breln @ A @ B @ S )
% 0.26/0.50 => ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.26/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.26/0.50 => ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ B )
% 0.26/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.26/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.26/0.50 => ( R = S ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(funcext_type,type,
% 0.26/0.50 funcext: $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcext,definition,
% 0.26/0.50 ( funcext
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ! [Xg: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xg )
% 0.26/0.50 => ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.50 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.26/0.50 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(funcext2_type,type,
% 0.26/0.50 funcext2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(funcext2,definition,
% 0.26/0.50 ( funcext2
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.50 => ! [Xg: $i] :
% 0.26/0.50 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.26/0.50 => ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.50 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.26/0.50 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ap2apEq1_type,type,
% 0.26/0.50 ap2apEq1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ap2apEq1,definition,
% 0.26/0.50 ( ap2apEq1
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.50 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ap2apEq2_type,type,
% 0.26/0.50 ap2apEq2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ap2apEq2,definition,
% 0.26/0.50 ( ap2apEq2
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.26/0.50 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(beta1_type,type,
% 0.26/0.50 beta1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(beta1,definition,
% 0.26/0.50 ( beta1
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( ( ap @ A @ B
% 0.26/0.50 @ ( lam @ A @ B
% 0.26/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.26/0.50 @ Xx )
% 0.26/0.50 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(eta1_type,type,
% 0.26/0.50 eta1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(eta1,definition,
% 0.26/0.50 ( eta1
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( func @ A @ B @ Xf )
% 0.26/0.50 => ( ( lam @ A @ B
% 0.26/0.50 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.26/0.50 = Xf ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(lam2lamEq_type,type,
% 0.26/0.50 lam2lamEq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(lam2lamEq,definition,
% 0.26/0.50 ( lam2lamEq
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.50 => ( ( lam @ A @ B
% 0.26/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.26/0.50 = ( lam @ A @ B
% 0.26/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(beta2_type,type,
% 0.26/0.50 beta2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(beta2,definition,
% 0.26/0.50 ( beta2
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.26/0.50 ( ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ( ( ap @ A @ B
% 0.26/0.50 @ ( lam @ A @ B
% 0.26/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.26/0.50 @ Xx )
% 0.26/0.50 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(eta2_type,type,
% 0.26/0.50 eta2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(eta2,definition,
% 0.26/0.50 ( eta2
% 0.26/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.26/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.26/0.50 => ( ( lam @ A @ B
% 0.26/0.50 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.26/0.50 = Xf ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(iffalseProp1_type,type,
% 0.26/0.50 iffalseProp1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(iffalseProp1,definition,
% 0.26/0.50 ( iffalseProp1
% 0.26/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( ~ Xphi
% 0.26/0.50 => ( in @ Xy
% 0.26/0.50 @ ( dsetconstr @ A
% 0.26/0.50 @ ^ [Xz: $i] :
% 0.26/0.50 ( ( Xphi
% 0.26/0.50 & ( Xz = Xx ) )
% 0.26/0.50 | ( ~ Xphi
% 0.26/0.50 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(iffalseProp2_type,type,
% 0.26/0.50 iffalseProp2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(iffalseProp2,definition,
% 0.26/0.50 ( iffalseProp2
% 0.26/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( ~ Xphi
% 0.26/0.50 => ( ( dsetconstr @ A
% 0.26/0.50 @ ^ [Xz: $i] :
% 0.26/0.50 ( ( Xphi
% 0.26/0.50 & ( Xz = Xx ) )
% 0.26/0.50 | ( ~ Xphi
% 0.26/0.50 & ( Xz = Xy ) ) ) )
% 0.26/0.50 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(iftrueProp1_type,type,
% 0.26/0.50 iftrueProp1: $o ).
% 0.26/0.50
% 0.26/0.50 thf(iftrueProp1,definition,
% 0.26/0.50 ( iftrueProp1
% 0.26/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( Xphi
% 0.26/0.50 => ( in @ Xx
% 0.26/0.50 @ ( dsetconstr @ A
% 0.26/0.50 @ ^ [Xz: $i] :
% 0.26/0.50 ( ( Xphi
% 0.26/0.50 & ( Xz = Xx ) )
% 0.26/0.50 | ( ~ Xphi
% 0.26/0.50 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(iftrueProp2_type,type,
% 0.26/0.50 iftrueProp2: $o ).
% 0.26/0.50
% 0.26/0.50 thf(iftrueProp2,definition,
% 0.26/0.50 ( iftrueProp2
% 0.26/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( Xphi
% 0.26/0.50 => ( ( dsetconstr @ A
% 0.26/0.50 @ ^ [Xz: $i] :
% 0.26/0.50 ( ( Xphi
% 0.26/0.50 & ( Xz = Xx ) )
% 0.26/0.50 | ( ~ Xphi
% 0.26/0.50 & ( Xz = Xy ) ) ) )
% 0.26/0.50 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(ifSingleton_type,type,
% 0.26/0.50 ifSingleton: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ifSingleton,definition,
% 0.26/0.50 ( ifSingleton
% 0.26/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( singleton
% 0.26/0.50 @ ( dsetconstr @ A
% 0.26/0.50 @ ^ [Xz: $i] :
% 0.26/0.50 ( ( Xphi
% 0.26/0.50 & ( Xz = Xx ) )
% 0.26/0.50 | ( ~ Xphi
% 0.26/0.50 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(if_type,type,
% 0.26/0.50 if: $i > $o > $i > $i > $i ).
% 0.26/0.50
% 0.26/0.50 thf(ifp_type,type,
% 0.26/0.50 ifp: $o ).
% 0.26/0.50
% 0.26/0.50 thf(ifp,definition,
% 0.26/0.50 ( ifp
% 0.26/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(theeq_type,type,
% 0.26/0.50 theeq: $o ).
% 0.26/0.50
% 0.26/0.50 thf(theeq,definition,
% 0.26/0.50 ( theeq
% 0.26/0.50 = ( ! [X: $i] :
% 0.26/0.50 ( ( singleton @ X )
% 0.26/0.50 => ! [Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ X )
% 0.26/0.50 => ( ( setunion @ X )
% 0.26/0.50 = Xx ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(iftrue_type,type,
% 0.26/0.50 iftrue: $o ).
% 0.26/0.50
% 0.26/0.50 thf(iftrue,definition,
% 0.26/0.50 ( iftrue
% 0.26/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( Xphi
% 0.26/0.50 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.26/0.50 = Xx ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(iffalse_type,type,
% 0.26/0.50 iffalse: $o ).
% 0.26/0.50
% 0.26/0.50 thf(iffalse,definition,
% 0.26/0.50 ( iffalse
% 0.26/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( ~ Xphi
% 0.26/0.50 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.26/0.50 = Xy ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(iftrueorfalse_type,type,
% 0.26/0.50 iftrueorfalse: $o ).
% 0.26/0.50
% 0.26/0.50 thf(iftrueorfalse,definition,
% 0.26/0.50 ( iftrueorfalse
% 0.26/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.26/0.50 ( ( in @ Xx @ A )
% 0.26/0.50 => ! [Xy: $i] :
% 0.26/0.50 ( ( in @ Xy @ A )
% 0.26/0.50 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(binintersectT_lem_type,type,
% 0.26/0.50 binintersectT_lem: $o ).
% 0.26/0.50
% 0.26/0.50 thf(binintersectT_lem,definition,
% 0.26/0.50 ( binintersectT_lem
% 0.26/0.50 = ( ! [A: $i,X: $i] :
% 0.26/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.50 => ! [Y: $i] :
% 0.26/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.26/0.50 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(binunionT_lem_type,type,
% 0.26/0.50 binunionT_lem: $o ).
% 0.26/0.50
% 0.26/0.50 thf(binunionT_lem,definition,
% 0.26/0.50 ( binunionT_lem
% 0.26/0.50 = ( ! [A: $i,X: $i] :
% 0.26/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.50 => ! [Y: $i] :
% 0.26/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.26/0.50 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 thf(powersetT_lem,conjecture,
% 0.26/0.50 ( setextAx
% 0.26/0.50 => ( emptysetAx
% 0.26/0.50 => ( setadjoinAx
% 0.26/0.50 => ( powersetAx
% 0.26/0.50 => ( setunionAx
% 0.26/0.50 => ( omega0Ax
% 0.26/0.50 => ( omegaSAx
% 0.26/0.50 => ( omegaIndAx
% 0.26/0.50 => ( replAx
% 0.26/0.50 => ( foundationAx
% 0.26/0.50 => ( wellorderingAx
% 0.26/0.50 => ( descrp
% 0.26/0.50 => ( dsetconstrI
% 0.26/0.50 => ( dsetconstrEL
% 0.26/0.50 => ( dsetconstrER
% 0.26/0.50 => ( exuE1
% 0.26/0.50 => ( prop2setE
% 0.26/0.50 => ( emptysetE
% 0.26/0.50 => ( emptysetimpfalse
% 0.26/0.50 => ( notinemptyset
% 0.26/0.50 => ( exuE3e
% 0.26/0.50 => ( setext
% 0.26/0.50 => ( emptyI
% 0.26/0.50 => ( noeltsimpempty
% 0.26/0.50 => ( setbeta
% 0.26/0.50 => ( nonemptyE1
% 0.26/0.50 => ( nonemptyI
% 0.26/0.50 => ( nonemptyI1
% 0.26/0.50 => ( setadjoinIL
% 0.26/0.50 => ( emptyinunitempty
% 0.26/0.50 => ( setadjoinIR
% 0.26/0.50 => ( setadjoinE
% 0.26/0.50 => ( setadjoinOr
% 0.26/0.50 => ( setoftrueEq
% 0.26/0.50 => ( powersetI
% 0.26/0.50 => ( emptyinPowerset
% 0.26/0.50 => ( emptyInPowerset
% 0.26/0.50 => ( powersetE
% 0.26/0.50 => ( setunionI
% 0.26/0.50 => ( setunionE
% 0.26/0.50 => ( subPowSU
% 0.26/0.50 => ( exuE2
% 0.26/0.50 => ( nonemptyImpWitness
% 0.26/0.50 => ( uniqinunit
% 0.26/0.50 => ( notinsingleton
% 0.26/0.50 => ( eqinunit
% 0.26/0.50 => ( singletonsswitch
% 0.26/0.50 => ( upairsetE
% 0.26/0.50 => ( upairsetIL
% 0.26/0.50 => ( upairsetIR
% 0.26/0.50 => ( emptyE1
% 0.26/0.50 => ( vacuousDall
% 0.26/0.50 => ( quantDeMorgan1
% 0.26/0.50 => ( quantDeMorgan2
% 0.26/0.50 => ( quantDeMorgan3
% 0.26/0.50 => ( quantDeMorgan4
% 0.26/0.50 => ( prop2setI
% 0.26/0.50 => ( prop2set2propI
% 0.26/0.50 => ( notdexE
% 0.26/0.50 => ( notdallE
% 0.26/0.50 => ( exuI1
% 0.26/0.50 => ( exuI3
% 0.26/0.50 => ( exuI2
% 0.26/0.50 => ( inCongP
% 0.26/0.50 => ( in__Cong
% 0.26/0.50 => ( exuE3u
% 0.26/0.50 => ( exu__Cong
% 0.26/0.50 => ( emptyset__Cong
% 0.26/0.50 => ( setadjoin__Cong
% 0.26/0.50 => ( powerset__Cong
% 0.26/0.50 => ( setunion__Cong
% 0.26/0.50 => ( omega__Cong
% 0.26/0.50 => ( exuEu
% 0.26/0.50 => ( descr__Cong
% 0.26/0.50 => ( dsetconstr__Cong
% 0.26/0.50 => ( subsetI1
% 0.26/0.50 => ( eqimpsubset2
% 0.26/0.50 => ( eqimpsubset1
% 0.26/0.50 => ( subsetI2
% 0.26/0.50 => ( emptysetsubset
% 0.26/0.50 => ( subsetE
% 0.26/0.50 => ( subsetE2
% 0.26/0.50 => ( notsubsetI
% 0.26/0.50 => ( notequalI1
% 0.26/0.50 => ( notequalI2
% 0.26/0.50 => ( subsetRefl
% 0.26/0.50 => ( subsetTrans
% 0.26/0.50 => ( setadjoinSub
% 0.26/0.50 => ( setadjoinSub2
% 0.26/0.50 => ( subset2powerset
% 0.26/0.50 => ( setextsub
% 0.26/0.50 => ( subsetemptysetimpeq
% 0.26/0.50 => ( powersetI1
% 0.26/0.50 => ( powersetE1
% 0.26/0.50 => ( inPowerset
% 0.26/0.50 => ( powersetsubset
% 0.26/0.50 => ( sepInPowerset
% 0.26/0.50 => ( sepSubset
% 0.26/0.50 => ( binunionIL
% 0.26/0.50 => ( upairset2IR
% 0.26/0.50 => ( binunionIR
% 0.26/0.50 => ( binunionEcases
% 0.26/0.50 => ( binunionE
% 0.26/0.50 => ( binunionLsub
% 0.26/0.50 => ( binunionRsub
% 0.26/0.50 => ( binintersectI
% 0.26/0.50 => ( binintersectSubset5
% 0.26/0.50 => ( binintersectEL
% 0.26/0.50 => ( binintersectLsub
% 0.26/0.50 => ( binintersectSubset2
% 0.26/0.50 => ( binintersectSubset3
% 0.26/0.50 => ( binintersectER
% 0.26/0.50 => ( disjointsetsI1
% 0.26/0.50 => ( binintersectRsub
% 0.26/0.50 => ( binintersectSubset4
% 0.26/0.50 => ( binintersectSubset1
% 0.26/0.50 => ( bs114d
% 0.26/0.50 => ( setminusI
% 0.26/0.50 => ( setminusEL
% 0.26/0.50 => ( setminusER
% 0.26/0.50 => ( setminusSubset2
% 0.26/0.50 => ( setminusERneg
% 0.26/0.50 => ( setminusELneg
% 0.26/0.50 => ( setminusILneg
% 0.26/0.50 => ( setminusIRneg
% 0.26/0.50 => ( setminusLsub
% 0.26/0.50 => ( setminusSubset1
% 0.26/0.50 => ( symdiffE
% 0.26/0.50 => ( symdiffI1
% 0.26/0.50 => ( symdiffI2
% 0.26/0.50 => ( symdiffIneg1
% 0.26/0.50 => ( symdiffIneg2
% 0.26/0.50 => ( secondinupair
% 0.26/0.50 => ( setukpairIL
% 0.26/0.50 => ( setukpairIR
% 0.26/0.50 => ( kpairiskpair
% 0.26/0.50 => ( kpairp
% 0.26/0.50 => ( singletonsubset
% 0.26/0.50 => ( singletoninpowerset
% 0.26/0.50 => ( singletoninpowunion
% 0.26/0.50 => ( upairset2E
% 0.26/0.50 => ( upairsubunion
% 0.26/0.50 => ( upairinpowunion
% 0.26/0.50 => ( ubforcartprodlem1
% 0.26/0.50 => ( ubforcartprodlem2
% 0.26/0.50 => ( ubforcartprodlem3
% 0.26/0.50 => ( cartprodpairin
% 0.26/0.50 => ( cartprodmempair1
% 0.26/0.50 => ( cartprodmempair
% 0.26/0.50 => ( setunionE2
% 0.26/0.50 => ( setunionsingleton1
% 0.26/0.50 => ( setunionsingleton2
% 0.26/0.50 => ( setunionsingleton
% 0.26/0.50 => ( singletonprop
% 0.26/0.50 => ( ex1E1
% 0.26/0.50 => ( ex1I
% 0.26/0.50 => ( ex1I2
% 0.26/0.50 => ( singletonsuniq
% 0.26/0.50 => ( setukpairinjL1
% 0.26/0.50 => ( kfstsingleton
% 0.26/0.50 => ( theprop
% 0.26/0.50 => ( kfstpairEq
% 0.26/0.50 => ( cartprodfstin
% 0.26/0.50 => ( setukpairinjL2
% 0.26/0.50 => ( setukpairinjL
% 0.26/0.50 => ( setukpairinjR11
% 0.26/0.50 => ( setukpairinjR12
% 0.26/0.50 => ( setukpairinjR1
% 0.26/0.50 => ( upairequniteq
% 0.26/0.50 => ( setukpairinjR2
% 0.26/0.50 => ( setukpairinjR
% 0.26/0.50 => ( ksndsingleton
% 0.26/0.50 => ( ksndpairEq
% 0.26/0.50 => ( kpairsurjEq
% 0.26/0.50 => ( cartprodsndin
% 0.26/0.50 => ( cartprodpairmemEL
% 0.26/0.50 => ( cartprodpairmemER
% 0.26/0.50 => ( cartprodmempaircEq
% 0.26/0.50 => ( cartprodfstpairEq
% 0.26/0.50 => ( cartprodsndpairEq
% 0.26/0.50 => ( cartprodpairsurjEq
% 0.26/0.50 => ( dpsetconstrI
% 0.26/0.50 => ( dpsetconstrSub
% 0.26/0.50 => ( setOfPairsIsBReln
% 0.26/0.50 => ( dpsetconstrERa
% 0.26/0.50 => ( dpsetconstrEL1
% 0.26/0.50 => ( dpsetconstrEL2
% 0.26/0.50 => ( dpsetconstrER
% 0.26/0.50 => ( funcImageSingleton
% 0.26/0.50 => ( apProp
% 0.26/0.50 => ( app
% 0.26/0.50 => ( infuncsetfunc
% 0.26/0.50 => ( ap2p
% 0.26/0.50 => ( funcinfuncset
% 0.26/0.50 => ( lamProp
% 0.26/0.50 => ( lamp
% 0.26/0.50 => ( lam2p
% 0.26/0.50 => ( brelnall1
% 0.26/0.50 => ( brelnall2
% 0.26/0.50 => ( ex1E2
% 0.26/0.50 => ( funcGraphProp1
% 0.26/0.50 => ( funcGraphProp3
% 0.26/0.50 => ( funcGraphProp2
% 0.26/0.50 => ( funcextLem
% 0.26/0.50 => ( funcGraphProp4
% 0.26/0.50 => ( subbreln
% 0.26/0.50 => ( eqbreln
% 0.26/0.50 => ( funcext
% 0.26/0.50 => ( funcext2
% 0.26/0.50 => ( ap2apEq1
% 0.26/0.50 => ( ap2apEq2
% 0.26/0.50 => ( beta1
% 0.26/0.50 => ( eta1
% 0.26/0.50 => ( lam2lamEq
% 0.26/0.50 => ( beta2
% 0.26/0.50 => ( eta2
% 0.26/0.50 => ( iffalseProp1
% 0.26/0.50 => ( iffalseProp2
% 0.26/0.50 => ( iftrueProp1
% 0.26/0.50 => ( iftrueProp2
% 0.26/0.50 => ( ifSingleton
% 0.26/0.50 => ( ifp
% 0.26/0.50 => ( theeq
% 0.26/0.50 => ( iftrue
% 0.26/0.50 => ( iffalse
% 0.26/0.50 => ( iftrueorfalse
% 0.26/0.50 => ( binintersectT_lem
% 0.26/0.50 => ( binunionT_lem
% 0.26/0.50 => ! [A: $i,X: $i] :
% 0.26/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.26/0.50 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.26/0.50
% 0.26/0.50 %------------------------------------------------------------------------------
% 0.26/0.50 ------------------------
% 2.34/1.49 ---- Embedded in THF ---
% 2.34/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.34/1.49 %%% Generated on Tue Feb 24 12:32:36 EST 2026
% 2.34/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.34/1.49 %%% Logic specification used:
% 2.34/1.49 %%% thf(spec, logic, $$dhol).
% 2.34/1.49
% 2.34/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.PfJgBtH2Qy/DTF2DTF5724.p
% See solution above
% 2.34/1.49 ------------------------
% 2.65/1.52 ---- Cleaned THF ---
% 2.65/1.52 thf(in_type,type,
% 2.65/1.52 in: $i > $i > $o ).
% 2.65/1.52
% 2.65/1.52 thf(exu_type,type,
% 2.65/1.52 exu: ( $i > $o ) > $o ).
% 2.65/1.52
% 2.65/1.52 thf(exu,definition,
% 2.65/1.52 ( exu
% 2.65/1.52 = ( ^ [Xphi: $i > $o] :
% 2.65/1.52 ? [Xx: $i] :
% 2.65/1.52 ( ( Xphi @ Xx )
% 2.65/1.52 & ! [Xy: $i] :
% 2.65/1.52 ( ( Xphi @ Xy )
% 2.65/1.52 => ( Xx = Xy ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setextAx_type,type,
% 2.65/1.52 setextAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setextAx,definition,
% 2.65/1.52 ( setextAx
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 <=> ( in @ Xx @ B ) )
% 2.65/1.52 => ( A = B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptyset_type,type,
% 2.65/1.52 emptyset: $i ).
% 2.65/1.52
% 2.65/1.52 thf(emptysetAx_type,type,
% 2.65/1.52 emptysetAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptysetAx,definition,
% 2.65/1.52 ( emptysetAx
% 2.65/1.52 = ( ! [Xx: $i] :
% 2.65/1.52 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoin_type,type,
% 2.65/1.52 setadjoin: $i > $i > $i ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinAx_type,type,
% 2.65/1.52 setadjoinAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinAx,definition,
% 2.65/1.52 ( setadjoinAx
% 2.65/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.65/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.65/1.52 <=> ( ( Xy = Xx )
% 2.65/1.52 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(powerset_type,type,
% 2.65/1.52 powerset: $i > $i ).
% 2.65/1.52
% 2.65/1.52 thf(powersetAx_type,type,
% 2.65/1.52 powersetAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(powersetAx,definition,
% 2.65/1.52 ( powersetAx
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( in @ B @ ( powerset @ A ) )
% 2.65/1.52 <=> ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ B )
% 2.65/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setunion_type,type,
% 2.65/1.52 setunion: $i > $i ).
% 2.65/1.52
% 2.65/1.52 thf(setunionAx_type,type,
% 2.65/1.52 setunionAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setunionAx,definition,
% 2.65/1.52 ( setunionAx
% 2.65/1.52 = ( ! [A: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 2.65/1.52 <=> ? [B: $i] :
% 2.65/1.52 ( ( in @ Xx @ B )
% 2.65/1.52 & ( in @ B @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(omega_type,type,
% 2.65/1.52 omega: $i ).
% 2.65/1.52
% 2.65/1.52 thf(omega0Ax_type,type,
% 2.65/1.52 omega0Ax: $o ).
% 2.65/1.52
% 2.65/1.52 thf(omega0Ax,definition,
% 2.65/1.52 ( omega0Ax
% 2.65/1.52 = ( in @ emptyset @ omega ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(omegaSAx_type,type,
% 2.65/1.52 omegaSAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(omegaSAx,definition,
% 2.65/1.52 ( omegaSAx
% 2.65/1.52 = ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ omega )
% 2.65/1.52 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(omegaIndAx_type,type,
% 2.65/1.52 omegaIndAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(omegaIndAx,definition,
% 2.65/1.52 ( omegaIndAx
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ( ( ( in @ emptyset @ A )
% 2.65/1.52 & ! [Xx: $i] :
% 2.65/1.52 ( ( ( in @ Xx @ omega )
% 2.65/1.52 & ( in @ Xx @ A ) )
% 2.65/1.52 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.65/1.52 => ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ omega )
% 2.65/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(replAx_type,type,
% 2.65/1.52 replAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(replAx,definition,
% 2.65/1.52 ( replAx
% 2.65/1.52 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( exu
% 2.65/1.52 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.65/1.52 => ? [B: $i] :
% 2.65/1.52 ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ B )
% 2.65/1.52 <=> ? [Xy: $i] :
% 2.65/1.52 ( ( in @ Xy @ A )
% 2.65/1.52 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(foundationAx_type,type,
% 2.65/1.52 foundationAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(foundationAx,definition,
% 2.65/1.52 ( foundationAx
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.65/1.52 => ? [B: $i] :
% 2.65/1.52 ( ( in @ B @ A )
% 2.65/1.52 & ~ ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ B )
% 2.65/1.52 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(wellorderingAx_type,type,
% 2.65/1.52 wellorderingAx: $o ).
% 2.65/1.52
% 2.65/1.52 thf(wellorderingAx,definition,
% 2.65/1.52 ( wellorderingAx
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ? [B: $i] :
% 2.65/1.52 ( ! [C: $i] :
% 2.65/1.52 ( ( in @ C @ B )
% 2.65/1.52 => ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ C )
% 2.65/1.52 => ( in @ Xx @ A ) ) )
% 2.65/1.52 & ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( ( in @ Xx @ A )
% 2.65/1.52 & ( in @ Xy @ A ) )
% 2.65/1.52 => ( ! [C: $i] :
% 2.65/1.52 ( ( in @ C @ B )
% 2.65/1.52 => ( ( in @ Xx @ C )
% 2.65/1.52 <=> ( in @ Xy @ C ) ) )
% 2.65/1.52 => ( Xx = Xy ) ) )
% 2.65/1.52 & ! [C: $i,D: $i] :
% 2.65/1.52 ( ( ( in @ C @ B )
% 2.65/1.52 & ( in @ D @ B ) )
% 2.65/1.52 => ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ C )
% 2.65/1.52 => ( in @ Xx @ D ) )
% 2.65/1.52 | ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ D )
% 2.65/1.52 => ( in @ Xx @ C ) ) ) )
% 2.65/1.52 & ! [C: $i] :
% 2.65/1.52 ( ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ C )
% 2.65/1.52 => ( in @ Xx @ A ) )
% 2.65/1.52 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.65/1.52 => ? [D: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ D @ B )
% 2.65/1.52 & ( in @ Xx @ C )
% 2.65/1.52 & ~ ? [Xy: $i] :
% 2.65/1.52 ( ( in @ Xy @ D )
% 2.65/1.52 & ( in @ Xy @ C ) )
% 2.65/1.52 & ! [E: $i] :
% 2.65/1.52 ( ( in @ E @ B )
% 2.65/1.52 => ( ! [Xy: $i] :
% 2.65/1.52 ( ( in @ Xy @ E )
% 2.65/1.52 => ( in @ Xy @ D ) )
% 2.65/1.52 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(descr_type,type,
% 2.65/1.52 descr: ( $i > $o ) > $i ).
% 2.65/1.52
% 2.65/1.52 thf(descrp_type,type,
% 2.65/1.52 descrp: $o ).
% 2.65/1.52
% 2.65/1.52 thf(descrp,definition,
% 2.65/1.52 ( descrp
% 2.65/1.52 = ( ! [Xphi: $i > $o] :
% 2.65/1.52 ( ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 => ( Xphi
% 2.65/1.52 @ ( descr
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(dsetconstr_type,type,
% 2.65/1.52 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.65/1.52
% 2.65/1.52 thf(dsetconstrI_type,type,
% 2.65/1.52 dsetconstrI: $o ).
% 2.65/1.52
% 2.65/1.52 thf(dsetconstrI,definition,
% 2.65/1.52 ( dsetconstrI
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( ( Xphi @ Xx )
% 2.65/1.52 => ( in @ Xx
% 2.65/1.52 @ ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(dsetconstrEL_type,type,
% 2.65/1.52 dsetconstrEL: $o ).
% 2.65/1.52
% 2.65/1.52 thf(dsetconstrEL,definition,
% 2.65/1.52 ( dsetconstrEL
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx
% 2.65/1.52 @ ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.65/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(dsetconstrER_type,type,
% 2.65/1.52 dsetconstrER: $o ).
% 2.65/1.52
% 2.65/1.52 thf(dsetconstrER,definition,
% 2.65/1.52 ( dsetconstrER
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx
% 2.65/1.52 @ ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.65/1.52 => ( Xphi @ Xx ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(exuE1_type,type,
% 2.65/1.52 exuE1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(exuE1,definition,
% 2.65/1.52 ( exuE1
% 2.65/1.52 = ( ! [Xphi: $i > $o] :
% 2.65/1.52 ( ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 => ? [Xx: $i] :
% 2.65/1.52 ( ( Xphi @ Xx )
% 2.65/1.52 & ! [Xy: $i] :
% 2.65/1.52 ( ( Xphi @ Xy )
% 2.65/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(prop2set_type,type,
% 2.65/1.52 prop2set: $o > $i ).
% 2.65/1.52
% 2.65/1.52 thf(prop2setE_type,type,
% 2.65/1.52 prop2setE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(prop2setE,definition,
% 2.65/1.52 ( prop2setE
% 2.65/1.52 = ( ! [Xphi: $o,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.65/1.52 => Xphi ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptysetE_type,type,
% 2.65/1.52 emptysetE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptysetE,definition,
% 2.65/1.52 ( emptysetE
% 2.65/1.52 = ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ emptyset )
% 2.65/1.52 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptysetimpfalse_type,type,
% 2.65/1.52 emptysetimpfalse: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptysetimpfalse,definition,
% 2.65/1.52 ( emptysetimpfalse
% 2.65/1.52 = ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ emptyset )
% 2.65/1.52 => $false ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(notinemptyset_type,type,
% 2.65/1.52 notinemptyset: $o ).
% 2.65/1.52
% 2.65/1.52 thf(notinemptyset,definition,
% 2.65/1.52 ( notinemptyset
% 2.65/1.52 = ( ! [Xx: $i] :
% 2.65/1.52 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(exuE3e_type,type,
% 2.65/1.52 exuE3e: $o ).
% 2.65/1.52
% 2.65/1.52 thf(exuE3e,definition,
% 2.65/1.52 ( exuE3e
% 2.65/1.52 = ( ! [Xphi: $i > $o] :
% 2.65/1.52 ( ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setext_type,type,
% 2.65/1.52 setext: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setext,definition,
% 2.65/1.52 ( setext
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( in @ Xx @ B ) )
% 2.65/1.52 => ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ B )
% 2.65/1.52 => ( in @ Xx @ A ) )
% 2.65/1.52 => ( A = B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptyI_type,type,
% 2.65/1.52 emptyI: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptyI,definition,
% 2.65/1.52 ( emptyI
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ~ ( in @ Xx @ A )
% 2.65/1.52 => ( A = emptyset ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(noeltsimpempty_type,type,
% 2.65/1.52 noeltsimpempty: $o ).
% 2.65/1.52
% 2.65/1.52 thf(noeltsimpempty,definition,
% 2.65/1.52 ( noeltsimpempty
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ~ ( in @ Xx @ A )
% 2.65/1.52 => ( A = emptyset ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setbeta_type,type,
% 2.65/1.52 setbeta: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setbeta,definition,
% 2.65/1.52 ( setbeta
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( ( in @ Xx
% 2.65/1.52 @ ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.65/1.52 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(nonempty_type,type,
% 2.65/1.52 nonempty: $i > $o ).
% 2.65/1.52
% 2.65/1.52 thf(nonempty,definition,
% 2.65/1.52 ( nonempty
% 2.65/1.52 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(nonemptyE1_type,type,
% 2.65/1.52 nonemptyE1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(nonemptyE1,definition,
% 2.65/1.52 ( nonemptyE1
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ( ( nonempty @ A )
% 2.65/1.52 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(nonemptyI_type,type,
% 2.65/1.52 nonemptyI: $o ).
% 2.65/1.52
% 2.65/1.52 thf(nonemptyI,definition,
% 2.65/1.52 ( nonemptyI
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( ( Xphi @ Xx )
% 2.65/1.52 => ( nonempty
% 2.65/1.52 @ ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(nonemptyI1_type,type,
% 2.65/1.52 nonemptyI1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(nonemptyI1,definition,
% 2.65/1.52 ( nonemptyI1
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.65/1.52 => ( nonempty @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinIL_type,type,
% 2.65/1.52 setadjoinIL: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinIL,definition,
% 2.65/1.52 ( setadjoinIL
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptyinunitempty_type,type,
% 2.65/1.52 emptyinunitempty: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptyinunitempty,definition,
% 2.65/1.52 ( emptyinunitempty
% 2.65/1.52 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinIR_type,type,
% 2.65/1.52 setadjoinIR: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinIR,definition,
% 2.65/1.52 ( setadjoinIR
% 2.65/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.65/1.52 ( ( in @ Xy @ A )
% 2.65/1.52 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinE_type,type,
% 2.65/1.52 setadjoinE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinE,definition,
% 2.65/1.52 ( setadjoinE
% 2.65/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.65/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.65/1.52 => ! [Xphi: $o] :
% 2.65/1.52 ( ( ( Xy = Xx )
% 2.65/1.52 => Xphi )
% 2.65/1.52 => ( ( ( in @ Xy @ A )
% 2.65/1.52 => Xphi )
% 2.65/1.52 => Xphi ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinOr_type,type,
% 2.65/1.52 setadjoinOr: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinOr,definition,
% 2.65/1.52 ( setadjoinOr
% 2.65/1.52 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.65/1.52 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.65/1.52 => ( ( Xy = Xx )
% 2.65/1.52 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setoftrueEq_type,type,
% 2.65/1.52 setoftrueEq: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setoftrueEq,definition,
% 2.65/1.52 ( setoftrueEq
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ( ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xx: $i] : $true )
% 2.65/1.52 = A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(powersetI_type,type,
% 2.65/1.52 powersetI: $o ).
% 2.65/1.52
% 2.65/1.52 thf(powersetI,definition,
% 2.65/1.52 ( powersetI
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ B )
% 2.65/1.52 => ( in @ Xx @ A ) )
% 2.65/1.52 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptyinPowerset_type,type,
% 2.65/1.52 emptyinPowerset: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptyinPowerset,definition,
% 2.65/1.52 ( emptyinPowerset
% 2.65/1.52 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptyInPowerset_type,type,
% 2.65/1.52 emptyInPowerset: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptyInPowerset,definition,
% 2.65/1.52 ( emptyInPowerset
% 2.65/1.52 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(powersetE_type,type,
% 2.65/1.52 powersetE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(powersetE,definition,
% 2.65/1.52 ( powersetE
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ B @ ( powerset @ A ) )
% 2.65/1.52 => ( ( in @ Xx @ B )
% 2.65/1.52 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setunionI_type,type,
% 2.65/1.52 setunionI: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setunionI,definition,
% 2.65/1.52 ( setunionI
% 2.65/1.52 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.65/1.52 ( ( in @ Xx @ B )
% 2.65/1.52 => ( ( in @ B @ A )
% 2.65/1.52 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setunionE_type,type,
% 2.65/1.52 setunionE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setunionE,definition,
% 2.65/1.52 ( setunionE
% 2.65/1.52 = ( ! [A: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ ( setunion @ A ) )
% 2.65/1.52 => ! [Xphi: $o] :
% 2.65/1.52 ( ! [B: $i] :
% 2.65/1.52 ( ( in @ Xx @ B )
% 2.65/1.52 => ( ( in @ B @ A )
% 2.65/1.52 => Xphi ) )
% 2.65/1.52 => Xphi ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(subPowSU_type,type,
% 2.65/1.52 subPowSU: $o ).
% 2.65/1.52
% 2.65/1.52 thf(subPowSU,definition,
% 2.65/1.52 ( subPowSU
% 2.65/1.52 = ( ! [A: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(exuE2_type,type,
% 2.65/1.52 exuE2: $o ).
% 2.65/1.52
% 2.65/1.52 thf(exuE2,definition,
% 2.65/1.52 ( exuE2
% 2.65/1.52 = ( ! [Xphi: $i > $o] :
% 2.65/1.52 ( ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 => ? [Xx: $i] :
% 2.65/1.52 ! [Xy: $i] :
% 2.65/1.52 ( ( Xphi @ Xy )
% 2.65/1.52 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(nonemptyImpWitness_type,type,
% 2.65/1.52 nonemptyImpWitness: $o ).
% 2.65/1.52
% 2.65/1.52 thf(nonemptyImpWitness,definition,
% 2.65/1.52 ( nonemptyImpWitness
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ( ( nonempty @ A )
% 2.65/1.52 => ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 & $true ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(uniqinunit_type,type,
% 2.65/1.52 uniqinunit: $o ).
% 2.65/1.52
% 2.65/1.52 thf(uniqinunit,definition,
% 2.65/1.52 ( uniqinunit
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.52 => ( Xx = Xy ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(notinsingleton_type,type,
% 2.65/1.52 notinsingleton: $o ).
% 2.65/1.52
% 2.65/1.52 thf(notinsingleton,definition,
% 2.65/1.52 ( notinsingleton
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xx != Xy )
% 2.65/1.52 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(eqinunit_type,type,
% 2.65/1.52 eqinunit: $o ).
% 2.65/1.52
% 2.65/1.52 thf(eqinunit,definition,
% 2.65/1.52 ( eqinunit
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xx = Xy )
% 2.65/1.52 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(singletonsswitch_type,type,
% 2.65/1.52 singletonsswitch: $o ).
% 2.65/1.52
% 2.65/1.52 thf(singletonsswitch,definition,
% 2.65/1.52 ( singletonsswitch
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.52 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(upairsetE_type,type,
% 2.65/1.52 upairsetE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(upairsetE,definition,
% 2.65/1.52 ( upairsetE
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.65/1.52 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.65/1.52 => ( ( Xz = Xx )
% 2.65/1.52 | ( Xz = Xy ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(upairsetIL_type,type,
% 2.65/1.52 upairsetIL: $o ).
% 2.65/1.52
% 2.65/1.52 thf(upairsetIL,definition,
% 2.65/1.52 ( upairsetIL
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(upairsetIR_type,type,
% 2.65/1.52 upairsetIR: $o ).
% 2.65/1.52
% 2.65/1.52 thf(upairsetIR,definition,
% 2.65/1.52 ( upairsetIR
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptyE1_type,type,
% 2.65/1.52 emptyE1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptyE1,definition,
% 2.65/1.52 ( emptyE1
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.52 ( ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 & ( Xphi @ Xx ) )
% 2.65/1.52 => ( ( ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 = emptyset )
% 2.65/1.52 => $false ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(vacuousDall_type,type,
% 2.65/1.52 vacuousDall: $o ).
% 2.65/1.52
% 2.65/1.52 thf(vacuousDall,definition,
% 2.65/1.52 ( vacuousDall
% 2.65/1.52 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ emptyset )
% 2.65/1.52 => ( Xphi @ Xx ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(quantDeMorgan1_type,type,
% 2.65/1.52 quantDeMorgan1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(quantDeMorgan1,definition,
% 2.65/1.52 ( quantDeMorgan1
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.52 ( ~ ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( Xphi @ Xx ) )
% 2.65/1.52 => ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(quantDeMorgan2_type,type,
% 2.65/1.52 quantDeMorgan2: $o ).
% 2.65/1.52
% 2.65/1.52 thf(quantDeMorgan2,definition,
% 2.65/1.52 ( quantDeMorgan2
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ~ ( Xphi @ Xx ) )
% 2.65/1.52 => ~ ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(quantDeMorgan3_type,type,
% 2.65/1.52 quantDeMorgan3: $o ).
% 2.65/1.52
% 2.65/1.52 thf(quantDeMorgan3,definition,
% 2.65/1.52 ( quantDeMorgan3
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.52 ( ~ ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 & ( Xphi @ Xx ) )
% 2.65/1.52 => ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(quantDeMorgan4_type,type,
% 2.65/1.52 quantDeMorgan4: $o ).
% 2.65/1.52
% 2.65/1.52 thf(quantDeMorgan4,definition,
% 2.65/1.52 ( quantDeMorgan4
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.52 ( ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 & ~ ( Xphi @ Xx ) )
% 2.65/1.52 => ~ ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(prop2setI_type,type,
% 2.65/1.52 prop2setI: $o ).
% 2.65/1.52
% 2.65/1.52 thf(prop2setI,definition,
% 2.65/1.52 ( prop2setI
% 2.65/1.52 = ( ! [Xphi: $o] :
% 2.65/1.52 ( Xphi
% 2.65/1.52 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(set2prop_type,type,
% 2.65/1.52 set2prop: $i > $o ).
% 2.65/1.52
% 2.65/1.52 thf(prop2set2propI_type,type,
% 2.65/1.52 prop2set2propI: $o ).
% 2.65/1.52
% 2.65/1.52 thf(prop2set2propI,definition,
% 2.65/1.52 ( prop2set2propI
% 2.65/1.52 = ( ! [Xphi: $o] :
% 2.65/1.52 ( Xphi
% 2.65/1.52 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(notdexE_type,type,
% 2.65/1.52 notdexE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(notdexE,definition,
% 2.65/1.52 ( notdexE
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.52 ( ~ ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 & ( Xphi @ Xx ) )
% 2.65/1.52 => ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(notdallE_type,type,
% 2.65/1.52 notdallE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(notdallE,definition,
% 2.65/1.52 ( notdallE
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.52 ( ~ ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( Xphi @ Xx ) )
% 2.65/1.52 => ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(exuI1_type,type,
% 2.65/1.52 exuI1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(exuI1,definition,
% 2.65/1.52 ( exuI1
% 2.65/1.52 = ( ! [Xphi: $i > $o] :
% 2.65/1.52 ( ? [Xx: $i] :
% 2.65/1.52 ( ( Xphi @ Xx )
% 2.65/1.52 & ! [Xy: $i] :
% 2.65/1.52 ( ( Xphi @ Xy )
% 2.65/1.52 => ( Xx = Xy ) ) )
% 2.65/1.52 => ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(exuI3_type,type,
% 2.65/1.52 exuI3: $o ).
% 2.65/1.52
% 2.65/1.52 thf(exuI3,definition,
% 2.65/1.52 ( exuI3
% 2.65/1.52 = ( ! [Xphi: $i > $o] :
% 2.65/1.52 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.65/1.52 => ( ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xphi @ Xx )
% 2.65/1.52 => ( ( Xphi @ Xy )
% 2.65/1.52 => ( Xx = Xy ) ) )
% 2.65/1.52 => ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(exuI2_type,type,
% 2.65/1.52 exuI2: $o ).
% 2.65/1.52
% 2.65/1.52 thf(exuI2,definition,
% 2.65/1.52 ( exuI2
% 2.65/1.52 = ( ! [Xphi: $i > $o] :
% 2.65/1.52 ( ? [Xx: $i] :
% 2.65/1.52 ! [Xy: $i] :
% 2.65/1.52 ( ( Xphi @ Xy )
% 2.65/1.52 <=> ( Xy = Xx ) )
% 2.65/1.52 => ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(inCongP_type,type,
% 2.65/1.52 inCongP: $o ).
% 2.65/1.52
% 2.65/1.52 thf(inCongP,definition,
% 2.65/1.52 ( inCongP
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( A = B )
% 2.65/1.52 => ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xx = Xy )
% 2.65/1.52 => ( ( in @ Xx @ A )
% 2.65/1.52 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(in__Cong_type,type,
% 2.65/1.52 in__Cong: $o ).
% 2.65/1.52
% 2.65/1.52 thf(in__Cong,definition,
% 2.65/1.52 ( in__Cong
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( A = B )
% 2.65/1.52 => ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xx = Xy )
% 2.65/1.52 => ( ( in @ Xx @ A )
% 2.65/1.52 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(exuE3u_type,type,
% 2.65/1.52 exuE3u: $o ).
% 2.65/1.52
% 2.65/1.52 thf(exuE3u,definition,
% 2.65/1.52 ( exuE3u
% 2.65/1.52 = ( ! [Xphi: $i > $o] :
% 2.65/1.52 ( ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 => ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xphi @ Xx )
% 2.65/1.52 => ( ( Xphi @ Xy )
% 2.65/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(exu__Cong_type,type,
% 2.65/1.52 exu__Cong: $o ).
% 2.65/1.52
% 2.65/1.52 thf(exu__Cong,definition,
% 2.65/1.52 ( exu__Cong
% 2.65/1.52 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.65/1.52 ( ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xx = Xy )
% 2.65/1.52 => ( ( Xphi @ Xx )
% 2.65/1.52 <=> ( Xpsi @ Xy ) ) )
% 2.65/1.52 => ( ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 <=> ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptyset__Cong_type,type,
% 2.65/1.52 emptyset__Cong: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptyset__Cong,definition,
% 2.65/1.52 ( emptyset__Cong
% 2.65/1.52 = ( emptyset = emptyset ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoin__Cong_type,type,
% 2.65/1.52 setadjoin__Cong: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoin__Cong,definition,
% 2.65/1.52 ( setadjoin__Cong
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xx = Xy )
% 2.65/1.52 => ! [Xz: $i,Xu: $i] :
% 2.65/1.52 ( ( Xz = Xu )
% 2.65/1.52 => ( ( setadjoin @ Xx @ Xz )
% 2.65/1.52 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(powerset__Cong_type,type,
% 2.65/1.52 powerset__Cong: $o ).
% 2.65/1.52
% 2.65/1.52 thf(powerset__Cong,definition,
% 2.65/1.52 ( powerset__Cong
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( A = B )
% 2.65/1.52 => ( ( powerset @ A )
% 2.65/1.52 = ( powerset @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setunion__Cong_type,type,
% 2.65/1.52 setunion__Cong: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setunion__Cong,definition,
% 2.65/1.52 ( setunion__Cong
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( A = B )
% 2.65/1.52 => ( ( setunion @ A )
% 2.65/1.52 = ( setunion @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(omega__Cong_type,type,
% 2.65/1.52 omega__Cong: $o ).
% 2.65/1.52
% 2.65/1.52 thf(omega__Cong,definition,
% 2.65/1.52 ( omega__Cong
% 2.65/1.52 = ( omega = omega ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(exuEu_type,type,
% 2.65/1.52 exuEu: $o ).
% 2.65/1.52
% 2.65/1.52 thf(exuEu,definition,
% 2.65/1.52 ( exuEu
% 2.65/1.52 = ( ! [Xphi: $i > $o] :
% 2.65/1.52 ( ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 => ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xphi @ Xx )
% 2.65/1.52 => ( ( Xphi @ Xy )
% 2.65/1.52 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(descr__Cong_type,type,
% 2.65/1.52 descr__Cong: $o ).
% 2.65/1.52
% 2.65/1.52 thf(descr__Cong,definition,
% 2.65/1.52 ( descr__Cong
% 2.65/1.52 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.65/1.52 ( ! [Xx: $i,Xy: $i] :
% 2.65/1.52 ( ( Xx = Xy )
% 2.65/1.52 => ( ( Xphi @ Xx )
% 2.65/1.52 <=> ( Xpsi @ Xy ) ) )
% 2.65/1.52 => ( ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 => ( ( exu
% 2.65/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.65/1.52 => ( ( descr
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 = ( descr
% 2.65/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(dsetconstr__Cong_type,type,
% 2.65/1.52 dsetconstr__Cong: $o ).
% 2.65/1.52
% 2.65/1.52 thf(dsetconstr__Cong,definition,
% 2.65/1.52 ( dsetconstr__Cong
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( A = B )
% 2.65/1.52 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ! [Xy: $i] :
% 2.65/1.52 ( ( in @ Xy @ B )
% 2.65/1.52 => ( ( Xx = Xy )
% 2.65/1.52 => ( ( Xphi @ Xx )
% 2.65/1.52 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.65/1.52 => ( ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 = ( dsetconstr @ B
% 2.65/1.52 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(subset_type,type,
% 2.65/1.52 subset: $i > $i > $o ).
% 2.65/1.52
% 2.65/1.52 thf(disjoint_type,type,
% 2.65/1.52 disjoint: $i > $i > $o ).
% 2.65/1.52
% 2.65/1.52 thf(setsmeet_type,type,
% 2.65/1.52 setsmeet: $i > $i > $o ).
% 2.65/1.52
% 2.65/1.52 thf(subsetI1_type,type,
% 2.65/1.52 subsetI1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(subsetI1,definition,
% 2.65/1.52 ( subsetI1
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( in @ Xx @ B ) )
% 2.65/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(eqimpsubset2_type,type,
% 2.65/1.52 eqimpsubset2: $o ).
% 2.65/1.52
% 2.65/1.52 thf(eqimpsubset2,definition,
% 2.65/1.52 ( eqimpsubset2
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( A = B )
% 2.65/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(eqimpsubset1_type,type,
% 2.65/1.52 eqimpsubset1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(eqimpsubset1,definition,
% 2.65/1.52 ( eqimpsubset1
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( A = B )
% 2.65/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(subsetI2_type,type,
% 2.65/1.52 subsetI2: $o ).
% 2.65/1.52
% 2.65/1.52 thf(subsetI2,definition,
% 2.65/1.52 ( subsetI2
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ! [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( in @ Xx @ B ) )
% 2.65/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(emptysetsubset_type,type,
% 2.65/1.52 emptysetsubset: $o ).
% 2.65/1.52
% 2.65/1.52 thf(emptysetsubset,definition,
% 2.65/1.52 ( emptysetsubset
% 2.65/1.52 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(subsetE_type,type,
% 2.65/1.52 subsetE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(subsetE,definition,
% 2.65/1.52 ( subsetE
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( subset @ A @ B )
% 2.65/1.52 => ( ( in @ Xx @ A )
% 2.65/1.52 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(subsetE2_type,type,
% 2.65/1.52 subsetE2: $o ).
% 2.65/1.52
% 2.65/1.52 thf(subsetE2,definition,
% 2.65/1.52 ( subsetE2
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( subset @ A @ B )
% 2.65/1.52 => ( ~ ( in @ Xx @ B )
% 2.65/1.52 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(notsubsetI_type,type,
% 2.65/1.52 notsubsetI: $o ).
% 2.65/1.52
% 2.65/1.52 thf(notsubsetI,definition,
% 2.65/1.52 ( notsubsetI
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( ~ ( in @ Xx @ B )
% 2.65/1.52 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(notequalI1_type,type,
% 2.65/1.52 notequalI1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(notequalI1,definition,
% 2.65/1.52 ( notequalI1
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ~ ( subset @ A @ B )
% 2.65/1.52 => ( A != B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(notequalI2_type,type,
% 2.65/1.52 notequalI2: $o ).
% 2.65/1.52
% 2.65/1.52 thf(notequalI2,definition,
% 2.65/1.52 ( notequalI2
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( ~ ( in @ Xx @ B )
% 2.65/1.52 => ( A != B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(subsetRefl_type,type,
% 2.65/1.52 subsetRefl: $o ).
% 2.65/1.52
% 2.65/1.52 thf(subsetRefl,definition,
% 2.65/1.52 ( subsetRefl
% 2.65/1.52 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(subsetTrans_type,type,
% 2.65/1.52 subsetTrans: $o ).
% 2.65/1.52
% 2.65/1.52 thf(subsetTrans,definition,
% 2.65/1.52 ( subsetTrans
% 2.65/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.65/1.52 ( ( subset @ A @ B )
% 2.65/1.52 => ( ( subset @ B @ C )
% 2.65/1.52 => ( subset @ A @ C ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinSub_type,type,
% 2.65/1.52 setadjoinSub: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinSub,definition,
% 2.65/1.52 ( setadjoinSub
% 2.65/1.52 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinSub2_type,type,
% 2.65/1.52 setadjoinSub2: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setadjoinSub2,definition,
% 2.65/1.52 ( setadjoinSub2
% 2.65/1.52 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.65/1.52 ( ( subset @ A @ B )
% 2.65/1.52 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(subset2powerset_type,type,
% 2.65/1.52 subset2powerset: $o ).
% 2.65/1.52
% 2.65/1.52 thf(subset2powerset,definition,
% 2.65/1.52 ( subset2powerset
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( subset @ A @ B )
% 2.65/1.52 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(setextsub_type,type,
% 2.65/1.52 setextsub: $o ).
% 2.65/1.52
% 2.65/1.52 thf(setextsub,definition,
% 2.65/1.52 ( setextsub
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( subset @ A @ B )
% 2.65/1.52 => ( ( subset @ B @ A )
% 2.65/1.52 => ( A = B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(subsetemptysetimpeq_type,type,
% 2.65/1.52 subsetemptysetimpeq: $o ).
% 2.65/1.52
% 2.65/1.52 thf(subsetemptysetimpeq,definition,
% 2.65/1.52 ( subsetemptysetimpeq
% 2.65/1.52 = ( ! [A: $i] :
% 2.65/1.52 ( ( subset @ A @ emptyset )
% 2.65/1.52 => ( A = emptyset ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(powersetI1_type,type,
% 2.65/1.52 powersetI1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(powersetI1,definition,
% 2.65/1.52 ( powersetI1
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( subset @ B @ A )
% 2.65/1.52 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(powersetE1_type,type,
% 2.65/1.52 powersetE1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(powersetE1,definition,
% 2.65/1.52 ( powersetE1
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( in @ B @ ( powerset @ A ) )
% 2.65/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(inPowerset_type,type,
% 2.65/1.52 inPowerset: $o ).
% 2.65/1.52
% 2.65/1.52 thf(inPowerset,definition,
% 2.65/1.52 ( inPowerset
% 2.65/1.52 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(powersetsubset_type,type,
% 2.65/1.52 powersetsubset: $o ).
% 2.65/1.52
% 2.65/1.52 thf(powersetsubset,definition,
% 2.65/1.52 ( powersetsubset
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( subset @ A @ B )
% 2.65/1.52 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sepInPowerset_type,type,
% 2.65/1.52 sepInPowerset: $o ).
% 2.65/1.52
% 2.65/1.52 thf(sepInPowerset,definition,
% 2.65/1.52 ( sepInPowerset
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.52 ( in
% 2.65/1.52 @ ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 @ ( powerset @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(sepSubset_type,type,
% 2.65/1.52 sepSubset: $o ).
% 2.65/1.52
% 2.65/1.52 thf(sepSubset,definition,
% 2.65/1.52 ( sepSubset
% 2.65/1.52 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.52 ( subset
% 2.65/1.52 @ ( dsetconstr @ A
% 2.65/1.52 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.52 @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binunion_type,type,
% 2.65/1.52 binunion: $i > $i > $i ).
% 2.65/1.52
% 2.65/1.52 thf(binunionIL_type,type,
% 2.65/1.52 binunionIL: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binunionIL,definition,
% 2.65/1.52 ( binunionIL
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(upairset2IR_type,type,
% 2.65/1.52 upairset2IR: $o ).
% 2.65/1.52
% 2.65/1.52 thf(upairset2IR,definition,
% 2.65/1.52 ( upairset2IR
% 2.65/1.52 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binunionIR_type,type,
% 2.65/1.52 binunionIR: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binunionIR,definition,
% 2.65/1.52 ( binunionIR
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ B )
% 2.65/1.52 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binunionEcases_type,type,
% 2.65/1.52 binunionEcases: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binunionEcases,definition,
% 2.65/1.52 ( binunionEcases
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.65/1.52 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.65/1.52 => ( ( ( in @ Xx @ A )
% 2.65/1.52 => Xphi )
% 2.65/1.52 => ( ( ( in @ Xx @ B )
% 2.65/1.52 => Xphi )
% 2.65/1.52 => Xphi ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binunionE_type,type,
% 2.65/1.52 binunionE: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binunionE,definition,
% 2.65/1.52 ( binunionE
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.65/1.52 => ( ( in @ Xx @ A )
% 2.65/1.52 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binunionLsub_type,type,
% 2.65/1.52 binunionLsub: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binunionLsub,definition,
% 2.65/1.52 ( binunionLsub
% 2.65/1.52 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binunionRsub_type,type,
% 2.65/1.52 binunionRsub: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binunionRsub,definition,
% 2.65/1.52 ( binunionRsub
% 2.65/1.52 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersect_type,type,
% 2.65/1.52 binintersect: $i > $i > $i ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectI_type,type,
% 2.65/1.52 binintersectI: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectI,definition,
% 2.65/1.52 ( binintersectI
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 => ( ( in @ Xx @ B )
% 2.65/1.52 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset5_type,type,
% 2.65/1.52 binintersectSubset5: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset5,definition,
% 2.65/1.52 ( binintersectSubset5
% 2.65/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.65/1.52 ( ( subset @ C @ A )
% 2.65/1.52 => ( ( subset @ C @ B )
% 2.65/1.52 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectEL_type,type,
% 2.65/1.52 binintersectEL: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectEL,definition,
% 2.65/1.52 ( binintersectEL
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.65/1.52 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectLsub_type,type,
% 2.65/1.52 binintersectLsub: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectLsub,definition,
% 2.65/1.52 ( binintersectLsub
% 2.65/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset2_type,type,
% 2.65/1.52 binintersectSubset2: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset2,definition,
% 2.65/1.52 ( binintersectSubset2
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( subset @ A @ B )
% 2.65/1.52 => ( ( binintersect @ A @ B )
% 2.65/1.52 = A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset3_type,type,
% 2.65/1.52 binintersectSubset3: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset3,definition,
% 2.65/1.52 ( binintersectSubset3
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( ( binintersect @ A @ B )
% 2.65/1.52 = B )
% 2.65/1.52 => ( subset @ B @ A ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectER_type,type,
% 2.65/1.52 binintersectER: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectER,definition,
% 2.65/1.52 ( binintersectER
% 2.65/1.52 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.65/1.52 => ( in @ Xx @ B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(disjointsetsI1_type,type,
% 2.65/1.52 disjointsetsI1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(disjointsetsI1,definition,
% 2.65/1.52 ( disjointsetsI1
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ~ ? [Xx: $i] :
% 2.65/1.52 ( ( in @ Xx @ A )
% 2.65/1.52 & ( in @ Xx @ B ) )
% 2.65/1.52 => ( ( binintersect @ A @ B )
% 2.65/1.52 = emptyset ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectRsub_type,type,
% 2.65/1.52 binintersectRsub: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectRsub,definition,
% 2.65/1.52 ( binintersectRsub
% 2.65/1.52 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset4_type,type,
% 2.65/1.52 binintersectSubset4: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset4,definition,
% 2.65/1.52 ( binintersectSubset4
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( subset @ B @ A )
% 2.65/1.52 => ( ( binintersect @ A @ B )
% 2.65/1.52 = B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset1_type,type,
% 2.65/1.52 binintersectSubset1: $o ).
% 2.65/1.52
% 2.65/1.52 thf(binintersectSubset1,definition,
% 2.65/1.52 ( binintersectSubset1
% 2.65/1.52 = ( ! [A: $i,B: $i] :
% 2.65/1.52 ( ( ( binintersect @ A @ B )
% 2.65/1.52 = A )
% 2.65/1.52 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.52
% 2.65/1.52 thf(bs114d_type,type,
% 2.65/1.52 bs114d: $o ).
% 2.65/1.52
% 2.65/1.52 thf(bs114d,definition,
% 2.65/1.52 ( bs114d
% 2.65/1.52 = ( ! [A: $i,B: $i,C: $i] :
% 2.65/1.52 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.65/1.53 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(regular_type,type,
% 2.65/1.53 regular: $i > $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminus_type,type,
% 2.65/1.53 setminus: $i > $i > $i ).
% 2.65/1.53
% 2.65/1.53 thf(setminusI_type,type,
% 2.65/1.53 setminusI: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusI,definition,
% 2.65/1.53 ( setminusI
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ~ ( in @ Xx @ B )
% 2.65/1.53 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setminusEL_type,type,
% 2.65/1.53 setminusEL: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusEL,definition,
% 2.65/1.53 ( setminusEL
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.65/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setminusER_type,type,
% 2.65/1.53 setminusER: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusER,definition,
% 2.65/1.53 ( setminusER
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.65/1.53 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setminusSubset2_type,type,
% 2.65/1.53 setminusSubset2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusSubset2,definition,
% 2.65/1.53 ( setminusSubset2
% 2.65/1.53 = ( ! [A: $i,B: $i] :
% 2.65/1.53 ( ( subset @ A @ B )
% 2.65/1.53 => ( ( setminus @ A @ B )
% 2.65/1.53 = emptyset ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setminusERneg_type,type,
% 2.65/1.53 setminusERneg: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusERneg,definition,
% 2.65/1.53 ( setminusERneg
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.65/1.53 => ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setminusELneg_type,type,
% 2.65/1.53 setminusELneg: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusELneg,definition,
% 2.65/1.53 ( setminusELneg
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.65/1.53 => ( ~ ( in @ Xx @ B )
% 2.65/1.53 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setminusILneg_type,type,
% 2.65/1.53 setminusILneg: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusILneg,definition,
% 2.65/1.53 ( setminusILneg
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ~ ( in @ Xx @ A )
% 2.65/1.53 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setminusIRneg_type,type,
% 2.65/1.53 setminusIRneg: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusIRneg,definition,
% 2.65/1.53 ( setminusIRneg
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ B )
% 2.65/1.53 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setminusLsub_type,type,
% 2.65/1.53 setminusLsub: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusLsub,definition,
% 2.65/1.53 ( setminusLsub
% 2.65/1.53 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setminusSubset1_type,type,
% 2.65/1.53 setminusSubset1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setminusSubset1,definition,
% 2.65/1.53 ( setminusSubset1
% 2.65/1.53 = ( ! [A: $i,B: $i] :
% 2.65/1.53 ( ( ( setminus @ A @ B )
% 2.65/1.53 = emptyset )
% 2.65/1.53 => ( subset @ A @ B ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(symdiff_type,type,
% 2.65/1.53 symdiff: $i > $i > $i ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffE_type,type,
% 2.65/1.53 symdiffE: $o ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffE,definition,
% 2.65/1.53 ( symdiffE
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.65/1.53 => ! [Xphi: $o] :
% 2.65/1.53 ( ( ( in @ Xx @ A )
% 2.65/1.53 => ( ~ ( in @ Xx @ B )
% 2.65/1.53 => Xphi ) )
% 2.65/1.53 => ( ( ~ ( in @ Xx @ A )
% 2.65/1.53 => ( ( in @ Xx @ B )
% 2.65/1.53 => Xphi ) )
% 2.65/1.53 => Xphi ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffI1_type,type,
% 2.65/1.53 symdiffI1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffI1,definition,
% 2.65/1.53 ( symdiffI1
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ~ ( in @ Xx @ B )
% 2.65/1.53 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffI2_type,type,
% 2.65/1.53 symdiffI2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffI2,definition,
% 2.65/1.53 ( symdiffI2
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ~ ( in @ Xx @ A )
% 2.65/1.53 => ( ( in @ Xx @ B )
% 2.65/1.53 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffIneg1_type,type,
% 2.65/1.53 symdiffIneg1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffIneg1,definition,
% 2.65/1.53 ( symdiffIneg1
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ( in @ Xx @ B )
% 2.65/1.53 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffIneg2_type,type,
% 2.65/1.53 symdiffIneg2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(symdiffIneg2,definition,
% 2.65/1.53 ( symdiffIneg2
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ~ ( in @ Xx @ A )
% 2.65/1.53 => ( ~ ( in @ Xx @ B )
% 2.65/1.53 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(iskpair_type,type,
% 2.65/1.53 iskpair: $i > $o ).
% 2.65/1.53
% 2.65/1.53 thf(secondinupair_type,type,
% 2.65/1.53 secondinupair: $o ).
% 2.65/1.53
% 2.65/1.53 thf(secondinupair,definition,
% 2.65/1.53 ( secondinupair
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairIL_type,type,
% 2.65/1.53 setukpairIL: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairIL,definition,
% 2.65/1.53 ( setukpairIL
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairIR_type,type,
% 2.65/1.53 setukpairIR: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairIR,definition,
% 2.65/1.53 ( setukpairIR
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(kpairiskpair_type,type,
% 2.65/1.53 kpairiskpair: $o ).
% 2.65/1.53
% 2.65/1.53 thf(kpairiskpair,definition,
% 2.65/1.53 ( kpairiskpair
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(kpair_type,type,
% 2.65/1.53 kpair: $i > $i > $i ).
% 2.65/1.53
% 2.65/1.53 thf(kpairp_type,type,
% 2.65/1.53 kpairp: $o ).
% 2.65/1.53
% 2.65/1.53 thf(kpairp,definition,
% 2.65/1.53 ( kpairp
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprod_type,type,
% 2.65/1.53 cartprod: $i > $i > $i ).
% 2.65/1.53
% 2.65/1.53 thf(singletonsubset_type,type,
% 2.65/1.53 singletonsubset: $o ).
% 2.65/1.53
% 2.65/1.53 thf(singletonsubset,definition,
% 2.65/1.53 ( singletonsubset
% 2.65/1.53 = ( ! [A: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(singletoninpowerset_type,type,
% 2.65/1.53 singletoninpowerset: $o ).
% 2.65/1.53
% 2.65/1.53 thf(singletoninpowerset,definition,
% 2.65/1.53 ( singletoninpowerset
% 2.65/1.53 = ( ! [A: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(singletoninpowunion_type,type,
% 2.65/1.53 singletoninpowunion: $o ).
% 2.65/1.53
% 2.65/1.53 thf(singletoninpowunion,definition,
% 2.65/1.53 ( singletoninpowunion
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(upairset2E_type,type,
% 2.65/1.53 upairset2E: $o ).
% 2.65/1.53
% 2.65/1.53 thf(upairset2E,definition,
% 2.65/1.53 ( upairset2E
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.65/1.53 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.65/1.53 => ( ( Xz = Xx )
% 2.65/1.53 | ( Xz = Xy ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(upairsubunion_type,type,
% 2.65/1.53 upairsubunion: $o ).
% 2.65/1.53
% 2.65/1.53 thf(upairsubunion,definition,
% 2.65/1.53 ( upairsubunion
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(upairinpowunion_type,type,
% 2.65/1.53 upairinpowunion: $o ).
% 2.65/1.53
% 2.65/1.53 thf(upairinpowunion,definition,
% 2.65/1.53 ( upairinpowunion
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ubforcartprodlem1_type,type,
% 2.65/1.53 ubforcartprodlem1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ubforcartprodlem1,definition,
% 2.65/1.53 ( ubforcartprodlem1
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ubforcartprodlem2_type,type,
% 2.65/1.53 ubforcartprodlem2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ubforcartprodlem2,definition,
% 2.65/1.53 ( ubforcartprodlem2
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ubforcartprodlem3_type,type,
% 2.65/1.53 ubforcartprodlem3: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ubforcartprodlem3,definition,
% 2.65/1.53 ( ubforcartprodlem3
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodpairin_type,type,
% 2.65/1.53 cartprodpairin: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodpairin,definition,
% 2.65/1.53 ( cartprodpairin
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodmempair1_type,type,
% 2.65/1.53 cartprodmempair1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodmempair1,definition,
% 2.65/1.53 ( cartprodmempair1
% 2.65/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.53 => ? [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 & ? [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 & ( Xu
% 2.65/1.53 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodmempair_type,type,
% 2.65/1.53 cartprodmempair: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodmempair,definition,
% 2.65/1.53 ( cartprodmempair
% 2.65/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.53 => ( iskpair @ Xu ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setunionE2_type,type,
% 2.65/1.53 setunionE2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setunionE2,definition,
% 2.65/1.53 ( setunionE2
% 2.65/1.53 = ( ! [A: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.65/1.53 => ? [X: $i] :
% 2.65/1.53 ( ( in @ X @ A )
% 2.65/1.53 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setunionsingleton1_type,type,
% 2.65/1.53 setunionsingleton1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setunionsingleton1,definition,
% 2.65/1.53 ( setunionsingleton1
% 2.65/1.53 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setunionsingleton2_type,type,
% 2.65/1.53 setunionsingleton2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setunionsingleton2,definition,
% 2.65/1.53 ( setunionsingleton2
% 2.65/1.53 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setunionsingleton_type,type,
% 2.65/1.53 setunionsingleton: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setunionsingleton,definition,
% 2.65/1.53 ( setunionsingleton
% 2.65/1.53 = ( ! [Xx: $i] :
% 2.65/1.53 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.65/1.53 = Xx ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(singleton_type,type,
% 2.65/1.53 singleton: $i > $o ).
% 2.65/1.53
% 2.65/1.53 thf(singletonprop_type,type,
% 2.65/1.53 singletonprop: $o ).
% 2.65/1.53
% 2.65/1.53 thf(singletonprop,definition,
% 2.65/1.53 ( singletonprop
% 2.65/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( ( Xphi @ Xx )
% 2.65/1.53 => ( ( Xphi @ Xy )
% 2.65/1.53 => ( Xx = Xy ) ) ) ) )
% 2.65/1.53 => ( ? [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 & ( Xphi @ Xx ) )
% 2.65/1.53 => ( singleton
% 2.65/1.53 @ ( dsetconstr @ A
% 2.65/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ex1_type,type,
% 2.65/1.53 ex1: $i > ( $i > $o ) > $o ).
% 2.65/1.53
% 2.65/1.53 thf(ex1,definition,
% 2.65/1.53 ( ex1
% 2.65/1.53 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.65/1.53 ( singleton
% 2.65/1.53 @ ( dsetconstr @ A
% 2.65/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ex1E1_type,type,
% 2.65/1.53 ex1E1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ex1E1,definition,
% 2.65/1.53 ( ex1E1
% 2.65/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.53 ( ( ex1 @ A
% 2.65/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.53 => ? [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ex1I_type,type,
% 2.65/1.53 ex1I: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ex1I,definition,
% 2.65/1.53 ( ex1I
% 2.65/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ( Xphi @ Xx )
% 2.65/1.53 => ( ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( ( Xphi @ Xy )
% 2.65/1.53 => ( Xy = Xx ) ) )
% 2.65/1.53 => ( ex1 @ A
% 2.65/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ex1I2_type,type,
% 2.65/1.53 ex1I2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ex1I2,definition,
% 2.65/1.53 ( ex1I2
% 2.65/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( ( Xphi @ Xx )
% 2.65/1.53 => ( ( Xphi @ Xy )
% 2.65/1.53 => ( Xx = Xy ) ) ) ) )
% 2.65/1.53 => ( ? [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 & ( Xphi @ Xx ) )
% 2.65/1.53 => ( ex1 @ A
% 2.65/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(singletonsuniq_type,type,
% 2.65/1.53 singletonsuniq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(singletonsuniq,definition,
% 2.65/1.53 ( singletonsuniq
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( ( setadjoin @ Xx @ emptyset )
% 2.65/1.53 = ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(atmost1p_type,type,
% 2.65/1.53 atmost1p: $i > $o ).
% 2.65/1.53
% 2.65/1.53 thf(atleast2p_type,type,
% 2.65/1.53 atleast2p: $i > $o ).
% 2.65/1.53
% 2.65/1.53 thf(atmost2p_type,type,
% 2.65/1.53 atmost2p: $i > $o ).
% 2.65/1.53
% 2.65/1.53 thf(upairsetp_type,type,
% 2.65/1.53 upairsetp: $i > $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjL1_type,type,
% 2.65/1.53 setukpairinjL1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjL1,definition,
% 2.65/1.53 ( setukpairinjL1
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.65/1.53 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.65/1.53 => ( Xx = Xz ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(kfstsingleton_type,type,
% 2.65/1.53 kfstsingleton: $o ).
% 2.65/1.53
% 2.65/1.53 thf(kfstsingleton,definition,
% 2.65/1.53 ( kfstsingleton
% 2.65/1.53 = ( ! [Xu: $i] :
% 2.65/1.53 ( ( iskpair @ Xu )
% 2.65/1.53 => ( singleton
% 2.65/1.53 @ ( dsetconstr @ ( setunion @ Xu )
% 2.65/1.53 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(theprop_type,type,
% 2.65/1.53 theprop: $o ).
% 2.65/1.53
% 2.65/1.53 thf(theprop,definition,
% 2.65/1.53 ( theprop
% 2.65/1.53 = ( ! [X: $i] :
% 2.65/1.53 ( ( singleton @ X )
% 2.65/1.53 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(kfst_type,type,
% 2.65/1.53 kfst: $i > $i ).
% 2.65/1.53
% 2.65/1.53 thf(kfstpairEq_type,type,
% 2.65/1.53 kfstpairEq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(kfstpairEq,definition,
% 2.65/1.53 ( kfstpairEq
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.65/1.53 = Xx ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodfstin_type,type,
% 2.65/1.53 cartprodfstin: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodfstin,definition,
% 2.65/1.53 ( cartprodfstin
% 2.65/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.53 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjL2_type,type,
% 2.65/1.53 setukpairinjL2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjL2,definition,
% 2.65/1.53 ( setukpairinjL2
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.53 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.65/1.53 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.65/1.53 => ( Xx = Xz ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjL_type,type,
% 2.65/1.53 setukpairinjL: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjL,definition,
% 2.65/1.53 ( setukpairinjL
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.53 ( ( ( kpair @ Xx @ Xy )
% 2.65/1.53 = ( kpair @ Xz @ Xu ) )
% 2.65/1.53 => ( Xx = Xz ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR11_type,type,
% 2.65/1.53 setukpairinjR11: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR11,definition,
% 2.65/1.53 ( setukpairinjR11
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( Xx = Xy )
% 2.65/1.53 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.53 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR12_type,type,
% 2.65/1.53 setukpairinjR12: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR12,definition,
% 2.65/1.53 ( setukpairinjR12
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( Xx = Xy )
% 2.65/1.53 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.65/1.53 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR1_type,type,
% 2.65/1.53 setukpairinjR1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR1,definition,
% 2.65/1.53 ( setukpairinjR1
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.53 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.65/1.53 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.65/1.53 => ( ( Xz = Xu )
% 2.65/1.53 => ( Xy = Xu ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(upairequniteq_type,type,
% 2.65/1.53 upairequniteq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(upairequniteq,definition,
% 2.65/1.53 ( upairequniteq
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.65/1.53 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.65/1.53 = ( setadjoin @ Xz @ emptyset ) )
% 2.65/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR2_type,type,
% 2.65/1.53 setukpairinjR2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR2,definition,
% 2.65/1.53 ( setukpairinjR2
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.53 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.65/1.53 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.65/1.53 => ( Xy = Xu ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR_type,type,
% 2.65/1.53 setukpairinjR: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setukpairinjR,definition,
% 2.65/1.53 ( setukpairinjR
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.65/1.53 ( ( ( kpair @ Xx @ Xy )
% 2.65/1.53 = ( kpair @ Xz @ Xu ) )
% 2.65/1.53 => ( Xy = Xu ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ksndsingleton_type,type,
% 2.65/1.53 ksndsingleton: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ksndsingleton,definition,
% 2.65/1.53 ( ksndsingleton
% 2.65/1.53 = ( ! [Xu: $i] :
% 2.65/1.53 ( ( iskpair @ Xu )
% 2.65/1.53 => ( singleton
% 2.65/1.53 @ ( dsetconstr @ ( setunion @ Xu )
% 2.65/1.53 @ ^ [Xx: $i] :
% 2.65/1.53 ( Xu
% 2.65/1.53 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ksnd_type,type,
% 2.65/1.53 ksnd: $i > $i ).
% 2.65/1.53
% 2.65/1.53 thf(ksndpairEq_type,type,
% 2.65/1.53 ksndpairEq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ksndpairEq,definition,
% 2.65/1.53 ( ksndpairEq
% 2.65/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.65/1.53 = Xy ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(kpairsurjEq_type,type,
% 2.65/1.53 kpairsurjEq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(kpairsurjEq,definition,
% 2.65/1.53 ( kpairsurjEq
% 2.65/1.53 = ( ! [Xu: $i] :
% 2.65/1.53 ( ( iskpair @ Xu )
% 2.65/1.53 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.65/1.53 = Xu ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodsndin_type,type,
% 2.65/1.53 cartprodsndin: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodsndin,definition,
% 2.65/1.53 ( cartprodsndin
% 2.65/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.53 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodpairmemEL_type,type,
% 2.65/1.53 cartprodpairmemEL: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodpairmemEL,definition,
% 2.65/1.53 ( cartprodpairmemEL
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.65/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodpairmemER_type,type,
% 2.65/1.53 cartprodpairmemER: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodpairmemER,definition,
% 2.65/1.53 ( cartprodpairmemER
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.65/1.53 => ( in @ Xy @ B ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodmempaircEq_type,type,
% 2.65/1.53 cartprodmempaircEq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodmempaircEq,definition,
% 2.65/1.53 ( cartprodmempaircEq
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( kpair @ Xx @ Xy )
% 2.65/1.53 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodfstpairEq_type,type,
% 2.65/1.53 cartprodfstpairEq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodfstpairEq,definition,
% 2.65/1.53 ( cartprodfstpairEq
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.65/1.53 = Xx ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodsndpairEq_type,type,
% 2.65/1.53 cartprodsndpairEq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodsndpairEq,definition,
% 2.65/1.53 ( cartprodsndpairEq
% 2.65/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.65/1.53 = Xy ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodpairsurjEq_type,type,
% 2.65/1.53 cartprodpairsurjEq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(cartprodpairsurjEq,definition,
% 2.65/1.53 ( cartprodpairsurjEq
% 2.65/1.53 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.65/1.53 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.65/1.53 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.65/1.53 = Xu ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(breln_type,type,
% 2.65/1.53 breln: $i > $i > $i > $o ).
% 2.65/1.53
% 2.65/1.53 thf(breln,definition,
% 2.65/1.53 ( breln
% 2.65/1.53 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstr_type,type,
% 2.65/1.53 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrI_type,type,
% 2.65/1.53 dpsetconstrI: $o ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrI,definition,
% 2.65/1.53 ( dpsetconstrI
% 2.65/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( Xphi @ Xx @ Xy )
% 2.65/1.53 => ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.53 @ ( dpsetconstr @ A @ B
% 2.65/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrSub_type,type,
% 2.65/1.53 dpsetconstrSub: $o ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrSub,definition,
% 2.65/1.53 ( dpsetconstrSub
% 2.65/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.65/1.53 ( subset
% 2.65/1.53 @ ( dpsetconstr @ A @ B
% 2.65/1.53 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.65/1.53 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(setOfPairsIsBReln_type,type,
% 2.65/1.53 setOfPairsIsBReln: $o ).
% 2.65/1.53
% 2.65/1.53 thf(setOfPairsIsBReln,definition,
% 2.65/1.53 ( setOfPairsIsBReln
% 2.65/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.65/1.53 ( breln @ A @ B
% 2.65/1.53 @ ( dpsetconstr @ A @ B
% 2.65/1.53 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrERa_type,type,
% 2.65/1.53 dpsetconstrERa: $o ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrERa,definition,
% 2.65/1.53 ( dpsetconstrERa
% 2.65/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.53 @ ( dpsetconstr @ A @ B
% 2.65/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.65/1.53 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrEL1_type,type,
% 2.65/1.53 dpsetconstrEL1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrEL1,definition,
% 2.65/1.53 ( dpsetconstrEL1
% 2.65/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.53 @ ( dpsetconstr @ A @ B
% 2.65/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.65/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrEL2_type,type,
% 2.65/1.53 dpsetconstrEL2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrEL2,definition,
% 2.65/1.53 ( dpsetconstrEL2
% 2.65/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.53 @ ( dpsetconstr @ A @ B
% 2.65/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.65/1.53 => ( in @ Xy @ B ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrER_type,type,
% 2.65/1.53 dpsetconstrER: $o ).
% 2.65/1.53
% 2.65/1.53 thf(dpsetconstrER,definition,
% 2.65/1.53 ( dpsetconstrER
% 2.65/1.53 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( in @ ( kpair @ Xx @ Xy )
% 2.65/1.53 @ ( dpsetconstr @ A @ B
% 2.65/1.53 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.65/1.53 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(func_type,type,
% 2.65/1.53 func: $i > $i > $i > $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcSet_type,type,
% 2.65/1.53 funcSet: $i > $i > $i ).
% 2.65/1.53
% 2.65/1.53 thf(funcImageSingleton_type,type,
% 2.65/1.53 funcImageSingleton: $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcImageSingleton,definition,
% 2.65/1.53 ( funcImageSingleton
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( singleton
% 2.65/1.53 @ ( dsetconstr @ B
% 2.65/1.53 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(apProp_type,type,
% 2.65/1.53 apProp: $o ).
% 2.65/1.53
% 2.65/1.53 thf(apProp,definition,
% 2.65/1.53 ( apProp
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in
% 2.65/1.53 @ ( setunion
% 2.65/1.53 @ ( dsetconstr @ B
% 2.65/1.53 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.65/1.53 @ B ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ap_type,type,
% 2.65/1.53 ap: $i > $i > $i > $i > $i ).
% 2.65/1.53
% 2.65/1.53 thf(app_type,type,
% 2.65/1.53 app: $o ).
% 2.65/1.53
% 2.65/1.53 thf(app,definition,
% 2.65/1.53 ( app
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(infuncsetfunc_type,type,
% 2.65/1.53 infuncsetfunc: $o ).
% 2.65/1.53
% 2.65/1.53 thf(infuncsetfunc,definition,
% 2.65/1.53 ( infuncsetfunc
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.53 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ap2p_type,type,
% 2.65/1.53 ap2p: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ap2p,definition,
% 2.65/1.53 ( ap2p
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(funcinfuncset_type,type,
% 2.65/1.53 funcinfuncset: $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcinfuncset,definition,
% 2.65/1.53 ( funcinfuncset
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(lamProp_type,type,
% 2.65/1.53 lamProp: $o ).
% 2.65/1.53
% 2.65/1.53 thf(lamProp,definition,
% 2.65/1.53 ( lamProp
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.53 => ( func @ A @ B
% 2.65/1.53 @ ( dpsetconstr @ A @ B
% 2.65/1.53 @ ^ [Xx: $i,Xy: $i] :
% 2.65/1.53 ( ( Xf @ Xx )
% 2.65/1.53 = Xy ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(lam_type,type,
% 2.65/1.53 lam: $i > $i > ( $i > $i ) > $i ).
% 2.65/1.53
% 2.65/1.53 thf(lamp_type,type,
% 2.65/1.53 lamp: $o ).
% 2.65/1.53
% 2.65/1.53 thf(lamp,definition,
% 2.65/1.53 ( lamp
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.53 => ( func @ A @ B
% 2.65/1.53 @ ( lam @ A @ B
% 2.65/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(lam2p_type,type,
% 2.65/1.53 lam2p: $o ).
% 2.65/1.53
% 2.65/1.53 thf(lam2p,definition,
% 2.65/1.53 ( lam2p
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.53 => ( in
% 2.65/1.53 @ ( lam @ A @ B
% 2.65/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.65/1.53 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(brelnall1_type,type,
% 2.65/1.53 brelnall1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(brelnall1,definition,
% 2.65/1.53 ( brelnall1
% 2.65/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.65/1.53 ( ( breln @ A @ B @ R )
% 2.65/1.53 => ! [Xphi: $i > $o] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.65/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ R )
% 2.65/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(brelnall2_type,type,
% 2.65/1.53 brelnall2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(brelnall2,definition,
% 2.65/1.53 ( brelnall2
% 2.65/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.65/1.53 ( ( breln @ A @ B @ R )
% 2.65/1.53 => ! [Xphi: $i > $o] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.65/1.53 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ R )
% 2.65/1.53 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ex1E2_type,type,
% 2.65/1.53 ex1E2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ex1E2,definition,
% 2.65/1.53 ( ex1E2
% 2.65/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.65/1.53 ( ( ex1 @ A
% 2.65/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( ( Xphi @ Xx )
% 2.65/1.53 => ( ( Xphi @ Xy )
% 2.65/1.53 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(funcGraphProp1_type,type,
% 2.65/1.53 funcGraphProp1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcGraphProp1,definition,
% 2.65/1.53 ( funcGraphProp1
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(funcGraphProp3_type,type,
% 2.65/1.53 funcGraphProp3: $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcGraphProp3,definition,
% 2.65/1.53 ( funcGraphProp3
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(funcGraphProp2_type,type,
% 2.65/1.53 funcGraphProp2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcGraphProp2,definition,
% 2.65/1.53 ( funcGraphProp2
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.65/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.53 = Xy ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(funcextLem_type,type,
% 2.65/1.53 funcextLem: $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcextLem,definition,
% 2.65/1.53 ( funcextLem
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ! [Xg: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xg )
% 2.65/1.53 => ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.53 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.65/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(funcGraphProp4_type,type,
% 2.65/1.53 funcGraphProp4: $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcGraphProp4,definition,
% 2.65/1.53 ( funcGraphProp4
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.65/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.53 = Xy ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(subbreln_type,type,
% 2.65/1.53 subbreln: $o ).
% 2.65/1.53
% 2.65/1.53 thf(subbreln,definition,
% 2.65/1.53 ( subbreln
% 2.65/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.65/1.53 ( ( breln @ A @ B @ R )
% 2.65/1.53 => ! [S: $i] :
% 2.65/1.53 ( ( breln @ A @ B @ S )
% 2.65/1.53 => ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.65/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.65/1.53 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(eqbreln_type,type,
% 2.65/1.53 eqbreln: $o ).
% 2.65/1.53
% 2.65/1.53 thf(eqbreln,definition,
% 2.65/1.53 ( eqbreln
% 2.65/1.53 = ( ! [A: $i,B: $i,R: $i] :
% 2.65/1.53 ( ( breln @ A @ B @ R )
% 2.65/1.53 => ! [S: $i] :
% 2.65/1.53 ( ( breln @ A @ B @ S )
% 2.65/1.53 => ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.65/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.65/1.53 => ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ B )
% 2.65/1.53 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.65/1.53 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.65/1.53 => ( R = S ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(funcext_type,type,
% 2.65/1.53 funcext: $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcext,definition,
% 2.65/1.53 ( funcext
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ! [Xg: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xg )
% 2.65/1.53 => ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.53 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.65/1.53 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(funcext2_type,type,
% 2.65/1.53 funcext2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(funcext2,definition,
% 2.65/1.53 ( funcext2
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.53 => ! [Xg: $i] :
% 2.65/1.53 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.65/1.53 => ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.53 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.65/1.53 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ap2apEq1_type,type,
% 2.65/1.53 ap2apEq1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ap2apEq1,definition,
% 2.65/1.53 ( ap2apEq1
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.53 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ap2apEq2_type,type,
% 2.65/1.53 ap2apEq2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ap2apEq2,definition,
% 2.65/1.53 ( ap2apEq2
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.65/1.53 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(beta1_type,type,
% 2.65/1.53 beta1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(beta1,definition,
% 2.65/1.53 ( beta1
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ( ap @ A @ B
% 2.65/1.53 @ ( lam @ A @ B
% 2.65/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.65/1.53 @ Xx )
% 2.65/1.53 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(eta1_type,type,
% 2.65/1.53 eta1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(eta1,definition,
% 2.65/1.53 ( eta1
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( func @ A @ B @ Xf )
% 2.65/1.53 => ( ( lam @ A @ B
% 2.65/1.53 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.65/1.53 = Xf ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(lam2lamEq_type,type,
% 2.65/1.53 lam2lamEq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(lam2lamEq,definition,
% 2.65/1.53 ( lam2lamEq
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.53 => ( ( lam @ A @ B
% 2.65/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.65/1.53 = ( lam @ A @ B
% 2.65/1.53 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(beta2_type,type,
% 2.65/1.53 beta2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(beta2,definition,
% 2.65/1.53 ( beta2
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.65/1.53 ( ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ( ( ap @ A @ B
% 2.65/1.53 @ ( lam @ A @ B
% 2.65/1.53 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.65/1.53 @ Xx )
% 2.65/1.53 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(eta2_type,type,
% 2.65/1.53 eta2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(eta2,definition,
% 2.65/1.53 ( eta2
% 2.65/1.53 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.65/1.53 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.65/1.53 => ( ( lam @ A @ B
% 2.65/1.53 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.65/1.53 = Xf ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(iffalseProp1_type,type,
% 2.65/1.53 iffalseProp1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(iffalseProp1,definition,
% 2.65/1.53 ( iffalseProp1
% 2.65/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( ~ Xphi
% 2.65/1.53 => ( in @ Xy
% 2.65/1.53 @ ( dsetconstr @ A
% 2.65/1.53 @ ^ [Xz: $i] :
% 2.65/1.53 ( ( Xphi
% 2.65/1.53 & ( Xz = Xx ) )
% 2.65/1.53 | ( ~ Xphi
% 2.65/1.53 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(iffalseProp2_type,type,
% 2.65/1.53 iffalseProp2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(iffalseProp2,definition,
% 2.65/1.53 ( iffalseProp2
% 2.65/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( ~ Xphi
% 2.65/1.53 => ( ( dsetconstr @ A
% 2.65/1.53 @ ^ [Xz: $i] :
% 2.65/1.53 ( ( Xphi
% 2.65/1.53 & ( Xz = Xx ) )
% 2.65/1.53 | ( ~ Xphi
% 2.65/1.53 & ( Xz = Xy ) ) ) )
% 2.65/1.53 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(iftrueProp1_type,type,
% 2.65/1.53 iftrueProp1: $o ).
% 2.65/1.53
% 2.65/1.53 thf(iftrueProp1,definition,
% 2.65/1.53 ( iftrueProp1
% 2.65/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( Xphi
% 2.65/1.53 => ( in @ Xx
% 2.65/1.53 @ ( dsetconstr @ A
% 2.65/1.53 @ ^ [Xz: $i] :
% 2.65/1.53 ( ( Xphi
% 2.65/1.53 & ( Xz = Xx ) )
% 2.65/1.53 | ( ~ Xphi
% 2.65/1.53 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(iftrueProp2_type,type,
% 2.65/1.53 iftrueProp2: $o ).
% 2.65/1.53
% 2.65/1.53 thf(iftrueProp2,definition,
% 2.65/1.53 ( iftrueProp2
% 2.65/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( Xphi
% 2.65/1.53 => ( ( dsetconstr @ A
% 2.65/1.53 @ ^ [Xz: $i] :
% 2.65/1.53 ( ( Xphi
% 2.65/1.53 & ( Xz = Xx ) )
% 2.65/1.53 | ( ~ Xphi
% 2.65/1.53 & ( Xz = Xy ) ) ) )
% 2.65/1.53 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(ifSingleton_type,type,
% 2.65/1.53 ifSingleton: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ifSingleton,definition,
% 2.65/1.53 ( ifSingleton
% 2.65/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( singleton
% 2.65/1.53 @ ( dsetconstr @ A
% 2.65/1.53 @ ^ [Xz: $i] :
% 2.65/1.53 ( ( Xphi
% 2.65/1.53 & ( Xz = Xx ) )
% 2.65/1.53 | ( ~ Xphi
% 2.65/1.53 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(if_type,type,
% 2.65/1.53 if: $i > $o > $i > $i > $i ).
% 2.65/1.53
% 2.65/1.53 thf(ifp_type,type,
% 2.65/1.53 ifp: $o ).
% 2.65/1.53
% 2.65/1.53 thf(ifp,definition,
% 2.65/1.53 ( ifp
% 2.65/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(theeq_type,type,
% 2.65/1.53 theeq: $o ).
% 2.65/1.53
% 2.65/1.53 thf(theeq,definition,
% 2.65/1.53 ( theeq
% 2.65/1.53 = ( ! [X: $i] :
% 2.65/1.53 ( ( singleton @ X )
% 2.65/1.53 => ! [Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ X )
% 2.65/1.53 => ( ( setunion @ X )
% 2.65/1.53 = Xx ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(iftrue_type,type,
% 2.65/1.53 iftrue: $o ).
% 2.65/1.53
% 2.65/1.53 thf(iftrue,definition,
% 2.65/1.53 ( iftrue
% 2.65/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( Xphi
% 2.65/1.53 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.65/1.53 = Xx ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(iffalse_type,type,
% 2.65/1.53 iffalse: $o ).
% 2.65/1.53
% 2.65/1.53 thf(iffalse,definition,
% 2.65/1.53 ( iffalse
% 2.65/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( ~ Xphi
% 2.65/1.53 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.65/1.53 = Xy ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(iftrueorfalse_type,type,
% 2.65/1.53 iftrueorfalse: $o ).
% 2.65/1.53
% 2.65/1.53 thf(iftrueorfalse,definition,
% 2.65/1.53 ( iftrueorfalse
% 2.65/1.53 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.65/1.53 ( ( in @ Xx @ A )
% 2.65/1.53 => ! [Xy: $i] :
% 2.65/1.53 ( ( in @ Xy @ A )
% 2.65/1.53 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(binintersectT_lem_type,type,
% 2.65/1.53 binintersectT_lem: $o ).
% 2.65/1.53
% 2.65/1.53 thf(binintersectT_lem,definition,
% 2.65/1.53 ( binintersectT_lem
% 2.65/1.53 = ( ! [A: $i,X: $i] :
% 2.65/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.53 => ! [Y: $i] :
% 2.65/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.65/1.53 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(binunionT_lem_type,type,
% 2.65/1.53 binunionT_lem: $o ).
% 2.65/1.53
% 2.65/1.53 thf(binunionT_lem,definition,
% 2.65/1.53 ( binunionT_lem
% 2.65/1.53 = ( ! [A: $i,X: $i] :
% 2.65/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.53 => ! [Y: $i] :
% 2.65/1.53 ( ( in @ Y @ ( powerset @ A ) )
% 2.65/1.53 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.65/1.53
% 2.65/1.53 thf(powersetT_lem,conjecture,
% 2.65/1.53 ( setextAx
% 2.65/1.53 => ( emptysetAx
% 2.65/1.53 => ( setadjoinAx
% 2.65/1.53 => ( powersetAx
% 2.65/1.53 => ( setunionAx
% 2.65/1.53 => ( omega0Ax
% 2.65/1.53 => ( omegaSAx
% 2.65/1.53 => ( omegaIndAx
% 2.65/1.53 => ( replAx
% 2.65/1.53 => ( foundationAx
% 2.65/1.53 => ( wellorderingAx
% 2.65/1.53 => ( descrp
% 2.65/1.53 => ( dsetconstrI
% 2.65/1.53 => ( dsetconstrEL
% 2.65/1.53 => ( dsetconstrER
% 2.65/1.53 => ( exuE1
% 2.65/1.53 => ( prop2setE
% 2.65/1.53 => ( emptysetE
% 2.65/1.53 => ( emptysetimpfalse
% 2.65/1.53 => ( notinemptyset
% 2.65/1.53 => ( exuE3e
% 2.65/1.53 => ( setext
% 2.65/1.53 => ( emptyI
% 2.65/1.53 => ( noeltsimpempty
% 2.65/1.53 => ( setbeta
% 2.65/1.53 => ( nonemptyE1
% 2.65/1.53 => ( nonemptyI
% 2.65/1.53 => ( nonemptyI1
% 2.65/1.53 => ( setadjoinIL
% 2.65/1.53 => ( emptyinunitempty
% 2.65/1.53 => ( setadjoinIR
% 2.65/1.53 => ( setadjoinE
% 2.65/1.53 => ( setadjoinOr
% 2.65/1.53 => ( setoftrueEq
% 2.65/1.53 => ( powersetI
% 2.65/1.53 => ( emptyinPowerset
% 2.65/1.53 => ( emptyInPowerset
% 2.65/1.53 => ( powersetE
% 2.65/1.53 => ( setunionI
% 2.65/1.53 => ( setunionE
% 2.65/1.53 => ( subPowSU
% 2.65/1.53 => ( exuE2
% 2.65/1.53 => ( nonemptyImpWitness
% 2.65/1.53 => ( uniqinunit
% 2.65/1.53 => ( notinsingleton
% 2.65/1.53 => ( eqinunit
% 2.65/1.53 => ( singletonsswitch
% 2.65/1.53 => ( upairsetE
% 2.65/1.53 => ( upairsetIL
% 2.65/1.53 => ( upairsetIR
% 2.65/1.53 => ( emptyE1
% 2.65/1.53 => ( vacuousDall
% 2.65/1.53 => ( quantDeMorgan1
% 2.65/1.53 => ( quantDeMorgan2
% 2.65/1.53 => ( quantDeMorgan3
% 2.65/1.53 => ( quantDeMorgan4
% 2.65/1.53 => ( prop2setI
% 2.65/1.53 => ( prop2set2propI
% 2.65/1.53 => ( notdexE
% 2.65/1.53 => ( notdallE
% 2.65/1.53 => ( exuI1
% 2.65/1.53 => ( exuI3
% 2.65/1.53 => ( exuI2
% 2.65/1.53 => ( inCongP
% 2.65/1.53 => ( in__Cong
% 2.65/1.53 => ( exuE3u
% 2.65/1.53 => ( exu__Cong
% 2.65/1.53 => ( emptyset__Cong
% 2.65/1.53 => ( setadjoin__Cong
% 2.65/1.53 => ( powerset__Cong
% 2.65/1.53 => ( setunion__Cong
% 2.65/1.53 => ( omega__Cong
% 2.65/1.53 => ( exuEu
% 2.65/1.53 => ( descr__Cong
% 2.65/1.53 => ( dsetconstr__Cong
% 2.65/1.53 => ( subsetI1
% 2.65/1.53 => ( eqimpsubset2
% 2.65/1.53 => ( eqimpsubset1
% 2.65/1.53 => ( subsetI2
% 2.65/1.53 => ( emptysetsubset
% 2.65/1.53 => ( subsetE
% 2.65/1.53 => ( subsetE2
% 2.65/1.53 => ( notsubsetI
% 2.65/1.53 => ( notequalI1
% 2.65/1.53 => ( notequalI2
% 2.65/1.53 => ( subsetRefl
% 2.65/1.53 => ( subsetTrans
% 2.65/1.53 => ( setadjoinSub
% 2.65/1.53 => ( setadjoinSub2
% 2.65/1.53 => ( subset2powerset
% 2.65/1.53 => ( setextsub
% 2.65/1.53 => ( subsetemptysetimpeq
% 2.65/1.53 => ( powersetI1
% 2.65/1.53 => ( powersetE1
% 2.65/1.53 => ( inPowerset
% 2.65/1.53 => ( powersetsubset
% 2.65/1.53 => ( sepInPowerset
% 2.65/1.53 => ( sepSubset
% 2.65/1.53 => ( binunionIL
% 2.65/1.53 => ( upairset2IR
% 2.65/1.53 => ( binunionIR
% 2.65/1.53 => ( binunionEcases
% 2.65/1.53 => ( binunionE
% 2.65/1.53 => ( binunionLsub
% 2.65/1.53 => ( binunionRsub
% 2.65/1.53 => ( binintersectI
% 2.65/1.53 => ( binintersectSubset5
% 2.65/1.53 => ( binintersectEL
% 2.65/1.53 => ( binintersectLsub
% 2.65/1.53 => ( binintersectSubset2
% 2.65/1.53 => ( binintersectSubset3
% 2.65/1.53 => ( binintersectER
% 2.65/1.53 => ( disjointsetsI1
% 2.65/1.53 => ( binintersectRsub
% 2.65/1.53 => ( binintersectSubset4
% 2.65/1.53 => ( binintersectSubset1
% 2.65/1.53 => ( bs114d
% 2.65/1.53 => ( setminusI
% 2.65/1.53 => ( setminusEL
% 2.65/1.53 => ( setminusER
% 2.65/1.53 => ( setminusSubset2
% 2.65/1.53 => ( setminusERneg
% 2.65/1.53 => ( setminusELneg
% 2.65/1.53 => ( setminusILneg
% 2.65/1.53 => ( setminusIRneg
% 2.65/1.53 => ( setminusLsub
% 2.65/1.53 => ( setminusSubset1
% 2.65/1.53 => ( symdiffE
% 2.65/1.53 => ( symdiffI1
% 2.65/1.53 => ( symdiffI2
% 2.65/1.53 => ( symdiffIneg1
% 2.65/1.53 => ( symdiffIneg2
% 2.65/1.53 => ( secondinupair
% 2.65/1.53 => ( setukpairIL
% 2.65/1.53 => ( setukpairIR
% 2.65/1.53 => ( kpairiskpair
% 2.65/1.53 => ( kpairp
% 2.65/1.53 => ( singletonsubset
% 2.65/1.53 => ( singletoninpowerset
% 2.65/1.53 => ( singletoninpowunion
% 2.65/1.53 => ( upairset2E
% 2.65/1.53 => ( upairsubunion
% 2.65/1.53 => ( upairinpowunion
% 2.65/1.53 => ( ubforcartprodlem1
% 2.65/1.53 => ( ubforcartprodlem2
% 2.65/1.53 => ( ubforcartprodlem3
% 2.65/1.53 => ( cartprodpairin
% 2.65/1.53 => ( cartprodmempair1
% 2.65/1.53 => ( cartprodmempair
% 2.65/1.53 => ( setunionE2
% 2.65/1.53 => ( setunionsingleton1
% 2.65/1.53 => ( setunionsingleton2
% 2.65/1.53 => ( setunionsingleton
% 2.65/1.53 => ( singletonprop
% 2.65/1.53 => ( ex1E1
% 2.65/1.53 => ( ex1I
% 2.65/1.53 => ( ex1I2
% 2.65/1.53 => ( singletonsuniq
% 2.65/1.53 => ( setukpairinjL1
% 2.65/1.53 => ( kfstsingleton
% 2.65/1.53 => ( theprop
% 2.65/1.53 => ( kfstpairEq
% 2.65/1.53 => ( cartprodfstin
% 2.65/1.53 => ( setukpairinjL2
% 2.65/1.53 => ( setukpairinjL
% 2.65/1.53 => ( setukpairinjR11
% 2.65/1.53 => ( setukpairinjR12
% 2.65/1.53 => ( setukpairinjR1
% 2.65/1.53 => ( upairequniteq
% 2.65/1.53 => ( setukpairinjR2
% 2.65/1.53 => ( setukpairinjR
% 2.65/1.53 => ( ksndsingleton
% 2.65/1.53 => ( ksndpairEq
% 2.65/1.53 => ( kpairsurjEq
% 2.65/1.53 => ( cartprodsndin
% 2.65/1.53 => ( cartprodpairmemEL
% 2.65/1.53 => ( cartprodpairmemER
% 2.65/1.53 => ( cartprodmempaircEq
% 2.65/1.53 => ( cartprodfstpairEq
% 2.65/1.53 => ( cartprodsndpairEq
% 2.65/1.53 => ( cartprodpairsurjEq
% 2.65/1.53 => ( dpsetconstrI
% 2.65/1.53 => ( dpsetconstrSub
% 2.65/1.53 => ( setOfPairsIsBReln
% 2.65/1.53 => ( dpsetconstrERa
% 2.65/1.53 => ( dpsetconstrEL1
% 2.65/1.53 => ( dpsetconstrEL2
% 2.65/1.53 => ( dpsetconstrER
% 2.65/1.53 => ( funcImageSingleton
% 2.65/1.53 => ( apProp
% 2.65/1.53 => ( app
% 2.65/1.53 => ( infuncsetfunc
% 2.65/1.53 => ( ap2p
% 2.65/1.53 => ( funcinfuncset
% 2.65/1.53 => ( lamProp
% 2.65/1.53 => ( lamp
% 2.65/1.53 => ( lam2p
% 2.65/1.53 => ( brelnall1
% 2.65/1.53 => ( brelnall2
% 2.65/1.53 => ( ex1E2
% 2.65/1.53 => ( funcGraphProp1
% 2.65/1.53 => ( funcGraphProp3
% 2.65/1.53 => ( funcGraphProp2
% 2.65/1.53 => ( funcextLem
% 2.65/1.53 => ( funcGraphProp4
% 2.65/1.53 => ( subbreln
% 2.65/1.53 => ( eqbreln
% 2.65/1.53 => ( funcext
% 2.65/1.53 => ( funcext2
% 2.65/1.53 => ( ap2apEq1
% 2.65/1.53 => ( ap2apEq2
% 2.65/1.53 => ( beta1
% 2.65/1.53 => ( eta1
% 2.65/1.53 => ( lam2lamEq
% 2.65/1.53 => ( beta2
% 2.65/1.53 => ( eta2
% 2.65/1.53 => ( iffalseProp1
% 2.65/1.53 => ( iffalseProp2
% 2.65/1.53 => ( iftrueProp1
% 2.65/1.53 => ( iftrueProp2
% 2.65/1.53 => ( ifSingleton
% 2.65/1.53 => ( ifp
% 2.65/1.53 => ( theeq
% 2.65/1.53 => ( iftrue
% 2.65/1.53 => ( iffalse
% 2.65/1.53 => ( iftrueorfalse
% 2.65/1.53 => ( binintersectT_lem
% 2.65/1.53 => ( binunionT_lem
% 2.65/1.53 => ! [A: $i,X: $i] :
% 2.65/1.53 ( ( in @ X @ ( powerset @ A ) )
% 2.65/1.53 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.65/1.53 ------------------------
% 2.65/1.55 % (5998)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_5724 for (2999ds/27Mi)
% 2.65/1.55 % (5997)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_5724 for (2999ds/4Mi)
% 2.65/1.56 % (5997)Instruction limit reached!
% 2.65/1.56 % (5997)------------------------------
% 2.65/1.56 % (5997)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.56 % (5997)Termination reason: Unknown
% 2.65/1.56 % (5997)Termination phase: shuffling
% 2.65/1.56
% 2.65/1.56 % (5997)Memory used [KB]: 1407
% 2.65/1.56 % (5997)Time elapsed: 0.003 s
% 2.65/1.56 % (5997)Instructions burned: 6 (million)
% 2.65/1.56 % (5997)------------------------------
% 2.65/1.56 % (5997)------------------------------
% 2.65/1.56 % (6000)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_5724 for (2999ds/2Mi)
% 2.65/1.56 % (6000)Instruction limit reached!
% 2.65/1.56 % (6000)------------------------------
% 2.65/1.56 % (6000)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.56 % (6000)Termination reason: Unknown
% 2.65/1.56 % (6000)Termination phase: shuffling
% 2.65/1.56
% 2.65/1.56 % (6000)Memory used [KB]: 1407
% 2.65/1.56 % (6000)Time elapsed: 0.003 s
% 2.65/1.56 % (6000)Instructions burned: 2 (million)
% 2.65/1.56 % (6000)------------------------------
% 2.65/1.56 % (6000)------------------------------
% 2.65/1.56 % (5996)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/183Mi)
% 2.65/1.56 % (5999)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/2Mi)
% 2.65/1.56 % (6001)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_5724 for (2999ds/275Mi)
% 2.65/1.56 % (6002)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/18Mi)
% 2.65/1.56 % (5998)Instruction limit reached!
% 2.65/1.56 % (5998)------------------------------
% 2.65/1.56 % (5998)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.56 % (5998)Termination reason: Unknown
% 2.65/1.56 % (5998)Termination phase: shuffling
% 2.65/1.56
% 2.65/1.56 % (5998)Memory used [KB]: 1918
% 2.65/1.56 % (5998)Time elapsed: 0.010 s
% 2.65/1.56 % (5998)Instructions burned: 30 (million)
% 2.65/1.56 % (5998)------------------------------
% 2.65/1.56 % (5998)------------------------------
% 2.65/1.56 % (5999)Instruction limit reached!
% 2.65/1.56 % (5999)------------------------------
% 2.65/1.56 % (5999)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.56 % (5999)Termination reason: Unknown
% 2.65/1.56 % (5999)Termination phase: shuffling
% 2.65/1.56
% 2.65/1.56 % (5999)Memory used [KB]: 1407
% 2.65/1.56 % (5999)Time elapsed: 0.004 s
% 2.65/1.56 % (5999)Instructions burned: 3 (million)
% 2.65/1.56 % (5999)------------------------------
% 2.65/1.56 % (5999)------------------------------
% 2.65/1.57 % (6003)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/3Mi)
% 2.65/1.57 % (6003)Instruction limit reached!
% 2.65/1.57 % (6003)------------------------------
% 2.65/1.57 % (6003)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.57 % (6003)Termination reason: Unknown
% 2.65/1.57 % (6003)Termination phase: shuffling
% 2.65/1.57
% 2.65/1.57 % (6003)Memory used [KB]: 1407
% 2.65/1.57 % (6003)Time elapsed: 0.002 s
% 2.65/1.57 % (6003)Instructions burned: 4 (million)
% 2.65/1.57 % (6003)------------------------------
% 2.65/1.57 % (6003)------------------------------
% 2.65/1.57 % (6002)Instruction limit reached!
% 2.65/1.57 % (6002)------------------------------
% 2.65/1.57 % (6002)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.57 % (6002)Termination reason: Unknown
% 2.65/1.57 % (6002)Termination phase: shuffling
% 2.65/1.57
% 2.65/1.57 % (6002)Memory used [KB]: 1791
% 2.65/1.57 % (6002)Time elapsed: 0.013 s
% 2.65/1.57 % (6002)Instructions burned: 18 (million)
% 2.65/1.57 % (6002)------------------------------
% 2.65/1.57 % (6002)------------------------------
% 2.65/1.57 % (6005)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_5724 for (2999ds/15Mi)
% 2.65/1.57 % (6004)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_5724 for (2999ds/37Mi)
% 2.65/1.57 % (6005)Instruction limit reached!
% 2.65/1.57 % (6005)------------------------------
% 2.65/1.57 % (6005)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.57 % (6005)Termination reason: Unknown
% 2.65/1.57 % (6005)Termination phase: shuffling
% 2.65/1.57
% 2.65/1.57 % (6005)Memory used [KB]: 1663
% 2.65/1.57 % (6005)Time elapsed: 0.007 s
% 2.65/1.57 % (6005)Instructions burned: 15 (million)
% 2.65/1.57 % (6005)------------------------------
% 2.65/1.57 % (6005)------------------------------
% 2.65/1.58 % (6006)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_5724 for (2999ds/3Mi)
% 2.65/1.58 % (6007)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_5724 for (2999ds/1041Mi)
% 2.65/1.58 % (6006)Instruction limit reached!
% 2.65/1.58 % (6006)------------------------------
% 2.65/1.58 % (6006)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.58 % (6006)Termination reason: Unknown
% 2.65/1.58 % (6006)Termination phase: shuffling
% 2.65/1.58
% 2.65/1.58 % (6006)Memory used [KB]: 1407
% 2.65/1.58 % (6006)Time elapsed: 0.004 s
% 2.65/1.58 % (6006)Instructions burned: 4 (million)
% 2.65/1.58 % (6006)------------------------------
% 2.65/1.58 % (6006)------------------------------
% 2.65/1.58 % (6009)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/16Mi)
% 2.65/1.59 % (6008)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/7Mi)
% 2.65/1.59 % (6004)Instruction limit reached!
% 2.65/1.59 % (6004)------------------------------
% 2.65/1.59 % (6004)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.59 % (6004)Termination reason: Unknown
% 2.65/1.59 % (6004)Termination phase: shuffling
% 2.65/1.59
% 2.65/1.59 % (6004)Memory used [KB]: 2174
% 2.65/1.59 % (6004)Time elapsed: 0.015 s
% 2.65/1.59 % (6004)Instructions burned: 38 (million)
% 2.65/1.59 % (6004)------------------------------
% 2.65/1.59 % (6004)------------------------------
% 2.65/1.59 % (6009)Instruction limit reached!
% 2.65/1.59 % (6009)------------------------------
% 2.65/1.59 % (6009)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.59 % (6009)Termination reason: Unknown
% 2.65/1.59 % (6009)Termination phase: shuffling
% 2.65/1.59
% 2.65/1.59 % (6009)Memory used [KB]: 1791
% 2.65/1.59 % (6009)Time elapsed: 0.007 s
% 2.65/1.59 % (6009)Instructions burned: 17 (million)
% 2.65/1.59 % (6009)------------------------------
% 2.65/1.59 % (6009)------------------------------
% 2.65/1.59 % (6008)Instruction limit reached!
% 2.65/1.59 % (6008)------------------------------
% 2.65/1.59 % (6008)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.65/1.59 % (6008)Termination reason: Unknown
% 2.65/1.59 % (6008)Termination phase: shuffling
% 2.65/1.59
% 2.65/1.59 % (6008)Memory used [KB]: 1535
% 2.65/1.59 % (6008)Time elapsed: 0.006 s
% 2.65/1.59 % (6008)Instructions burned: 7 (million)
% 2.65/1.59 % (6008)------------------------------
% 2.65/1.59 % (6008)------------------------------
% 2.65/1.59 % (6010)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_5724 for (2999ds/3Mi)
% 2.68/1.60 % (6011)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_5724 for (2999ds/3Mi)
% 2.68/1.60 % (6010)Instruction limit reached!
% 2.68/1.60 % (6010)------------------------------
% 2.68/1.60 % (6010)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.60 % (6010)Termination reason: Unknown
% 2.68/1.60 % (6010)Termination phase: shuffling
% 2.68/1.60
% 2.68/1.60 % (6010)Memory used [KB]: 1407
% 2.68/1.60 % (6010)Time elapsed: 0.004 s
% 2.68/1.60 % (6010)Instructions burned: 5 (million)
% 2.68/1.60 % (6010)------------------------------
% 2.68/1.60 % (6010)------------------------------
% 2.68/1.60 % (6011)Instruction limit reached!
% 2.68/1.60 % (6011)------------------------------
% 2.68/1.60 % (6011)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.60 % (6011)Termination reason: Unknown
% 2.68/1.60 % (6011)Termination phase: shuffling
% 2.68/1.60
% 2.68/1.60 % (6011)Memory used [KB]: 1407
% 2.68/1.60 % (6011)Time elapsed: 0.003 s
% 2.68/1.60 % (6011)Instructions burned: 4 (million)
% 2.68/1.60 % (6011)------------------------------
% 2.68/1.60 % (6011)------------------------------
% 2.68/1.60 % (6012)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/7Mi)
% 2.68/1.60 % (6012)Instruction limit reached!
% 2.68/1.60 % (6012)------------------------------
% 2.68/1.60 % (6012)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.60 % (6012)Termination reason: Unknown
% 2.68/1.60 % (6012)Termination phase: shuffling
% 2.68/1.60
% 2.68/1.60 % (6012)Memory used [KB]: 1535
% 2.68/1.60 % (6012)Time elapsed: 0.004 s
% 2.68/1.60 % (6012)Instructions burned: 7 (million)
% 2.68/1.60 % (6012)------------------------------
% 2.68/1.60 % (6012)------------------------------
% 2.68/1.61 % (6013)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/3Mi)
% 2.68/1.61 % (6014)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/4Mi)
% 2.68/1.61 % (6015)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/18Mi)
% 2.68/1.61 % (6013)Instruction limit reached!
% 2.68/1.61 % (6013)------------------------------
% 2.68/1.61 % (6013)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.61 % (6013)Termination reason: Unknown
% 2.68/1.61 % (6013)Termination phase: shuffling
% 2.68/1.61
% 2.68/1.61 % (6013)Memory used [KB]: 1407
% 2.68/1.61 % (6013)Time elapsed: 0.003 s
% 2.68/1.61 % (6013)Instructions burned: 3 (million)
% 2.68/1.61 % (6013)------------------------------
% 2.68/1.61 % (6013)------------------------------
% 2.68/1.61 % (6014)Instruction limit reached!
% 2.68/1.61 % (6014)------------------------------
% 2.68/1.61 % (6014)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.61 % (6014)Termination reason: Unknown
% 2.68/1.61 % (6014)Termination phase: shuffling
% 2.68/1.61
% 2.68/1.61 % (6014)Memory used [KB]: 1407
% 2.68/1.61 % (6014)Time elapsed: 0.003 s
% 2.68/1.61 % (6014)Instructions burned: 4 (million)
% 2.68/1.61 % (6014)------------------------------
% 2.68/1.61 % (6014)------------------------------
% 2.68/1.61 % (6016)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/710Mi)
% 2.68/1.61 % (6015)Instruction limit reached!
% 2.68/1.61 % (6015)------------------------------
% 2.68/1.61 % (6015)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.61 % (6015)Termination reason: Unknown
% 2.68/1.61 % (6015)Termination phase: shuffling
% 2.68/1.61
% 2.68/1.61 % (6015)Memory used [KB]: 1791
% 2.68/1.61 % (6015)Time elapsed: 0.007 s
% 2.68/1.61 % (6015)Instructions burned: 21 (million)
% 2.68/1.61 % (6015)------------------------------
% 2.68/1.61 % (6015)------------------------------
% 2.68/1.62 % (6017)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/6Mi)
% 2.68/1.62 % (6017)Instruction limit reached!
% 2.68/1.62 % (6017)------------------------------
% 2.68/1.62 % (6017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.62 % (6017)Termination reason: Unknown
% 2.68/1.62 % (6017)Termination phase: shuffling
% 2.68/1.62
% 2.68/1.62 % (6017)Memory used [KB]: 1535
% 2.68/1.62 % (6017)Time elapsed: 0.004 s
% 2.68/1.62 % (6017)Instructions burned: 7 (million)
% 2.68/1.62 % (6017)------------------------------
% 2.68/1.62 % (6017)------------------------------
% 2.68/1.62 % (6019)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_5724 for (2999ds/21Mi)
% 2.68/1.63 % (6018)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_5724 for (2999ds/902Mi)
% 2.68/1.63 % (6019)Instruction limit reached!
% 2.68/1.63 % (6019)------------------------------
% 2.68/1.63 % (6019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.63 % (6019)Termination reason: Unknown
% 2.68/1.63 % (6019)Termination phase: shuffling
% 2.68/1.63
% 2.68/1.63 % (6019)Memory used [KB]: 1791
% 2.68/1.63 % (6019)Time elapsed: 0.007 s
% 2.68/1.63 % (6019)Instructions burned: 21 (million)
% 2.68/1.63 % (6019)------------------------------
% 2.68/1.63 % (6019)------------------------------
% 2.68/1.63 % (6020)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/5Mi)
% 2.68/1.63 % (6020)Instruction limit reached!
% 2.68/1.63 % (6020)------------------------------
% 2.68/1.63 % (6020)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.63 % (6020)Termination reason: Unknown
% 2.68/1.63 % (6020)Termination phase: shuffling
% 2.68/1.63
% 2.68/1.63 % (6020)Memory used [KB]: 1535
% 2.68/1.63 % (6020)Time elapsed: 0.004 s
% 2.68/1.63 % (6020)Instructions burned: 8 (million)
% 2.68/1.63 % (6020)------------------------------
% 2.68/1.63 % (6020)------------------------------
% 2.68/1.64 % (6021)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_5724 for (2999ds/6Mi)
% 2.68/1.64 % (6021)Instruction limit reached!
% 2.68/1.64 % (6021)------------------------------
% 2.68/1.64 % (6021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.64 % (6021)Termination reason: Unknown
% 2.68/1.64 % (6021)Termination phase: shuffling
% 2.68/1.64
% 2.68/1.64 % (6021)Memory used [KB]: 1535
% 2.68/1.64 % (6021)Time elapsed: 0.004 s
% 2.68/1.64 % (6021)Instructions burned: 7 (million)
% 2.68/1.64 % (6021)------------------------------
% 2.68/1.64 % (6021)------------------------------
% 2.68/1.64 % (6022)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_5724 for (2999ds/377Mi)
% 2.68/1.64 % (5996)Instruction limit reached!
% 2.68/1.64 % (5996)------------------------------
% 2.68/1.64 % (5996)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.64 % (5996)Termination reason: Unknown
% 2.68/1.64 % (5996)Termination phase: Saturation
% 2.68/1.64
% 2.68/1.64 % (5996)Memory used [KB]: 7547
% 2.68/1.64 % (5996)Time elapsed: 0.090 s
% 2.68/1.64 % (5996)Instructions burned: 185 (million)
% 2.68/1.64 % (5996)------------------------------
% 2.68/1.64 % (5996)------------------------------
% 2.68/1.65 % (6023)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_5724 for (2999ds/779Mi)
% 2.68/1.66 % (6024)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_5724 for (2998ds/19Mi)
% 2.68/1.66 % (6024)Instruction limit reached!
% 2.68/1.66 % (6024)------------------------------
% 2.68/1.66 % (6024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.66 % (6024)Termination reason: Unknown
% 2.68/1.66 % (6024)Termination phase: shuffling
% 2.68/1.66
% 2.68/1.66 % (6024)Memory used [KB]: 1791
% 2.68/1.66 % (6024)Time elapsed: 0.007 s
% 2.68/1.66 % (6024)Instructions burned: 21 (million)
% 2.68/1.66 % (6024)------------------------------
% 2.68/1.66 % (6024)------------------------------
% 2.68/1.67 % (6025)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_5724 for (2998ds/879Mi)
% 2.68/1.68 % (6001)Instruction limit reached!
% 2.68/1.68 % (6001)------------------------------
% 2.68/1.68 % (6001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.68 % (6001)Termination reason: Unknown
% 2.68/1.68 % (6001)Termination phase: Saturation
% 2.68/1.68
% 2.68/1.68 % (6001)Memory used [KB]: 9083
% 2.68/1.68 % (6001)Time elapsed: 0.123 s
% 2.68/1.68 % (6001)Instructions burned: 276 (million)
% 2.68/1.68 % (6001)------------------------------
% 2.68/1.68 % (6001)------------------------------
% 2.68/1.69 % (6026)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_5724 for (2998ds/17Mi)
% 2.68/1.69 % (6026)Instruction limit reached!
% 2.68/1.69 % (6026)------------------------------
% 2.68/1.69 % (6026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.69 % (6026)Termination reason: Unknown
% 2.68/1.69 % (6026)Termination phase: shuffling
% 2.68/1.69
% 2.68/1.69 % (6026)Memory used [KB]: 1791
% 2.68/1.69 % (6026)Time elapsed: 0.006 s
% 2.68/1.69 % (6026)Instructions burned: 17 (million)
% 2.68/1.69 % (6026)------------------------------
% 2.68/1.69 % (6026)------------------------------
% 2.68/1.70 % (6027)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_5724 for (2998ds/3Mi)
% 2.68/1.71 % (6027)Instruction limit reached!
% 2.68/1.71 % (6027)------------------------------
% 2.68/1.71 % (6027)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.71 % (6027)Termination reason: Unknown
% 2.68/1.71 % (6027)Termination phase: shuffling
% 2.68/1.71
% 2.68/1.71 % (6027)Memory used [KB]: 1407
% 2.68/1.71 % (6027)Time elapsed: 0.003 s
% 2.68/1.71 % (6027)Instructions burned: 4 (million)
% 2.68/1.71 % (6027)------------------------------
% 2.68/1.71 % (6027)------------------------------
% 2.68/1.72 % (6028)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_5724 for (2998ds/30Mi)
% 2.68/1.72 % (6028)Instruction limit reached!
% 2.68/1.72 % (6028)------------------------------
% 2.68/1.72 % (6028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.72 % (6028)Termination reason: Unknown
% 2.68/1.72 % (6028)Termination phase: shuffling
% 2.68/1.72
% 2.68/1.72 % (6028)Memory used [KB]: 1918
% 2.68/1.72 % (6028)Time elapsed: 0.011 s
% 2.68/1.72 % (6028)Instructions burned: 31 (million)
% 2.68/1.72 % (6028)------------------------------
% 2.68/1.72 % (6028)------------------------------
% 2.68/1.73 % (6029)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_5724 for (2998ds/127Mi)
% 2.68/1.76 % (6022)Instruction limit reached!
% 2.68/1.76 % (6022)------------------------------
% 2.68/1.76 % (6022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.76 % (6022)Termination reason: Unknown
% 2.68/1.76 % (6022)Termination phase: Saturation
% 2.68/1.76
% 2.68/1.76 % (6022)Memory used [KB]: 8699
% 2.68/1.76 % (6022)Time elapsed: 0.116 s
% 2.68/1.76 % (6022)Instructions burned: 378 (million)
% 2.68/1.76 % (6022)------------------------------
% 2.68/1.76 % (6022)------------------------------
% 2.68/1.76 % (6018)First to succeed.
% 2.68/1.77 % (6018)Refutation found. Thanks to Tanya!
% 2.68/1.77 % SZS status Theorem for DTF2THF_5724
% 2.68/1.77 % SZS output start Proof for DTF2THF_5724
% 2.68/1.77 thf(func_def_0, type, in: $i > $i > $o).
% 2.68/1.77 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.68/1.77 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.68/1.77 thf(func_def_8, type, powerset: $i > $i).
% 2.68/1.77 thf(func_def_10, type, setunion: $i > $i).
% 2.68/1.77 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.68/1.77 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_26, type, prop2set: $o > $i).
% 2.68/1.77 thf(func_def_36, type, nonempty: $i > $o).
% 2.68/1.77 thf(func_def_69, type, set2prop: $i > $o).
% 2.68/1.77 thf(func_def_88, type, subset: $i > $i > $o).
% 2.68/1.77 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.68/1.77 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.68/1.77 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.68/1.77 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.68/1.77 thf(func_def_135, type, regular: $i > $o).
% 2.68/1.77 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.68/1.77 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.68/1.77 thf(func_def_153, type, iskpair: $i > $o).
% 2.68/1.77 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.68/1.77 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.68/1.77 thf(func_def_177, type, singleton: $i > $o).
% 2.68/1.77 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.68/1.77 thf(func_def_184, type, atmost1p: $i > $o).
% 2.68/1.77 thf(func_def_185, type, atleast2p: $i > $o).
% 2.68/1.77 thf(func_def_186, type, atmost2p: $i > $o).
% 2.68/1.77 thf(func_def_187, type, upairsetp: $i > $o).
% 2.68/1.77 thf(func_def_191, type, kfst: $i > $i).
% 2.68/1.77 thf(func_def_203, type, ksnd: $i > $i).
% 2.68/1.77 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.68/1.77 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.68/1.77 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.68/1.77 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.68/1.77 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.68/1.77 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.68/1.77 thf(func_def_283, type, sP3: $i > $o > $i > $i > $o).
% 2.68/1.77 thf(func_def_284, type, sP4: $i > $i > $o).
% 2.68/1.77 thf(func_def_285, type, sP5: $i > $o).
% 2.68/1.77 thf(func_def_286, type, sP6: $i > $i > $o).
% 2.68/1.77 thf(func_def_287, type, sP7: $i > $i > $o).
% 2.68/1.77 thf(func_def_289, type, sP9: $i > $i > $i > $i > $o).
% 2.68/1.77 thf(func_def_295, type, sK15: $i > $i > ($i > $i) > $i).
% 2.68/1.77 thf(func_def_296, type, sK16: $i > $i).
% 2.68/1.77 thf(func_def_303, type, sK23: $i > $i > $i).
% 2.68/1.77 thf(func_def_310, type, sK30: $i > $o).
% 2.68/1.77 thf(func_def_311, type, sK31: ($i > $o) > $i).
% 2.68/1.77 thf(func_def_313, type, sK33: $i > $i).
% 2.68/1.77 thf(func_def_315, type, sK35: $i > ($i > $i) > $i > $i).
% 2.68/1.77 thf(func_def_322, type, sK42: $i > $o).
% 2.68/1.77 thf(func_def_340, type, sK60: $i > $o).
% 2.68/1.77 thf(func_def_341, type, sK61: $i > $i).
% 2.68/1.77 thf(func_def_342, type, sK62: ($i > $o) > $i).
% 2.68/1.77 thf(func_def_350, type, sK70: ($i > $o) > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_351, type, sK71: ($i > $o) > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_352, type, sK72: $i > $o).
% 2.68/1.77 thf(func_def_353, type, sK73: $i > $o).
% 2.68/1.77 thf(func_def_358, type, sK78: $i > $i).
% 2.68/1.77 thf(func_def_360, type, sK80: $i > $i).
% 2.68/1.77 thf(func_def_361, type, sK81: $i > $i).
% 2.68/1.77 thf(func_def_368, type, sK88: $i > $o).
% 2.68/1.77 thf(func_def_370, type, sK90: $i > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_378, type, sK98: $i > $o).
% 2.68/1.77 thf(func_def_413, type, sK133: $o > $i > $i > $i).
% 2.68/1.77 thf(func_def_414, type, sK134: ($i > $o) > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_415, type, sK135: ($i > $o) > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_416, type, sK136: $i > $o).
% 2.68/1.77 thf(func_def_417, type, sK137: $i > $o).
% 2.68/1.77 thf(func_def_420, type, sK140: $i > $i > $o).
% 2.68/1.77 thf(func_def_424, type, sK144: $i > $o).
% 2.68/1.77 thf(func_def_426, type, sK146: ($i > $o) > $i > $i).
% 2.68/1.77 thf(func_def_427, type, sK147: ($i > $o) > $i > $i).
% 2.68/1.77 thf(func_def_431, type, sK151: $i > $o).
% 2.68/1.77 thf(func_def_440, type, sK160: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_441, type, sK161: $i > $o).
% 2.68/1.77 thf(func_def_452, type, sK172: $i > $i).
% 2.68/1.77 thf(func_def_482, type, sK202: $i > $i > $o).
% 2.68/1.77 thf(func_def_483, type, sK203: $i > $i).
% 2.68/1.77 thf(func_def_484, type, sK204: $i > $i).
% 2.68/1.77 thf(func_def_485, type, sK205: ($i > $i > $o) > $i > $i).
% 2.68/1.77 thf(func_def_486, type, sK206: $i > ($i > $i > $o) > $i > $i).
% 2.68/1.77 thf(func_def_487, type, sK207: ($i > $i > $o) > $i > $i).
% 2.68/1.77 thf(func_def_510, type, sK230: $i > $i > $i).
% 2.68/1.77 thf(func_def_526, type, sK246: $i > $o).
% 2.68/1.77 thf(func_def_528, type, sK248: ($i > $o) > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_529, type, sK249: ($i > $o) > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_530, type, sK250: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_536, type, sK256: $i > $i).
% 2.68/1.77 thf(func_def_539, type, sK259: $i > ($i > $i) > $i > $i).
% 2.68/1.77 thf(func_def_546, type, sK266: $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_547, type, sK267: $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_551, type, sK271: $i > $o).
% 2.68/1.77 thf(func_def_553, type, sK273: $i > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_554, type, sK274: $i > $o).
% 2.68/1.77 thf(func_def_557, type, sK277: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_564, type, sK284: $i > $i > $i).
% 2.68/1.77 thf(func_def_565, type, sK285: $i > $i > $i).
% 2.68/1.77 thf(func_def_566, type, sK286: $i > $i > $i).
% 2.68/1.77 thf(func_def_567, type, sK287: $i > $i > $i).
% 2.68/1.77 thf(func_def_568, type, sK288: $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_569, type, sK289: $i > $i).
% 2.68/1.77 thf(func_def_570, type, sK290: $i > $i).
% 2.68/1.77 thf(func_def_571, type, sK291: $i > $i).
% 2.68/1.77 thf(func_def_572, type, sK292: $i > $i).
% 2.68/1.77 thf(func_def_573, type, sK293: $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_574, type, sK294: $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_575, type, sK295: $i > $i > $i).
% 2.68/1.77 thf(func_def_576, type, sK296: $i > $i > $i).
% 2.68/1.77 thf(func_def_577, type, sK297: $i > $i > $i).
% 2.68/1.77 thf(func_def_579, type, sK299: $i > $i).
% 2.68/1.77 thf(func_def_580, type, sK300: $i > $i).
% 2.68/1.77 thf(func_def_581, type, sK301: $i > $i).
% 2.68/1.77 thf(func_def_582, type, sK302: $i > $i > $i).
% 2.68/1.77 thf(func_def_586, type, sK306: $i > $o).
% 2.68/1.77 thf(func_def_599, type, sK319: ($i > $o) > $i > $i).
% 2.68/1.77 thf(func_def_601, type, sK321: $i > $o).
% 2.68/1.77 thf(func_def_610, type, sK330: $i > $o).
% 2.68/1.77 thf(func_def_626, type, sK346: $i > $o).
% 2.68/1.77 thf(func_def_629, type, sK349: $i > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_632, type, sK352: $i > $i > $o).
% 2.68/1.77 thf(func_def_643, type, sK363: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_644, type, sK364: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_649, type, sK369: $i > $o).
% 2.68/1.77 thf(func_def_651, type, sK371: ($i > $o) > $i).
% 2.68/1.77 thf(func_def_652, type, sK372: ($i > $o) > $i).
% 2.68/1.77 thf(func_def_659, type, sK379: $i > $i).
% 2.68/1.77 thf(func_def_661, type, sK381: $i > $o).
% 2.68/1.77 thf(func_def_669, type, sK389: $i > $i > $o).
% 2.68/1.77 thf(func_def_670, type, sK390: $i > $o).
% 2.68/1.77 thf(func_def_673, type, sK393: $i > $i > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_684, type, sK404: $i > $o).
% 2.68/1.77 thf(func_def_690, type, sK410: $i > $i > $o).
% 2.68/1.77 thf(func_def_693, type, sK413: $i > $o).
% 2.68/1.77 thf(func_def_695, type, sK415: $i > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_701, type, sK421: $i > $i > $o).
% 2.68/1.77 thf(func_def_706, type, sK426: $i > $o).
% 2.68/1.77 thf(func_def_709, type, sK429: $i > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_710, type, sK430: $i > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_719, type, sK439: $i > $i).
% 2.68/1.77 thf(func_def_722, type, sK442: $i > ($i > $i) > $i > $i).
% 2.68/1.77 thf(func_def_724, type, sK444: $i > $o).
% 2.68/1.77 thf(func_def_726, type, sK446: ($i > $o) > $i > $i).
% 2.68/1.77 thf(func_def_733, type, sK453: $i > $i > $i).
% 2.68/1.77 thf(func_def_734, type, sK454: $i > $i > $i).
% 2.68/1.77 thf(func_def_757, type, sK477: $i > ($i > $o) > $i).
% 2.68/1.77 thf(func_def_758, type, sK478: $i > $o).
% 2.68/1.77 thf(func_def_761, type, sK481: ($i > $o) > $i > $i).
% 2.68/1.77 thf(func_def_763, type, sK483: $i > $o).
% 2.68/1.77 thf(func_def_774, type, sK494: $i > $i > $i).
% 2.68/1.77 thf(func_def_784, type, sK504: $i > $i > $i).
% 2.68/1.77 thf(func_def_800, type, sK520: $i > $i > $o).
% 2.68/1.77 thf(func_def_806, type, sK526: $i > $i > $i).
% 2.68/1.77 thf(func_def_820, type, sK540: $i > $i > $o).
% 2.68/1.77 thf(func_def_826, type, sK546: $i > $o).
% 2.68/1.77 thf(func_def_835, type, sK555: $i > $o).
% 2.68/1.77 thf(func_def_838, type, sK558: $i > $i).
% 2.68/1.77 thf(func_def_850, type, sK570: $i > $i).
% 2.68/1.77 thf(func_def_859, type, sK579: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_860, type, sK580: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_861, type, sK581: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_862, type, sK582: $i > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_876, type, sK596: $i > $i > $i).
% 2.68/1.77 thf(func_def_878, type, sK598: $i > $o).
% 2.68/1.77 thf(func_def_906, type, sK626: $i > $o).
% 2.68/1.77 thf(func_def_907, type, sK627: ($i > $o) > $i).
% 2.68/1.77 thf(func_def_908, type, sK628: $i > $o).
% 2.68/1.77 thf(func_def_909, type, sK629: $i > $i).
% 2.68/1.77 thf(func_def_910, type, sK630: $i > ($i > $i) > $i > $i).
% 2.68/1.77 thf(func_def_912, type, sK632: $i > $i).
% 2.68/1.77 thf(func_def_914, type, sK634: ($i > $o) > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_915, type, sK635: ($i > $o) > $i > $i > $i > $i).
% 2.68/1.77 thf(func_def_919, type, sK639: $i > $o).
% 2.68/1.77 thf(func_def_924, type, sK644: $i > $o).
% 2.68/1.77 thf(func_def_925, type, sK645: $i > $o).
% 2.68/1.77 thf(func_def_926, type, sK646: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.68/1.77 thf(func_def_927, type, sK647: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.68/1.77 thf(func_def_930, type, sK650: $i > $i > $i).
% 2.68/1.77 thf(func_def_945, type, sK665: $i > ($i > $i) > $i > $i).
% 2.68/1.77 thf(func_def_947, type, sK667: $i > $i).
% 2.68/1.77 thf(func_def_954, type, sK674: $i > $i > $i).
% 2.68/1.77 thf(func_def_958, type, sK678: ($i > $o) > $i > $i).
% 2.68/1.77 thf(func_def_960, type, sK680: $i > $o).
% 2.68/1.77 thf(func_def_999, type, ph719: !>[X0: $tType]:(X0)).
% 2.68/1.77 thf(f4368,plain,(
% 2.68/1.77 $false),
% 2.68/1.77 inference(trivial_inequality_removal,[status(thm)],[f4361])).
% 2.68/1.77 thf(f4361,plain,(
% 2.68/1.77 ($true != $true)),
% 2.68/1.77 inference(superposition,[status(thm)],[f3488,f3426])).
% 2.68/1.77 thf(f3426,plain,(
% 2.68/1.77 ((subset @ sK698 @ sK699) = $true)),
% 2.68/1.77 inference(trivial_inequality_removal,[status(thm)],[f3425])).
% 2.68/1.77 thf(f3425,plain,(
% 2.68/1.77 ((subset @ sK698 @ sK699) = $true) | ($true != $true)),
% 2.68/1.77 inference(forward_demodulation,[status(thm)],[f3415,f3064])).
% 2.68/1.77 thf(f3064,plain,(
% 2.68/1.77 (powersetE1 = $true)),
% 2.68/1.77 inference(cnf_transformation,[status(thm)],[f2063])).
% 2.68/1.77 thf(f2063,plain,(
% 2.68/1.77 (binintersectER = $true) & (dpsetconstrEL1 = $true) & (upairsetIL = $true) & (in__Cong = $true) & (subbreln = $true) & (dpsetconstrEL2 = $true) & (inCongP = $true) & (omega__Cong = $true) & (powersetI1 = $true) & (binunionLsub = $true) & (upairsetIR = $true) & (exu__Cong = $true) & (funcinfuncset = $true) & (setbeta = $true) & (notdallE = $true) & (dsetconstr__Cong = $true) & (setminusLsub = $true) & (subsetRefl = $true) & (beta1 = $true) & (funcext2 = $true) & (setukpairinjL2 = $true) & (wellorderingAx = $true) & (apProp = $true) & (setoftrueEq = $true) & (binintersectSubset4 = $true) & (setukpairinjR1 = $true) & (ap2p = $true) & (setextAx = $true) & (ex1I2 = $true) & (lam2lamEq = $true) & (dsetconstrEL = $true) & (binintersectI = $true) & (funcGraphProp1 = $true) & (setunionsingleton2 = $true) & (dsetconstrER = $true) & (ksndpairEq = $true) & (cartprodpairin = $true) & (upairinpowunion = $true) & (brelnall2 = $true) & (setukpairIL = $true) & (cartprodpairmemEL = $true) & (ap2apEq1 = $true) & (setminusI = $true) & (singletonsuniq = $true) & (binunionIR = $true) & (emptyinPowerset = $true) & (cartprodfstin = $true) & (setadjoinIR = $true) & (iffalseProp1 = $true) & (ubforcartprodlem3 = $true) & (prop2setI = $true) & (setOfPairsIsBReln = $true) & (funcImageSingleton = $true) & (notequalI1 = $true) & (cartprodmempair = $true) & (iffalseProp2 = $true) & (ubforcartprodlem2 = $true) & (binintersectSubset1 = $true) & (setunionE = $true) & (app = $true) & (ex1I = $true) & (eta1 = $true) & (setminusER = $true) & (setext = $true) & (upairsetE = $true) & (cartprodsndin = $true) & (binunionIL = $true) & (setunionI = $true) & (setunionE2 = $true) & (notinemptyset = $true) & (binunionE = $true) & (omegaIndAx = $true) & (upairset2IR = $true) & (subsetI2 = $true) & (subPowSU = $true) & (symdiffI2 = $true) & (powersetI = $true) & (infuncsetfunc = $true) & (quantDeMorgan4 = $true) & (setukpairinjR2 = $true) & (dpsetconstrI = $true) & (eqimpsubset2 = $true) & (eqimpsubset1 = $true) & (exuE3e = $true) & (disjointsetsI1 = $true) & (funcext = $true) & (exuE1 = $true) & (setminusERneg = $true) & (kfstsingleton = $true) & (setadjoin__Cong = $true) & (quantDeMorgan2 = $true) & (nonemptyI = $true) & (symdiffIneg2 = $true) & (setunionsingleton1 = $true) & (setukpairIR = $true) & (nonemptyImpWitness = $true) & (singletoninpowerset = $true) & (((in @ sK698 @ (powerset @ sK699)) = $true) & ((in @ (powerset @ sK698) @ (powerset @ (powerset @ sK699))) != $true)) & (emptyset__Cong = $true) & (replAx = $true) & (setadjoinE = $true) & (binunionT_lem = $true) & (ap2apEq2 = $true) & (sepSubset = $true) & (setminusEL = $true) & (secondinupair = $true) & (subsetTrans = $true) & (funcGraphProp4 = $true) & (singletonsubset = $true) & (emptyInPowerset = $true) & (subsetI1 = $true) & (uniqinunit = $true) & (symdiffE = $true) & (dpsetconstrSub = $true) & (iftrueProp2 = $true) & (theeq = $true) & (dsetconstrI = $true) & (binintersectT_lem = $true) & (lam2p = $true) & (setadjoinSub2 = $true) & (iffalse = $true) & (iftrueProp1 = $true) & (powersetsubset = $true) & (binintersectLsub = $true) & (singletonsswitch = $true) & (omega0Ax = $true) & (setunionAx = $true) & (binunionRsub = $true) & (exuE3u = $true) & (emptysetAx = $true) & (exuEu = $true) & (upairsubunion = $true) & (cartprodmempaircEq = $true) & (vacuousDall = $true) & (notequalI2 = $true) & (eqinunit = $true) & (kpairsurjEq = $true) & (prop2set2propI = $true) & (eta2 = $true) & (binintersectSubset3 = $true) & (binunionEcases = $true) & (exuI2 = $true) & (cartprodpairsurjEq = $true) & (cartprodfstpairEq = $true) & (inPowerset = $true) & (noeltsimpempty = $true) & (subsetE2 = $true) & (emptysetimpfalse = $true) & (foundationAx = $true) & (cartprodmempair1 = $true) & (prop2setE = $true) & (kfstpairEq = $true) & (notsubsetI = $true) & (powersetE1 = $true) & (setadjoinSub = $true) & (dpsetconstrER = $true) & (emptyinunitempty = $true) & (symdiffIneg1 = $true) & (quantDeMorgan1 = $true) & (dpsetconstrERa = $true) & (iftrue = $true) & (ksndsingleton = $true) & (emptyI = $true) & (ifSingleton = $true) & (emptyE1 = $true) & (subsetemptysetimpeq = $true) & (funcGraphProp2 = $true) & (setukpairinjL = $true) & (binintersectSubset2 = $true) & (emptysetE = $true) & (setukpairinjR11 = $true) & (exuI1 = $true) & (iftrueorfalse = $true) & (lamProp = $true) & (setminusILneg = $true) & (descr__Cong = $true) & (nonemptyI1 = $true) & (descrp = $true) & (kpairiskpair = $true) & (singletonprop = $true) & (theprop = $true) & (upairset2E = $true) & (setunion__Cong = $true) & (ubforcartprodlem1 = $true) & (setukpairinjR = $true) & (subsetE = $true) & (eqbreln = $true) & (bs114d = $true) & (ifp = $true) & (setminusELneg = $true) & (setadjoinAx = $true) & (exuI3 = $true) & (symdiffI1 = $true) & (powerset__Cong = $true) & (funcGraphProp3 = $true) & (kpairp = $true) & (powersetE = $true) & (setextsub = $true) & (lamp = $true) & (notdexE = $true) & (subset2powerset = $true) & (emptysetsubset = $true) & (setminusSubset1 = $true) & (omegaSAx = $true) & (setukpairinjR12 = $true) & (sepInPowerset = $true) & (binintersectSubset5 = $true) & (powersetAx = $true) & (setadjoinIL = $true) & (cartprodsndpairEq = $true) & (notinsingleton = $true) & (binintersectEL = $true) & (nonemptyE1 = $true) & (setminusSubset2 = $true) & (quantDeMorgan3 = $true) & (brelnall1 = $true) & (upairequniteq = $true) & (beta2 = $true) & (exuE2 = $true) & (setukpairinjL1 = $true) & (setadjoinOr = $true) & (cartprodpairmemER = $true) & (binintersectRsub = $true) & (funcextLem = $true) & (setminusIRneg = $true) & (setunionsingleton = $true) & (ex1E1 = $true) & (ex1E2 = $true) & (singletoninpowunion = $true)),
% 2.68/1.77 inference(skolemisation,[status(esa),new_symbols(skolem,[sK698,sK699])],[f856,f2062])).
% 2.68/1.77 thf(f2062,plain,(
% 2.68/1.77 ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ($true != (in @ (powerset @ X0) @ (powerset @ (powerset @ X1))))) => (((in @ sK698 @ (powerset @ sK699)) = $true) & ((in @ (powerset @ sK698) @ (powerset @ (powerset @ sK699))) != $true))),
% 2.68/1.77 introduced(definition,[],[choice_axiom])).
% 2.68/1.77 thf(f856,plain,(
% 2.68/1.77 (binintersectER = $true) & (dpsetconstrEL1 = $true) & (upairsetIL = $true) & (in__Cong = $true) & (subbreln = $true) & (dpsetconstrEL2 = $true) & (inCongP = $true) & (omega__Cong = $true) & (powersetI1 = $true) & (binunionLsub = $true) & (upairsetIR = $true) & (exu__Cong = $true) & (funcinfuncset = $true) & (setbeta = $true) & (notdallE = $true) & (dsetconstr__Cong = $true) & (setminusLsub = $true) & (subsetRefl = $true) & (beta1 = $true) & (funcext2 = $true) & (setukpairinjL2 = $true) & (wellorderingAx = $true) & (apProp = $true) & (setoftrueEq = $true) & (binintersectSubset4 = $true) & (setukpairinjR1 = $true) & (ap2p = $true) & (setextAx = $true) & (ex1I2 = $true) & (lam2lamEq = $true) & (dsetconstrEL = $true) & (binintersectI = $true) & (funcGraphProp1 = $true) & (setunionsingleton2 = $true) & (dsetconstrER = $true) & (ksndpairEq = $true) & (cartprodpairin = $true) & (upairinpowunion = $true) & (brelnall2 = $true) & (setukpairIL = $true) & (cartprodpairmemEL = $true) & (ap2apEq1 = $true) & (setminusI = $true) & (singletonsuniq = $true) & (binunionIR = $true) & (emptyinPowerset = $true) & (cartprodfstin = $true) & (setadjoinIR = $true) & (iffalseProp1 = $true) & (ubforcartprodlem3 = $true) & (prop2setI = $true) & (setOfPairsIsBReln = $true) & (funcImageSingleton = $true) & (notequalI1 = $true) & (cartprodmempair = $true) & (iffalseProp2 = $true) & (ubforcartprodlem2 = $true) & (binintersectSubset1 = $true) & (setunionE = $true) & (app = $true) & (ex1I = $true) & (eta1 = $true) & (setminusER = $true) & (setext = $true) & (upairsetE = $true) & (cartprodsndin = $true) & (binunionIL = $true) & (setunionI = $true) & (setunionE2 = $true) & (notinemptyset = $true) & (binunionE = $true) & (omegaIndAx = $true) & (upairset2IR = $true) & (subsetI2 = $true) & (subPowSU = $true) & (symdiffI2 = $true) & (powersetI = $true) & (infuncsetfunc = $true) & (quantDeMorgan4 = $true) & (setukpairinjR2 = $true) & (dpsetconstrI = $true) & (eqimpsubset2 = $true) & (eqimpsubset1 = $true) & (exuE3e = $true) & (disjointsetsI1 = $true) & (funcext = $true) & (exuE1 = $true) & (setminusERneg = $true) & (kfstsingleton = $true) & (setadjoin__Cong = $true) & (quantDeMorgan2 = $true) & (nonemptyI = $true) & (symdiffIneg2 = $true) & (setunionsingleton1 = $true) & (setukpairIR = $true) & (nonemptyImpWitness = $true) & (singletoninpowerset = $true) & ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ($true != (in @ (powerset @ X0) @ (powerset @ (powerset @ X1))))) & (emptyset__Cong = $true) & (replAx = $true) & (setadjoinE = $true) & (binunionT_lem = $true) & (ap2apEq2 = $true) & (sepSubset = $true) & (setminusEL = $true) & (secondinupair = $true) & (subsetTrans = $true) & (funcGraphProp4 = $true) & (singletonsubset = $true) & (emptyInPowerset = $true) & (subsetI1 = $true) & (uniqinunit = $true) & (symdiffE = $true) & (dpsetconstrSub = $true) & (iftrueProp2 = $true) & (theeq = $true) & (dsetconstrI = $true) & (binintersectT_lem = $true) & (lam2p = $true) & (setadjoinSub2 = $true) & (iffalse = $true) & (iftrueProp1 = $true) & (powersetsubset = $true) & (binintersectLsub = $true) & (singletonsswitch = $true) & (omega0Ax = $true) & (setunionAx = $true) & (binunionRsub = $true) & (exuE3u = $true) & (emptysetAx = $true) & (exuEu = $true) & (upairsubunion = $true) & (cartprodmempaircEq = $true) & (vacuousDall = $true) & (notequalI2 = $true) & (eqinunit = $true) & (kpairsurjEq = $true) & (prop2set2propI = $true) & (eta2 = $true) & (binintersectSubset3 = $true) & (binunionEcases = $true) & (exuI2 = $true) & (cartprodpairsurjEq = $true) & (cartprodfstpairEq = $true) & (inPowerset = $true) & (noeltsimpempty = $true) & (subsetE2 = $true) & (emptysetimpfalse = $true) & (foundationAx = $true) & (cartprodmempair1 = $true) & (prop2setE = $true) & (kfstpairEq = $true) & (notsubsetI = $true) & (powersetE1 = $true) & (setadjoinSub = $true) & (dpsetconstrER = $true) & (emptyinunitempty = $true) & (symdiffIneg1 = $true) & (quantDeMorgan1 = $true) & (dpsetconstrERa = $true) & (iftrue = $true) & (ksndsingleton = $true) & (emptyI = $true) & (ifSingleton = $true) & (emptyE1 = $true) & (subsetemptysetimpeq = $true) & (funcGraphProp2 = $true) & (setukpairinjL = $true) & (binintersectSubset2 = $true) & (emptysetE = $true) & (setukpairinjR11 = $true) & (exuI1 = $true) & (iftrueorfalse = $true) & (lamProp = $true) & (setminusILneg = $true) & (descr__Cong = $true) & (nonemptyI1 = $true) & (descrp = $true) & (kpairiskpair = $true) & (singletonprop = $true) & (theprop = $true) & (upairset2E = $true) & (setunion__Cong = $true) & (ubforcartprodlem1 = $true) & (setukpairinjR = $true) & (subsetE = $true) & (eqbreln = $true) & (bs114d = $true) & (ifp = $true) & (setminusELneg = $true) & (setadjoinAx = $true) & (exuI3 = $true) & (symdiffI1 = $true) & (powerset__Cong = $true) & (funcGraphProp3 = $true) & (kpairp = $true) & (powersetE = $true) & (setextsub = $true) & (lamp = $true) & (notdexE = $true) & (subset2powerset = $true) & (emptysetsubset = $true) & (setminusSubset1 = $true) & (omegaSAx = $true) & (setukpairinjR12 = $true) & (sepInPowerset = $true) & (binintersectSubset5 = $true) & (powersetAx = $true) & (setadjoinIL = $true) & (cartprodsndpairEq = $true) & (notinsingleton = $true) & (binintersectEL = $true) & (nonemptyE1 = $true) & (setminusSubset2 = $true) & (quantDeMorgan3 = $true) & (brelnall1 = $true) & (upairequniteq = $true) & (beta2 = $true) & (exuE2 = $true) & (setukpairinjL1 = $true) & (setadjoinOr = $true) & (cartprodpairmemER = $true) & (binintersectRsub = $true) & (funcextLem = $true) & (setminusIRneg = $true) & (setunionsingleton = $true) & (ex1E1 = $true) & (ex1E2 = $true) & (singletoninpowunion = $true)),
% 2.68/1.77 inference(flattening,[status(thm)],[f855])).
% 2.68/1.77 thf(f855,plain,(
% 2.68/1.77 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ($true != (in @ (powerset @ X0) @ (powerset @ (powerset @ X1))))) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.68/1.77 inference(ennf_transformation,[status(thm)],[f476])).
% 2.68/1.77 thf(f476,plain,(
% 2.68/1.77 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ($true = (in @ (powerset @ X0) @ (powerset @ (powerset @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.68/1.77 inference(fool_elimination,[status(thm)],[f475])).
% 2.68/1.77 thf(f475,plain,(
% 2.68/1.77 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => (in @ (powerset @ X0) @ (powerset @ (powerset @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.68/1.77 inference(rectify,[status(thm)],[f234])).
% 2.68/1.77 thf(f234,negated_conjecture,(
% 2.68/1.77 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => (in @ (powerset @ X11) @ (powerset @ (powerset @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.68/1.77 inference(negated_conjecture,[status(cth)],[f233])).
% 2.68/1.77 thf(f233,conjecture,(
% 2.68/1.77 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => (in @ (powerset @ X11) @ (powerset @ (powerset @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.68/1.77 file('/export/starexec/sandbox2/tmp/tmp.PfJgBtH2Qy/DTF2THF_5724.p',powersetT_lem)).
% 2.68/1.77 thf(f3415,plain,(
% 2.68/1.77 ((subset @ sK698 @ sK699) = $true) | (powersetE1 != $true)),
% 2.68/1.77 inference(trivial_inequality_removal,[status(thm)],[f3411])).
% 2.68/1.77 thf(f3411,plain,(
% 2.68/1.77 ((subset @ sK698 @ sK699) = $true) | (powersetE1 != $true) | ($true != $true)),
% 2.68/1.77 inference(superposition,[status(thm)],[f2285,f3121])).
% 2.68/1.77 thf(f3121,plain,(
% 2.68/1.77 ((in @ sK698 @ (powerset @ sK699)) = $true)),
% 2.68/1.77 inference(cnf_transformation,[status(thm)],[f2063])).
% 2.68/1.77 thf(f2285,plain,(
% 2.68/1.77 ( ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | (powersetE1 != $true) | ((subset @ X1 @ X0) = $true)) )),
% 2.68/1.77 inference(cnf_transformation,[status(thm)],[f1240])).
% 2.68/1.77 thf(f1240,plain,(
% 2.68/1.77 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | ((subset @ X1 @ X0) = $true)) | (powersetE1 != $true)) & ((powersetE1 = $true) | (((in @ sK165 @ (powerset @ sK164)) = $true) & ((subset @ sK165 @ sK164) != $true)))),
% 2.68/1.77 inference(skolemisation,[status(esa),new_symbols(skolem,[sK164,sK165])],[f1238,f1239])).
% 2.68/1.77 thf(f1239,plain,(
% 2.68/1.77 ? [X2 : $i,X3 : $i] : (($true = (in @ X3 @ (powerset @ X2))) & ((subset @ X3 @ X2) != $true)) => (((in @ sK165 @ (powerset @ sK164)) = $true) & ((subset @ sK165 @ sK164) != $true))),
% 2.68/1.77 introduced(definition,[],[choice_axiom])).
% 2.68/1.77 thf(f1238,plain,(
% 2.68/1.77 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | ((subset @ X1 @ X0) = $true)) | (powersetE1 != $true)) & ((powersetE1 = $true) | ? [X2 : $i,X3 : $i] : (($true = (in @ X3 @ (powerset @ X2))) & ((subset @ X3 @ X2) != $true)))),
% 2.68/1.77 inference(rectify,[status(thm)],[f1237])).
% 2.68/1.77 thf(f1237,plain,(
% 2.68/1.77 (! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | ((subset @ X1 @ X0) = $true)) | (powersetE1 != $true)) & ((powersetE1 = $true) | ? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) & ((subset @ X1 @ X0) != $true)))),
% 2.68/1.77 inference(nnf_transformation,[status(thm)],[f871])).
% 2.68/1.77 thf(f871,plain,(
% 2.68/1.77 ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) != $true) | ((subset @ X1 @ X0) = $true)) <=> (powersetE1 = $true)),
% 2.68/1.77 inference(ennf_transformation,[status(thm)],[f432])).
% 2.68/1.77 thf(f432,plain,(
% 2.68/1.77 (powersetE1 = $true) <=> ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ((subset @ X1 @ X0) = $true))),
% 2.68/1.77 inference(fool_elimination,[status(thm)],[f431])).
% 2.68/1.77 thf(f431,plain,(
% 2.68/1.77 (powersetE1 = ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => (subset @ X1 @ X0)))),
% 2.68/1.77 inference(rectify,[status(thm)],[f96])).
% 2.68/1.77 thf(f96,axiom,(
% 2.68/1.77 (powersetE1 = ! [X3 : $i,X4 : $i] : ((in @ X4 @ (powerset @ X3)) => (subset @ X4 @ X3)))),
% 2.68/1.77 file('/export/starexec/sandbox2/tmp/tmp.PfJgBtH2Qy/DTF2THF_5724.p',powersetE1)).
% 2.68/1.77 thf(f3488,plain,(
% 2.68/1.77 ((subset @ sK698 @ sK699) != $true)),
% 2.68/1.77 inference(trivial_inequality_removal,[status(thm)],[f3487])).
% 2.68/1.77 thf(f3487,plain,(
% 2.68/1.77 ((subset @ sK698 @ sK699) != $true) | ($true != $true)),
% 2.68/1.77 inference(forward_demodulation,[status(thm)],[f3483,f3095])).
% 2.68/1.77 thf(f3095,plain,(
% 2.68/1.77 (powersetsubset = $true)),
% 2.68/1.77 inference(cnf_transformation,[status(thm)],[f2063])).
% 2.68/1.77 thf(f3483,plain,(
% 2.68/1.77 (powersetsubset != $true) | ((subset @ sK698 @ sK699) != $true)),
% 2.68/1.77 inference(trivial_inequality_removal,[status(thm)],[f3475])).
% 2.68/1.77 thf(f3475,plain,(
% 2.68/1.77 ($true != $true) | (powersetsubset != $true) | ((subset @ sK698 @ sK699) != $true)),
% 2.68/1.77 inference(superposition,[status(thm)],[f3457,f2127])).
% 2.68/1.77 thf(f2127,plain,(
% 2.68/1.77 ( ! [X2 : $i,X3 : $i] : (((subset @ (powerset @ X2) @ (powerset @ X3)) = $true) | ((subset @ X2 @ X3) != $true) | (powersetsubset != $true)) )),
% 2.68/1.77 inference(cnf_transformation,[status(thm)],[f1060])).
% 2.68/1.77 thf(f1060,plain,(
% 2.68/1.77 ((powersetsubset = $true) | (((subset @ sK44 @ sK45) = $true) & ((subset @ (powerset @ sK44) @ (powerset @ sK45)) != $true))) & (! [X2 : $i,X3 : $i] : (((subset @ X2 @ X3) != $true) | ((subset @ (powerset @ X2) @ (powerset @ X3)) = $true)) | (powersetsubset != $true))),
% 2.68/1.77 inference(skolemisation,[status(esa),new_symbols(skolem,[sK44,sK45])],[f1058,f1059])).
% 2.68/1.77 thf(f1059,plain,(
% 2.68/1.77 ? [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) = $true) & ((subset @ (powerset @ X0) @ (powerset @ X1)) != $true)) => (((subset @ sK44 @ sK45) = $true) & ((subset @ (powerset @ sK44) @ (powerset @ sK45)) != $true))),
% 2.68/1.77 introduced(definition,[],[choice_axiom])).
% 2.68/1.77 thf(f1058,plain,(
% 2.68/1.77 ((powersetsubset = $true) | ? [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) = $true) & ((subset @ (powerset @ X0) @ (powerset @ X1)) != $true))) & (! [X2 : $i,X3 : $i] : (((subset @ X2 @ X3) != $true) | ((subset @ (powerset @ X2) @ (powerset @ X3)) = $true)) | (powersetsubset != $true))),
% 2.68/1.77 inference(rectify,[status(thm)],[f1057])).
% 2.68/1.77 thf(f1057,plain,(
% 2.68/1.77 ((powersetsubset = $true) | ? [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) = $true) & ((subset @ (powerset @ X0) @ (powerset @ X1)) != $true))) & (! [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) != $true) | ((subset @ (powerset @ X0) @ (powerset @ X1)) = $true)) | (powersetsubset != $true))),
% 2.68/1.77 inference(nnf_transformation,[status(thm)],[f860])).
% 2.68/1.77 thf(f860,plain,(
% 2.68/1.77 (powersetsubset = $true) <=> ! [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) != $true) | ((subset @ (powerset @ X0) @ (powerset @ X1)) = $true))),
% 2.68/1.77 inference(ennf_transformation,[status(thm)],[f494])).
% 2.68/1.77 thf(f494,plain,(
% 2.68/1.77 (powersetsubset = $true) <=> ! [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) = $true) => ((subset @ (powerset @ X0) @ (powerset @ X1)) = $true))),
% 2.68/1.77 inference(fool_elimination,[status(thm)],[f493])).
% 2.68/1.77 thf(f493,plain,(
% 2.68/1.77 (! [X0 : $i,X1 : $i] : ((subset @ X0 @ X1) => (subset @ (powerset @ X0) @ (powerset @ X1))) = powersetsubset)),
% 2.68/1.77 inference(rectify,[status(thm)],[f98])).
% 2.68/1.77 thf(f98,axiom,(
% 2.68/1.77 (! [X3 : $i,X4 : $i] : ((subset @ X3 @ X4) => (subset @ (powerset @ X3) @ (powerset @ X4))) = powersetsubset)),
% 2.68/1.77 file('/export/starexec/sandbox2/tmp/tmp.PfJgBtH2Qy/DTF2THF_5724.p',powersetsubset)).
% 2.68/1.77 thf(f3457,plain,(
% 2.68/1.77 ((subset @ (powerset @ sK698) @ (powerset @ sK699)) != $true)),
% 2.68/1.77 inference(trivial_inequality_removal,[status(thm)],[f3456])).
% 2.68/1.77 thf(f3456,plain,(
% 2.68/1.77 ($true != $true) | ((subset @ (powerset @ sK698) @ (powerset @ sK699)) != $true)),
% 2.68/1.77 inference(forward_demodulation,[status(thm)],[f3447,f3017])).
% 2.68/1.77 thf(f3017,plain,(
% 2.68/1.77 (subset2powerset = $true)),
% 2.68/1.77 inference(cnf_transformation,[status(thm)],[f2063])).
% 2.68/1.77 thf(f3447,plain,(
% 2.68/1.77 ((subset @ (powerset @ sK698) @ (powerset @ sK699)) != $true) | (subset2powerset != $true)),
% 2.68/1.77 inference(trivial_inequality_removal,[status(thm)],[f3439])).
% 2.68/1.77 thf(f3439,plain,(
% 2.68/1.77 ($true != $true) | (subset2powerset != $true) | ((subset @ (powerset @ sK698) @ (powerset @ sK699)) != $true)),
% 2.68/1.77 inference(superposition,[status(thm)],[f3120,f2565])).
% 2.68/1.77 thf(f2565,plain,(
% 2.68/1.77 ( ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) | ((subset @ X1 @ X0) != $true) | (subset2powerset != $true)) )),
% 2.68/1.77 inference(cnf_transformation,[status(thm)],[f1552])).
% 2.68/1.77 thf(f1552,plain,(
% 2.68/1.77 (! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) != $true) | ((in @ X1 @ (powerset @ X0)) = $true)) | (subset2powerset != $true)) & ((subset2powerset = $true) | (((subset @ sK374 @ sK373) = $true) & ((in @ sK374 @ (powerset @ sK373)) != $true)))),
% 2.68/1.77 inference(skolemisation,[status(esa),new_symbols(skolem,[sK373,sK374])],[f1550,f1551])).
% 2.68/1.77 thf(f1551,plain,(
% 2.68/1.77 ? [X2 : $i,X3 : $i] : (((subset @ X3 @ X2) = $true) & ($true != (in @ X3 @ (powerset @ X2)))) => (((subset @ sK374 @ sK373) = $true) & ((in @ sK374 @ (powerset @ sK373)) != $true))),
% 2.68/1.77 introduced(definition,[],[choice_axiom])).
% 2.68/1.77 thf(f1550,plain,(
% 2.68/1.77 (! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) != $true) | ((in @ X1 @ (powerset @ X0)) = $true)) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X2 : $i,X3 : $i] : (((subset @ X3 @ X2) = $true) & ($true != (in @ X3 @ (powerset @ X2)))))),
% 2.68/1.77 inference(rectify,[status(thm)],[f1549])).
% 2.68/1.77 thf(f1549,plain,(
% 2.68/1.77 (! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) != $true) | ((in @ X1 @ (powerset @ X0)) = $true)) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) & ((in @ X1 @ (powerset @ X0)) != $true)))),
% 2.68/1.77 inference(nnf_transformation,[status(thm)],[f977])).
% 2.68/1.77 thf(f977,plain,(
% 2.68/1.77 ! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) != $true) | ((in @ X1 @ (powerset @ X0)) = $true)) <=> (subset2powerset = $true)),
% 2.68/1.77 inference(ennf_transformation,[status(thm)],[f354])).
% 2.68/1.77 thf(f354,plain,(
% 2.68/1.77 ! [X0 : $i,X1 : $i] : (((subset @ X1 @ X0) = $true) => ((in @ X1 @ (powerset @ X0)) = $true)) <=> (subset2powerset = $true)),
% 2.68/1.77 inference(fool_elimination,[status(thm)],[f353])).
% 2.68/1.77 thf(f353,plain,(
% 2.68/1.77 (! [X0 : $i,X1 : $i] : ((subset @ X1 @ X0) => (in @ X1 @ (powerset @ X0))) = subset2powerset)),
% 2.68/1.77 inference(rectify,[status(thm)],[f92])).
% 2.68/1.77 thf(f92,axiom,(
% 2.68/1.77 (! [X4 : $i,X3 : $i] : ((subset @ X3 @ X4) => (in @ X3 @ (powerset @ X4))) = subset2powerset)),
% 2.68/1.77 file('/export/starexec/sandbox2/tmp/tmp.PfJgBtH2Qy/DTF2THF_5724.p',subset2powerset)).
% 2.68/1.77 thf(f3120,plain,(
% 2.68/1.77 ((in @ (powerset @ sK698) @ (powerset @ (powerset @ sK699))) != $true)),
% 2.68/1.77 inference(cnf_transformation,[status(thm)],[f2063])).
% 2.68/1.77 % SZS output end Proof for DTF2THF_5724
% 2.68/1.77 % (6018)------------------------------
% 2.68/1.77 % (6018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.68/1.77 % (6018)Termination reason: Refutation
% 2.68/1.77
% 2.68/1.77 % (6018)Memory used [KB]: 9978
% 2.68/1.77 % (6018)Time elapsed: 0.141 s
% 2.68/1.77 % (6018)Instructions burned: 392 (million)
% 2.68/1.77 % (6018)------------------------------
% 2.68/1.77 % (6018)------------------------------
% 2.68/1.77 % (5995)Success in time 0.22 s
%------------------------------------------------------------------------------