%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU660^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n028.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 09:36:01 AM UTC 2026
% Result : Theorem 2.96s 1.96s
% Output : Refutation 2.96s
% Verified :
% SZS Type : Refutation
% Derivation depth : 19
% Number of leaves : 11
% Syntax : Number of formulae : 71 ( 20 unt; 0 typ; 1 def)
% Number of atoms : 3282 ( 900 equ; 0 cnn)
% Maximal formula atoms : 178 ( 46 avg)
% Number of connectives : 1851 ( 103 ~; 69 |; 565 &; 377 @)
% ( 10 <=>; 727 =>; 0 <=; 0 <~>)
% Maximal formula depth : 187 ( 22 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 334 ( 331 usr; 193 con; 0-4 aty)
% Number of variables : 147 ( 0 ^; 105 !; 42 ?; 147 :)
% 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_222,type,
sP2: $i > $i > $o ).
thf(func_def_223,type,
sP3: $i > $o ).
thf(func_def_224,type,
sP4: $i > $i > $o ).
thf(func_def_225,type,
sP5: $i > $i > $o ).
thf(func_def_227,type,
sP7: $o > $i > $i > $i > $o ).
thf(func_def_238,type,
sK18: $i > $i > $o ).
thf(func_def_240,type,
sK20: $i > $i ).
thf(func_def_241,type,
sK21: $i > $i ).
thf(func_def_242,type,
sK22: $i > ( $i > $i > $o ) > $i ).
thf(func_def_243,type,
sK23: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_244,type,
sK24: $i > ( $i > $i > $o ) > $i ).
thf(func_def_255,type,
sK35: ( $i > $o ) > $i ).
thf(func_def_256,type,
sK36: $i > $o ).
thf(func_def_257,type,
sK37: $i > $i ).
thf(func_def_265,type,
sK45: $i > $i ).
thf(func_def_281,type,
sK61: $i > $o ).
thf(func_def_282,type,
sK62: ( $i > $o ) > $i > $i ).
thf(func_def_288,type,
sK68: $i > $o ).
thf(func_def_289,type,
sK69: $i > $o ).
thf(func_def_290,type,
sK70: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_291,type,
sK71: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_303,type,
sK83: $i > $o ).
thf(func_def_304,type,
sK84: ( $i > $o ) > $i > $i > $i ).
thf(func_def_307,type,
sK87: $i > $o ).
thf(func_def_309,type,
sK89: $o > $i > $i > $i ).
thf(func_def_319,type,
sK99: $i > $o ).
thf(func_def_322,type,
sK102: $i > $o ).
thf(func_def_324,type,
sK104: $i > $o ).
thf(func_def_325,type,
sK105: $i > $o ).
thf(func_def_326,type,
sK106: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_327,type,
sK107: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_331,type,
sK111: $i > $o ).
thf(func_def_334,type,
sK114: $i > $o ).
thf(func_def_337,type,
sK117: $i > ( $i > $o ) > $i ).
thf(func_def_340,type,
sK120: $i > $i ).
thf(func_def_343,type,
sK123: $i > $o ).
thf(func_def_346,type,
sK126: $i > ( $i > $o ) > $i ).
thf(func_def_347,type,
sK127: $i > ( $i > $o ) > $i ).
thf(func_def_350,type,
sK130: $i > $o ).
thf(func_def_352,type,
sK132: $i > ( $i > $o ) > $i ).
thf(func_def_353,type,
sK133: $i > ( $i > $o ) > $i ).
thf(func_def_354,type,
sK134: $i > $o ).
thf(func_def_375,type,
sK155: $i > $i > $i ).
thf(func_def_404,type,
sK184: $i > $o ).
thf(func_def_413,type,
sK193: ( $i > $o ) > $i ).
thf(func_def_414,type,
sK194: $i > $o ).
thf(func_def_422,type,
sK202: $i > $i > $i ).
thf(func_def_423,type,
sK203: $i > $i > $i ).
thf(func_def_424,type,
sK204: $i > $i > $i ).
thf(func_def_425,type,
sK205: $i > $i > $i ).
thf(func_def_426,type,
sK206: $i > $i > $i ).
thf(func_def_427,type,
sK207: $i > $i > $i > $i ).
thf(func_def_428,type,
sK208: $i > $i ).
thf(func_def_429,type,
sK209: $i > $i ).
thf(func_def_430,type,
sK210: $i > $i ).
thf(func_def_431,type,
sK211: $i > $i ).
thf(func_def_432,type,
sK212: $i > $i > $i ).
thf(func_def_433,type,
sK213: $i > $i > $i > $i ).
thf(func_def_434,type,
sK214: $i > $i > $i > $i ).
thf(func_def_435,type,
sK215: $i > $i > $i ).
thf(func_def_436,type,
sK216: $i > $i > $i ).
thf(func_def_437,type,
sK217: $i > $i ).
thf(func_def_439,type,
sK219: $i > $i ).
thf(func_def_440,type,
sK220: $i > $i ).
thf(func_def_442,type,
sK222: $i > $o ).
thf(func_def_444,type,
sK224: ( $i > $o ) > $i > $i ).
thf(func_def_445,type,
sK225: ( $i > $o ) > $i > $i ).
thf(func_def_446,type,
sK226: $i > $i ).
thf(func_def_449,type,
sK229: $i > $i ).
thf(func_def_474,type,
sK254: $i > $i ).
thf(func_def_478,type,
sK258: $i > $i > $i > $i ).
thf(func_def_479,type,
sK259: $i > $i > $i > $i ).
thf(func_def_488,type,
sK268: $i > $o ).
thf(func_def_490,type,
sK270: $i > $o ).
thf(func_def_491,type,
sK271: $i > $i > $i ).
thf(func_def_497,type,
sK277: $i > $i > $i ).
thf(func_def_501,type,
sK281: $i > ( $i > $o ) > $i ).
thf(func_def_502,type,
sK282: $i > $o ).
thf(func_def_523,type,
sK303: $i > ( $i > $o ) > $i ).
thf(func_def_524,type,
sK304: $i > $o ).
thf(func_def_529,type,
sK309: $i > $i > $i ).
thf(func_def_530,type,
sK310: $i > $i > $i ).
thf(func_def_531,type,
sK311: $i > $o ).
thf(func_def_533,type,
sK313: $i > ( $i > $o ) > $i ).
thf(func_def_545,type,
sK325: $i > $i ).
thf(func_def_558,type,
sK338: $i > $i > $i ).
thf(func_def_582,type,
sK362: $i > $o ).
thf(func_def_598,type,
sK378: $i > $i > $i ).
thf(func_def_623,type,
sK403: $i > $i > $i ).
thf(func_def_632,type,
sK412: $i > $i > $i ).
thf(func_def_648,type,
sK428: $i > $o ).
thf(func_def_649,type,
sK429: $i > $o ).
thf(func_def_650,type,
sK430: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_651,type,
sK431: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_653,type,
sK433: $i > $o ).
thf(func_def_670,type,
sK450: $i > $o ).
thf(func_def_674,type,
sK454: $i > $o ).
thf(func_def_676,type,
sK456: ( $i > $o ) > $i ).
thf(func_def_677,type,
sK457: ( $i > $o ) > $i ).
thf(func_def_679,type,
sK459: $i > $o ).
thf(func_def_683,type,
sK463: $i > $o ).
thf(func_def_685,type,
sK465: ( $i > $o ) > $i > $i ).
thf(func_def_686,type,
sK466: $i > $o ).
thf(func_def_692,type,
sK472: $i > $i > $i ).
thf(func_def_698,type,
sK478: $i > $o ).
thf(func_def_701,type,
sK481: $i > ( $i > $o ) > $i ).
thf(func_def_702,type,
sK482: $i > $o ).
thf(func_def_703,type,
sK483: $i > $i ).
thf(func_def_704,type,
sK484: ( $i > $o ) > $i ).
thf(func_def_713,type,
sK493: $i > $i ).
thf(func_def_715,type,
ph495:
!>[X0: $tType] : X0 ).
thf(f3336,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3329,f3335]) ).
thf(f3335,plain,
spl494_15,
inference(avatar_split_clause,[status(thm)],[f2808,f3194]) ).
thf(f3194,definition,
( spl494_15
<=> ( $true
= ( in @ sK178 @ sK180 ) ) ),
introduced(definition,[new_symbols(naming,[spl494_15])],[avatar_definition]) ).
thf(f2808,plain,
( $true
= ( in @ sK178 @ sK180 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2807]) ).
thf(f2807,plain,
( ( $true
= ( in @ sK178 @ sK180 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2802,f1855]) ).
thf(f1855,plain,
ksndpairEq = $true,
inference(cnf_transformation,[status(thm)],[f1054]) ).
thf(f1054,plain,
( ( upairsetE = $true )
& ( exuEu = $true )
& ( setminusERneg = $true )
& ( emptyInPowerset = $true )
& ( setadjoinSub2 = $true )
& ( binintersectRsub = $true )
& ( eqinunit = $true )
& ( emptyI = $true )
& ( subsetI2 = $true )
& ( binintersectSubset5 = $true )
& ( setminusILneg = $true )
& ( setbeta = $true )
& ( upairset2E = $true )
& ( ksndsingleton = $true )
& ( quantDeMorgan1 = $true )
& ( notsubsetI = $true )
& ( notdallE = $true )
& ( setoftrueEq = $true )
& ( binintersectI = $true )
& ( setunionI = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyI = $true )
& ( upairsetIL = $true )
& ( kpairsurjEq = $true )
& ( notinemptyset = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinE = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairmemEL = $true )
& ( binintersectSubset1 = $true )
& ( ubforcartprodlem3 = $true )
& ( notinsingleton = $true )
& ( nonemptyE1 = $true )
& ( descr__Cong = $true )
& ( binintersectSubset2 = $true )
& ( setextAx = $true )
& ( setunionsingleton = $true )
& ( setukpairinjL = $true )
& ( exuI1 = $true )
& ( quantDeMorgan4 = $true )
& ( setunionsingleton2 = $true )
& ( setukpairinjL1 = $true )
& ( binintersectSubset4 = $true )
& ( secondinupair = $true )
& ( notequalI1 = $true )
& ( setukpairinjR2 = $true )
& ( upairsetIR = $true )
& ( powerset__Cong = $true )
& ( setminusELneg = $true )
& ( setextsub = $true )
& ( setunionsingleton1 = $true )
& ( setukpairinjR12 = $true )
& ( setadjoin__Cong = $true )
& ( exuE1 = $true )
& ( prop2set2propI = $true )
& ( singletonsubset = $true )
& ( subsetE2 = $true )
& ( ex1I2 = $true )
& ( symdiffI1 = $true )
& ( singletoninpowunion = $true )
& ( subsetI1 = $true )
& ( subset2powerset = $true )
& ( ex1I = $true )
& ( powersetE = $true )
& ( kfstpairEq = $true )
& ( setukpairinjL2 = $true )
& ( prop2setE = $true )
& ( binunionE = $true )
& ( exuE2 = $true )
& ( binintersectEL = $true )
& ( emptysetE = $true )
& ( setadjoinOr = $true )
& ( setukpairinjR = $true )
& ( omegaSAx = $true )
& ( setminusIRneg = $true )
& ( exu__Cong = $true )
& ( setunionE = $true )
& ( exuE3u = $true )
& ( exuE3e = $true )
& ( omega0Ax = $true )
& ( vacuousDall = $true )
& ( kpairp = $true )
& ( emptyset__Cong = $true )
& ( ubforcartprodlem1 = $true )
& ( noeltsimpempty = $true )
& ( sepInPowerset = $true )
& ( inPowerset = $true )
& ( replAx = $true )
& ( setukpairinjR1 = $true )
& ( setunionE2 = $true )
& ( wellorderingAx = $true )
& ( dsetconstr__Cong = $true )
& ( emptyinunitempty = $true )
& ( upairsubunion = $true )
& ( notdexE = $true )
& ( singletonprop = $true )
& ( descrp = $true )
& ( setminusSubset1 = $true )
& ( binintersectER = $true )
& ( powersetI = $true )
& ( setadjoinSub = $true )
& ( setukpairinjR11 = $true )
& ( emptysetAx = $true )
& ( eqimpsubset1 = $true )
& ( binunionLsub = $true )
& ( setukpairIL = $true )
& ( setukpairIR = $true )
& ( cartprodpairin = $true )
& ( dsetconstrI = $true )
& ( symdiffIneg1 = $true )
& ( singletonsswitch = $true )
& ( cartprodfstin = $true )
& ( exuI3 = $true )
& ( binunionIL = $true )
& ( binunionEcases = $true )
& ( cartprodmempair1 = $true )
& ( subsetRefl = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetE1 = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstrER = $true )
& ( setext = $true )
& ( setadjoinAx = $true )
& ( setminusSubset2 = $true )
& ( dsetconstrEL = $true )
& ( kfstsingleton = $true )
& ( binintersectSubset3 = $true )
& ( powersetsubset = $true )
& ( emptysetimpfalse = $true )
& ( binintersectLsub = $true )
& ( theprop = $true )
& ( disjointsetsI1 = $true )
& ( emptyE1 = $true )
& ( prop2setI = $true )
& ( subsetTrans = $true )
& ( ksndpairEq = $true )
& ( binunionIR = $true )
& ( cartprodsndin = $true )
& ( cartprodmempair = $true )
& ( upairequniteq = $true )
& ( emptysetsubset = $true )
& ( setminusEL = $true )
& ( foundationAx = $true )
& ( subsetE = $true )
& ( setminusI = $true )
& ( in__Cong = $true )
& ( uniqinunit = $true )
& ( setadjoinIR = $true )
& ( symdiffE = $true )
& ( notequalI2 = $true )
& ( setadjoinIL = $true )
& ( setminusER = $true )
& ( symdiffI2 = $true )
& ( upairinpowunion = $true )
& ( singletoninpowerset = $true )
& ( setunionAx = $true )
& ( powersetI1 = $true )
& ( powersetAx = $true )
& ( ex1E1 = $true )
& ( bs114d = $true )
& ( $true
!= ( in @ sK178 @ sK180 ) )
& ( $true
= ( in @ ( kpair @ sK181 @ sK178 ) @ ( cartprod @ sK179 @ sK180 ) ) )
& ( omegaIndAx = $true )
& ( sepSubset = $true )
& ( upairset2IR = $true )
& ( exuI2 = $true )
& ( subPowSU = $true )
& ( setunion__Cong = $true )
& ( omega__Cong = $true )
& ( singletonsuniq = $true )
& ( kpairiskpair = $true )
& ( nonemptyI1 = $true )
& ( binunionRsub = $true )
& ( eqimpsubset2 = $true )
& ( nonemptyImpWitness = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusLsub = $true )
& ( inCongP = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK178,sK179,sK180,sK181])],[f585,f1053]) ).
thf(f1053,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
!= ( in @ X0 @ X2 ) )
& ( $true
= ( in @ ( kpair @ X3 @ X0 ) @ ( cartprod @ X1 @ X2 ) ) ) )
=> ( ( $true
!= ( in @ sK178 @ sK180 ) )
& ( $true
= ( in @ ( kpair @ sK181 @ sK178 ) @ ( cartprod @ sK179 @ sK180 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f585,plain,
( ( upairsetE = $true )
& ( exuEu = $true )
& ( setminusERneg = $true )
& ( emptyInPowerset = $true )
& ( setadjoinSub2 = $true )
& ( binintersectRsub = $true )
& ( eqinunit = $true )
& ( emptyI = $true )
& ( subsetI2 = $true )
& ( binintersectSubset5 = $true )
& ( setminusILneg = $true )
& ( setbeta = $true )
& ( upairset2E = $true )
& ( ksndsingleton = $true )
& ( quantDeMorgan1 = $true )
& ( notsubsetI = $true )
& ( notdallE = $true )
& ( setoftrueEq = $true )
& ( binintersectI = $true )
& ( setunionI = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyI = $true )
& ( upairsetIL = $true )
& ( kpairsurjEq = $true )
& ( notinemptyset = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinE = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairmemEL = $true )
& ( binintersectSubset1 = $true )
& ( ubforcartprodlem3 = $true )
& ( notinsingleton = $true )
& ( nonemptyE1 = $true )
& ( descr__Cong = $true )
& ( binintersectSubset2 = $true )
& ( setextAx = $true )
& ( setunionsingleton = $true )
& ( setukpairinjL = $true )
& ( exuI1 = $true )
& ( quantDeMorgan4 = $true )
& ( setunionsingleton2 = $true )
& ( setukpairinjL1 = $true )
& ( binintersectSubset4 = $true )
& ( secondinupair = $true )
& ( notequalI1 = $true )
& ( setukpairinjR2 = $true )
& ( upairsetIR = $true )
& ( powerset__Cong = $true )
& ( setminusELneg = $true )
& ( setextsub = $true )
& ( setunionsingleton1 = $true )
& ( setukpairinjR12 = $true )
& ( setadjoin__Cong = $true )
& ( exuE1 = $true )
& ( prop2set2propI = $true )
& ( singletonsubset = $true )
& ( subsetE2 = $true )
& ( ex1I2 = $true )
& ( symdiffI1 = $true )
& ( singletoninpowunion = $true )
& ( subsetI1 = $true )
& ( subset2powerset = $true )
& ( ex1I = $true )
& ( powersetE = $true )
& ( kfstpairEq = $true )
& ( setukpairinjL2 = $true )
& ( prop2setE = $true )
& ( binunionE = $true )
& ( exuE2 = $true )
& ( binintersectEL = $true )
& ( emptysetE = $true )
& ( setadjoinOr = $true )
& ( setukpairinjR = $true )
& ( omegaSAx = $true )
& ( setminusIRneg = $true )
& ( exu__Cong = $true )
& ( setunionE = $true )
& ( exuE3u = $true )
& ( exuE3e = $true )
& ( omega0Ax = $true )
& ( vacuousDall = $true )
& ( kpairp = $true )
& ( emptyset__Cong = $true )
& ( ubforcartprodlem1 = $true )
& ( noeltsimpempty = $true )
& ( sepInPowerset = $true )
& ( inPowerset = $true )
& ( replAx = $true )
& ( setukpairinjR1 = $true )
& ( setunionE2 = $true )
& ( wellorderingAx = $true )
& ( dsetconstr__Cong = $true )
& ( emptyinunitempty = $true )
& ( upairsubunion = $true )
& ( notdexE = $true )
& ( singletonprop = $true )
& ( descrp = $true )
& ( setminusSubset1 = $true )
& ( binintersectER = $true )
& ( powersetI = $true )
& ( setadjoinSub = $true )
& ( setukpairinjR11 = $true )
& ( emptysetAx = $true )
& ( eqimpsubset1 = $true )
& ( binunionLsub = $true )
& ( setukpairIL = $true )
& ( setukpairIR = $true )
& ( cartprodpairin = $true )
& ( dsetconstrI = $true )
& ( symdiffIneg1 = $true )
& ( singletonsswitch = $true )
& ( cartprodfstin = $true )
& ( exuI3 = $true )
& ( binunionIL = $true )
& ( binunionEcases = $true )
& ( cartprodmempair1 = $true )
& ( subsetRefl = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetE1 = $true )
& ( quantDeMorgan2 = $true )
& ( dsetconstrER = $true )
& ( setext = $true )
& ( setadjoinAx = $true )
& ( setminusSubset2 = $true )
& ( dsetconstrEL = $true )
& ( kfstsingleton = $true )
& ( binintersectSubset3 = $true )
& ( powersetsubset = $true )
& ( emptysetimpfalse = $true )
& ( binintersectLsub = $true )
& ( theprop = $true )
& ( disjointsetsI1 = $true )
& ( emptyE1 = $true )
& ( prop2setI = $true )
& ( subsetTrans = $true )
& ( ksndpairEq = $true )
& ( binunionIR = $true )
& ( cartprodsndin = $true )
& ( cartprodmempair = $true )
& ( upairequniteq = $true )
& ( emptysetsubset = $true )
& ( setminusEL = $true )
& ( foundationAx = $true )
& ( subsetE = $true )
& ( setminusI = $true )
& ( in__Cong = $true )
& ( uniqinunit = $true )
& ( setadjoinIR = $true )
& ( symdiffE = $true )
& ( notequalI2 = $true )
& ( setadjoinIL = $true )
& ( setminusER = $true )
& ( symdiffI2 = $true )
& ( upairinpowunion = $true )
& ( singletoninpowerset = $true )
& ( setunionAx = $true )
& ( powersetI1 = $true )
& ( powersetAx = $true )
& ( ex1E1 = $true )
& ( bs114d = $true )
& ? [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
!= ( in @ X0 @ X2 ) )
& ( $true
= ( in @ ( kpair @ X3 @ X0 ) @ ( cartprod @ X1 @ X2 ) ) ) )
& ( omegaIndAx = $true )
& ( sepSubset = $true )
& ( upairset2IR = $true )
& ( exuI2 = $true )
& ( subPowSU = $true )
& ( setunion__Cong = $true )
& ( omega__Cong = $true )
& ( singletonsuniq = $true )
& ( kpairiskpair = $true )
& ( nonemptyI1 = $true )
& ( binunionRsub = $true )
& ( eqimpsubset2 = $true )
& ( nonemptyImpWitness = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusLsub = $true )
& ( inCongP = $true ) ),
inference(flattening,[status(thm)],[f584]) ).
thf(f584,plain,
( ? [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
!= ( in @ X0 @ X2 ) )
& ( $true
= ( in @ ( kpair @ X3 @ X0 ) @ ( cartprod @ X1 @ X2 ) ) ) )
& ( 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)],[f452]) ).
thf(f452,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 )
=> ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( $true
= ( in @ ( kpair @ X3 @ X0 ) @ ( cartprod @ X1 @ X2 ) ) )
=> ( $true
= ( in @ X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f451]) ).
thf(f451,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
=> ! [X0: $i,X1: $i,X2: $i,X3: $i] :
( ( in @ ( kpair @ X3 @ X0 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f182]) ).
thf(f182,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
=> ! [X2: $i,X3: $i,X4: $i,X1: $i] :
( ( in @ ( kpair @ X1 @ X2 ) @ ( cartprod @ X3 @ X4 ) )
=> ( in @ X2 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f181]) ).
thf(f181,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
=> ! [X2: $i,X3: $i,X4: $i,X1: $i] :
( ( in @ ( kpair @ X1 @ X2 ) @ ( cartprod @ X3 @ X4 ) )
=> ( in @ X2 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',cartprodpairmemER) ).
thf(f2802,plain,
( ( ksndpairEq != $true )
| ( $true
= ( in @ sK178 @ sK180 ) ) ),
inference(superposition,[status(thm)],[f2578,f1794]) ).
thf(f1794,plain,
! [X0: $i,X1: $i] :
( ( ( ksnd @ ( kpair @ X1 @ X0 ) )
= X0 )
| ( ksndpairEq != $true ) ),
inference(cnf_transformation,[status(thm)],[f1023]) ).
thf(f1023,plain,
( ( ! [X0: $i,X1: $i] :
( ( ksnd @ ( kpair @ X1 @ X0 ) )
= X0 )
| ( ksndpairEq != $true ) )
& ( ( ksndpairEq = $true )
| ( ( ksnd @ ( kpair @ sK161 @ sK160 ) )
!= sK160 ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK160,sK161])],[f1021,f1022]) ).
thf(f1022,plain,
( ? [X2: $i,X3: $i] :
( ( ksnd @ ( kpair @ X3 @ X2 ) )
!= X2 )
=> ( ( ksnd @ ( kpair @ sK161 @ sK160 ) )
!= sK160 ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1021,plain,
( ( ! [X0: $i,X1: $i] :
( ( ksnd @ ( kpair @ X1 @ X0 ) )
= X0 )
| ( ksndpairEq != $true ) )
& ( ( ksndpairEq = $true )
| ? [X2: $i,X3: $i] :
( ( ksnd @ ( kpair @ X3 @ X2 ) )
!= X2 ) ) ),
inference(rectify,[status(thm)],[f1020]) ).
thf(f1020,plain,
( ( ! [X0: $i,X1: $i] :
( ( ksnd @ ( kpair @ X1 @ X0 ) )
= X0 )
| ( ksndpairEq != $true ) )
& ( ( ksndpairEq = $true )
| ? [X0: $i,X1: $i] :
( ( ksnd @ ( kpair @ X1 @ X0 ) )
!= X0 ) ) ),
inference(nnf_transformation,[status(thm)],[f510]) ).
thf(f510,plain,
( ! [X0: $i,X1: $i] :
( ( ksnd @ ( kpair @ X1 @ X0 ) )
= X0 )
<=> ( ksndpairEq = $true ) ),
inference(fool_elimination,[status(thm)],[f509]) ).
thf(f509,plain,
( ksndpairEq
= ( ! [X0: $i,X1: $i] :
( ( ksnd @ ( kpair @ X1 @ X0 ) )
= X0 ) ) ),
inference(rectify,[status(thm)],[f177]) ).
thf(f177,axiom,
( ksndpairEq
= ( ! [X2: $i,X1: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',ksndpairEq) ).
thf(f2578,plain,
( ( in @ ( ksnd @ ( kpair @ sK181 @ sK178 ) ) @ sK180 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2577]) ).
thf(f2577,plain,
( ( ( in @ ( ksnd @ ( kpair @ sK181 @ sK178 ) ) @ sK180 )
= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2570,f1853]) ).
thf(f1853,plain,
cartprodsndin = $true,
inference(cnf_transformation,[status(thm)],[f1054]) ).
thf(f2570,plain,
( ( cartprodsndin != $true )
| ( ( in @ ( ksnd @ ( kpair @ sK181 @ sK178 ) ) @ sK180 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2566]) ).
thf(f2566,plain,
( ( $true != $true )
| ( cartprodsndin != $true )
| ( ( in @ ( ksnd @ ( kpair @ sK181 @ sK178 ) ) @ sK180 )
= $true ) ),
inference(superposition,[status(thm)],[f2296,f1829]) ).
thf(f1829,plain,
( $true
= ( in @ ( kpair @ sK181 @ sK178 ) @ ( cartprod @ sK179 @ sK180 ) ) ),
inference(cnf_transformation,[status(thm)],[f1054]) ).
thf(f2296,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ ( cartprod @ X4 @ X3 ) )
!= $true )
| ( cartprodsndin != $true )
| ( $true
= ( in @ ( ksnd @ X5 ) @ X3 ) ) ),
inference(cnf_transformation,[status(thm)],[f1423]) ).
thf(f1423,plain,
( ( ( cartprodsndin = $true )
| ( ( $true
!= ( in @ ( ksnd @ sK399 ) @ sK397 ) )
& ( $true
= ( in @ sK399 @ ( cartprod @ sK398 @ sK397 ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ ( ksnd @ X5 ) @ X3 ) )
| ( ( in @ X5 @ ( cartprod @ X4 @ X3 ) )
!= $true ) )
| ( cartprodsndin != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK397,sK398,sK399])],[f1421,f1422]) ).
thf(f1422,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( ksnd @ X2 ) @ X0 )
!= $true )
& ( $true
= ( in @ X2 @ ( cartprod @ X1 @ X0 ) ) ) )
=> ( ( $true
!= ( in @ ( ksnd @ sK399 ) @ sK397 ) )
& ( $true
= ( in @ sK399 @ ( cartprod @ sK398 @ sK397 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1421,plain,
( ( ( cartprodsndin = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( ksnd @ X2 ) @ X0 )
!= $true )
& ( $true
= ( in @ X2 @ ( cartprod @ X1 @ X0 ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ ( ksnd @ X5 ) @ X3 ) )
| ( ( in @ X5 @ ( cartprod @ X4 @ X3 ) )
!= $true ) )
| ( cartprodsndin != $true ) ) ),
inference(rectify,[status(thm)],[f1420]) ).
thf(f1420,plain,
( ( ( cartprodsndin = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( ksnd @ X2 ) @ X0 )
!= $true )
& ( $true
= ( in @ X2 @ ( cartprod @ X1 @ X0 ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( ksnd @ X2 ) @ X0 )
= $true )
| ( $true
!= ( in @ X2 @ ( cartprod @ X1 @ X0 ) ) ) )
| ( cartprodsndin != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f602]) ).
thf(f602,plain,
( ( cartprodsndin = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ ( ksnd @ X2 ) @ X0 )
= $true )
| ( $true
!= ( in @ X2 @ ( cartprod @ X1 @ X0 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f392]) ).
thf(f392,plain,
( ( cartprodsndin = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( cartprod @ X1 @ X0 ) ) )
=> ( ( in @ ( ksnd @ X2 ) @ X0 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f391]) ).
thf(f391,plain,
( cartprodsndin
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( cartprod @ X1 @ X0 ) )
=> ( in @ ( ksnd @ X2 ) @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f179]) ).
thf(f179,axiom,
( cartprodsndin
= ( ! [X4: $i,X3: $i,X10: $i] :
( ( in @ X10 @ ( cartprod @ X3 @ X4 ) )
=> ( in @ ( ksnd @ X10 ) @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',cartprodsndin) ).
thf(f3329,plain,
~ spl494_15,
inference(avatar_split_clause,[status(thm)],[f2783,f3194]) ).
thf(f2783,plain,
( $true
!= ( in @ sK178 @ sK180 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2782]) ).
thf(f2782,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK178 @ sK180 ) ) ),
inference(forward_demodulation,[status(thm)],[f2752,f1887]) ).
thf(f1887,plain,
eqimpsubset1 = $true,
inference(cnf_transformation,[status(thm)],[f1054]) ).
thf(f2752,plain,
( ( $true
!= ( in @ sK178 @ sK180 ) )
| ( eqimpsubset1 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2735]) ).
thf(f2735,plain,
( ( $true
!= ( in @ sK178 @ sK180 ) )
| ( $true != $true )
| ( eqimpsubset1 != $true ) ),
inference(superposition,[status(thm)],[f2562,f2439]) ).
thf(f2439,plain,
! [X3: $i] :
( ( ( subset @ X3 @ X3 )
= $true )
| ( eqimpsubset1 != $true ) ),
inference(equality_resolution,[status(thm)],[f1656]) ).
thf(f1656,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
| ( X2 != X3 )
| ( eqimpsubset1 != $true ) ),
inference(cnf_transformation,[status(thm)],[f868]) ).
thf(f868,plain,
( ( ( eqimpsubset1 = $true )
| ( ( $true
!= ( subset @ sK63 @ sK64 ) )
& ( sK64 = sK63 ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
| ( X2 != X3 ) )
| ( eqimpsubset1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK63,sK64])],[f866,f867]) ).
thf(f867,plain,
( ? [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
& ( X0 = X1 ) )
=> ( ( $true
!= ( subset @ sK63 @ sK64 ) )
& ( sK64 = sK63 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f866,plain,
( ( ( eqimpsubset1 = $true )
| ? [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
& ( X0 = X1 ) ) )
& ( ! [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
| ( X2 != X3 ) )
| ( eqimpsubset1 != $true ) ) ),
inference(rectify,[status(thm)],[f865]) ).
thf(f865,plain,
( ( ( eqimpsubset1 = $true )
| ? [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
& ( X0 = X1 ) ) )
& ( ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
| ( X0 != X1 ) )
| ( eqimpsubset1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f638]) ).
thf(f638,plain,
( ( eqimpsubset1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
= $true )
| ( X0 != X1 ) ) ),
inference(ennf_transformation,[status(thm)],[f490]) ).
thf(f490,plain,
( ( eqimpsubset1 = $true )
<=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( subset @ X0 @ X1 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f489]) ).
thf(f489,plain,
! [X0: $i,X1: $i] :
( ( ( X0 = X1 )
=> ( subset @ X0 @ X1 ) )
= eqimpsubset1 ),
inference(rectify,[status(thm)],[f80]) ).
thf(f80,axiom,
! [X3: $i,X4: $i] :
( ( ( X3 = X4 )
=> ( subset @ X3 @ X4 ) )
= eqimpsubset1 ),
file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',eqimpsubset1) ).
thf(f2562,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ sK180 ) )
| ( ( in @ sK178 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2561]) ).
thf(f2561,plain,
! [X0: $i] :
( ( $true != $true )
| ( $true
!= ( subset @ X0 @ sK180 ) )
| ( ( in @ sK178 @ X0 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f2554,f1975]) ).
thf(f1975,plain,
notsubsetI = $true,
inference(cnf_transformation,[status(thm)],[f1054]) ).
thf(f2554,plain,
! [X0: $i] :
( ( ( in @ sK178 @ X0 )
!= $true )
| ( $true
!= ( subset @ X0 @ sK180 ) )
| ( notsubsetI != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2552]) ).
thf(f2552,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ sK180 ) )
| ( notsubsetI != $true )
| ( ( in @ sK178 @ X0 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f1830,f1703]) ).
thf(f1703,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ X4 )
= $true )
| ( ( subset @ X5 @ X4 )
!= $true )
| ( notsubsetI != $true )
| ( ( in @ X3 @ X5 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f920]) ).
thf(f920,plain,
( ( ( notsubsetI = $true )
| ( ( ( in @ sK95 @ sK97 )
= $true )
& ( $true
!= ( in @ sK95 @ sK96 ) )
& ( ( subset @ sK97 @ sK96 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ X5 )
!= $true )
| ( ( in @ X3 @ X4 )
= $true )
| ( ( subset @ X5 @ X4 )
!= $true ) )
| ( notsubsetI != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK95,sK96,sK97])],[f918,f919]) ).
thf(f919,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X2 ) )
& ( $true
!= ( in @ X0 @ X1 ) )
& ( ( subset @ X2 @ X1 )
= $true ) )
=> ( ( ( in @ sK95 @ sK97 )
= $true )
& ( $true
!= ( in @ sK95 @ sK96 ) )
& ( ( subset @ sK97 @ sK96 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f918,plain,
( ( ( notsubsetI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X2 ) )
& ( $true
!= ( in @ X0 @ X1 ) )
& ( ( subset @ X2 @ X1 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ X5 )
!= $true )
| ( ( in @ X3 @ X4 )
= $true )
| ( ( subset @ X5 @ X4 )
!= $true ) )
| ( notsubsetI != $true ) ) ),
inference(rectify,[status(thm)],[f917]) ).
thf(f917,plain,
( ( ( notsubsetI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X2 ) )
& ( $true
!= ( in @ X0 @ X1 ) )
& ( ( subset @ X2 @ X1 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X2 ) )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( subset @ X2 @ X1 )
!= $true ) )
| ( notsubsetI != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f713]) ).
thf(f713,plain,
( ( notsubsetI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X2 ) )
| ( $true
= ( in @ X0 @ X1 ) )
| ( ( subset @ X2 @ X1 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f712]) ).
thf(f712,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X1 )
!= $true )
| ( $true
= ( in @ X0 @ X1 ) )
| ( $true
!= ( in @ X0 @ X2 ) ) )
<=> ( notsubsetI = $true ) ),
inference(ennf_transformation,[status(thm)],[f567]) ).
thf(f567,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X2 ) )
=> ( ( $true
!= ( in @ X0 @ X1 ) )
=> ( ( subset @ X2 @ X1 )
!= $true ) ) )
<=> ( notsubsetI = $true ) ),
inference(flattening,[status(thm)],[f442]) ).
thf(f442,plain,
( ( notsubsetI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X2 ) )
=> ( ( $true
!= ( in @ X0 @ X1 ) )
=> ( ( subset @ X2 @ X1 )
!= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f441]) ).
thf(f441,plain,
( notsubsetI
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ( ~ ( in @ X0 @ X1 )
=> ~ ( subset @ X2 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f85]) ).
thf(f85,axiom,
( notsubsetI
= ( ! [X1: $i,X4: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ X4 )
=> ~ ( subset @ X3 @ X4 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',notsubsetI) ).
thf(f1830,plain,
( $true
!= ( in @ sK178 @ sK180 ) ),
inference(cnf_transformation,[status(thm)],[f1054]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : SEU660^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.11 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.13/0.31 % Computer : n028.cluster.edu
% 0.13/0.31 % Model : x86_64 x86_64
% 0.13/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.31 % Memory : 8042.1875MB
% 0.13/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.31 % CPULimit : 300
% 0.13/0.31 % WCLimit : 300
% 0.13/0.31 % DateTime : Tue Feb 24 12:29:42 EST 2026
% 0.13/0.31 % CPUTime :
% 0.13/0.31 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.22/0.44 ---- Original DTF file ---
% 0.22/0.44 thf(spec,logic,$$dhol).
% 0.22/0.44 %------------------------------------------------------------------------------
% 0.22/0.44 % File : SEU660^1 : TPTP v9.2.1. Released v3.7.0.
% 0.22/0.44 % Domain : Set Theory
% 0.22/0.44 % Problem : Ordered Pairs - Properties of Pairs
% 0.22/0.44 % Version : Especial.
% 0.22/0.44 % English : (! A:i.! B:i.! x:i.! y:i.in (kpair x y) (cartprod A B) -> in y B)
% 0.22/0.44
% 0.22/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.22/0.44 % Source : [Bro08]
% 0.22/0.44 % Names : ZFC162g [Bro08]
% 0.22/0.44
% 0.22/0.44 % Status : Theorem
% 0.22/0.44 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.22/0.44 % Syntax : Number of formulae : 388 ( 180 unt; 207 typ; 180 def)
% 0.22/0.44 % Number of atoms : 956 ( 269 equ; 0 cnn)
% 0.22/0.44 % Maximal formula atoms : 178 ( 5 avg)
% 0.22/0.44 % Number of connectives : 1677 ( 47 ~; 7 |; 35 &;1152 @)
% 0.22/0.44 % ( 14 <=>; 422 =>; 0 <=; 0 <~>)
% 0.22/0.44 % Maximal formula depth : 186 ( 2 avg)
% 0.22/0.44 % Number of types : 2 ( 0 usr)
% 0.22/0.44 % Number of type conns : 84 ( 84 >; 0 *; 0 +; 0 <<)
% 0.22/0.44 % Number of symbols : 210 ( 207 usr; 180 con; 0-2 aty)
% 0.22/0.44 % Number of variables : 526 ( 40 ^; 450 !; 36 ?; 526 :)
% 0.22/0.44 % SPC : TH0_THM_EQU_NAR
% 0.22/0.44
% 0.22/0.44 % Comments : http://mathgate.info/detsetitem.php?id=338
% 0.22/0.44 %------------------------------------------------------------------------------
% 0.22/0.44 thf(in_type,type,
% 0.22/0.44 in: $i > $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(exu_type,type,
% 0.22/0.44 exu: ( $i > $o ) > $o ).
% 0.22/0.44
% 0.22/0.44 thf(exu,definition,
% 0.22/0.44 ( exu
% 0.22/0.44 = ( ^ [Xphi: $i > $o] :
% 0.22/0.44 ? [Xx: $i] :
% 0.22/0.44 ( ( Xphi @ Xx )
% 0.22/0.44 & ! [Xy: $i] :
% 0.22/0.44 ( ( Xphi @ Xy )
% 0.22/0.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setextAx_type,type,
% 0.22/0.44 setextAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setextAx,definition,
% 0.22/0.44 ( setextAx
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 <=> ( in @ Xx @ B ) )
% 0.22/0.44 => ( A = B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptyset_type,type,
% 0.22/0.44 emptyset: $i ).
% 0.22/0.44
% 0.22/0.44 thf(emptysetAx_type,type,
% 0.22/0.44 emptysetAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptysetAx,definition,
% 0.22/0.44 ( emptysetAx
% 0.22/0.44 = ( ! [Xx: $i] :
% 0.22/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoin_type,type,
% 0.22/0.44 setadjoin: $i > $i > $i ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinAx_type,type,
% 0.22/0.44 setadjoinAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinAx,definition,
% 0.22/0.44 ( setadjoinAx
% 0.22/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.22/0.44 <=> ( ( Xy = Xx )
% 0.22/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(powerset_type,type,
% 0.22/0.44 powerset: $i > $i ).
% 0.22/0.44
% 0.22/0.44 thf(powersetAx_type,type,
% 0.22/0.44 powersetAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(powersetAx,definition,
% 0.22/0.44 ( powersetAx
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.22/0.44 <=> ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setunion_type,type,
% 0.22/0.44 setunion: $i > $i ).
% 0.22/0.44
% 0.22/0.44 thf(setunionAx_type,type,
% 0.22/0.44 setunionAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setunionAx,definition,
% 0.22/0.44 ( setunionAx
% 0.22/0.44 = ( ! [A: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.22/0.44 <=> ? [B: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(omega_type,type,
% 0.22/0.44 omega: $i ).
% 0.22/0.44
% 0.22/0.44 thf(omega0Ax_type,type,
% 0.22/0.44 omega0Ax: $o ).
% 0.22/0.44
% 0.22/0.44 thf(omega0Ax,definition,
% 0.22/0.44 ( omega0Ax
% 0.22/0.44 = ( in @ emptyset @ omega ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(omegaSAx_type,type,
% 0.22/0.44 omegaSAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(omegaSAx,definition,
% 0.22/0.44 ( omegaSAx
% 0.22/0.44 = ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ omega )
% 0.22/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(omegaIndAx_type,type,
% 0.22/0.44 omegaIndAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(omegaIndAx,definition,
% 0.22/0.44 ( omegaIndAx
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ( ( ( in @ emptyset @ A )
% 0.22/0.44 & ! [Xx: $i] :
% 0.22/0.44 ( ( ( in @ Xx @ omega )
% 0.22/0.44 & ( in @ Xx @ A ) )
% 0.22/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.22/0.44 => ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ omega )
% 0.22/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(replAx_type,type,
% 0.22/0.44 replAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(replAx,definition,
% 0.22/0.44 ( replAx
% 0.22/0.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( exu
% 0.22/0.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.22/0.44 => ? [B: $i] :
% 0.22/0.44 ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 <=> ? [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ A )
% 0.22/0.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(foundationAx_type,type,
% 0.22/0.44 foundationAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(foundationAx,definition,
% 0.22/0.44 ( foundationAx
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.22/0.44 => ? [B: $i] :
% 0.22/0.44 ( ( in @ B @ A )
% 0.22/0.44 & ~ ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(wellorderingAx_type,type,
% 0.22/0.44 wellorderingAx: $o ).
% 0.22/0.44
% 0.22/0.44 thf(wellorderingAx,definition,
% 0.22/0.44 ( wellorderingAx
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ? [B: $i] :
% 0.22/0.44 ( ! [C: $i] :
% 0.22/0.44 ( ( in @ C @ B )
% 0.22/0.44 => ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ C )
% 0.22/0.44 => ( in @ Xx @ A ) ) )
% 0.22/0.44 & ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( ( in @ Xx @ A )
% 0.22/0.44 & ( in @ Xy @ A ) )
% 0.22/0.44 => ( ! [C: $i] :
% 0.22/0.44 ( ( in @ C @ B )
% 0.22/0.44 => ( ( in @ Xx @ C )
% 0.22/0.44 <=> ( in @ Xy @ C ) ) )
% 0.22/0.44 => ( Xx = Xy ) ) )
% 0.22/0.44 & ! [C: $i,D: $i] :
% 0.22/0.44 ( ( ( in @ C @ B )
% 0.22/0.44 & ( in @ D @ B ) )
% 0.22/0.44 => ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ C )
% 0.22/0.44 => ( in @ Xx @ D ) )
% 0.22/0.44 | ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ D )
% 0.22/0.44 => ( in @ Xx @ C ) ) ) )
% 0.22/0.44 & ! [C: $i] :
% 0.22/0.44 ( ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ C )
% 0.22/0.44 => ( in @ Xx @ A ) )
% 0.22/0.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.22/0.44 => ? [D: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ D @ B )
% 0.22/0.44 & ( in @ Xx @ C )
% 0.22/0.44 & ~ ? [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ D )
% 0.22/0.44 & ( in @ Xy @ C ) )
% 0.22/0.44 & ! [E: $i] :
% 0.22/0.44 ( ( in @ E @ B )
% 0.22/0.44 => ( ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ E )
% 0.22/0.44 => ( in @ Xy @ D ) )
% 0.22/0.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(descr_type,type,
% 0.22/0.44 descr: ( $i > $o ) > $i ).
% 0.22/0.44
% 0.22/0.44 thf(descrp_type,type,
% 0.22/0.44 descrp: $o ).
% 0.22/0.44
% 0.22/0.44 thf(descrp,definition,
% 0.22/0.44 ( descrp
% 0.22/0.44 = ( ! [Xphi: $i > $o] :
% 0.22/0.44 ( ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 => ( Xphi
% 0.22/0.44 @ ( descr
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(dsetconstr_type,type,
% 0.22/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.22/0.44
% 0.22/0.44 thf(dsetconstrI_type,type,
% 0.22/0.44 dsetconstrI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(dsetconstrI,definition,
% 0.22/0.44 ( dsetconstrI
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ( Xphi @ Xx )
% 0.22/0.44 => ( in @ Xx
% 0.22/0.44 @ ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(dsetconstrEL_type,type,
% 0.22/0.44 dsetconstrEL: $o ).
% 0.22/0.44
% 0.22/0.44 thf(dsetconstrEL,definition,
% 0.22/0.44 ( dsetconstrEL
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx
% 0.22/0.44 @ ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.22/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(dsetconstrER_type,type,
% 0.22/0.44 dsetconstrER: $o ).
% 0.22/0.44
% 0.22/0.44 thf(dsetconstrER,definition,
% 0.22/0.44 ( dsetconstrER
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx
% 0.22/0.44 @ ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.22/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(exuE1_type,type,
% 0.22/0.44 exuE1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(exuE1,definition,
% 0.22/0.44 ( exuE1
% 0.22/0.44 = ( ! [Xphi: $i > $o] :
% 0.22/0.44 ( ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 => ? [Xx: $i] :
% 0.22/0.44 ( ( Xphi @ Xx )
% 0.22/0.44 & ! [Xy: $i] :
% 0.22/0.44 ( ( Xphi @ Xy )
% 0.22/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(prop2set_type,type,
% 0.22/0.44 prop2set: $o > $i ).
% 0.22/0.44
% 0.22/0.44 thf(prop2setE_type,type,
% 0.22/0.44 prop2setE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(prop2setE,definition,
% 0.22/0.44 ( prop2setE
% 0.22/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.22/0.44 => Xphi ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptysetE_type,type,
% 0.22/0.44 emptysetE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptysetE,definition,
% 0.22/0.44 ( emptysetE
% 0.22/0.44 = ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ emptyset )
% 0.22/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptysetimpfalse_type,type,
% 0.22/0.44 emptysetimpfalse: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptysetimpfalse,definition,
% 0.22/0.44 ( emptysetimpfalse
% 0.22/0.44 = ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ emptyset )
% 0.22/0.44 => $false ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(notinemptyset_type,type,
% 0.22/0.44 notinemptyset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(notinemptyset,definition,
% 0.22/0.44 ( notinemptyset
% 0.22/0.44 = ( ! [Xx: $i] :
% 0.22/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(exuE3e_type,type,
% 0.22/0.44 exuE3e: $o ).
% 0.22/0.44
% 0.22/0.44 thf(exuE3e,definition,
% 0.22/0.44 ( exuE3e
% 0.22/0.44 = ( ! [Xphi: $i > $o] :
% 0.22/0.44 ( ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setext_type,type,
% 0.22/0.44 setext: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setext,definition,
% 0.22/0.44 ( setext
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ Xx @ B ) )
% 0.22/0.44 => ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 => ( in @ Xx @ A ) )
% 0.22/0.44 => ( A = B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptyI_type,type,
% 0.22/0.44 emptyI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptyI,definition,
% 0.22/0.44 ( emptyI
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ~ ( in @ Xx @ A )
% 0.22/0.44 => ( A = emptyset ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(noeltsimpempty_type,type,
% 0.22/0.44 noeltsimpempty: $o ).
% 0.22/0.44
% 0.22/0.44 thf(noeltsimpempty,definition,
% 0.22/0.44 ( noeltsimpempty
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ~ ( in @ Xx @ A )
% 0.22/0.44 => ( A = emptyset ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setbeta_type,type,
% 0.22/0.44 setbeta: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setbeta,definition,
% 0.22/0.44 ( setbeta
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ( in @ Xx
% 0.22/0.44 @ ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.22/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(nonempty_type,type,
% 0.22/0.44 nonempty: $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(nonempty,definition,
% 0.22/0.44 ( nonempty
% 0.22/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(nonemptyE1_type,type,
% 0.22/0.44 nonemptyE1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(nonemptyE1,definition,
% 0.22/0.44 ( nonemptyE1
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ( ( nonempty @ A )
% 0.22/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(nonemptyI_type,type,
% 0.22/0.44 nonemptyI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(nonemptyI,definition,
% 0.22/0.44 ( nonemptyI
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ( Xphi @ Xx )
% 0.22/0.44 => ( nonempty
% 0.22/0.44 @ ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(nonemptyI1_type,type,
% 0.22/0.44 nonemptyI1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(nonemptyI1,definition,
% 0.22/0.44 ( nonemptyI1
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.22/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinIL_type,type,
% 0.22/0.44 setadjoinIL: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinIL,definition,
% 0.22/0.44 ( setadjoinIL
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptyinunitempty_type,type,
% 0.22/0.44 emptyinunitempty: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptyinunitempty,definition,
% 0.22/0.44 ( emptyinunitempty
% 0.22/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinIR_type,type,
% 0.22/0.44 setadjoinIR: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinIR,definition,
% 0.22/0.44 ( setadjoinIR
% 0.22/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ A )
% 0.22/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinE_type,type,
% 0.22/0.44 setadjoinE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinE,definition,
% 0.22/0.44 ( setadjoinE
% 0.22/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.22/0.44 => ! [Xphi: $o] :
% 0.22/0.44 ( ( ( Xy = Xx )
% 0.22/0.44 => Xphi )
% 0.22/0.44 => ( ( ( in @ Xy @ A )
% 0.22/0.44 => Xphi )
% 0.22/0.44 => Xphi ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinOr_type,type,
% 0.22/0.44 setadjoinOr: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinOr,definition,
% 0.22/0.44 ( setadjoinOr
% 0.22/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.22/0.44 => ( ( Xy = Xx )
% 0.22/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setoftrueEq_type,type,
% 0.22/0.44 setoftrueEq: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setoftrueEq,definition,
% 0.22/0.44 ( setoftrueEq
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ( ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xx: $i] : $true )
% 0.22/0.44 = A ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(powersetI_type,type,
% 0.22/0.44 powersetI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(powersetI,definition,
% 0.22/0.44 ( powersetI
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 => ( in @ Xx @ A ) )
% 0.22/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptyinPowerset_type,type,
% 0.22/0.44 emptyinPowerset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptyinPowerset,definition,
% 0.22/0.44 ( emptyinPowerset
% 0.22/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptyInPowerset_type,type,
% 0.22/0.44 emptyInPowerset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptyInPowerset,definition,
% 0.22/0.44 ( emptyInPowerset
% 0.22/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(powersetE_type,type,
% 0.22/0.44 powersetE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(powersetE,definition,
% 0.22/0.44 ( powersetE
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.22/0.44 => ( ( in @ Xx @ B )
% 0.22/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setunionI_type,type,
% 0.22/0.44 setunionI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setunionI,definition,
% 0.22/0.44 ( setunionI
% 0.22/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 => ( ( in @ B @ A )
% 0.22/0.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setunionE_type,type,
% 0.22/0.44 setunionE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setunionE,definition,
% 0.22/0.44 ( setunionE
% 0.22/0.44 = ( ! [A: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.22/0.44 => ! [Xphi: $o] :
% 0.22/0.44 ( ! [B: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 => ( ( in @ B @ A )
% 0.22/0.44 => Xphi ) )
% 0.22/0.44 => Xphi ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(subPowSU_type,type,
% 0.22/0.44 subPowSU: $o ).
% 0.22/0.44
% 0.22/0.44 thf(subPowSU,definition,
% 0.22/0.44 ( subPowSU
% 0.22/0.44 = ( ! [A: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(exuE2_type,type,
% 0.22/0.44 exuE2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(exuE2,definition,
% 0.22/0.44 ( exuE2
% 0.22/0.44 = ( ! [Xphi: $i > $o] :
% 0.22/0.44 ( ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 => ? [Xx: $i] :
% 0.22/0.44 ! [Xy: $i] :
% 0.22/0.44 ( ( Xphi @ Xy )
% 0.22/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(nonemptyImpWitness_type,type,
% 0.22/0.44 nonemptyImpWitness: $o ).
% 0.22/0.44
% 0.22/0.44 thf(nonemptyImpWitness,definition,
% 0.22/0.44 ( nonemptyImpWitness
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ( ( nonempty @ A )
% 0.22/0.44 => ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & $true ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(uniqinunit_type,type,
% 0.22/0.44 uniqinunit: $o ).
% 0.22/0.44
% 0.22/0.44 thf(uniqinunit,definition,
% 0.22/0.44 ( uniqinunit
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.22/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(notinsingleton_type,type,
% 0.22/0.44 notinsingleton: $o ).
% 0.22/0.44
% 0.22/0.44 thf(notinsingleton,definition,
% 0.22/0.44 ( notinsingleton
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xx != Xy )
% 0.22/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(eqinunit_type,type,
% 0.22/0.44 eqinunit: $o ).
% 0.22/0.44
% 0.22/0.44 thf(eqinunit,definition,
% 0.22/0.44 ( eqinunit
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xx = Xy )
% 0.22/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(singletonsswitch_type,type,
% 0.22/0.44 singletonsswitch: $o ).
% 0.22/0.44
% 0.22/0.44 thf(singletonsswitch,definition,
% 0.22/0.44 ( singletonsswitch
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.22/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(upairsetE_type,type,
% 0.22/0.44 upairsetE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(upairsetE,definition,
% 0.22/0.44 ( upairsetE
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.22/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.22/0.44 => ( ( Xz = Xx )
% 0.22/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(upairsetIL_type,type,
% 0.22/0.44 upairsetIL: $o ).
% 0.22/0.44
% 0.22/0.44 thf(upairsetIL,definition,
% 0.22/0.44 ( upairsetIL
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(upairsetIR_type,type,
% 0.22/0.44 upairsetIR: $o ).
% 0.22/0.44
% 0.22/0.44 thf(upairsetIR,definition,
% 0.22/0.44 ( upairsetIR
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptyE1_type,type,
% 0.22/0.44 emptyE1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptyE1,definition,
% 0.22/0.44 ( emptyE1
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ( Xphi @ Xx ) )
% 0.22/0.44 => ( ( ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 = emptyset )
% 0.22/0.44 => $false ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(vacuousDall_type,type,
% 0.22/0.44 vacuousDall: $o ).
% 0.22/0.44
% 0.22/0.44 thf(vacuousDall,definition,
% 0.22/0.44 ( vacuousDall
% 0.22/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ emptyset )
% 0.22/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(quantDeMorgan1_type,type,
% 0.22/0.44 quantDeMorgan1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(quantDeMorgan1,definition,
% 0.22/0.44 ( quantDeMorgan1
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ~ ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( Xphi @ Xx ) )
% 0.22/0.44 => ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(quantDeMorgan2_type,type,
% 0.22/0.44 quantDeMorgan2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(quantDeMorgan2,definition,
% 0.22/0.44 ( quantDeMorgan2
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ~ ( Xphi @ Xx ) )
% 0.22/0.44 => ~ ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(quantDeMorgan3_type,type,
% 0.22/0.44 quantDeMorgan3: $o ).
% 0.22/0.44
% 0.22/0.44 thf(quantDeMorgan3,definition,
% 0.22/0.44 ( quantDeMorgan3
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ~ ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ( Xphi @ Xx ) )
% 0.22/0.44 => ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(quantDeMorgan4_type,type,
% 0.22/0.44 quantDeMorgan4: $o ).
% 0.22/0.44
% 0.22/0.44 thf(quantDeMorgan4,definition,
% 0.22/0.44 ( quantDeMorgan4
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ~ ( Xphi @ Xx ) )
% 0.22/0.44 => ~ ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(prop2setI_type,type,
% 0.22/0.44 prop2setI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(prop2setI,definition,
% 0.22/0.44 ( prop2setI
% 0.22/0.44 = ( ! [Xphi: $o] :
% 0.22/0.44 ( Xphi
% 0.22/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(set2prop_type,type,
% 0.22/0.44 set2prop: $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(prop2set2propI_type,type,
% 0.22/0.44 prop2set2propI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(prop2set2propI,definition,
% 0.22/0.44 ( prop2set2propI
% 0.22/0.44 = ( ! [Xphi: $o] :
% 0.22/0.44 ( Xphi
% 0.22/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(notdexE_type,type,
% 0.22/0.44 notdexE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(notdexE,definition,
% 0.22/0.44 ( notdexE
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ~ ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ( Xphi @ Xx ) )
% 0.22/0.44 => ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(notdallE_type,type,
% 0.22/0.44 notdallE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(notdallE,definition,
% 0.22/0.44 ( notdallE
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ~ ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( Xphi @ Xx ) )
% 0.22/0.44 => ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(exuI1_type,type,
% 0.22/0.44 exuI1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(exuI1,definition,
% 0.22/0.44 ( exuI1
% 0.22/0.44 = ( ! [Xphi: $i > $o] :
% 0.22/0.44 ( ? [Xx: $i] :
% 0.22/0.44 ( ( Xphi @ Xx )
% 0.22/0.44 & ! [Xy: $i] :
% 0.22/0.44 ( ( Xphi @ Xy )
% 0.22/0.44 => ( Xx = Xy ) ) )
% 0.22/0.44 => ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(exuI3_type,type,
% 0.22/0.44 exuI3: $o ).
% 0.22/0.44
% 0.22/0.44 thf(exuI3,definition,
% 0.22/0.44 ( exuI3
% 0.22/0.44 = ( ! [Xphi: $i > $o] :
% 0.22/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.22/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xphi @ Xx )
% 0.22/0.44 => ( ( Xphi @ Xy )
% 0.22/0.44 => ( Xx = Xy ) ) )
% 0.22/0.44 => ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(exuI2_type,type,
% 0.22/0.44 exuI2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(exuI2,definition,
% 0.22/0.44 ( exuI2
% 0.22/0.44 = ( ! [Xphi: $i > $o] :
% 0.22/0.44 ( ? [Xx: $i] :
% 0.22/0.44 ! [Xy: $i] :
% 0.22/0.44 ( ( Xphi @ Xy )
% 0.22/0.44 <=> ( Xy = Xx ) )
% 0.22/0.44 => ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(inCongP_type,type,
% 0.22/0.44 inCongP: $o ).
% 0.22/0.44
% 0.22/0.44 thf(inCongP,definition,
% 0.22/0.44 ( inCongP
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( A = B )
% 0.22/0.44 => ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xx = Xy )
% 0.22/0.44 => ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(in__Cong_type,type,
% 0.22/0.44 in__Cong: $o ).
% 0.22/0.44
% 0.22/0.44 thf(in__Cong,definition,
% 0.22/0.44 ( in__Cong
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( A = B )
% 0.22/0.44 => ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xx = Xy )
% 0.22/0.44 => ( ( in @ Xx @ A )
% 0.22/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(exuE3u_type,type,
% 0.22/0.44 exuE3u: $o ).
% 0.22/0.44
% 0.22/0.44 thf(exuE3u,definition,
% 0.22/0.44 ( exuE3u
% 0.22/0.44 = ( ! [Xphi: $i > $o] :
% 0.22/0.44 ( ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 => ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xphi @ Xx )
% 0.22/0.44 => ( ( Xphi @ Xy )
% 0.22/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(exu__Cong_type,type,
% 0.22/0.44 exu__Cong: $o ).
% 0.22/0.44
% 0.22/0.44 thf(exu__Cong,definition,
% 0.22/0.44 ( exu__Cong
% 0.22/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.22/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xx = Xy )
% 0.22/0.44 => ( ( Xphi @ Xx )
% 0.22/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.22/0.44 => ( ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 <=> ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptyset__Cong_type,type,
% 0.22/0.44 emptyset__Cong: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptyset__Cong,definition,
% 0.22/0.44 ( emptyset__Cong
% 0.22/0.44 = ( emptyset = emptyset ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoin__Cong_type,type,
% 0.22/0.44 setadjoin__Cong: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoin__Cong,definition,
% 0.22/0.44 ( setadjoin__Cong
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xx = Xy )
% 0.22/0.44 => ! [Xz: $i,Xu: $i] :
% 0.22/0.44 ( ( Xz = Xu )
% 0.22/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.22/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(powerset__Cong_type,type,
% 0.22/0.44 powerset__Cong: $o ).
% 0.22/0.44
% 0.22/0.44 thf(powerset__Cong,definition,
% 0.22/0.44 ( powerset__Cong
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( A = B )
% 0.22/0.44 => ( ( powerset @ A )
% 0.22/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setunion__Cong_type,type,
% 0.22/0.44 setunion__Cong: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setunion__Cong,definition,
% 0.22/0.44 ( setunion__Cong
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( A = B )
% 0.22/0.44 => ( ( setunion @ A )
% 0.22/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(omega__Cong_type,type,
% 0.22/0.44 omega__Cong: $o ).
% 0.22/0.44
% 0.22/0.44 thf(omega__Cong,definition,
% 0.22/0.44 ( omega__Cong
% 0.22/0.44 = ( omega = omega ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(exuEu_type,type,
% 0.22/0.44 exuEu: $o ).
% 0.22/0.44
% 0.22/0.44 thf(exuEu,definition,
% 0.22/0.44 ( exuEu
% 0.22/0.44 = ( ! [Xphi: $i > $o] :
% 0.22/0.44 ( ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 => ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xphi @ Xx )
% 0.22/0.44 => ( ( Xphi @ Xy )
% 0.22/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(descr__Cong_type,type,
% 0.22/0.44 descr__Cong: $o ).
% 0.22/0.44
% 0.22/0.44 thf(descr__Cong,definition,
% 0.22/0.44 ( descr__Cong
% 0.22/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.22/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( Xx = Xy )
% 0.22/0.44 => ( ( Xphi @ Xx )
% 0.22/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.22/0.44 => ( ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 => ( ( exu
% 0.22/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.22/0.44 => ( ( descr
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 = ( descr
% 0.22/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(dsetconstr__Cong_type,type,
% 0.22/0.44 dsetconstr__Cong: $o ).
% 0.22/0.44
% 0.22/0.44 thf(dsetconstr__Cong,definition,
% 0.22/0.44 ( dsetconstr__Cong
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( A = B )
% 0.22/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ B )
% 0.22/0.44 => ( ( Xx = Xy )
% 0.22/0.44 => ( ( Xphi @ Xx )
% 0.22/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.22/0.44 => ( ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 = ( dsetconstr @ B
% 0.22/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(subset_type,type,
% 0.22/0.44 subset: $i > $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(disjoint_type,type,
% 0.22/0.44 disjoint: $i > $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(setsmeet_type,type,
% 0.22/0.44 setsmeet: $i > $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(subsetI1_type,type,
% 0.22/0.44 subsetI1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(subsetI1,definition,
% 0.22/0.44 ( subsetI1
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ Xx @ B ) )
% 0.22/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(eqimpsubset2_type,type,
% 0.22/0.44 eqimpsubset2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(eqimpsubset2,definition,
% 0.22/0.44 ( eqimpsubset2
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( A = B )
% 0.22/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(eqimpsubset1_type,type,
% 0.22/0.44 eqimpsubset1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(eqimpsubset1,definition,
% 0.22/0.44 ( eqimpsubset1
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( A = B )
% 0.22/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(subsetI2_type,type,
% 0.22/0.44 subsetI2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(subsetI2,definition,
% 0.22/0.44 ( subsetI2
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ Xx @ B ) )
% 0.22/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(emptysetsubset_type,type,
% 0.22/0.44 emptysetsubset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(emptysetsubset,definition,
% 0.22/0.44 ( emptysetsubset
% 0.22/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(subsetE_type,type,
% 0.22/0.44 subsetE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(subsetE,definition,
% 0.22/0.44 ( subsetE
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( subset @ A @ B )
% 0.22/0.44 => ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(subsetE2_type,type,
% 0.22/0.44 subsetE2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(subsetE2,definition,
% 0.22/0.44 ( subsetE2
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( subset @ A @ B )
% 0.22/0.44 => ( ~ ( in @ Xx @ B )
% 0.22/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(notsubsetI_type,type,
% 0.22/0.44 notsubsetI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(notsubsetI,definition,
% 0.22/0.44 ( notsubsetI
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ~ ( in @ Xx @ B )
% 0.22/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(notequalI1_type,type,
% 0.22/0.44 notequalI1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(notequalI1,definition,
% 0.22/0.44 ( notequalI1
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ~ ( subset @ A @ B )
% 0.22/0.44 => ( A != B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(notequalI2_type,type,
% 0.22/0.44 notequalI2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(notequalI2,definition,
% 0.22/0.44 ( notequalI2
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ~ ( in @ Xx @ B )
% 0.22/0.44 => ( A != B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(subsetRefl_type,type,
% 0.22/0.44 subsetRefl: $o ).
% 0.22/0.44
% 0.22/0.44 thf(subsetRefl,definition,
% 0.22/0.44 ( subsetRefl
% 0.22/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(subsetTrans_type,type,
% 0.22/0.44 subsetTrans: $o ).
% 0.22/0.44
% 0.22/0.44 thf(subsetTrans,definition,
% 0.22/0.44 ( subsetTrans
% 0.22/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.22/0.44 ( ( subset @ A @ B )
% 0.22/0.44 => ( ( subset @ B @ C )
% 0.22/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinSub_type,type,
% 0.22/0.44 setadjoinSub: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinSub,definition,
% 0.22/0.44 ( setadjoinSub
% 0.22/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinSub2_type,type,
% 0.22/0.44 setadjoinSub2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setadjoinSub2,definition,
% 0.22/0.44 ( setadjoinSub2
% 0.22/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.22/0.44 ( ( subset @ A @ B )
% 0.22/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(subset2powerset_type,type,
% 0.22/0.44 subset2powerset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(subset2powerset,definition,
% 0.22/0.44 ( subset2powerset
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( subset @ A @ B )
% 0.22/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setextsub_type,type,
% 0.22/0.44 setextsub: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setextsub,definition,
% 0.22/0.44 ( setextsub
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( subset @ A @ B )
% 0.22/0.44 => ( ( subset @ B @ A )
% 0.22/0.44 => ( A = B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(subsetemptysetimpeq_type,type,
% 0.22/0.44 subsetemptysetimpeq: $o ).
% 0.22/0.44
% 0.22/0.44 thf(subsetemptysetimpeq,definition,
% 0.22/0.44 ( subsetemptysetimpeq
% 0.22/0.44 = ( ! [A: $i] :
% 0.22/0.44 ( ( subset @ A @ emptyset )
% 0.22/0.44 => ( A = emptyset ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(powersetI1_type,type,
% 0.22/0.44 powersetI1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(powersetI1,definition,
% 0.22/0.44 ( powersetI1
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( subset @ B @ A )
% 0.22/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(powersetE1_type,type,
% 0.22/0.44 powersetE1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(powersetE1,definition,
% 0.22/0.44 ( powersetE1
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.22/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(inPowerset_type,type,
% 0.22/0.44 inPowerset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(inPowerset,definition,
% 0.22/0.44 ( inPowerset
% 0.22/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(powersetsubset_type,type,
% 0.22/0.44 powersetsubset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(powersetsubset,definition,
% 0.22/0.44 ( powersetsubset
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( subset @ A @ B )
% 0.22/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(sepInPowerset_type,type,
% 0.22/0.44 sepInPowerset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(sepInPowerset,definition,
% 0.22/0.44 ( sepInPowerset
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( in
% 0.22/0.44 @ ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(sepSubset_type,type,
% 0.22/0.44 sepSubset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(sepSubset,definition,
% 0.22/0.44 ( sepSubset
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( subset
% 0.22/0.44 @ ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 @ A ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binunion_type,type,
% 0.22/0.44 binunion: $i > $i > $i ).
% 0.22/0.44
% 0.22/0.44 thf(binunionIL_type,type,
% 0.22/0.44 binunionIL: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binunionIL,definition,
% 0.22/0.44 ( binunionIL
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(upairset2IR_type,type,
% 0.22/0.44 upairset2IR: $o ).
% 0.22/0.44
% 0.22/0.44 thf(upairset2IR,definition,
% 0.22/0.44 ( upairset2IR
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binunionIR_type,type,
% 0.22/0.44 binunionIR: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binunionIR,definition,
% 0.22/0.44 ( binunionIR
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binunionEcases_type,type,
% 0.22/0.44 binunionEcases: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binunionEcases,definition,
% 0.22/0.44 ( binunionEcases
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.22/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.22/0.44 => ( ( ( in @ Xx @ A )
% 0.22/0.44 => Xphi )
% 0.22/0.44 => ( ( ( in @ Xx @ B )
% 0.22/0.44 => Xphi )
% 0.22/0.44 => Xphi ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binunionE_type,type,
% 0.22/0.44 binunionE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binunionE,definition,
% 0.22/0.44 ( binunionE
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.22/0.44 => ( ( in @ Xx @ A )
% 0.22/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binunionLsub_type,type,
% 0.22/0.44 binunionLsub: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binunionLsub,definition,
% 0.22/0.44 ( binunionLsub
% 0.22/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binunionRsub_type,type,
% 0.22/0.44 binunionRsub: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binunionRsub,definition,
% 0.22/0.44 ( binunionRsub
% 0.22/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersect_type,type,
% 0.22/0.44 binintersect: $i > $i > $i ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectI_type,type,
% 0.22/0.44 binintersectI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectI,definition,
% 0.22/0.44 ( binintersectI
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ( in @ Xx @ B )
% 0.22/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset5_type,type,
% 0.22/0.44 binintersectSubset5: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset5,definition,
% 0.22/0.44 ( binintersectSubset5
% 0.22/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.22/0.44 ( ( subset @ C @ A )
% 0.22/0.44 => ( ( subset @ C @ B )
% 0.22/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectEL_type,type,
% 0.22/0.44 binintersectEL: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectEL,definition,
% 0.22/0.44 ( binintersectEL
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.22/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectLsub_type,type,
% 0.22/0.44 binintersectLsub: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectLsub,definition,
% 0.22/0.44 ( binintersectLsub
% 0.22/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset2_type,type,
% 0.22/0.44 binintersectSubset2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset2,definition,
% 0.22/0.44 ( binintersectSubset2
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( subset @ A @ B )
% 0.22/0.44 => ( ( binintersect @ A @ B )
% 0.22/0.44 = A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset3_type,type,
% 0.22/0.44 binintersectSubset3: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset3,definition,
% 0.22/0.44 ( binintersectSubset3
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( ( binintersect @ A @ B )
% 0.22/0.44 = B )
% 0.22/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectER_type,type,
% 0.22/0.44 binintersectER: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectER,definition,
% 0.22/0.44 ( binintersectER
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.22/0.44 => ( in @ Xx @ B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(disjointsetsI1_type,type,
% 0.22/0.44 disjointsetsI1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(disjointsetsI1,definition,
% 0.22/0.44 ( disjointsetsI1
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ~ ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ( in @ Xx @ B ) )
% 0.22/0.44 => ( ( binintersect @ A @ B )
% 0.22/0.44 = emptyset ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectRsub_type,type,
% 0.22/0.44 binintersectRsub: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectRsub,definition,
% 0.22/0.44 ( binintersectRsub
% 0.22/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset4_type,type,
% 0.22/0.44 binintersectSubset4: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset4,definition,
% 0.22/0.44 ( binintersectSubset4
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( subset @ B @ A )
% 0.22/0.44 => ( ( binintersect @ A @ B )
% 0.22/0.44 = B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset1_type,type,
% 0.22/0.44 binintersectSubset1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(binintersectSubset1,definition,
% 0.22/0.44 ( binintersectSubset1
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( ( binintersect @ A @ B )
% 0.22/0.44 = A )
% 0.22/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(bs114d_type,type,
% 0.22/0.44 bs114d: $o ).
% 0.22/0.44
% 0.22/0.44 thf(bs114d,definition,
% 0.22/0.44 ( bs114d
% 0.22/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.22/0.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.22/0.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(regular_type,type,
% 0.22/0.44 regular: $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminus_type,type,
% 0.22/0.44 setminus: $i > $i > $i ).
% 0.22/0.44
% 0.22/0.44 thf(setminusI_type,type,
% 0.22/0.44 setminusI: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusI,definition,
% 0.22/0.44 ( setminusI
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ~ ( in @ Xx @ B )
% 0.22/0.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setminusEL_type,type,
% 0.22/0.44 setminusEL: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusEL,definition,
% 0.22/0.44 ( setminusEL
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.22/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setminusER_type,type,
% 0.22/0.44 setminusER: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusER,definition,
% 0.22/0.44 ( setminusER
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.22/0.44 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setminusSubset2_type,type,
% 0.22/0.44 setminusSubset2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusSubset2,definition,
% 0.22/0.44 ( setminusSubset2
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( subset @ A @ B )
% 0.22/0.44 => ( ( setminus @ A @ B )
% 0.22/0.44 = emptyset ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setminusERneg_type,type,
% 0.22/0.44 setminusERneg: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusERneg,definition,
% 0.22/0.44 ( setminusERneg
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.22/0.44 => ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setminusELneg_type,type,
% 0.22/0.44 setminusELneg: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusELneg,definition,
% 0.22/0.44 ( setminusELneg
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.22/0.44 => ( ~ ( in @ Xx @ B )
% 0.22/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setminusILneg_type,type,
% 0.22/0.44 setminusILneg: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusILneg,definition,
% 0.22/0.44 ( setminusILneg
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ~ ( in @ Xx @ A )
% 0.22/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setminusIRneg_type,type,
% 0.22/0.44 setminusIRneg: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusIRneg,definition,
% 0.22/0.44 ( setminusIRneg
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ B )
% 0.22/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setminusLsub_type,type,
% 0.22/0.44 setminusLsub: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusLsub,definition,
% 0.22/0.44 ( setminusLsub
% 0.22/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setminusSubset1_type,type,
% 0.22/0.44 setminusSubset1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setminusSubset1,definition,
% 0.22/0.44 ( setminusSubset1
% 0.22/0.44 = ( ! [A: $i,B: $i] :
% 0.22/0.44 ( ( ( setminus @ A @ B )
% 0.22/0.44 = emptyset )
% 0.22/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(symdiff_type,type,
% 0.22/0.44 symdiff: $i > $i > $i ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffE_type,type,
% 0.22/0.44 symdiffE: $o ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffE,definition,
% 0.22/0.44 ( symdiffE
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.22/0.44 => ! [Xphi: $o] :
% 0.22/0.44 ( ( ( in @ Xx @ A )
% 0.22/0.44 => ( ~ ( in @ Xx @ B )
% 0.22/0.44 => Xphi ) )
% 0.22/0.44 => ( ( ~ ( in @ Xx @ A )
% 0.22/0.44 => ( ( in @ Xx @ B )
% 0.22/0.44 => Xphi ) )
% 0.22/0.44 => Xphi ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffI1_type,type,
% 0.22/0.44 symdiffI1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffI1,definition,
% 0.22/0.44 ( symdiffI1
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ~ ( in @ Xx @ B )
% 0.22/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffI2_type,type,
% 0.22/0.44 symdiffI2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffI2,definition,
% 0.22/0.44 ( symdiffI2
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ~ ( in @ Xx @ A )
% 0.22/0.44 => ( ( in @ Xx @ B )
% 0.22/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffIneg1_type,type,
% 0.22/0.44 symdiffIneg1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffIneg1,definition,
% 0.22/0.44 ( symdiffIneg1
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ( in @ Xx @ B )
% 0.22/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffIneg2_type,type,
% 0.22/0.44 symdiffIneg2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(symdiffIneg2,definition,
% 0.22/0.44 ( symdiffIneg2
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ~ ( in @ Xx @ A )
% 0.22/0.44 => ( ~ ( in @ Xx @ B )
% 0.22/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(iskpair_type,type,
% 0.22/0.44 iskpair: $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(secondinupair_type,type,
% 0.22/0.44 secondinupair: $o ).
% 0.22/0.44
% 0.22/0.44 thf(secondinupair,definition,
% 0.22/0.44 ( secondinupair
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setukpairIL_type,type,
% 0.22/0.44 setukpairIL: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setukpairIL,definition,
% 0.22/0.44 ( setukpairIL
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setukpairIR_type,type,
% 0.22/0.44 setukpairIR: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setukpairIR,definition,
% 0.22/0.44 ( setukpairIR
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(kpairiskpair_type,type,
% 0.22/0.44 kpairiskpair: $o ).
% 0.22/0.44
% 0.22/0.44 thf(kpairiskpair,definition,
% 0.22/0.44 ( kpairiskpair
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(kpair_type,type,
% 0.22/0.44 kpair: $i > $i > $i ).
% 0.22/0.44
% 0.22/0.44 thf(kpairp_type,type,
% 0.22/0.44 kpairp: $o ).
% 0.22/0.44
% 0.22/0.44 thf(kpairp,definition,
% 0.22/0.44 ( kpairp
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(cartprod_type,type,
% 0.22/0.44 cartprod: $i > $i > $i ).
% 0.22/0.44
% 0.22/0.44 thf(singletonsubset_type,type,
% 0.22/0.44 singletonsubset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(singletonsubset,definition,
% 0.22/0.44 ( singletonsubset
% 0.22/0.44 = ( ! [A: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(singletoninpowerset_type,type,
% 0.22/0.44 singletoninpowerset: $o ).
% 0.22/0.44
% 0.22/0.44 thf(singletoninpowerset,definition,
% 0.22/0.44 ( singletoninpowerset
% 0.22/0.44 = ( ! [A: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(singletoninpowunion_type,type,
% 0.22/0.44 singletoninpowunion: $o ).
% 0.22/0.44
% 0.22/0.44 thf(singletoninpowunion,definition,
% 0.22/0.44 ( singletoninpowunion
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(upairset2E_type,type,
% 0.22/0.44 upairset2E: $o ).
% 0.22/0.44
% 0.22/0.44 thf(upairset2E,definition,
% 0.22/0.44 ( upairset2E
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.22/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.22/0.44 => ( ( Xz = Xx )
% 0.22/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(upairsubunion_type,type,
% 0.22/0.44 upairsubunion: $o ).
% 0.22/0.44
% 0.22/0.44 thf(upairsubunion,definition,
% 0.22/0.44 ( upairsubunion
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ B )
% 0.22/0.44 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(upairinpowunion_type,type,
% 0.22/0.44 upairinpowunion: $o ).
% 0.22/0.44
% 0.22/0.44 thf(upairinpowunion,definition,
% 0.22/0.44 ( upairinpowunion
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ B )
% 0.22/0.44 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(ubforcartprodlem1_type,type,
% 0.22/0.44 ubforcartprodlem1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(ubforcartprodlem1,definition,
% 0.22/0.44 ( ubforcartprodlem1
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ B )
% 0.22/0.44 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(ubforcartprodlem2_type,type,
% 0.22/0.44 ubforcartprodlem2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(ubforcartprodlem2,definition,
% 0.22/0.44 ( ubforcartprodlem2
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ B )
% 0.22/0.44 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(ubforcartprodlem3_type,type,
% 0.22/0.44 ubforcartprodlem3: $o ).
% 0.22/0.44
% 0.22/0.44 thf(ubforcartprodlem3,definition,
% 0.22/0.44 ( ubforcartprodlem3
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ B )
% 0.22/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(cartprodpairin_type,type,
% 0.22/0.44 cartprodpairin: $o ).
% 0.22/0.44
% 0.22/0.44 thf(cartprodpairin,definition,
% 0.22/0.44 ( cartprodpairin
% 0.22/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ B )
% 0.22/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(cartprodmempair1_type,type,
% 0.22/0.44 cartprodmempair1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(cartprodmempair1,definition,
% 0.22/0.44 ( cartprodmempair1
% 0.22/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.22/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.22/0.44 => ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ? [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ B )
% 0.22/0.44 & ( Xu
% 0.22/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(cartprodmempair_type,type,
% 0.22/0.44 cartprodmempair: $o ).
% 0.22/0.44
% 0.22/0.44 thf(cartprodmempair,definition,
% 0.22/0.44 ( cartprodmempair
% 0.22/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.22/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.22/0.44 => ( iskpair @ Xu ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setunionE2_type,type,
% 0.22/0.44 setunionE2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setunionE2,definition,
% 0.22/0.44 ( setunionE2
% 0.22/0.44 = ( ! [A: $i,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.22/0.44 => ? [X: $i] :
% 0.22/0.44 ( ( in @ X @ A )
% 0.22/0.44 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setunionsingleton1_type,type,
% 0.22/0.44 setunionsingleton1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setunionsingleton1,definition,
% 0.22/0.44 ( setunionsingleton1
% 0.22/0.44 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setunionsingleton2_type,type,
% 0.22/0.44 setunionsingleton2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setunionsingleton2,definition,
% 0.22/0.44 ( setunionsingleton2
% 0.22/0.44 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(setunionsingleton_type,type,
% 0.22/0.44 setunionsingleton: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setunionsingleton,definition,
% 0.22/0.44 ( setunionsingleton
% 0.22/0.44 = ( ! [Xx: $i] :
% 0.22/0.44 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.22/0.44 = Xx ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(singleton_type,type,
% 0.22/0.44 singleton: $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(singleton,definition,
% 0.22/0.44 ( singleton
% 0.22/0.44 = ( ^ [A: $i] :
% 0.22/0.44 ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ( A
% 0.22/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(singletonprop_type,type,
% 0.22/0.44 singletonprop: $o ).
% 0.22/0.44
% 0.22/0.44 thf(singletonprop,definition,
% 0.22/0.44 ( singletonprop
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ A )
% 0.22/0.44 => ( ( Xphi @ Xx )
% 0.22/0.44 => ( ( Xphi @ Xy )
% 0.22/0.44 => ( Xx = Xy ) ) ) ) )
% 0.22/0.44 => ( ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ( Xphi @ Xx ) )
% 0.22/0.44 => ( singleton
% 0.22/0.44 @ ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(ex1_type,type,
% 0.22/0.44 ex1: $i > ( $i > $o ) > $o ).
% 0.22/0.44
% 0.22/0.44 thf(ex1,definition,
% 0.22/0.44 ( ex1
% 0.22/0.44 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( singleton
% 0.22/0.44 @ ( dsetconstr @ A
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(ex1E1_type,type,
% 0.22/0.44 ex1E1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(ex1E1,definition,
% 0.22/0.44 ( ex1E1
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ( ex1 @ A
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.22/0.44 => ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(ex1I_type,type,
% 0.22/0.44 ex1I: $o ).
% 0.22/0.44
% 0.22/0.44 thf(ex1I,definition,
% 0.22/0.44 ( ex1I
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ( ( Xphi @ Xx )
% 0.22/0.44 => ( ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ A )
% 0.22/0.44 => ( ( Xphi @ Xy )
% 0.22/0.44 => ( Xy = Xx ) ) )
% 0.22/0.44 => ( ex1 @ A
% 0.22/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(ex1I2_type,type,
% 0.22/0.44 ex1I2: $o ).
% 0.22/0.44
% 0.22/0.44 thf(ex1I2,definition,
% 0.22/0.44 ( ex1I2
% 0.22/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.22/0.44 ( ! [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 => ! [Xy: $i] :
% 0.22/0.44 ( ( in @ Xy @ A )
% 0.22/0.44 => ( ( Xphi @ Xx )
% 0.22/0.44 => ( ( Xphi @ Xy )
% 0.22/0.44 => ( Xx = Xy ) ) ) ) )
% 0.22/0.44 => ( ? [Xx: $i] :
% 0.22/0.44 ( ( in @ Xx @ A )
% 0.22/0.44 & ( Xphi @ Xx ) )
% 0.22/0.44 => ( ex1 @ A
% 0.22/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(singletonsuniq_type,type,
% 0.22/0.44 singletonsuniq: $o ).
% 0.22/0.44
% 0.22/0.44 thf(singletonsuniq,definition,
% 0.22/0.44 ( singletonsuniq
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.44 ( ( ( setadjoin @ Xx @ emptyset )
% 0.22/0.44 = ( setadjoin @ Xy @ emptyset ) )
% 0.22/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(atmost1p_type,type,
% 0.22/0.44 atmost1p: $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(atleast2p_type,type,
% 0.22/0.44 atleast2p: $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(atmost2p_type,type,
% 0.22/0.44 atmost2p: $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(upairsetp_type,type,
% 0.22/0.44 upairsetp: $i > $o ).
% 0.22/0.44
% 0.22/0.44 thf(setukpairinjL1_type,type,
% 0.22/0.44 setukpairinjL1: $o ).
% 0.22/0.44
% 0.22/0.44 thf(setukpairinjL1,definition,
% 0.22/0.44 ( setukpairinjL1
% 0.22/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.22/0.44 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.22/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(kfstsingleton_type,type,
% 0.22/0.44 kfstsingleton: $o ).
% 0.22/0.44
% 0.22/0.44 thf(kfstsingleton,definition,
% 0.22/0.44 ( kfstsingleton
% 0.22/0.44 = ( ! [Xu: $i] :
% 0.22/0.44 ( ( iskpair @ Xu )
% 0.22/0.44 => ( singleton
% 0.22/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.22/0.44 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(theprop_type,type,
% 0.22/0.44 theprop: $o ).
% 0.22/0.44
% 0.22/0.44 thf(theprop,definition,
% 0.22/0.44 ( theprop
% 0.22/0.44 = ( ! [X: $i] :
% 0.22/0.44 ( ( singleton @ X )
% 0.22/0.44 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.22/0.44
% 0.22/0.44 thf(kfst_type,type,
% 0.22/0.45 kfst: $i > $i ).
% 0.22/0.45
% 0.22/0.45 thf(kfstpairEq_type,type,
% 0.22/0.45 kfstpairEq: $o ).
% 0.22/0.45
% 0.22/0.45 thf(kfstpairEq,definition,
% 0.22/0.45 ( kfstpairEq
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.45 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.22/0.45 = Xx ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(cartprodfstin_type,type,
% 0.22/0.45 cartprodfstin: $o ).
% 0.22/0.45
% 0.22/0.45 thf(cartprodfstin,definition,
% 0.22/0.45 ( cartprodfstin
% 0.22/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.22/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.22/0.45 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjL2_type,type,
% 0.22/0.45 setukpairinjL2: $o ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjL2,definition,
% 0.22/0.45 ( setukpairinjL2
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.22/0.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.22/0.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.22/0.45 => ( Xx = Xz ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjL_type,type,
% 0.22/0.45 setukpairinjL: $o ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjL,definition,
% 0.22/0.45 ( setukpairinjL
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.22/0.45 ( ( ( kpair @ Xx @ Xy )
% 0.22/0.45 = ( kpair @ Xz @ Xu ) )
% 0.22/0.45 => ( Xx = Xz ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR11_type,type,
% 0.22/0.45 setukpairinjR11: $o ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR11,definition,
% 0.22/0.45 ( setukpairinjR11
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.45 ( ( Xx = Xy )
% 0.22/0.45 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.22/0.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR12_type,type,
% 0.22/0.45 setukpairinjR12: $o ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR12,definition,
% 0.22/0.45 ( setukpairinjR12
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.45 ( ( Xx = Xy )
% 0.22/0.45 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.22/0.45 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR1_type,type,
% 0.22/0.45 setukpairinjR1: $o ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR1,definition,
% 0.22/0.45 ( setukpairinjR1
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.22/0.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.22/0.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.22/0.45 => ( ( Xz = Xu )
% 0.22/0.45 => ( Xy = Xu ) ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(upairequniteq_type,type,
% 0.22/0.45 upairequniteq: $o ).
% 0.22/0.45
% 0.22/0.45 thf(upairequniteq,definition,
% 0.22/0.45 ( upairequniteq
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.22/0.45 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.22/0.45 = ( setadjoin @ Xz @ emptyset ) )
% 0.22/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR2_type,type,
% 0.22/0.45 setukpairinjR2: $o ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR2,definition,
% 0.22/0.45 ( setukpairinjR2
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.22/0.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.22/0.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.22/0.45 => ( Xy = Xu ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR_type,type,
% 0.22/0.45 setukpairinjR: $o ).
% 0.22/0.45
% 0.22/0.45 thf(setukpairinjR,definition,
% 0.22/0.45 ( setukpairinjR
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.22/0.45 ( ( ( kpair @ Xx @ Xy )
% 0.22/0.45 = ( kpair @ Xz @ Xu ) )
% 0.22/0.45 => ( Xy = Xu ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(ksndsingleton_type,type,
% 0.22/0.45 ksndsingleton: $o ).
% 0.22/0.45
% 0.22/0.45 thf(ksndsingleton,definition,
% 0.22/0.45 ( ksndsingleton
% 0.22/0.45 = ( ! [Xu: $i] :
% 0.22/0.45 ( ( iskpair @ Xu )
% 0.22/0.45 => ( singleton
% 0.22/0.45 @ ( dsetconstr @ ( setunion @ Xu )
% 0.22/0.45 @ ^ [Xx: $i] :
% 0.22/0.45 ( Xu
% 0.22/0.45 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(ksnd_type,type,
% 0.22/0.45 ksnd: $i > $i ).
% 0.22/0.45
% 0.22/0.45 thf(ksndpairEq_type,type,
% 0.22/0.45 ksndpairEq: $o ).
% 0.22/0.45
% 0.22/0.45 thf(ksndpairEq,definition,
% 0.22/0.45 ( ksndpairEq
% 0.22/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.22/0.45 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.22/0.45 = Xy ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(kpairsurjEq_type,type,
% 0.22/0.45 kpairsurjEq: $o ).
% 0.22/0.45
% 0.22/0.45 thf(kpairsurjEq,definition,
% 0.22/0.45 ( kpairsurjEq
% 0.22/0.45 = ( ! [Xu: $i] :
% 0.22/0.45 ( ( iskpair @ Xu )
% 0.22/0.45 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.22/0.45 = Xu ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(cartprodsndin_type,type,
% 0.22/0.45 cartprodsndin: $o ).
% 0.22/0.45
% 0.22/0.45 thf(cartprodsndin,definition,
% 0.22/0.45 ( cartprodsndin
% 0.22/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.22/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.22/0.45 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(cartprodpairmemEL_type,type,
% 0.22/0.45 cartprodpairmemEL: $o ).
% 0.22/0.45
% 0.22/0.45 thf(cartprodpairmemEL,definition,
% 0.22/0.45 ( cartprodpairmemEL
% 0.22/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.22/0.45 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.22/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 thf(cartprodpairmemER,conjecture,
% 0.22/0.45 ( setextAx
% 0.22/0.45 => ( emptysetAx
% 0.22/0.45 => ( setadjoinAx
% 0.22/0.45 => ( powersetAx
% 0.22/0.45 => ( setunionAx
% 0.22/0.45 => ( omega0Ax
% 0.22/0.45 => ( omegaSAx
% 0.22/0.45 => ( omegaIndAx
% 0.22/0.45 => ( replAx
% 0.22/0.45 => ( foundationAx
% 0.22/0.45 => ( wellorderingAx
% 0.22/0.45 => ( descrp
% 0.22/0.45 => ( dsetconstrI
% 0.22/0.45 => ( dsetconstrEL
% 0.22/0.45 => ( dsetconstrER
% 0.22/0.45 => ( exuE1
% 0.22/0.45 => ( prop2setE
% 0.22/0.45 => ( emptysetE
% 0.22/0.45 => ( emptysetimpfalse
% 0.22/0.45 => ( notinemptyset
% 0.22/0.45 => ( exuE3e
% 0.22/0.45 => ( setext
% 0.22/0.45 => ( emptyI
% 0.22/0.45 => ( noeltsimpempty
% 0.22/0.45 => ( setbeta
% 0.22/0.45 => ( nonemptyE1
% 0.22/0.45 => ( nonemptyI
% 0.22/0.45 => ( nonemptyI1
% 0.22/0.45 => ( setadjoinIL
% 0.22/0.45 => ( emptyinunitempty
% 0.22/0.45 => ( setadjoinIR
% 0.22/0.45 => ( setadjoinE
% 0.22/0.45 => ( setadjoinOr
% 0.22/0.45 => ( setoftrueEq
% 0.22/0.45 => ( powersetI
% 0.22/0.45 => ( emptyinPowerset
% 0.22/0.45 => ( emptyInPowerset
% 0.22/0.45 => ( powersetE
% 0.22/0.45 => ( setunionI
% 0.22/0.45 => ( setunionE
% 0.22/0.45 => ( subPowSU
% 0.22/0.45 => ( exuE2
% 0.22/0.45 => ( nonemptyImpWitness
% 0.22/0.45 => ( uniqinunit
% 0.22/0.45 => ( notinsingleton
% 0.22/0.45 => ( eqinunit
% 0.22/0.45 => ( singletonsswitch
% 0.22/0.45 => ( upairsetE
% 0.22/0.45 => ( upairsetIL
% 0.22/0.45 => ( upairsetIR
% 0.22/0.45 => ( emptyE1
% 0.22/0.45 => ( vacuousDall
% 0.22/0.45 => ( quantDeMorgan1
% 0.22/0.45 => ( quantDeMorgan2
% 0.22/0.45 => ( quantDeMorgan3
% 0.22/0.45 => ( quantDeMorgan4
% 0.22/0.45 => ( prop2setI
% 0.22/0.45 => ( prop2set2propI
% 0.22/0.45 => ( notdexE
% 0.22/0.45 => ( notdallE
% 0.22/0.45 => ( exuI1
% 0.22/0.45 => ( exuI3
% 0.22/0.45 => ( exuI2
% 0.22/0.45 => ( inCongP
% 0.22/0.45 => ( in__Cong
% 0.22/0.45 => ( exuE3u
% 0.22/0.45 => ( exu__Cong
% 0.22/0.45 => ( emptyset__Cong
% 0.22/0.45 => ( setadjoin__Cong
% 0.22/0.45 => ( powerset__Cong
% 0.22/0.45 => ( setunion__Cong
% 0.22/0.45 => ( omega__Cong
% 0.22/0.45 => ( exuEu
% 0.22/0.45 => ( descr__Cong
% 0.22/0.45 => ( dsetconstr__Cong
% 0.22/0.45 => ( subsetI1
% 0.22/0.45 => ( eqimpsubset2
% 0.22/0.45 => ( eqimpsubset1
% 0.22/0.45 => ( subsetI2
% 0.22/0.45 => ( emptysetsubset
% 0.22/0.45 => ( subsetE
% 0.22/0.45 => ( subsetE2
% 0.22/0.45 => ( notsubsetI
% 0.22/0.45 => ( notequalI1
% 0.22/0.45 => ( notequalI2
% 0.22/0.45 => ( subsetRefl
% 0.22/0.45 => ( subsetTrans
% 0.22/0.45 => ( setadjoinSub
% 0.22/0.45 => ( setadjoinSub2
% 0.22/0.45 => ( subset2powerset
% 0.22/0.45 => ( setextsub
% 0.22/0.45 => ( subsetemptysetimpeq
% 0.22/0.45 => ( powersetI1
% 0.22/0.45 => ( powersetE1
% 0.22/0.45 => ( inPowerset
% 0.22/0.45 => ( powersetsubset
% 0.22/0.45 => ( sepInPowerset
% 0.22/0.45 => ( sepSubset
% 0.22/0.45 => ( binunionIL
% 0.22/0.45 => ( upairset2IR
% 0.22/0.45 => ( binunionIR
% 0.22/0.45 => ( binunionEcases
% 0.22/0.45 => ( binunionE
% 0.22/0.45 => ( binunionLsub
% 0.22/0.45 => ( binunionRsub
% 0.22/0.45 => ( binintersectI
% 0.22/0.45 => ( binintersectSubset5
% 0.22/0.45 => ( binintersectEL
% 0.22/0.45 => ( binintersectLsub
% 0.22/0.45 => ( binintersectSubset2
% 0.22/0.45 => ( binintersectSubset3
% 0.22/0.45 => ( binintersectER
% 0.22/0.45 => ( disjointsetsI1
% 0.22/0.45 => ( binintersectRsub
% 0.22/0.45 => ( binintersectSubset4
% 0.22/0.45 => ( binintersectSubset1
% 0.22/0.45 => ( bs114d
% 0.22/0.45 => ( setminusI
% 0.22/0.45 => ( setminusEL
% 0.22/0.45 => ( setminusER
% 0.22/0.45 => ( setminusSubset2
% 0.22/0.45 => ( setminusERneg
% 0.22/0.45 => ( setminusELneg
% 0.22/0.45 => ( setminusILneg
% 0.22/0.45 => ( setminusIRneg
% 0.22/0.45 => ( setminusLsub
% 0.22/0.45 => ( setminusSubset1
% 0.22/0.45 => ( symdiffE
% 0.22/0.45 => ( symdiffI1
% 0.22/0.45 => ( symdiffI2
% 0.22/0.45 => ( symdiffIneg1
% 0.22/0.45 => ( symdiffIneg2
% 0.22/0.45 => ( secondinupair
% 0.22/0.45 => ( setukpairIL
% 0.22/0.45 => ( setukpairIR
% 0.22/0.45 => ( kpairiskpair
% 0.22/0.45 => ( kpairp
% 0.22/0.45 => ( singletonsubset
% 0.22/0.45 => ( singletoninpowerset
% 0.22/0.45 => ( singletoninpowunion
% 0.22/0.45 => ( upairset2E
% 0.22/0.45 => ( upairsubunion
% 0.22/0.45 => ( upairinpowunion
% 0.22/0.45 => ( ubforcartprodlem1
% 0.22/0.45 => ( ubforcartprodlem2
% 0.22/0.45 => ( ubforcartprodlem3
% 0.22/0.45 => ( cartprodpairin
% 0.22/0.45 => ( cartprodmempair1
% 0.22/0.45 => ( cartprodmempair
% 0.22/0.45 => ( setunionE2
% 0.22/0.45 => ( setunionsingleton1
% 0.22/0.45 => ( setunionsingleton2
% 0.22/0.45 => ( setunionsingleton
% 0.22/0.45 => ( singletonprop
% 0.22/0.45 => ( ex1E1
% 0.22/0.45 => ( ex1I
% 0.22/0.45 => ( ex1I2
% 0.22/0.45 => ( singletonsuniq
% 0.22/0.45 => ( setukpairinjL1
% 0.22/0.45 => ( kfstsingleton
% 0.22/0.45 => ( theprop
% 0.22/0.45 => ( kfstpairEq
% 0.22/0.45 => ( cartprodfstin
% 0.22/0.45 => ( setukpairinjL2
% 0.22/0.45 => ( setukpairinjL
% 0.22/0.45 => ( setukpairinjR11
% 0.22/0.45 => ( setukpairinjR12
% 0.22/0.45 => ( setukpairinjR1
% 0.22/0.45 => ( upairequniteq
% 0.22/0.45 => ( setukpairinjR2
% 0.22/0.45 => ( setukpairinjR
% 0.22/0.45 => ( ksndsingleton
% 0.22/0.45 => ( ksndpairEq
% 0.22/0.45 => ( kpairsurjEq
% 0.22/0.45 => ( cartprodsndin
% 0.22/0.45 => ( cartprodpairmemEL
% 0.22/0.45 => ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.22/0.45 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.22/0.45 => ( in @ Xy @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.22/0.45
% 0.22/0.45 %------------------------------------------------------------------------------
% 0.22/0.45 ------------------------
% 2.75/1.65 ---- Embedded in THF ---
% 2.75/1.65 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.75/1.65 %%% Generated on Tue Feb 24 12:29:43 EST 2026
% 2.75/1.65 %%% using '$$dhol' embedding, version 1.3.0.
% 2.75/1.65 %%% Logic specification used:
% 2.75/1.65 %%% thf(spec, logic, $$dhol).
% 2.75/1.65
% 2.75/1.65 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2DTF16138.p
% See solution above
% 2.75/1.66 ------------------------
% 2.75/1.69 ---- Cleaned THF ---
% 2.75/1.69 thf(in_type,type,
% 2.75/1.69 in: $i > $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(exu_type,type,
% 2.75/1.69 exu: ( $i > $o ) > $o ).
% 2.75/1.69
% 2.75/1.69 thf(exu,definition,
% 2.75/1.69 ( exu
% 2.75/1.69 = ( ^ [Xphi: $i > $o] :
% 2.75/1.69 ? [Xx: $i] :
% 2.75/1.69 ( ( Xphi @ Xx )
% 2.75/1.69 & ! [Xy: $i] :
% 2.75/1.69 ( ( Xphi @ Xy )
% 2.75/1.69 => ( Xx = Xy ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setextAx_type,type,
% 2.75/1.69 setextAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setextAx,definition,
% 2.75/1.69 ( setextAx
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 <=> ( in @ Xx @ B ) )
% 2.75/1.69 => ( A = B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptyset_type,type,
% 2.75/1.69 emptyset: $i ).
% 2.75/1.69
% 2.75/1.69 thf(emptysetAx_type,type,
% 2.75/1.69 emptysetAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptysetAx,definition,
% 2.75/1.69 ( emptysetAx
% 2.75/1.69 = ( ! [Xx: $i] :
% 2.75/1.69 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoin_type,type,
% 2.75/1.69 setadjoin: $i > $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinAx_type,type,
% 2.75/1.69 setadjoinAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinAx,definition,
% 2.75/1.69 ( setadjoinAx
% 2.75/1.69 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.75/1.69 <=> ( ( Xy = Xx )
% 2.75/1.69 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(powerset_type,type,
% 2.75/1.69 powerset: $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(powersetAx_type,type,
% 2.75/1.69 powersetAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(powersetAx,definition,
% 2.75/1.69 ( powersetAx
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( in @ B @ ( powerset @ A ) )
% 2.75/1.69 <=> ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setunion_type,type,
% 2.75/1.69 setunion: $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(setunionAx_type,type,
% 2.75/1.69 setunionAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setunionAx,definition,
% 2.75/1.69 ( setunionAx
% 2.75/1.69 = ( ! [A: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( setunion @ A ) )
% 2.75/1.69 <=> ? [B: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 & ( in @ B @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(omega_type,type,
% 2.75/1.69 omega: $i ).
% 2.75/1.69
% 2.75/1.69 thf(omega0Ax_type,type,
% 2.75/1.69 omega0Ax: $o ).
% 2.75/1.69
% 2.75/1.69 thf(omega0Ax,definition,
% 2.75/1.69 ( omega0Ax
% 2.75/1.69 = ( in @ emptyset @ omega ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(omegaSAx_type,type,
% 2.75/1.69 omegaSAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(omegaSAx,definition,
% 2.75/1.69 ( omegaSAx
% 2.75/1.69 = ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ omega )
% 2.75/1.69 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(omegaIndAx_type,type,
% 2.75/1.69 omegaIndAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(omegaIndAx,definition,
% 2.75/1.69 ( omegaIndAx
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ( ( ( in @ emptyset @ A )
% 2.75/1.69 & ! [Xx: $i] :
% 2.75/1.69 ( ( ( in @ Xx @ omega )
% 2.75/1.69 & ( in @ Xx @ A ) )
% 2.75/1.69 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.75/1.69 => ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ omega )
% 2.75/1.69 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(replAx_type,type,
% 2.75/1.69 replAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(replAx,definition,
% 2.75/1.69 ( replAx
% 2.75/1.69 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( exu
% 2.75/1.69 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.75/1.69 => ? [B: $i] :
% 2.75/1.69 ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 <=> ? [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ A )
% 2.75/1.69 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(foundationAx_type,type,
% 2.75/1.69 foundationAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(foundationAx,definition,
% 2.75/1.69 ( foundationAx
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.75/1.69 => ? [B: $i] :
% 2.75/1.69 ( ( in @ B @ A )
% 2.75/1.69 & ~ ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(wellorderingAx_type,type,
% 2.75/1.69 wellorderingAx: $o ).
% 2.75/1.69
% 2.75/1.69 thf(wellorderingAx,definition,
% 2.75/1.69 ( wellorderingAx
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ? [B: $i] :
% 2.75/1.69 ( ! [C: $i] :
% 2.75/1.69 ( ( in @ C @ B )
% 2.75/1.69 => ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ C )
% 2.75/1.69 => ( in @ Xx @ A ) ) )
% 2.75/1.69 & ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( ( in @ Xx @ A )
% 2.75/1.69 & ( in @ Xy @ A ) )
% 2.75/1.69 => ( ! [C: $i] :
% 2.75/1.69 ( ( in @ C @ B )
% 2.75/1.69 => ( ( in @ Xx @ C )
% 2.75/1.69 <=> ( in @ Xy @ C ) ) )
% 2.75/1.69 => ( Xx = Xy ) ) )
% 2.75/1.69 & ! [C: $i,D: $i] :
% 2.75/1.69 ( ( ( in @ C @ B )
% 2.75/1.69 & ( in @ D @ B ) )
% 2.75/1.69 => ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ C )
% 2.75/1.69 => ( in @ Xx @ D ) )
% 2.75/1.69 | ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ D )
% 2.75/1.69 => ( in @ Xx @ C ) ) ) )
% 2.75/1.69 & ! [C: $i] :
% 2.75/1.69 ( ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ C )
% 2.75/1.69 => ( in @ Xx @ A ) )
% 2.75/1.69 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.75/1.69 => ? [D: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ D @ B )
% 2.75/1.69 & ( in @ Xx @ C )
% 2.75/1.69 & ~ ? [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ D )
% 2.75/1.69 & ( in @ Xy @ C ) )
% 2.75/1.69 & ! [E: $i] :
% 2.75/1.69 ( ( in @ E @ B )
% 2.75/1.69 => ( ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ E )
% 2.75/1.69 => ( in @ Xy @ D ) )
% 2.75/1.69 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(descr_type,type,
% 2.75/1.69 descr: ( $i > $o ) > $i ).
% 2.75/1.69
% 2.75/1.69 thf(descrp_type,type,
% 2.75/1.69 descrp: $o ).
% 2.75/1.69
% 2.75/1.69 thf(descrp,definition,
% 2.75/1.69 ( descrp
% 2.75/1.69 = ( ! [Xphi: $i > $o] :
% 2.75/1.69 ( ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 => ( Xphi
% 2.75/1.69 @ ( descr
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(dsetconstr_type,type,
% 2.75/1.69 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.75/1.69
% 2.75/1.69 thf(dsetconstrI_type,type,
% 2.75/1.69 dsetconstrI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(dsetconstrI,definition,
% 2.75/1.69 ( dsetconstrI
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ( Xphi @ Xx )
% 2.75/1.69 => ( in @ Xx
% 2.75/1.69 @ ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(dsetconstrEL_type,type,
% 2.75/1.69 dsetconstrEL: $o ).
% 2.75/1.69
% 2.75/1.69 thf(dsetconstrEL,definition,
% 2.75/1.69 ( dsetconstrEL
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx
% 2.75/1.69 @ ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.75/1.69 => ( in @ Xx @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(dsetconstrER_type,type,
% 2.75/1.69 dsetconstrER: $o ).
% 2.75/1.69
% 2.75/1.69 thf(dsetconstrER,definition,
% 2.75/1.69 ( dsetconstrER
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx
% 2.75/1.69 @ ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.75/1.69 => ( Xphi @ Xx ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(exuE1_type,type,
% 2.75/1.69 exuE1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(exuE1,definition,
% 2.75/1.69 ( exuE1
% 2.75/1.69 = ( ! [Xphi: $i > $o] :
% 2.75/1.69 ( ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 => ? [Xx: $i] :
% 2.75/1.69 ( ( Xphi @ Xx )
% 2.75/1.69 & ! [Xy: $i] :
% 2.75/1.69 ( ( Xphi @ Xy )
% 2.75/1.69 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(prop2set_type,type,
% 2.75/1.69 prop2set: $o > $i ).
% 2.75/1.69
% 2.75/1.69 thf(prop2setE_type,type,
% 2.75/1.69 prop2setE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(prop2setE,definition,
% 2.75/1.69 ( prop2setE
% 2.75/1.69 = ( ! [Xphi: $o,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.75/1.69 => Xphi ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptysetE_type,type,
% 2.75/1.69 emptysetE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptysetE,definition,
% 2.75/1.69 ( emptysetE
% 2.75/1.69 = ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ emptyset )
% 2.75/1.69 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptysetimpfalse_type,type,
% 2.75/1.69 emptysetimpfalse: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptysetimpfalse,definition,
% 2.75/1.69 ( emptysetimpfalse
% 2.75/1.69 = ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ emptyset )
% 2.75/1.69 => $false ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(notinemptyset_type,type,
% 2.75/1.69 notinemptyset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(notinemptyset,definition,
% 2.75/1.69 ( notinemptyset
% 2.75/1.69 = ( ! [Xx: $i] :
% 2.75/1.69 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(exuE3e_type,type,
% 2.75/1.69 exuE3e: $o ).
% 2.75/1.69
% 2.75/1.69 thf(exuE3e,definition,
% 2.75/1.69 ( exuE3e
% 2.75/1.69 = ( ! [Xphi: $i > $o] :
% 2.75/1.69 ( ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setext_type,type,
% 2.75/1.69 setext: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setext,definition,
% 2.75/1.69 ( setext
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ Xx @ B ) )
% 2.75/1.69 => ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 => ( in @ Xx @ A ) )
% 2.75/1.69 => ( A = B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptyI_type,type,
% 2.75/1.69 emptyI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptyI,definition,
% 2.75/1.69 ( emptyI
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ~ ( in @ Xx @ A )
% 2.75/1.69 => ( A = emptyset ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(noeltsimpempty_type,type,
% 2.75/1.69 noeltsimpempty: $o ).
% 2.75/1.69
% 2.75/1.69 thf(noeltsimpempty,definition,
% 2.75/1.69 ( noeltsimpempty
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ~ ( in @ Xx @ A )
% 2.75/1.69 => ( A = emptyset ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setbeta_type,type,
% 2.75/1.69 setbeta: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setbeta,definition,
% 2.75/1.69 ( setbeta
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ( in @ Xx
% 2.75/1.69 @ ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.75/1.69 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(nonempty_type,type,
% 2.75/1.69 nonempty: $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(nonempty,definition,
% 2.75/1.69 ( nonempty
% 2.75/1.69 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(nonemptyE1_type,type,
% 2.75/1.69 nonemptyE1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(nonemptyE1,definition,
% 2.75/1.69 ( nonemptyE1
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ( ( nonempty @ A )
% 2.75/1.69 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(nonemptyI_type,type,
% 2.75/1.69 nonemptyI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(nonemptyI,definition,
% 2.75/1.69 ( nonemptyI
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ( Xphi @ Xx )
% 2.75/1.69 => ( nonempty
% 2.75/1.69 @ ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(nonemptyI1_type,type,
% 2.75/1.69 nonemptyI1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(nonemptyI1,definition,
% 2.75/1.69 ( nonemptyI1
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.75/1.69 => ( nonempty @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinIL_type,type,
% 2.75/1.69 setadjoinIL: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinIL,definition,
% 2.75/1.69 ( setadjoinIL
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptyinunitempty_type,type,
% 2.75/1.69 emptyinunitempty: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptyinunitempty,definition,
% 2.75/1.69 ( emptyinunitempty
% 2.75/1.69 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinIR_type,type,
% 2.75/1.69 setadjoinIR: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinIR,definition,
% 2.75/1.69 ( setadjoinIR
% 2.75/1.69 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ A )
% 2.75/1.69 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinE_type,type,
% 2.75/1.69 setadjoinE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinE,definition,
% 2.75/1.69 ( setadjoinE
% 2.75/1.69 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.75/1.69 => ! [Xphi: $o] :
% 2.75/1.69 ( ( ( Xy = Xx )
% 2.75/1.69 => Xphi )
% 2.75/1.69 => ( ( ( in @ Xy @ A )
% 2.75/1.69 => Xphi )
% 2.75/1.69 => Xphi ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinOr_type,type,
% 2.75/1.69 setadjoinOr: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinOr,definition,
% 2.75/1.69 ( setadjoinOr
% 2.75/1.69 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.75/1.69 => ( ( Xy = Xx )
% 2.75/1.69 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setoftrueEq_type,type,
% 2.75/1.69 setoftrueEq: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setoftrueEq,definition,
% 2.75/1.69 ( setoftrueEq
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ( ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xx: $i] : $true )
% 2.75/1.69 = A ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(powersetI_type,type,
% 2.75/1.69 powersetI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(powersetI,definition,
% 2.75/1.69 ( powersetI
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 => ( in @ Xx @ A ) )
% 2.75/1.69 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptyinPowerset_type,type,
% 2.75/1.69 emptyinPowerset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptyinPowerset,definition,
% 2.75/1.69 ( emptyinPowerset
% 2.75/1.69 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptyInPowerset_type,type,
% 2.75/1.69 emptyInPowerset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptyInPowerset,definition,
% 2.75/1.69 ( emptyInPowerset
% 2.75/1.69 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(powersetE_type,type,
% 2.75/1.69 powersetE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(powersetE,definition,
% 2.75/1.69 ( powersetE
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ B @ ( powerset @ A ) )
% 2.75/1.69 => ( ( in @ Xx @ B )
% 2.75/1.69 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setunionI_type,type,
% 2.75/1.69 setunionI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setunionI,definition,
% 2.75/1.69 ( setunionI
% 2.75/1.69 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 => ( ( in @ B @ A )
% 2.75/1.69 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setunionE_type,type,
% 2.75/1.69 setunionE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setunionE,definition,
% 2.75/1.69 ( setunionE
% 2.75/1.69 = ( ! [A: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( setunion @ A ) )
% 2.75/1.69 => ! [Xphi: $o] :
% 2.75/1.69 ( ! [B: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 => ( ( in @ B @ A )
% 2.75/1.69 => Xphi ) )
% 2.75/1.69 => Xphi ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(subPowSU_type,type,
% 2.75/1.69 subPowSU: $o ).
% 2.75/1.69
% 2.75/1.69 thf(subPowSU,definition,
% 2.75/1.69 ( subPowSU
% 2.75/1.69 = ( ! [A: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(exuE2_type,type,
% 2.75/1.69 exuE2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(exuE2,definition,
% 2.75/1.69 ( exuE2
% 2.75/1.69 = ( ! [Xphi: $i > $o] :
% 2.75/1.69 ( ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 => ? [Xx: $i] :
% 2.75/1.69 ! [Xy: $i] :
% 2.75/1.69 ( ( Xphi @ Xy )
% 2.75/1.69 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(nonemptyImpWitness_type,type,
% 2.75/1.69 nonemptyImpWitness: $o ).
% 2.75/1.69
% 2.75/1.69 thf(nonemptyImpWitness,definition,
% 2.75/1.69 ( nonemptyImpWitness
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ( ( nonempty @ A )
% 2.75/1.69 => ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & $true ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(uniqinunit_type,type,
% 2.75/1.69 uniqinunit: $o ).
% 2.75/1.69
% 2.75/1.69 thf(uniqinunit,definition,
% 2.75/1.69 ( uniqinunit
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.75/1.69 => ( Xx = Xy ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(notinsingleton_type,type,
% 2.75/1.69 notinsingleton: $o ).
% 2.75/1.69
% 2.75/1.69 thf(notinsingleton,definition,
% 2.75/1.69 ( notinsingleton
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xx != Xy )
% 2.75/1.69 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(eqinunit_type,type,
% 2.75/1.69 eqinunit: $o ).
% 2.75/1.69
% 2.75/1.69 thf(eqinunit,definition,
% 2.75/1.69 ( eqinunit
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xx = Xy )
% 2.75/1.69 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(singletonsswitch_type,type,
% 2.75/1.69 singletonsswitch: $o ).
% 2.75/1.69
% 2.75/1.69 thf(singletonsswitch,definition,
% 2.75/1.69 ( singletonsswitch
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.75/1.69 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(upairsetE_type,type,
% 2.75/1.69 upairsetE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(upairsetE,definition,
% 2.75/1.69 ( upairsetE
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.75/1.69 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.75/1.69 => ( ( Xz = Xx )
% 2.75/1.69 | ( Xz = Xy ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(upairsetIL_type,type,
% 2.75/1.69 upairsetIL: $o ).
% 2.75/1.69
% 2.75/1.69 thf(upairsetIL,definition,
% 2.75/1.69 ( upairsetIL
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(upairsetIR_type,type,
% 2.75/1.69 upairsetIR: $o ).
% 2.75/1.69
% 2.75/1.69 thf(upairsetIR,definition,
% 2.75/1.69 ( upairsetIR
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptyE1_type,type,
% 2.75/1.69 emptyE1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptyE1,definition,
% 2.75/1.69 ( emptyE1
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ( Xphi @ Xx ) )
% 2.75/1.69 => ( ( ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 = emptyset )
% 2.75/1.69 => $false ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(vacuousDall_type,type,
% 2.75/1.69 vacuousDall: $o ).
% 2.75/1.69
% 2.75/1.69 thf(vacuousDall,definition,
% 2.75/1.69 ( vacuousDall
% 2.75/1.69 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ emptyset )
% 2.75/1.69 => ( Xphi @ Xx ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(quantDeMorgan1_type,type,
% 2.75/1.69 quantDeMorgan1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(quantDeMorgan1,definition,
% 2.75/1.69 ( quantDeMorgan1
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ~ ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( Xphi @ Xx ) )
% 2.75/1.69 => ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(quantDeMorgan2_type,type,
% 2.75/1.69 quantDeMorgan2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(quantDeMorgan2,definition,
% 2.75/1.69 ( quantDeMorgan2
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ~ ( Xphi @ Xx ) )
% 2.75/1.69 => ~ ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(quantDeMorgan3_type,type,
% 2.75/1.69 quantDeMorgan3: $o ).
% 2.75/1.69
% 2.75/1.69 thf(quantDeMorgan3,definition,
% 2.75/1.69 ( quantDeMorgan3
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ~ ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ( Xphi @ Xx ) )
% 2.75/1.69 => ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(quantDeMorgan4_type,type,
% 2.75/1.69 quantDeMorgan4: $o ).
% 2.75/1.69
% 2.75/1.69 thf(quantDeMorgan4,definition,
% 2.75/1.69 ( quantDeMorgan4
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ~ ( Xphi @ Xx ) )
% 2.75/1.69 => ~ ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(prop2setI_type,type,
% 2.75/1.69 prop2setI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(prop2setI,definition,
% 2.75/1.69 ( prop2setI
% 2.75/1.69 = ( ! [Xphi: $o] :
% 2.75/1.69 ( Xphi
% 2.75/1.69 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(set2prop_type,type,
% 2.75/1.69 set2prop: $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(prop2set2propI_type,type,
% 2.75/1.69 prop2set2propI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(prop2set2propI,definition,
% 2.75/1.69 ( prop2set2propI
% 2.75/1.69 = ( ! [Xphi: $o] :
% 2.75/1.69 ( Xphi
% 2.75/1.69 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(notdexE_type,type,
% 2.75/1.69 notdexE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(notdexE,definition,
% 2.75/1.69 ( notdexE
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ~ ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ( Xphi @ Xx ) )
% 2.75/1.69 => ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(notdallE_type,type,
% 2.75/1.69 notdallE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(notdallE,definition,
% 2.75/1.69 ( notdallE
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ~ ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( Xphi @ Xx ) )
% 2.75/1.69 => ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(exuI1_type,type,
% 2.75/1.69 exuI1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(exuI1,definition,
% 2.75/1.69 ( exuI1
% 2.75/1.69 = ( ! [Xphi: $i > $o] :
% 2.75/1.69 ( ? [Xx: $i] :
% 2.75/1.69 ( ( Xphi @ Xx )
% 2.75/1.69 & ! [Xy: $i] :
% 2.75/1.69 ( ( Xphi @ Xy )
% 2.75/1.69 => ( Xx = Xy ) ) )
% 2.75/1.69 => ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(exuI3_type,type,
% 2.75/1.69 exuI3: $o ).
% 2.75/1.69
% 2.75/1.69 thf(exuI3,definition,
% 2.75/1.69 ( exuI3
% 2.75/1.69 = ( ! [Xphi: $i > $o] :
% 2.75/1.69 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.75/1.69 => ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xphi @ Xx )
% 2.75/1.69 => ( ( Xphi @ Xy )
% 2.75/1.69 => ( Xx = Xy ) ) )
% 2.75/1.69 => ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(exuI2_type,type,
% 2.75/1.69 exuI2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(exuI2,definition,
% 2.75/1.69 ( exuI2
% 2.75/1.69 = ( ! [Xphi: $i > $o] :
% 2.75/1.69 ( ? [Xx: $i] :
% 2.75/1.69 ! [Xy: $i] :
% 2.75/1.69 ( ( Xphi @ Xy )
% 2.75/1.69 <=> ( Xy = Xx ) )
% 2.75/1.69 => ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(inCongP_type,type,
% 2.75/1.69 inCongP: $o ).
% 2.75/1.69
% 2.75/1.69 thf(inCongP,definition,
% 2.75/1.69 ( inCongP
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( A = B )
% 2.75/1.69 => ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xx = Xy )
% 2.75/1.69 => ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(in__Cong_type,type,
% 2.75/1.69 in__Cong: $o ).
% 2.75/1.69
% 2.75/1.69 thf(in__Cong,definition,
% 2.75/1.69 ( in__Cong
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( A = B )
% 2.75/1.69 => ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xx = Xy )
% 2.75/1.69 => ( ( in @ Xx @ A )
% 2.75/1.69 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(exuE3u_type,type,
% 2.75/1.69 exuE3u: $o ).
% 2.75/1.69
% 2.75/1.69 thf(exuE3u,definition,
% 2.75/1.69 ( exuE3u
% 2.75/1.69 = ( ! [Xphi: $i > $o] :
% 2.75/1.69 ( ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 => ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xphi @ Xx )
% 2.75/1.69 => ( ( Xphi @ Xy )
% 2.75/1.69 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(exu__Cong_type,type,
% 2.75/1.69 exu__Cong: $o ).
% 2.75/1.69
% 2.75/1.69 thf(exu__Cong,definition,
% 2.75/1.69 ( exu__Cong
% 2.75/1.69 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.75/1.69 ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xx = Xy )
% 2.75/1.69 => ( ( Xphi @ Xx )
% 2.75/1.69 <=> ( Xpsi @ Xy ) ) )
% 2.75/1.69 => ( ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 <=> ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptyset__Cong_type,type,
% 2.75/1.69 emptyset__Cong: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptyset__Cong,definition,
% 2.75/1.69 ( emptyset__Cong
% 2.75/1.69 = ( emptyset = emptyset ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoin__Cong_type,type,
% 2.75/1.69 setadjoin__Cong: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoin__Cong,definition,
% 2.75/1.69 ( setadjoin__Cong
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xx = Xy )
% 2.75/1.69 => ! [Xz: $i,Xu: $i] :
% 2.75/1.69 ( ( Xz = Xu )
% 2.75/1.69 => ( ( setadjoin @ Xx @ Xz )
% 2.75/1.69 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(powerset__Cong_type,type,
% 2.75/1.69 powerset__Cong: $o ).
% 2.75/1.69
% 2.75/1.69 thf(powerset__Cong,definition,
% 2.75/1.69 ( powerset__Cong
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( A = B )
% 2.75/1.69 => ( ( powerset @ A )
% 2.75/1.69 = ( powerset @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setunion__Cong_type,type,
% 2.75/1.69 setunion__Cong: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setunion__Cong,definition,
% 2.75/1.69 ( setunion__Cong
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( A = B )
% 2.75/1.69 => ( ( setunion @ A )
% 2.75/1.69 = ( setunion @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(omega__Cong_type,type,
% 2.75/1.69 omega__Cong: $o ).
% 2.75/1.69
% 2.75/1.69 thf(omega__Cong,definition,
% 2.75/1.69 ( omega__Cong
% 2.75/1.69 = ( omega = omega ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(exuEu_type,type,
% 2.75/1.69 exuEu: $o ).
% 2.75/1.69
% 2.75/1.69 thf(exuEu,definition,
% 2.75/1.69 ( exuEu
% 2.75/1.69 = ( ! [Xphi: $i > $o] :
% 2.75/1.69 ( ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 => ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xphi @ Xx )
% 2.75/1.69 => ( ( Xphi @ Xy )
% 2.75/1.69 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(descr__Cong_type,type,
% 2.75/1.69 descr__Cong: $o ).
% 2.75/1.69
% 2.75/1.69 thf(descr__Cong,definition,
% 2.75/1.69 ( descr__Cong
% 2.75/1.69 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.75/1.69 ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xx = Xy )
% 2.75/1.69 => ( ( Xphi @ Xx )
% 2.75/1.69 <=> ( Xpsi @ Xy ) ) )
% 2.75/1.69 => ( ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 => ( ( exu
% 2.75/1.69 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.75/1.69 => ( ( descr
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 = ( descr
% 2.75/1.69 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(dsetconstr__Cong_type,type,
% 2.75/1.69 dsetconstr__Cong: $o ).
% 2.75/1.69
% 2.75/1.69 thf(dsetconstr__Cong,definition,
% 2.75/1.69 ( dsetconstr__Cong
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( A = B )
% 2.75/1.69 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ B )
% 2.75/1.69 => ( ( Xx = Xy )
% 2.75/1.69 => ( ( Xphi @ Xx )
% 2.75/1.69 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.75/1.69 => ( ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 = ( dsetconstr @ B
% 2.75/1.69 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(subset_type,type,
% 2.75/1.69 subset: $i > $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(disjoint_type,type,
% 2.75/1.69 disjoint: $i > $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(setsmeet_type,type,
% 2.75/1.69 setsmeet: $i > $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(subsetI1_type,type,
% 2.75/1.69 subsetI1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(subsetI1,definition,
% 2.75/1.69 ( subsetI1
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ Xx @ B ) )
% 2.75/1.69 => ( subset @ A @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(eqimpsubset2_type,type,
% 2.75/1.69 eqimpsubset2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(eqimpsubset2,definition,
% 2.75/1.69 ( eqimpsubset2
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( A = B )
% 2.75/1.69 => ( subset @ B @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(eqimpsubset1_type,type,
% 2.75/1.69 eqimpsubset1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(eqimpsubset1,definition,
% 2.75/1.69 ( eqimpsubset1
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( A = B )
% 2.75/1.69 => ( subset @ A @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(subsetI2_type,type,
% 2.75/1.69 subsetI2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(subsetI2,definition,
% 2.75/1.69 ( subsetI2
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ Xx @ B ) )
% 2.75/1.69 => ( subset @ A @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(emptysetsubset_type,type,
% 2.75/1.69 emptysetsubset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(emptysetsubset,definition,
% 2.75/1.69 ( emptysetsubset
% 2.75/1.69 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(subsetE_type,type,
% 2.75/1.69 subsetE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(subsetE,definition,
% 2.75/1.69 ( subsetE
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( subset @ A @ B )
% 2.75/1.69 => ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(subsetE2_type,type,
% 2.75/1.69 subsetE2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(subsetE2,definition,
% 2.75/1.69 ( subsetE2
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( subset @ A @ B )
% 2.75/1.69 => ( ~ ( in @ Xx @ B )
% 2.75/1.69 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(notsubsetI_type,type,
% 2.75/1.69 notsubsetI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(notsubsetI,definition,
% 2.75/1.69 ( notsubsetI
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ~ ( in @ Xx @ B )
% 2.75/1.69 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(notequalI1_type,type,
% 2.75/1.69 notequalI1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(notequalI1,definition,
% 2.75/1.69 ( notequalI1
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ~ ( subset @ A @ B )
% 2.75/1.69 => ( A != B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(notequalI2_type,type,
% 2.75/1.69 notequalI2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(notequalI2,definition,
% 2.75/1.69 ( notequalI2
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ~ ( in @ Xx @ B )
% 2.75/1.69 => ( A != B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(subsetRefl_type,type,
% 2.75/1.69 subsetRefl: $o ).
% 2.75/1.69
% 2.75/1.69 thf(subsetRefl,definition,
% 2.75/1.69 ( subsetRefl
% 2.75/1.69 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(subsetTrans_type,type,
% 2.75/1.69 subsetTrans: $o ).
% 2.75/1.69
% 2.75/1.69 thf(subsetTrans,definition,
% 2.75/1.69 ( subsetTrans
% 2.75/1.69 = ( ! [A: $i,B: $i,C: $i] :
% 2.75/1.69 ( ( subset @ A @ B )
% 2.75/1.69 => ( ( subset @ B @ C )
% 2.75/1.69 => ( subset @ A @ C ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinSub_type,type,
% 2.75/1.69 setadjoinSub: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinSub,definition,
% 2.75/1.69 ( setadjoinSub
% 2.75/1.69 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinSub2_type,type,
% 2.75/1.69 setadjoinSub2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setadjoinSub2,definition,
% 2.75/1.69 ( setadjoinSub2
% 2.75/1.69 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.75/1.69 ( ( subset @ A @ B )
% 2.75/1.69 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(subset2powerset_type,type,
% 2.75/1.69 subset2powerset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(subset2powerset,definition,
% 2.75/1.69 ( subset2powerset
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( subset @ A @ B )
% 2.75/1.69 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setextsub_type,type,
% 2.75/1.69 setextsub: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setextsub,definition,
% 2.75/1.69 ( setextsub
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( subset @ A @ B )
% 2.75/1.69 => ( ( subset @ B @ A )
% 2.75/1.69 => ( A = B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(subsetemptysetimpeq_type,type,
% 2.75/1.69 subsetemptysetimpeq: $o ).
% 2.75/1.69
% 2.75/1.69 thf(subsetemptysetimpeq,definition,
% 2.75/1.69 ( subsetemptysetimpeq
% 2.75/1.69 = ( ! [A: $i] :
% 2.75/1.69 ( ( subset @ A @ emptyset )
% 2.75/1.69 => ( A = emptyset ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(powersetI1_type,type,
% 2.75/1.69 powersetI1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(powersetI1,definition,
% 2.75/1.69 ( powersetI1
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( subset @ B @ A )
% 2.75/1.69 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(powersetE1_type,type,
% 2.75/1.69 powersetE1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(powersetE1,definition,
% 2.75/1.69 ( powersetE1
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( in @ B @ ( powerset @ A ) )
% 2.75/1.69 => ( subset @ B @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(inPowerset_type,type,
% 2.75/1.69 inPowerset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(inPowerset,definition,
% 2.75/1.69 ( inPowerset
% 2.75/1.69 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(powersetsubset_type,type,
% 2.75/1.69 powersetsubset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(powersetsubset,definition,
% 2.75/1.69 ( powersetsubset
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( subset @ A @ B )
% 2.75/1.69 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(sepInPowerset_type,type,
% 2.75/1.69 sepInPowerset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(sepInPowerset,definition,
% 2.75/1.69 ( sepInPowerset
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( in
% 2.75/1.69 @ ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 @ ( powerset @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(sepSubset_type,type,
% 2.75/1.69 sepSubset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(sepSubset,definition,
% 2.75/1.69 ( sepSubset
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( subset
% 2.75/1.69 @ ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 @ A ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binunion_type,type,
% 2.75/1.69 binunion: $i > $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(binunionIL_type,type,
% 2.75/1.69 binunionIL: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binunionIL,definition,
% 2.75/1.69 ( binunionIL
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(upairset2IR_type,type,
% 2.75/1.69 upairset2IR: $o ).
% 2.75/1.69
% 2.75/1.69 thf(upairset2IR,definition,
% 2.75/1.69 ( upairset2IR
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binunionIR_type,type,
% 2.75/1.69 binunionIR: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binunionIR,definition,
% 2.75/1.69 ( binunionIR
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binunionEcases_type,type,
% 2.75/1.69 binunionEcases: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binunionEcases,definition,
% 2.75/1.69 ( binunionEcases
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.75/1.69 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.75/1.69 => ( ( ( in @ Xx @ A )
% 2.75/1.69 => Xphi )
% 2.75/1.69 => ( ( ( in @ Xx @ B )
% 2.75/1.69 => Xphi )
% 2.75/1.69 => Xphi ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binunionE_type,type,
% 2.75/1.69 binunionE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binunionE,definition,
% 2.75/1.69 ( binunionE
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.75/1.69 => ( ( in @ Xx @ A )
% 2.75/1.69 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binunionLsub_type,type,
% 2.75/1.69 binunionLsub: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binunionLsub,definition,
% 2.75/1.69 ( binunionLsub
% 2.75/1.69 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binunionRsub_type,type,
% 2.75/1.69 binunionRsub: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binunionRsub,definition,
% 2.75/1.69 ( binunionRsub
% 2.75/1.69 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersect_type,type,
% 2.75/1.69 binintersect: $i > $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectI_type,type,
% 2.75/1.69 binintersectI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectI,definition,
% 2.75/1.69 ( binintersectI
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ( in @ Xx @ B )
% 2.75/1.69 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset5_type,type,
% 2.75/1.69 binintersectSubset5: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset5,definition,
% 2.75/1.69 ( binintersectSubset5
% 2.75/1.69 = ( ! [A: $i,B: $i,C: $i] :
% 2.75/1.69 ( ( subset @ C @ A )
% 2.75/1.69 => ( ( subset @ C @ B )
% 2.75/1.69 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectEL_type,type,
% 2.75/1.69 binintersectEL: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectEL,definition,
% 2.75/1.69 ( binintersectEL
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.75/1.69 => ( in @ Xx @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectLsub_type,type,
% 2.75/1.69 binintersectLsub: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectLsub,definition,
% 2.75/1.69 ( binintersectLsub
% 2.75/1.69 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset2_type,type,
% 2.75/1.69 binintersectSubset2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset2,definition,
% 2.75/1.69 ( binintersectSubset2
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( subset @ A @ B )
% 2.75/1.69 => ( ( binintersect @ A @ B )
% 2.75/1.69 = A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset3_type,type,
% 2.75/1.69 binintersectSubset3: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset3,definition,
% 2.75/1.69 ( binintersectSubset3
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( ( binintersect @ A @ B )
% 2.75/1.69 = B )
% 2.75/1.69 => ( subset @ B @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectER_type,type,
% 2.75/1.69 binintersectER: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectER,definition,
% 2.75/1.69 ( binintersectER
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.75/1.69 => ( in @ Xx @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(disjointsetsI1_type,type,
% 2.75/1.69 disjointsetsI1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(disjointsetsI1,definition,
% 2.75/1.69 ( disjointsetsI1
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ~ ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ( in @ Xx @ B ) )
% 2.75/1.69 => ( ( binintersect @ A @ B )
% 2.75/1.69 = emptyset ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectRsub_type,type,
% 2.75/1.69 binintersectRsub: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectRsub,definition,
% 2.75/1.69 ( binintersectRsub
% 2.75/1.69 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset4_type,type,
% 2.75/1.69 binintersectSubset4: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset4,definition,
% 2.75/1.69 ( binintersectSubset4
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( subset @ B @ A )
% 2.75/1.69 => ( ( binintersect @ A @ B )
% 2.75/1.69 = B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset1_type,type,
% 2.75/1.69 binintersectSubset1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(binintersectSubset1,definition,
% 2.75/1.69 ( binintersectSubset1
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( ( binintersect @ A @ B )
% 2.75/1.69 = A )
% 2.75/1.69 => ( subset @ A @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(bs114d_type,type,
% 2.75/1.69 bs114d: $o ).
% 2.75/1.69
% 2.75/1.69 thf(bs114d,definition,
% 2.75/1.69 ( bs114d
% 2.75/1.69 = ( ! [A: $i,B: $i,C: $i] :
% 2.75/1.69 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.75/1.69 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(regular_type,type,
% 2.75/1.69 regular: $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminus_type,type,
% 2.75/1.69 setminus: $i > $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(setminusI_type,type,
% 2.75/1.69 setminusI: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusI,definition,
% 2.75/1.69 ( setminusI
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ~ ( in @ Xx @ B )
% 2.75/1.69 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setminusEL_type,type,
% 2.75/1.69 setminusEL: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusEL,definition,
% 2.75/1.69 ( setminusEL
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.75/1.69 => ( in @ Xx @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setminusER_type,type,
% 2.75/1.69 setminusER: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusER,definition,
% 2.75/1.69 ( setminusER
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.75/1.69 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setminusSubset2_type,type,
% 2.75/1.69 setminusSubset2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusSubset2,definition,
% 2.75/1.69 ( setminusSubset2
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( subset @ A @ B )
% 2.75/1.69 => ( ( setminus @ A @ B )
% 2.75/1.69 = emptyset ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setminusERneg_type,type,
% 2.75/1.69 setminusERneg: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusERneg,definition,
% 2.75/1.69 ( setminusERneg
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.75/1.69 => ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setminusELneg_type,type,
% 2.75/1.69 setminusELneg: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusELneg,definition,
% 2.75/1.69 ( setminusELneg
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.75/1.69 => ( ~ ( in @ Xx @ B )
% 2.75/1.69 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setminusILneg_type,type,
% 2.75/1.69 setminusILneg: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusILneg,definition,
% 2.75/1.69 ( setminusILneg
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ~ ( in @ Xx @ A )
% 2.75/1.69 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setminusIRneg_type,type,
% 2.75/1.69 setminusIRneg: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusIRneg,definition,
% 2.75/1.69 ( setminusIRneg
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ B )
% 2.75/1.69 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setminusLsub_type,type,
% 2.75/1.69 setminusLsub: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusLsub,definition,
% 2.75/1.69 ( setminusLsub
% 2.75/1.69 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setminusSubset1_type,type,
% 2.75/1.69 setminusSubset1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setminusSubset1,definition,
% 2.75/1.69 ( setminusSubset1
% 2.75/1.69 = ( ! [A: $i,B: $i] :
% 2.75/1.69 ( ( ( setminus @ A @ B )
% 2.75/1.69 = emptyset )
% 2.75/1.69 => ( subset @ A @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(symdiff_type,type,
% 2.75/1.69 symdiff: $i > $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffE_type,type,
% 2.75/1.69 symdiffE: $o ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffE,definition,
% 2.75/1.69 ( symdiffE
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.75/1.69 => ! [Xphi: $o] :
% 2.75/1.69 ( ( ( in @ Xx @ A )
% 2.75/1.69 => ( ~ ( in @ Xx @ B )
% 2.75/1.69 => Xphi ) )
% 2.75/1.69 => ( ( ~ ( in @ Xx @ A )
% 2.75/1.69 => ( ( in @ Xx @ B )
% 2.75/1.69 => Xphi ) )
% 2.75/1.69 => Xphi ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffI1_type,type,
% 2.75/1.69 symdiffI1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffI1,definition,
% 2.75/1.69 ( symdiffI1
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ~ ( in @ Xx @ B )
% 2.75/1.69 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffI2_type,type,
% 2.75/1.69 symdiffI2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffI2,definition,
% 2.75/1.69 ( symdiffI2
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ~ ( in @ Xx @ A )
% 2.75/1.69 => ( ( in @ Xx @ B )
% 2.75/1.69 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffIneg1_type,type,
% 2.75/1.69 symdiffIneg1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffIneg1,definition,
% 2.75/1.69 ( symdiffIneg1
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ( in @ Xx @ B )
% 2.75/1.69 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffIneg2_type,type,
% 2.75/1.69 symdiffIneg2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(symdiffIneg2,definition,
% 2.75/1.69 ( symdiffIneg2
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ~ ( in @ Xx @ A )
% 2.75/1.69 => ( ~ ( in @ Xx @ B )
% 2.75/1.69 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(iskpair_type,type,
% 2.75/1.69 iskpair: $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(secondinupair_type,type,
% 2.75/1.69 secondinupair: $o ).
% 2.75/1.69
% 2.75/1.69 thf(secondinupair,definition,
% 2.75/1.69 ( secondinupair
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairIL_type,type,
% 2.75/1.69 setukpairIL: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairIL,definition,
% 2.75/1.69 ( setukpairIL
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairIR_type,type,
% 2.75/1.69 setukpairIR: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairIR,definition,
% 2.75/1.69 ( setukpairIR
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(kpairiskpair_type,type,
% 2.75/1.69 kpairiskpair: $o ).
% 2.75/1.69
% 2.75/1.69 thf(kpairiskpair,definition,
% 2.75/1.69 ( kpairiskpair
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(kpair_type,type,
% 2.75/1.69 kpair: $i > $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(kpairp_type,type,
% 2.75/1.69 kpairp: $o ).
% 2.75/1.69
% 2.75/1.69 thf(kpairp,definition,
% 2.75/1.69 ( kpairp
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(cartprod_type,type,
% 2.75/1.69 cartprod: $i > $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(singletonsubset_type,type,
% 2.75/1.69 singletonsubset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(singletonsubset,definition,
% 2.75/1.69 ( singletonsubset
% 2.75/1.69 = ( ! [A: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(singletoninpowerset_type,type,
% 2.75/1.69 singletoninpowerset: $o ).
% 2.75/1.69
% 2.75/1.69 thf(singletoninpowerset,definition,
% 2.75/1.69 ( singletoninpowerset
% 2.75/1.69 = ( ! [A: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(singletoninpowunion_type,type,
% 2.75/1.69 singletoninpowunion: $o ).
% 2.75/1.69
% 2.75/1.69 thf(singletoninpowunion,definition,
% 2.75/1.69 ( singletoninpowunion
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(upairset2E_type,type,
% 2.75/1.69 upairset2E: $o ).
% 2.75/1.69
% 2.75/1.69 thf(upairset2E,definition,
% 2.75/1.69 ( upairset2E
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.75/1.69 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.75/1.69 => ( ( Xz = Xx )
% 2.75/1.69 | ( Xz = Xy ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(upairsubunion_type,type,
% 2.75/1.69 upairsubunion: $o ).
% 2.75/1.69
% 2.75/1.69 thf(upairsubunion,definition,
% 2.75/1.69 ( upairsubunion
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ B )
% 2.75/1.69 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(upairinpowunion_type,type,
% 2.75/1.69 upairinpowunion: $o ).
% 2.75/1.69
% 2.75/1.69 thf(upairinpowunion,definition,
% 2.75/1.69 ( upairinpowunion
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ B )
% 2.75/1.69 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(ubforcartprodlem1_type,type,
% 2.75/1.69 ubforcartprodlem1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(ubforcartprodlem1,definition,
% 2.75/1.69 ( ubforcartprodlem1
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ B )
% 2.75/1.69 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(ubforcartprodlem2_type,type,
% 2.75/1.69 ubforcartprodlem2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(ubforcartprodlem2,definition,
% 2.75/1.69 ( ubforcartprodlem2
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ B )
% 2.75/1.69 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(ubforcartprodlem3_type,type,
% 2.75/1.69 ubforcartprodlem3: $o ).
% 2.75/1.69
% 2.75/1.69 thf(ubforcartprodlem3,definition,
% 2.75/1.69 ( ubforcartprodlem3
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ B )
% 2.75/1.69 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodpairin_type,type,
% 2.75/1.69 cartprodpairin: $o ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodpairin,definition,
% 2.75/1.69 ( cartprodpairin
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ B )
% 2.75/1.69 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodmempair1_type,type,
% 2.75/1.69 cartprodmempair1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodmempair1,definition,
% 2.75/1.69 ( cartprodmempair1
% 2.75/1.69 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.75/1.69 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.75/1.69 => ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ? [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ B )
% 2.75/1.69 & ( Xu
% 2.75/1.69 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodmempair_type,type,
% 2.75/1.69 cartprodmempair: $o ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodmempair,definition,
% 2.75/1.69 ( cartprodmempair
% 2.75/1.69 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.75/1.69 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.75/1.69 => ( iskpair @ Xu ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setunionE2_type,type,
% 2.75/1.69 setunionE2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setunionE2,definition,
% 2.75/1.69 ( setunionE2
% 2.75/1.69 = ( ! [A: $i,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ ( setunion @ A ) )
% 2.75/1.69 => ? [X: $i] :
% 2.75/1.69 ( ( in @ X @ A )
% 2.75/1.69 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setunionsingleton1_type,type,
% 2.75/1.69 setunionsingleton1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setunionsingleton1,definition,
% 2.75/1.69 ( setunionsingleton1
% 2.75/1.69 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setunionsingleton2_type,type,
% 2.75/1.69 setunionsingleton2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setunionsingleton2,definition,
% 2.75/1.69 ( setunionsingleton2
% 2.75/1.69 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setunionsingleton_type,type,
% 2.75/1.69 setunionsingleton: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setunionsingleton,definition,
% 2.75/1.69 ( setunionsingleton
% 2.75/1.69 = ( ! [Xx: $i] :
% 2.75/1.69 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.75/1.69 = Xx ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(singleton_type,type,
% 2.75/1.69 singleton: $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(singleton,definition,
% 2.75/1.69 ( singleton
% 2.75/1.69 = ( ^ [A: $i] :
% 2.75/1.69 ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ( A
% 2.75/1.69 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(singletonprop_type,type,
% 2.75/1.69 singletonprop: $o ).
% 2.75/1.69
% 2.75/1.69 thf(singletonprop,definition,
% 2.75/1.69 ( singletonprop
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ A )
% 2.75/1.69 => ( ( Xphi @ Xx )
% 2.75/1.69 => ( ( Xphi @ Xy )
% 2.75/1.69 => ( Xx = Xy ) ) ) ) )
% 2.75/1.69 => ( ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ( Xphi @ Xx ) )
% 2.75/1.69 => ( singleton
% 2.75/1.69 @ ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(ex1_type,type,
% 2.75/1.69 ex1: $i > ( $i > $o ) > $o ).
% 2.75/1.69
% 2.75/1.69 thf(ex1,definition,
% 2.75/1.69 ( ex1
% 2.75/1.69 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( singleton
% 2.75/1.69 @ ( dsetconstr @ A
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(ex1E1_type,type,
% 2.75/1.69 ex1E1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(ex1E1,definition,
% 2.75/1.69 ( ex1E1
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ( ex1 @ A
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.75/1.69 => ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(ex1I_type,type,
% 2.75/1.69 ex1I: $o ).
% 2.75/1.69
% 2.75/1.69 thf(ex1I,definition,
% 2.75/1.69 ( ex1I
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ( ( Xphi @ Xx )
% 2.75/1.69 => ( ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ A )
% 2.75/1.69 => ( ( Xphi @ Xy )
% 2.75/1.69 => ( Xy = Xx ) ) )
% 2.75/1.69 => ( ex1 @ A
% 2.75/1.69 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(ex1I2_type,type,
% 2.75/1.69 ex1I2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(ex1I2,definition,
% 2.75/1.69 ( ex1I2
% 2.75/1.69 = ( ! [A: $i,Xphi: $i > $o] :
% 2.75/1.69 ( ! [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 => ! [Xy: $i] :
% 2.75/1.69 ( ( in @ Xy @ A )
% 2.75/1.69 => ( ( Xphi @ Xx )
% 2.75/1.69 => ( ( Xphi @ Xy )
% 2.75/1.69 => ( Xx = Xy ) ) ) ) )
% 2.75/1.69 => ( ? [Xx: $i] :
% 2.75/1.69 ( ( in @ Xx @ A )
% 2.75/1.69 & ( Xphi @ Xx ) )
% 2.75/1.69 => ( ex1 @ A
% 2.75/1.69 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(singletonsuniq_type,type,
% 2.75/1.69 singletonsuniq: $o ).
% 2.75/1.69
% 2.75/1.69 thf(singletonsuniq,definition,
% 2.75/1.69 ( singletonsuniq
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( ( setadjoin @ Xx @ emptyset )
% 2.75/1.69 = ( setadjoin @ Xy @ emptyset ) )
% 2.75/1.69 => ( Xx = Xy ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(atmost1p_type,type,
% 2.75/1.69 atmost1p: $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(atleast2p_type,type,
% 2.75/1.69 atleast2p: $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(atmost2p_type,type,
% 2.75/1.69 atmost2p: $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(upairsetp_type,type,
% 2.75/1.69 upairsetp: $i > $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjL1_type,type,
% 2.75/1.69 setukpairinjL1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjL1,definition,
% 2.75/1.69 ( setukpairinjL1
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.75/1.69 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.75/1.69 => ( Xx = Xz ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(kfstsingleton_type,type,
% 2.75/1.69 kfstsingleton: $o ).
% 2.75/1.69
% 2.75/1.69 thf(kfstsingleton,definition,
% 2.75/1.69 ( kfstsingleton
% 2.75/1.69 = ( ! [Xu: $i] :
% 2.75/1.69 ( ( iskpair @ Xu )
% 2.75/1.69 => ( singleton
% 2.75/1.69 @ ( dsetconstr @ ( setunion @ Xu )
% 2.75/1.69 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(theprop_type,type,
% 2.75/1.69 theprop: $o ).
% 2.75/1.69
% 2.75/1.69 thf(theprop,definition,
% 2.75/1.69 ( theprop
% 2.75/1.69 = ( ! [X: $i] :
% 2.75/1.69 ( ( singleton @ X )
% 2.75/1.69 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(kfst_type,type,
% 2.75/1.69 kfst: $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(kfstpairEq_type,type,
% 2.75/1.69 kfstpairEq: $o ).
% 2.75/1.69
% 2.75/1.69 thf(kfstpairEq,definition,
% 2.75/1.69 ( kfstpairEq
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.75/1.69 = Xx ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodfstin_type,type,
% 2.75/1.69 cartprodfstin: $o ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodfstin,definition,
% 2.75/1.69 ( cartprodfstin
% 2.75/1.69 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.75/1.69 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.75/1.69 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjL2_type,type,
% 2.75/1.69 setukpairinjL2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjL2,definition,
% 2.75/1.69 ( setukpairinjL2
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.75/1.69 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.75/1.69 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.75/1.69 => ( Xx = Xz ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjL_type,type,
% 2.75/1.69 setukpairinjL: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjL,definition,
% 2.75/1.69 ( setukpairinjL
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.75/1.69 ( ( ( kpair @ Xx @ Xy )
% 2.75/1.69 = ( kpair @ Xz @ Xu ) )
% 2.75/1.69 => ( Xx = Xz ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR11_type,type,
% 2.75/1.69 setukpairinjR11: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR11,definition,
% 2.75/1.69 ( setukpairinjR11
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xx = Xy )
% 2.75/1.69 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.75/1.69 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR12_type,type,
% 2.75/1.69 setukpairinjR12: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR12,definition,
% 2.75/1.69 ( setukpairinjR12
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( Xx = Xy )
% 2.75/1.69 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.75/1.69 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR1_type,type,
% 2.75/1.69 setukpairinjR1: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR1,definition,
% 2.75/1.69 ( setukpairinjR1
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.75/1.69 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.75/1.69 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.75/1.69 => ( ( Xz = Xu )
% 2.75/1.69 => ( Xy = Xu ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(upairequniteq_type,type,
% 2.75/1.69 upairequniteq: $o ).
% 2.75/1.69
% 2.75/1.69 thf(upairequniteq,definition,
% 2.75/1.69 ( upairequniteq
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.75/1.69 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.75/1.69 = ( setadjoin @ Xz @ emptyset ) )
% 2.75/1.69 => ( Xx = Xy ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR2_type,type,
% 2.75/1.69 setukpairinjR2: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR2,definition,
% 2.75/1.69 ( setukpairinjR2
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.75/1.69 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.75/1.69 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.75/1.69 => ( Xy = Xu ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR_type,type,
% 2.75/1.69 setukpairinjR: $o ).
% 2.75/1.69
% 2.75/1.69 thf(setukpairinjR,definition,
% 2.75/1.69 ( setukpairinjR
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.75/1.69 ( ( ( kpair @ Xx @ Xy )
% 2.75/1.69 = ( kpair @ Xz @ Xu ) )
% 2.75/1.69 => ( Xy = Xu ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(ksndsingleton_type,type,
% 2.75/1.69 ksndsingleton: $o ).
% 2.75/1.69
% 2.75/1.69 thf(ksndsingleton,definition,
% 2.75/1.69 ( ksndsingleton
% 2.75/1.69 = ( ! [Xu: $i] :
% 2.75/1.69 ( ( iskpair @ Xu )
% 2.75/1.69 => ( singleton
% 2.75/1.69 @ ( dsetconstr @ ( setunion @ Xu )
% 2.75/1.69 @ ^ [Xx: $i] :
% 2.75/1.69 ( Xu
% 2.75/1.69 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(ksnd_type,type,
% 2.75/1.69 ksnd: $i > $i ).
% 2.75/1.69
% 2.75/1.69 thf(ksndpairEq_type,type,
% 2.75/1.69 ksndpairEq: $o ).
% 2.75/1.69
% 2.75/1.69 thf(ksndpairEq,definition,
% 2.75/1.69 ( ksndpairEq
% 2.75/1.69 = ( ! [Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.75/1.69 = Xy ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(kpairsurjEq_type,type,
% 2.75/1.69 kpairsurjEq: $o ).
% 2.75/1.69
% 2.75/1.69 thf(kpairsurjEq,definition,
% 2.75/1.69 ( kpairsurjEq
% 2.75/1.69 = ( ! [Xu: $i] :
% 2.75/1.69 ( ( iskpair @ Xu )
% 2.75/1.69 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.75/1.69 = Xu ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodsndin_type,type,
% 2.75/1.69 cartprodsndin: $o ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodsndin,definition,
% 2.75/1.69 ( cartprodsndin
% 2.75/1.69 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.75/1.69 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.75/1.69 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodpairmemEL_type,type,
% 2.75/1.69 cartprodpairmemEL: $o ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodpairmemEL,definition,
% 2.75/1.69 ( cartprodpairmemEL
% 2.75/1.69 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.75/1.69 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.75/1.69 => ( in @ Xx @ A ) ) ) ) ).
% 2.75/1.69
% 2.75/1.69 thf(cartprodpairmemER,conjecture,
% 2.75/1.70 ( setextAx
% 2.75/1.70 => ( emptysetAx
% 2.75/1.70 => ( setadjoinAx
% 2.75/1.70 => ( powersetAx
% 2.75/1.70 => ( setunionAx
% 2.75/1.70 => ( omega0Ax
% 2.75/1.70 => ( omegaSAx
% 2.75/1.70 => ( omegaIndAx
% 2.75/1.70 => ( replAx
% 2.75/1.70 => ( foundationAx
% 2.75/1.70 => ( wellorderingAx
% 2.75/1.70 => ( descrp
% 2.75/1.70 => ( dsetconstrI
% 2.75/1.70 => ( dsetconstrEL
% 2.75/1.70 => ( dsetconstrER
% 2.75/1.70 => ( exuE1
% 2.75/1.70 => ( prop2setE
% 2.75/1.70 => ( emptysetE
% 2.75/1.70 => ( emptysetimpfalse
% 2.75/1.70 => ( notinemptyset
% 2.75/1.70 => ( exuE3e
% 2.75/1.70 => ( setext
% 2.75/1.70 => ( emptyI
% 2.75/1.70 => ( noeltsimpempty
% 2.75/1.70 => ( setbeta
% 2.75/1.70 => ( nonemptyE1
% 2.75/1.70 => ( nonemptyI
% 2.75/1.70 => ( nonemptyI1
% 2.75/1.70 => ( setadjoinIL
% 2.75/1.70 => ( emptyinunitempty
% 2.75/1.70 => ( setadjoinIR
% 2.75/1.70 => ( setadjoinE
% 2.75/1.70 => ( setadjoinOr
% 2.75/1.70 => ( setoftrueEq
% 2.75/1.70 => ( powersetI
% 2.75/1.70 => ( emptyinPowerset
% 2.75/1.70 => ( emptyInPowerset
% 2.75/1.70 => ( powersetE
% 2.75/1.70 => ( setunionI
% 2.75/1.70 => ( setunionE
% 2.75/1.70 => ( subPowSU
% 2.75/1.70 => ( exuE2
% 2.75/1.70 => ( nonemptyImpWitness
% 2.75/1.70 => ( uniqinunit
% 2.75/1.70 => ( notinsingleton
% 2.75/1.70 => ( eqinunit
% 2.75/1.70 => ( singletonsswitch
% 2.75/1.70 => ( upairsetE
% 2.75/1.70 => ( upairsetIL
% 2.75/1.70 => ( upairsetIR
% 2.75/1.70 => ( emptyE1
% 2.75/1.70 => ( vacuousDall
% 2.75/1.70 => ( quantDeMorgan1
% 2.75/1.70 => ( quantDeMorgan2
% 2.75/1.70 => ( quantDeMorgan3
% 2.75/1.70 => ( quantDeMorgan4
% 2.75/1.70 => ( prop2setI
% 2.75/1.70 => ( prop2set2propI
% 2.75/1.70 => ( notdexE
% 2.75/1.70 => ( notdallE
% 2.75/1.70 => ( exuI1
% 2.75/1.70 => ( exuI3
% 2.75/1.70 => ( exuI2
% 2.75/1.70 => ( inCongP
% 2.75/1.70 => ( in__Cong
% 2.75/1.70 => ( exuE3u
% 2.75/1.70 => ( exu__Cong
% 2.75/1.70 => ( emptyset__Cong
% 2.75/1.70 => ( setadjoin__Cong
% 2.75/1.70 => ( powerset__Cong
% 2.75/1.70 => ( setunion__Cong
% 2.75/1.70 => ( omega__Cong
% 2.75/1.70 => ( exuEu
% 2.75/1.70 => ( descr__Cong
% 2.75/1.70 => ( dsetconstr__Cong
% 2.75/1.70 => ( subsetI1
% 2.75/1.70 => ( eqimpsubset2
% 2.75/1.70 => ( eqimpsubset1
% 2.75/1.70 => ( subsetI2
% 2.75/1.70 => ( emptysetsubset
% 2.75/1.70 => ( subsetE
% 2.75/1.70 => ( subsetE2
% 2.75/1.70 => ( notsubsetI
% 2.75/1.70 => ( notequalI1
% 2.75/1.70 => ( notequalI2
% 2.75/1.70 => ( subsetRefl
% 2.75/1.70 => ( subsetTrans
% 2.75/1.70 => ( setadjoinSub
% 2.75/1.70 => ( setadjoinSub2
% 2.75/1.70 => ( subset2powerset
% 2.75/1.70 => ( setextsub
% 2.75/1.70 => ( subsetemptysetimpeq
% 2.75/1.70 => ( powersetI1
% 2.75/1.70 => ( powersetE1
% 2.75/1.70 => ( inPowerset
% 2.75/1.70 => ( powersetsubset
% 2.75/1.70 => ( sepInPowerset
% 2.75/1.70 => ( sepSubset
% 2.75/1.70 => ( binunionIL
% 2.75/1.70 => ( upairset2IR
% 2.75/1.70 => ( binunionIR
% 2.75/1.70 => ( binunionEcases
% 2.75/1.70 => ( binunionE
% 2.75/1.70 => ( binunionLsub
% 2.75/1.70 => ( binunionRsub
% 2.75/1.70 => ( binintersectI
% 2.75/1.70 => ( binintersectSubset5
% 2.75/1.70 => ( binintersectEL
% 2.75/1.70 => ( binintersectLsub
% 2.75/1.70 => ( binintersectSubset2
% 2.75/1.70 => ( binintersectSubset3
% 2.75/1.70 => ( binintersectER
% 2.75/1.70 => ( disjointsetsI1
% 2.75/1.70 => ( binintersectRsub
% 2.75/1.70 => ( binintersectSubset4
% 2.75/1.70 => ( binintersectSubset1
% 2.75/1.70 => ( bs114d
% 2.75/1.70 => ( setminusI
% 2.75/1.70 => ( setminusEL
% 2.75/1.70 => ( setminusER
% 2.75/1.70 => ( setminusSubset2
% 2.75/1.70 => ( setminusERneg
% 2.75/1.70 => ( setminusELneg
% 2.75/1.70 => ( setminusILneg
% 2.75/1.70 => ( setminusIRneg
% 2.75/1.70 => ( setminusLsub
% 2.75/1.70 => ( setminusSubset1
% 2.75/1.70 => ( symdiffE
% 2.75/1.70 => ( symdiffI1
% 2.75/1.70 => ( symdiffI2
% 2.75/1.70 => ( symdiffIneg1
% 2.75/1.70 => ( symdiffIneg2
% 2.75/1.70 => ( secondinupair
% 2.75/1.70 => ( setukpairIL
% 2.75/1.70 => ( setukpairIR
% 2.75/1.70 => ( kpairiskpair
% 2.75/1.70 => ( kpairp
% 2.75/1.70 => ( singletonsubset
% 2.75/1.70 => ( singletoninpowerset
% 2.75/1.70 => ( singletoninpowunion
% 2.75/1.70 => ( upairset2E
% 2.75/1.70 => ( upairsubunion
% 2.75/1.70 => ( upairinpowunion
% 2.75/1.70 => ( ubforcartprodlem1
% 2.75/1.70 => ( ubforcartprodlem2
% 2.75/1.70 => ( ubforcartprodlem3
% 2.75/1.70 => ( cartprodpairin
% 2.75/1.70 => ( cartprodmempair1
% 2.75/1.70 => ( cartprodmempair
% 2.75/1.70 => ( setunionE2
% 2.75/1.70 => ( setunionsingleton1
% 2.75/1.70 => ( setunionsingleton2
% 2.75/1.70 => ( setunionsingleton
% 2.75/1.70 => ( singletonprop
% 2.75/1.70 => ( ex1E1
% 2.75/1.70 => ( ex1I
% 2.75/1.70 => ( ex1I2
% 2.75/1.70 => ( singletonsuniq
% 2.75/1.70 => ( setukpairinjL1
% 2.75/1.70 => ( kfstsingleton
% 2.75/1.70 => ( theprop
% 2.75/1.70 => ( kfstpairEq
% 2.75/1.70 => ( cartprodfstin
% 2.75/1.70 => ( setukpairinjL2
% 2.75/1.70 => ( setukpairinjL
% 2.75/1.70 => ( setukpairinjR11
% 2.75/1.70 => ( setukpairinjR12
% 2.75/1.70 => ( setukpairinjR1
% 2.75/1.70 => ( upairequniteq
% 2.75/1.70 => ( setukpairinjR2
% 2.75/1.70 => ( setukpairinjR
% 2.75/1.70 => ( ksndsingleton
% 2.75/1.70 => ( ksndpairEq
% 2.75/1.70 => ( kpairsurjEq
% 2.75/1.70 => ( cartprodsndin
% 2.75/1.70 => ( cartprodpairmemEL
% 2.75/1.70 => ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.75/1.70 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.75/1.70 => ( in @ Xy @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.75/1.70 ------------------------
% 2.96/1.72 % (16241)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_16138 for (2999ds/275Mi)
% 2.96/1.72 % (16242)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/18Mi)
% 2.96/1.72 % (16237)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_16138 for (2999ds/4Mi)
% 2.96/1.72 % (16239)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/2Mi)
% 2.96/1.72 % (16238)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_16138 for (2999ds/27Mi)
% 2.96/1.72 % (16236)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/183Mi)
% 2.96/1.72 % (16240)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_16138 for (2999ds/2Mi)
% 2.96/1.72 % (16239)Instruction limit reached!
% 2.96/1.72 % (16239)------------------------------
% 2.96/1.72 % (16239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.72 % (16239)Termination reason: Unknown
% 2.96/1.72 % (16239)Termination phase: shuffling
% 2.96/1.72
% 2.96/1.72 % (16240)Instruction limit reached!
% 2.96/1.72 % (16240)------------------------------
% 2.96/1.72 % (16240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.72 % (16240)Termination reason: Unknown
% 2.96/1.72 % (16240)Termination phase: shuffling
% 2.96/1.72
% 2.96/1.72 % (16240)Memory used [KB]: 1279
% 2.96/1.72 % (16240)Time elapsed: 0.003 s
% 2.96/1.72 % (16240)Instructions burned: 2 (million)
% 2.96/1.72 % (16240)------------------------------
% 2.96/1.72 % (16240)------------------------------
% 2.96/1.72 % (16239)Memory used [KB]: 1279
% 2.96/1.72 % (16239)Time elapsed: 0.003 s
% 2.96/1.72 % (16239)Instructions burned: 2 (million)
% 2.96/1.72 % (16239)------------------------------
% 2.96/1.72 % (16239)------------------------------
% 2.96/1.73 % (16237)Instruction limit reached!
% 2.96/1.73 % (16237)------------------------------
% 2.96/1.73 % (16237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.73 % (16237)Termination reason: Unknown
% 2.96/1.73 % (16237)Termination phase: shuffling
% 2.96/1.73
% 2.96/1.73 % (16237)Memory used [KB]: 1279
% 2.96/1.73 % (16237)Time elapsed: 0.004 s
% 2.96/1.73 % (16237)Instructions burned: 5 (million)
% 2.96/1.73 % (16237)------------------------------
% 2.96/1.73 % (16237)------------------------------
% 2.96/1.73 % (16242)Instruction limit reached!
% 2.96/1.73 % (16242)------------------------------
% 2.96/1.73 % (16242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.73 % (16242)Termination reason: Unknown
% 2.96/1.73 % (16242)Termination phase: Property scanning
% 2.96/1.73
% 2.96/1.73 % (16242)Memory used [KB]: 1663
% 2.96/1.73 % (16242)Time elapsed: 0.011 s
% 2.96/1.73 % (16242)Instructions burned: 19 (million)
% 2.96/1.73 % (16242)------------------------------
% 2.96/1.73 % (16242)------------------------------
% 2.96/1.74 % (16238)Instruction limit reached!
% 2.96/1.74 % (16238)------------------------------
% 2.96/1.74 % (16238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.74 % (16238)Termination reason: Unknown
% 2.96/1.74 % (16238)Termination phase: Property scanning
% 2.96/1.74
% 2.96/1.74 % (16238)Memory used [KB]: 1791
% 2.96/1.74 % (16238)Time elapsed: 0.015 s
% 2.96/1.74 % (16238)Instructions burned: 28 (million)
% 2.96/1.74 % (16238)------------------------------
% 2.96/1.74 % (16238)------------------------------
% 2.96/1.74 % (16243)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/3Mi)
% 2.96/1.74 % (16244)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_16138 for (2999ds/37Mi)
% 2.96/1.74 % (16245)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_16138 for (2999ds/15Mi)
% 2.96/1.74 % (16243)Instruction limit reached!
% 2.96/1.74 % (16243)------------------------------
% 2.96/1.74 % (16243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.74 % (16243)Termination reason: Unknown
% 2.96/1.74 % (16243)Termination phase: shuffling
% 2.96/1.74
% 2.96/1.74 % (16243)Memory used [KB]: 1279
% 2.96/1.74 % (16243)Time elapsed: 0.003 s
% 2.96/1.74 % (16243)Instructions burned: 3 (million)
% 2.96/1.74 % (16243)------------------------------
% 2.96/1.74 % (16243)------------------------------
% 2.96/1.75 % (16245)Instruction limit reached!
% 2.96/1.75 % (16245)------------------------------
% 2.96/1.75 % (16245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.75 % (16245)Termination reason: Unknown
% 2.96/1.75 % (16245)Termination phase: shuffling
% 2.96/1.75
% 2.96/1.75 % (16245)Memory used [KB]: 1663
% 2.96/1.75 % (16245)Time elapsed: 0.009 s
% 2.96/1.75 % (16245)Instructions burned: 15 (million)
% 2.96/1.75 % (16245)------------------------------
% 2.96/1.75 % (16245)------------------------------
% 2.96/1.75 % (16246)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_16138 for (2999ds/3Mi)
% 2.96/1.75 % (16246)Instruction limit reached!
% 2.96/1.75 % (16246)------------------------------
% 2.96/1.75 % (16246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.75 % (16246)Termination reason: Unknown
% 2.96/1.75 % (16246)Termination phase: shuffling
% 2.96/1.75
% 2.96/1.75 % (16246)Memory used [KB]: 1279
% 2.96/1.75 % (16246)Time elapsed: 0.003 s
% 2.96/1.75 % (16246)Instructions burned: 3 (million)
% 2.96/1.75 % (16246)------------------------------
% 2.96/1.75 % (16246)------------------------------
% 2.96/1.75 % (16247)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_16138 for (2999ds/1041Mi)
% 2.96/1.76 % (16244)Instruction limit reached!
% 2.96/1.76 % (16244)------------------------------
% 2.96/1.76 % (16244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.76 % (16244)Termination reason: Unknown
% 2.96/1.76 % (16244)Termination phase: Property scanning
% 2.96/1.76
% 2.96/1.76 % (16244)Memory used [KB]: 1918
% 2.96/1.76 % (16244)Time elapsed: 0.018 s
% 2.96/1.76 % (16244)Instructions burned: 37 (million)
% 2.96/1.76 % (16244)------------------------------
% 2.96/1.76 % (16244)------------------------------
% 2.96/1.76 % (16248)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/7Mi)
% 2.96/1.76 % (16248)Instruction limit reached!
% 2.96/1.76 % (16248)------------------------------
% 2.96/1.76 % (16248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.76 % (16248)Termination reason: Unknown
% 2.96/1.76 % (16248)Termination phase: shuffling
% 2.96/1.76
% 2.96/1.76 % (16248)Memory used [KB]: 1407
% 2.96/1.76 % (16248)Time elapsed: 0.005 s
% 2.96/1.76 % (16248)Instructions burned: 7 (million)
% 2.96/1.76 % (16248)------------------------------
% 2.96/1.76 % (16248)------------------------------
% 2.96/1.77 % (16249)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/16Mi)
% 2.96/1.77 % (16250)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_16138 for (2999ds/3Mi)
% 2.96/1.77 % (16250)Instruction limit reached!
% 2.96/1.77 % (16250)------------------------------
% 2.96/1.77 % (16250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.77 % (16250)Termination reason: Unknown
% 2.96/1.77 % (16250)Termination phase: shuffling
% 2.96/1.77
% 2.96/1.77 % (16250)Memory used [KB]: 1279
% 2.96/1.77 % (16250)Time elapsed: 0.003 s
% 2.96/1.77 % (16250)Instructions burned: 3 (million)
% 2.96/1.77 % (16250)------------------------------
% 2.96/1.77 % (16250)------------------------------
% 2.96/1.77 % (16249)Instruction limit reached!
% 2.96/1.77 % (16249)------------------------------
% 2.96/1.77 % (16249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.77 % (16249)Termination reason: Unknown
% 2.96/1.77 % (16249)Termination phase: shuffling
% 2.96/1.77
% 2.96/1.77 % (16249)Memory used [KB]: 1663
% 2.96/1.77 % (16249)Time elapsed: 0.009 s
% 2.96/1.77 % (16249)Instructions burned: 16 (million)
% 2.96/1.77 % (16249)------------------------------
% 2.96/1.77 % (16249)------------------------------
% 2.96/1.77 % (16251)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_16138 for (2999ds/3Mi)
% 2.96/1.78 % (16251)Instruction limit reached!
% 2.96/1.78 % (16251)------------------------------
% 2.96/1.78 % (16251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.78 % (16251)Termination reason: Unknown
% 2.96/1.78 % (16251)Termination phase: shuffling
% 2.96/1.78
% 2.96/1.78 % (16251)Memory used [KB]: 1279
% 2.96/1.78 % (16251)Time elapsed: 0.003 s
% 2.96/1.78 % (16251)Instructions burned: 3 (million)
% 2.96/1.78 % (16251)------------------------------
% 2.96/1.78 % (16251)------------------------------
% 2.96/1.78 % (16252)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/7Mi)
% 2.96/1.78 % (16252)Instruction limit reached!
% 2.96/1.78 % (16252)------------------------------
% 2.96/1.78 % (16252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.78 % (16252)Termination reason: Unknown
% 2.96/1.78 % (16252)Termination phase: shuffling
% 2.96/1.78
% 2.96/1.78 % (16252)Memory used [KB]: 1407
% 2.96/1.78 % (16252)Time elapsed: 0.005 s
% 2.96/1.78 % (16252)Instructions burned: 8 (million)
% 2.96/1.78 % (16252)------------------------------
% 2.96/1.78 % (16252)------------------------------
% 2.96/1.78 % (16253)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/3Mi)
% 2.96/1.79 % (16253)Instruction limit reached!
% 2.96/1.79 % (16253)------------------------------
% 2.96/1.79 % (16253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.79 % (16253)Termination reason: Unknown
% 2.96/1.79 % (16253)Termination phase: shuffling
% 2.96/1.79
% 2.96/1.79 % (16253)Memory used [KB]: 1279
% 2.96/1.79 % (16253)Time elapsed: 0.003 s
% 2.96/1.79 % (16253)Instructions burned: 3 (million)
% 2.96/1.79 % (16253)------------------------------
% 2.96/1.79 % (16253)------------------------------
% 2.96/1.79 % (16254)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/4Mi)
% 2.96/1.79 % (16254)Instruction limit reached!
% 2.96/1.79 % (16254)------------------------------
% 2.96/1.79 % (16254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.79 % (16254)Termination reason: Unknown
% 2.96/1.79 % (16254)Termination phase: shuffling
% 2.96/1.79
% 2.96/1.79 % (16254)Memory used [KB]: 1407
% 2.96/1.79 % (16254)Time elapsed: 0.004 s
% 2.96/1.79 % (16254)Instructions burned: 5 (million)
% 2.96/1.79 % (16254)------------------------------
% 2.96/1.79 % (16254)------------------------------
% 2.96/1.79 % (16255)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/18Mi)
% 2.96/1.80 % (16256)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/710Mi)
% 2.96/1.80 % (16255)Instruction limit reached!
% 2.96/1.80 % (16255)------------------------------
% 2.96/1.80 % (16255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.80 % (16255)Termination reason: Unknown
% 2.96/1.80 % (16255)Termination phase: shuffling
% 2.96/1.80
% 2.96/1.80 % (16255)Memory used [KB]: 1663
% 2.96/1.80 % (16255)Time elapsed: 0.010 s
% 2.96/1.80 % (16255)Instructions burned: 18 (million)
% 2.96/1.80 % (16255)------------------------------
% 2.96/1.80 % (16255)------------------------------
% 2.96/1.80 % (16257)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/6Mi)
% 2.96/1.81 % (16257)Instruction limit reached!
% 2.96/1.81 % (16257)------------------------------
% 2.96/1.81 % (16257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.81 % (16257)Termination reason: Unknown
% 2.96/1.81 % (16257)Termination phase: shuffling
% 2.96/1.81
% 2.96/1.81 % (16257)Memory used [KB]: 1407
% 2.96/1.81 % (16257)Time elapsed: 0.005 s
% 2.96/1.81 % (16257)Instructions burned: 8 (million)
% 2.96/1.81 % (16257)------------------------------
% 2.96/1.81 % (16257)------------------------------
% 2.96/1.81 % (16236)Instruction limit reached!
% 2.96/1.81 % (16236)------------------------------
% 2.96/1.81 % (16236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.81 % (16236)Termination reason: Unknown
% 2.96/1.81 % (16236)Termination phase: Saturation
% 2.96/1.81
% 2.96/1.81 % (16236)Memory used [KB]: 7547
% 2.96/1.81 % (16236)Time elapsed: 0.086 s
% 2.96/1.81 % (16236)Instructions burned: 184 (million)
% 2.96/1.81 % (16258)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_16138 for (2999ds/902Mi)
% 2.96/1.81 % (16236)------------------------------
% 2.96/1.81 % (16236)------------------------------
% 2.96/1.82 % (16259)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_16138 for (2999ds/21Mi)
% 2.96/1.82 % (16260)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_16138 for (2999ds/5Mi)
% 2.96/1.82 % (16260)Instruction limit reached!
% 2.96/1.82 % (16260)------------------------------
% 2.96/1.82 % (16260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.82 % (16260)Termination reason: Unknown
% 2.96/1.82 % (16260)Termination phase: shuffling
% 2.96/1.82
% 2.96/1.82 % (16260)Memory used [KB]: 1407
% 2.96/1.82 % (16260)Time elapsed: 0.004 s
% 2.96/1.82 % (16260)Instructions burned: 5 (million)
% 2.96/1.82 % (16260)------------------------------
% 2.96/1.82 % (16260)------------------------------
% 2.96/1.82 % (16261)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_16138 for (2998ds/6Mi)
% 2.96/1.83 % (16259)Instruction limit reached!
% 2.96/1.83 % (16259)------------------------------
% 2.96/1.83 % (16259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.83 % (16259)Termination reason: Unknown
% 2.96/1.83 % (16259)Termination phase: shuffling
% 2.96/1.83
% 2.96/1.83 % (16259)Memory used [KB]: 1791
% 2.96/1.83 % (16259)Time elapsed: 0.012 s
% 2.96/1.83 % (16259)Instructions burned: 22 (million)
% 2.96/1.83 % (16259)------------------------------
% 2.96/1.83 % (16259)------------------------------
% 2.96/1.83 % (16261)Instruction limit reached!
% 2.96/1.83 % (16261)------------------------------
% 2.96/1.83 % (16261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.83 % (16261)Termination reason: Unknown
% 2.96/1.83 % (16261)Termination phase: shuffling
% 2.96/1.83
% 2.96/1.83 % (16261)Memory used [KB]: 1407
% 2.96/1.83 % (16261)Time elapsed: 0.005 s
% 2.96/1.83 % (16261)Instructions burned: 7 (million)
% 2.96/1.83 % (16261)------------------------------
% 2.96/1.83 % (16261)------------------------------
% 2.96/1.84 % (16262)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_16138 for (2998ds/377Mi)
% 2.96/1.84 % (16263)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_16138 for (2998ds/779Mi)
% 2.96/1.84 % (16264)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_16138 for (2998ds/19Mi)
% 2.96/1.85 % (16264)Instruction limit reached!
% 2.96/1.85 % (16264)------------------------------
% 2.96/1.85 % (16264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.85 % (16264)Termination reason: Unknown
% 2.96/1.85 % (16264)Termination phase: shuffling
% 2.96/1.85
% 2.96/1.85 % (16264)Memory used [KB]: 1663
% 2.96/1.85 % (16264)Time elapsed: 0.011 s
% 2.96/1.85 % (16264)Instructions burned: 21 (million)
% 2.96/1.85 % (16264)------------------------------
% 2.96/1.85 % (16264)------------------------------
% 2.96/1.86 % (16241)Instruction limit reached!
% 2.96/1.86 % (16241)------------------------------
% 2.96/1.86 % (16241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.86 % (16241)Termination reason: Unknown
% 2.96/1.86 % (16241)Termination phase: Saturation
% 2.96/1.86
% 2.96/1.86 % (16241)Memory used [KB]: 8315
% 2.96/1.86 % (16241)Time elapsed: 0.137 s
% 2.96/1.86 % (16241)Instructions burned: 276 (million)
% 2.96/1.86 % (16241)------------------------------
% 2.96/1.86 % (16241)------------------------------
% 2.96/1.87 % (16265)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_16138 for (2998ds/879Mi)
% 2.96/1.88 % (16266)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_16138 for (2998ds/17Mi)
% 2.96/1.88 % (16266)Instruction limit reached!
% 2.96/1.88 % (16266)------------------------------
% 2.96/1.88 % (16266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.88 % (16266)Termination reason: Unknown
% 2.96/1.88 % (16266)Termination phase: shuffling
% 2.96/1.88
% 2.96/1.88 % (16266)Memory used [KB]: 1663
% 2.96/1.88 % (16266)Time elapsed: 0.009 s
% 2.96/1.88 % (16266)Instructions burned: 17 (million)
% 2.96/1.88 % (16266)------------------------------
% 2.96/1.88 % (16266)------------------------------
% 2.96/1.90 % (16267)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_16138 for (2998ds/3Mi)
% 2.96/1.90 % (16267)Instruction limit reached!
% 2.96/1.90 % (16267)------------------------------
% 2.96/1.90 % (16267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.90 % (16267)Termination reason: Unknown
% 2.96/1.90 % (16267)Termination phase: shuffling
% 2.96/1.90
% 2.96/1.90 % (16267)Memory used [KB]: 1279
% 2.96/1.90 % (16267)Time elapsed: 0.004 s
% 2.96/1.90 % (16267)Instructions burned: 4 (million)
% 2.96/1.90 % (16267)------------------------------
% 2.96/1.90 % (16267)------------------------------
% 2.96/1.92 % (16268)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_16138 for (2997ds/30Mi)
% 2.96/1.93 % (16268)Instruction limit reached!
% 2.96/1.93 % (16268)------------------------------
% 2.96/1.93 % (16268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.93 % (16268)Termination reason: Unknown
% 2.96/1.93 % (16268)Termination phase: Property scanning
% 2.96/1.93
% 2.96/1.93 % (16268)Memory used [KB]: 1791
% 2.96/1.93 % (16268)Time elapsed: 0.016 s
% 2.96/1.93 % (16268)Instructions burned: 31 (million)
% 2.96/1.93 % (16268)------------------------------
% 2.96/1.93 % (16268)------------------------------
% 2.96/1.95 % (16269)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_16138 for (2997ds/127Mi)
% 2.96/1.95 % (16258)First to succeed.
% 2.96/1.96 % (16258)Refutation found. Thanks to Tanya!
% 2.96/1.96 % SZS status Theorem for DTF2THF_16138
% 2.96/1.96 % SZS output start Proof for DTF2THF_16138
% 2.96/1.96 thf(func_def_0, type, in: $i > $i > $o).
% 2.96/1.96 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.96/1.96 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.96/1.96 thf(func_def_8, type, powerset: $i > $i).
% 2.96/1.96 thf(func_def_10, type, setunion: $i > $i).
% 2.96/1.96 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.96/1.96 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_26, type, prop2set: $o > $i).
% 2.96/1.96 thf(func_def_36, type, nonempty: $i > $o).
% 2.96/1.96 thf(func_def_69, type, set2prop: $i > $o).
% 2.96/1.96 thf(func_def_88, type, subset: $i > $i > $o).
% 2.96/1.96 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.96/1.96 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.96/1.96 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.96/1.96 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.96/1.96 thf(func_def_135, type, regular: $i > $o).
% 2.96/1.96 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.96/1.96 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.96/1.96 thf(func_def_153, type, iskpair: $i > $o).
% 2.96/1.96 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.96/1.96 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.96/1.96 thf(func_def_177, type, singleton: $i > $o).
% 2.96/1.96 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.96/1.96 thf(func_def_184, type, atmost1p: $i > $o).
% 2.96/1.96 thf(func_def_185, type, atleast2p: $i > $o).
% 2.96/1.96 thf(func_def_186, type, atmost2p: $i > $o).
% 2.96/1.96 thf(func_def_187, type, upairsetp: $i > $o).
% 2.96/1.96 thf(func_def_191, type, kfst: $i > $i).
% 2.96/1.96 thf(func_def_203, type, ksnd: $i > $i).
% 2.96/1.96 thf(func_def_222, type, sP2: $i > $i > $o).
% 2.96/1.96 thf(func_def_223, type, sP3: $i > $o).
% 2.96/1.96 thf(func_def_224, type, sP4: $i > $i > $o).
% 2.96/1.96 thf(func_def_225, type, sP5: $i > $i > $o).
% 2.96/1.96 thf(func_def_227, type, sP7: $o > $i > $i > $i > $o).
% 2.96/1.96 thf(func_def_238, type, sK18: $i > $i > $o).
% 2.96/1.96 thf(func_def_240, type, sK20: $i > $i).
% 2.96/1.96 thf(func_def_241, type, sK21: $i > $i).
% 2.96/1.96 thf(func_def_242, type, sK22: $i > ($i > $i > $o) > $i).
% 2.96/1.96 thf(func_def_243, type, sK23: $i > $i > ($i > $i > $o) > $i).
% 2.96/1.96 thf(func_def_244, type, sK24: $i > ($i > $i > $o) > $i).
% 2.96/1.96 thf(func_def_255, type, sK35: ($i > $o) > $i).
% 2.96/1.96 thf(func_def_256, type, sK36: $i > $o).
% 2.96/1.96 thf(func_def_257, type, sK37: $i > $i).
% 2.96/1.96 thf(func_def_265, type, sK45: $i > $i).
% 2.96/1.96 thf(func_def_281, type, sK61: $i > $o).
% 2.96/1.96 thf(func_def_282, type, sK62: ($i > $o) > $i > $i).
% 2.96/1.96 thf(func_def_288, type, sK68: $i > $o).
% 2.96/1.96 thf(func_def_289, type, sK69: $i > $o).
% 2.96/1.96 thf(func_def_290, type, sK70: ($i > $o) > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_291, type, sK71: ($i > $o) > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_303, type, sK83: $i > $o).
% 2.96/1.96 thf(func_def_304, type, sK84: ($i > $o) > $i > $i > $i).
% 2.96/1.96 thf(func_def_307, type, sK87: $i > $o).
% 2.96/1.96 thf(func_def_309, type, sK89: $o > $i > $i > $i).
% 2.96/1.96 thf(func_def_319, type, sK99: $i > $o).
% 2.96/1.96 thf(func_def_322, type, sK102: $i > $o).
% 2.96/1.96 thf(func_def_324, type, sK104: $i > $o).
% 2.96/1.96 thf(func_def_325, type, sK105: $i > $o).
% 2.96/1.96 thf(func_def_326, type, sK106: ($i > $o) > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_327, type, sK107: ($i > $o) > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_331, type, sK111: $i > $o).
% 2.96/1.96 thf(func_def_334, type, sK114: $i > $o).
% 2.96/1.96 thf(func_def_337, type, sK117: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_340, type, sK120: $i > $i).
% 2.96/1.96 thf(func_def_343, type, sK123: $i > $o).
% 2.96/1.96 thf(func_def_346, type, sK126: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_347, type, sK127: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_350, type, sK130: $i > $o).
% 2.96/1.96 thf(func_def_352, type, sK132: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_353, type, sK133: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_354, type, sK134: $i > $o).
% 2.96/1.96 thf(func_def_375, type, sK155: $i > $i > $i).
% 2.96/1.96 thf(func_def_404, type, sK184: $i > $o).
% 2.96/1.96 thf(func_def_413, type, sK193: ($i > $o) > $i).
% 2.96/1.96 thf(func_def_414, type, sK194: $i > $o).
% 2.96/1.96 thf(func_def_422, type, sK202: $i > $i > $i).
% 2.96/1.96 thf(func_def_423, type, sK203: $i > $i > $i).
% 2.96/1.96 thf(func_def_424, type, sK204: $i > $i > $i).
% 2.96/1.96 thf(func_def_425, type, sK205: $i > $i > $i).
% 2.96/1.96 thf(func_def_426, type, sK206: $i > $i > $i).
% 2.96/1.96 thf(func_def_427, type, sK207: $i > $i > $i > $i).
% 2.96/1.96 thf(func_def_428, type, sK208: $i > $i).
% 2.96/1.96 thf(func_def_429, type, sK209: $i > $i).
% 2.96/1.96 thf(func_def_430, type, sK210: $i > $i).
% 2.96/1.96 thf(func_def_431, type, sK211: $i > $i).
% 2.96/1.96 thf(func_def_432, type, sK212: $i > $i > $i).
% 2.96/1.96 thf(func_def_433, type, sK213: $i > $i > $i > $i).
% 2.96/1.96 thf(func_def_434, type, sK214: $i > $i > $i > $i).
% 2.96/1.96 thf(func_def_435, type, sK215: $i > $i > $i).
% 2.96/1.96 thf(func_def_436, type, sK216: $i > $i > $i).
% 2.96/1.96 thf(func_def_437, type, sK217: $i > $i).
% 2.96/1.96 thf(func_def_439, type, sK219: $i > $i).
% 2.96/1.96 thf(func_def_440, type, sK220: $i > $i).
% 2.96/1.96 thf(func_def_442, type, sK222: $i > $o).
% 2.96/1.96 thf(func_def_444, type, sK224: ($i > $o) > $i > $i).
% 2.96/1.96 thf(func_def_445, type, sK225: ($i > $o) > $i > $i).
% 2.96/1.96 thf(func_def_446, type, sK226: $i > $i).
% 2.96/1.96 thf(func_def_449, type, sK229: $i > $i).
% 2.96/1.96 thf(func_def_474, type, sK254: $i > $i).
% 2.96/1.96 thf(func_def_478, type, sK258: $i > $i > $i > $i).
% 2.96/1.96 thf(func_def_479, type, sK259: $i > $i > $i > $i).
% 2.96/1.96 thf(func_def_488, type, sK268: $i > $o).
% 2.96/1.96 thf(func_def_490, type, sK270: $i > $o).
% 2.96/1.96 thf(func_def_491, type, sK271: $i > $i > $i).
% 2.96/1.96 thf(func_def_497, type, sK277: $i > $i > $i).
% 2.96/1.96 thf(func_def_501, type, sK281: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_502, type, sK282: $i > $o).
% 2.96/1.96 thf(func_def_523, type, sK303: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_524, type, sK304: $i > $o).
% 2.96/1.96 thf(func_def_529, type, sK309: $i > $i > $i).
% 2.96/1.96 thf(func_def_530, type, sK310: $i > $i > $i).
% 2.96/1.96 thf(func_def_531, type, sK311: $i > $o).
% 2.96/1.96 thf(func_def_533, type, sK313: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_545, type, sK325: $i > $i).
% 2.96/1.96 thf(func_def_558, type, sK338: $i > $i > $i).
% 2.96/1.96 thf(func_def_582, type, sK362: $i > $o).
% 2.96/1.96 thf(func_def_598, type, sK378: $i > $i > $i).
% 2.96/1.96 thf(func_def_623, type, sK403: $i > $i > $i).
% 2.96/1.96 thf(func_def_632, type, sK412: $i > $i > $i).
% 2.96/1.96 thf(func_def_648, type, sK428: $i > $o).
% 2.96/1.96 thf(func_def_649, type, sK429: $i > $o).
% 2.96/1.96 thf(func_def_650, type, sK430: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.96/1.96 thf(func_def_651, type, sK431: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.96/1.96 thf(func_def_653, type, sK433: $i > $o).
% 2.96/1.96 thf(func_def_670, type, sK450: $i > $o).
% 2.96/1.96 thf(func_def_674, type, sK454: $i > $o).
% 2.96/1.96 thf(func_def_676, type, sK456: ($i > $o) > $i).
% 2.96/1.96 thf(func_def_677, type, sK457: ($i > $o) > $i).
% 2.96/1.96 thf(func_def_679, type, sK459: $i > $o).
% 2.96/1.96 thf(func_def_683, type, sK463: $i > $o).
% 2.96/1.96 thf(func_def_685, type, sK465: ($i > $o) > $i > $i).
% 2.96/1.96 thf(func_def_686, type, sK466: $i > $o).
% 2.96/1.96 thf(func_def_692, type, sK472: $i > $i > $i).
% 2.96/1.96 thf(func_def_698, type, sK478: $i > $o).
% 2.96/1.96 thf(func_def_701, type, sK481: $i > ($i > $o) > $i).
% 2.96/1.96 thf(func_def_702, type, sK482: $i > $o).
% 2.96/1.96 thf(func_def_703, type, sK483: $i > $i).
% 2.96/1.96 thf(func_def_704, type, sK484: ($i > $o) > $i).
% 2.96/1.96 thf(func_def_713, type, sK493: $i > $i).
% 2.96/1.96 thf(func_def_715, type, ph495: !>[X0: $tType]:(X0)).
% 2.96/1.96 thf(f3336,plain,(
% 2.96/1.96 $false),
% 2.96/1.96 inference(avatar_sat_refutation,[status(thm)],[f3329,f3335])).
% 2.96/1.96 thf(f3335,plain,(
% 2.96/1.96 spl494_15),
% 2.96/1.96 inference(avatar_split_clause,[status(thm)],[f2808,f3194])).
% 2.96/1.96 thf(f3194,definition,(
% 2.96/1.96 spl494_15 <=> ($true = (in @ sK178 @ sK180))),
% 2.96/1.96 introduced(definition,[new_symbols(naming,[spl494_15])],[avatar_definition])).
% 2.96/1.96 thf(f2808,plain,(
% 2.96/1.96 ($true = (in @ sK178 @ sK180))),
% 2.96/1.96 inference(trivial_inequality_removal,[status(thm)],[f2807])).
% 2.96/1.96 thf(f2807,plain,(
% 2.96/1.96 ($true = (in @ sK178 @ sK180)) | ($true != $true)),
% 2.96/1.96 inference(forward_demodulation,[status(thm)],[f2802,f1855])).
% 2.96/1.96 thf(f1855,plain,(
% 2.96/1.96 (ksndpairEq = $true)),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f1054])).
% 2.96/1.96 thf(f1054,plain,(
% 2.96/1.96 (upairsetE = $true) & (exuEu = $true) & (setminusERneg = $true) & (emptyInPowerset = $true) & (setadjoinSub2 = $true) & (binintersectRsub = $true) & (eqinunit = $true) & (emptyI = $true) & (subsetI2 = $true) & (binintersectSubset5 = $true) & (setminusILneg = $true) & (setbeta = $true) & (upairset2E = $true) & (ksndsingleton = $true) & (quantDeMorgan1 = $true) & (notsubsetI = $true) & (notdallE = $true) & (setoftrueEq = $true) & (binintersectI = $true) & (setunionI = $true) & (quantDeMorgan3 = $true) & (nonemptyI = $true) & (upairsetIL = $true) & (kpairsurjEq = $true) & (notinemptyset = $true) & (symdiffIneg2 = $true) & (setadjoinE = $true) & (emptyinPowerset = $true) & (cartprodpairmemEL = $true) & (binintersectSubset1 = $true) & (ubforcartprodlem3 = $true) & (notinsingleton = $true) & (nonemptyE1 = $true) & (descr__Cong = $true) & (binintersectSubset2 = $true) & (setextAx = $true) & (setunionsingleton = $true) & (setukpairinjL = $true) & (exuI1 = $true) & (quantDeMorgan4 = $true) & (setunionsingleton2 = $true) & (setukpairinjL1 = $true) & (binintersectSubset4 = $true) & (secondinupair = $true) & (notequalI1 = $true) & (setukpairinjR2 = $true) & (upairsetIR = $true) & (powerset__Cong = $true) & (setminusELneg = $true) & (setextsub = $true) & (setunionsingleton1 = $true) & (setukpairinjR12 = $true) & (setadjoin__Cong = $true) & (exuE1 = $true) & (prop2set2propI = $true) & (singletonsubset = $true) & (subsetE2 = $true) & (ex1I2 = $true) & (symdiffI1 = $true) & (singletoninpowunion = $true) & (subsetI1 = $true) & (subset2powerset = $true) & (ex1I = $true) & (powersetE = $true) & (kfstpairEq = $true) & (setukpairinjL2 = $true) & (prop2setE = $true) & (binunionE = $true) & (exuE2 = $true) & (binintersectEL = $true) & (emptysetE = $true) & (setadjoinOr = $true) & (setukpairinjR = $true) & (omegaSAx = $true) & (setminusIRneg = $true) & (exu__Cong = $true) & (setunionE = $true) & (exuE3u = $true) & (exuE3e = $true) & (omega0Ax = $true) & (vacuousDall = $true) & (kpairp = $true) & (emptyset__Cong = $true) & (ubforcartprodlem1 = $true) & (noeltsimpempty = $true) & (sepInPowerset = $true) & (inPowerset = $true) & (replAx = $true) & (setukpairinjR1 = $true) & (setunionE2 = $true) & (wellorderingAx = $true) & (dsetconstr__Cong = $true) & (emptyinunitempty = $true) & (upairsubunion = $true) & (notdexE = $true) & (singletonprop = $true) & (descrp = $true) & (setminusSubset1 = $true) & (binintersectER = $true) & (powersetI = $true) & (setadjoinSub = $true) & (setukpairinjR11 = $true) & (emptysetAx = $true) & (eqimpsubset1 = $true) & (binunionLsub = $true) & (setukpairIL = $true) & (setukpairIR = $true) & (cartprodpairin = $true) & (dsetconstrI = $true) & (symdiffIneg1 = $true) & (singletonsswitch = $true) & (cartprodfstin = $true) & (exuI3 = $true) & (binunionIL = $true) & (binunionEcases = $true) & (cartprodmempair1 = $true) & (subsetRefl = $true) & (ubforcartprodlem2 = $true) & (powersetE1 = $true) & (quantDeMorgan2 = $true) & (dsetconstrER = $true) & (setext = $true) & (setadjoinAx = $true) & (setminusSubset2 = $true) & (dsetconstrEL = $true) & (kfstsingleton = $true) & (binintersectSubset3 = $true) & (powersetsubset = $true) & (emptysetimpfalse = $true) & (binintersectLsub = $true) & (theprop = $true) & (disjointsetsI1 = $true) & (emptyE1 = $true) & (prop2setI = $true) & (subsetTrans = $true) & (ksndpairEq = $true) & (binunionIR = $true) & (cartprodsndin = $true) & (cartprodmempair = $true) & (upairequniteq = $true) & (emptysetsubset = $true) & (setminusEL = $true) & (foundationAx = $true) & (subsetE = $true) & (setminusI = $true) & (in__Cong = $true) & (uniqinunit = $true) & (setadjoinIR = $true) & (symdiffE = $true) & (notequalI2 = $true) & (setadjoinIL = $true) & (setminusER = $true) & (symdiffI2 = $true) & (upairinpowunion = $true) & (singletoninpowerset = $true) & (setunionAx = $true) & (powersetI1 = $true) & (powersetAx = $true) & (ex1E1 = $true) & (bs114d = $true) & (($true != (in @ sK178 @ sK180)) & ($true = (in @ (kpair @ sK181 @ sK178) @ (cartprod @ sK179 @ sK180)))) & (omegaIndAx = $true) & (sepSubset = $true) & (upairset2IR = $true) & (exuI2 = $true) & (subPowSU = $true) & (setunion__Cong = $true) & (omega__Cong = $true) & (singletonsuniq = $true) & (kpairiskpair = $true) & (nonemptyI1 = $true) & (binunionRsub = $true) & (eqimpsubset2 = $true) & (nonemptyImpWitness = $true) & (subsetemptysetimpeq = $true) & (setminusLsub = $true) & (inCongP = $true)),
% 2.96/1.96 inference(skolemisation,[status(esa),new_symbols(skolem,[sK178,sK179,sK180,sK181])],[f585,f1053])).
% 2.96/1.96 thf(f1053,plain,(
% 2.96/1.96 ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true != (in @ X0 @ X2)) & ($true = (in @ (kpair @ X3 @ X0) @ (cartprod @ X1 @ X2)))) => (($true != (in @ sK178 @ sK180)) & ($true = (in @ (kpair @ sK181 @ sK178) @ (cartprod @ sK179 @ sK180))))),
% 2.96/1.96 introduced(definition,[],[choice_axiom])).
% 2.96/1.96 thf(f585,plain,(
% 2.96/1.96 (upairsetE = $true) & (exuEu = $true) & (setminusERneg = $true) & (emptyInPowerset = $true) & (setadjoinSub2 = $true) & (binintersectRsub = $true) & (eqinunit = $true) & (emptyI = $true) & (subsetI2 = $true) & (binintersectSubset5 = $true) & (setminusILneg = $true) & (setbeta = $true) & (upairset2E = $true) & (ksndsingleton = $true) & (quantDeMorgan1 = $true) & (notsubsetI = $true) & (notdallE = $true) & (setoftrueEq = $true) & (binintersectI = $true) & (setunionI = $true) & (quantDeMorgan3 = $true) & (nonemptyI = $true) & (upairsetIL = $true) & (kpairsurjEq = $true) & (notinemptyset = $true) & (symdiffIneg2 = $true) & (setadjoinE = $true) & (emptyinPowerset = $true) & (cartprodpairmemEL = $true) & (binintersectSubset1 = $true) & (ubforcartprodlem3 = $true) & (notinsingleton = $true) & (nonemptyE1 = $true) & (descr__Cong = $true) & (binintersectSubset2 = $true) & (setextAx = $true) & (setunionsingleton = $true) & (setukpairinjL = $true) & (exuI1 = $true) & (quantDeMorgan4 = $true) & (setunionsingleton2 = $true) & (setukpairinjL1 = $true) & (binintersectSubset4 = $true) & (secondinupair = $true) & (notequalI1 = $true) & (setukpairinjR2 = $true) & (upairsetIR = $true) & (powerset__Cong = $true) & (setminusELneg = $true) & (setextsub = $true) & (setunionsingleton1 = $true) & (setukpairinjR12 = $true) & (setadjoin__Cong = $true) & (exuE1 = $true) & (prop2set2propI = $true) & (singletonsubset = $true) & (subsetE2 = $true) & (ex1I2 = $true) & (symdiffI1 = $true) & (singletoninpowunion = $true) & (subsetI1 = $true) & (subset2powerset = $true) & (ex1I = $true) & (powersetE = $true) & (kfstpairEq = $true) & (setukpairinjL2 = $true) & (prop2setE = $true) & (binunionE = $true) & (exuE2 = $true) & (binintersectEL = $true) & (emptysetE = $true) & (setadjoinOr = $true) & (setukpairinjR = $true) & (omegaSAx = $true) & (setminusIRneg = $true) & (exu__Cong = $true) & (setunionE = $true) & (exuE3u = $true) & (exuE3e = $true) & (omega0Ax = $true) & (vacuousDall = $true) & (kpairp = $true) & (emptyset__Cong = $true) & (ubforcartprodlem1 = $true) & (noeltsimpempty = $true) & (sepInPowerset = $true) & (inPowerset = $true) & (replAx = $true) & (setukpairinjR1 = $true) & (setunionE2 = $true) & (wellorderingAx = $true) & (dsetconstr__Cong = $true) & (emptyinunitempty = $true) & (upairsubunion = $true) & (notdexE = $true) & (singletonprop = $true) & (descrp = $true) & (setminusSubset1 = $true) & (binintersectER = $true) & (powersetI = $true) & (setadjoinSub = $true) & (setukpairinjR11 = $true) & (emptysetAx = $true) & (eqimpsubset1 = $true) & (binunionLsub = $true) & (setukpairIL = $true) & (setukpairIR = $true) & (cartprodpairin = $true) & (dsetconstrI = $true) & (symdiffIneg1 = $true) & (singletonsswitch = $true) & (cartprodfstin = $true) & (exuI3 = $true) & (binunionIL = $true) & (binunionEcases = $true) & (cartprodmempair1 = $true) & (subsetRefl = $true) & (ubforcartprodlem2 = $true) & (powersetE1 = $true) & (quantDeMorgan2 = $true) & (dsetconstrER = $true) & (setext = $true) & (setadjoinAx = $true) & (setminusSubset2 = $true) & (dsetconstrEL = $true) & (kfstsingleton = $true) & (binintersectSubset3 = $true) & (powersetsubset = $true) & (emptysetimpfalse = $true) & (binintersectLsub = $true) & (theprop = $true) & (disjointsetsI1 = $true) & (emptyE1 = $true) & (prop2setI = $true) & (subsetTrans = $true) & (ksndpairEq = $true) & (binunionIR = $true) & (cartprodsndin = $true) & (cartprodmempair = $true) & (upairequniteq = $true) & (emptysetsubset = $true) & (setminusEL = $true) & (foundationAx = $true) & (subsetE = $true) & (setminusI = $true) & (in__Cong = $true) & (uniqinunit = $true) & (setadjoinIR = $true) & (symdiffE = $true) & (notequalI2 = $true) & (setadjoinIL = $true) & (setminusER = $true) & (symdiffI2 = $true) & (upairinpowunion = $true) & (singletoninpowerset = $true) & (setunionAx = $true) & (powersetI1 = $true) & (powersetAx = $true) & (ex1E1 = $true) & (bs114d = $true) & ? [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true != (in @ X0 @ X2)) & ($true = (in @ (kpair @ X3 @ X0) @ (cartprod @ X1 @ X2)))) & (omegaIndAx = $true) & (sepSubset = $true) & (upairset2IR = $true) & (exuI2 = $true) & (subPowSU = $true) & (setunion__Cong = $true) & (omega__Cong = $true) & (singletonsuniq = $true) & (kpairiskpair = $true) & (nonemptyI1 = $true) & (binunionRsub = $true) & (eqimpsubset2 = $true) & (nonemptyImpWitness = $true) & (subsetemptysetimpeq = $true) & (setminusLsub = $true) & (inCongP = $true)),
% 2.96/1.96 inference(flattening,[status(thm)],[f584])).
% 2.96/1.96 thf(f584,plain,(
% 2.96/1.96 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true != (in @ X0 @ X2)) & ($true = (in @ (kpair @ X3 @ X0) @ (cartprod @ X1 @ X2)))) & (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)),
% 2.96/1.96 inference(ennf_transformation,[status(thm)],[f452])).
% 2.96/1.96 thf(f452,plain,(
% 2.96/1.96 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : (($true = (in @ (kpair @ X3 @ X0) @ (cartprod @ X1 @ X2))) => ($true = (in @ X0 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.96/1.96 inference(fool_elimination,[status(thm)],[f451])).
% 2.96/1.96 thf(f451,plain,(
% 2.96/1.96 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i,X3 : $i] : ((in @ (kpair @ X3 @ X0) @ (cartprod @ X1 @ X2)) => (in @ X0 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.96/1.96 inference(rectify,[status(thm)],[f182])).
% 2.96/1.96 thf(f182,negated_conjecture,(
% 2.96/1.96 ~(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 => ! [X2 : $i,X3 : $i,X4 : $i,X1 : $i] : ((in @ (kpair @ X1 @ X2) @ (cartprod @ X3 @ X4)) => (in @ X2 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.96/1.96 inference(negated_conjecture,[status(cth)],[f181])).
% 2.96/1.96 thf(f181,conjecture,(
% 2.96/1.96 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 => ! [X2 : $i,X3 : $i,X4 : $i,X1 : $i] : ((in @ (kpair @ X1 @ X2) @ (cartprod @ X3 @ X4)) => (in @ X2 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.96/1.96 file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',cartprodpairmemER)).
% 2.96/1.96 thf(f2802,plain,(
% 2.96/1.96 (ksndpairEq != $true) | ($true = (in @ sK178 @ sK180))),
% 2.96/1.96 inference(superposition,[status(thm)],[f2578,f1794])).
% 2.96/1.96 thf(f1794,plain,(
% 2.96/1.96 ( ! [X0 : $i,X1 : $i] : (((ksnd @ (kpair @ X1 @ X0)) = X0) | (ksndpairEq != $true)) )),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f1023])).
% 2.96/1.96 thf(f1023,plain,(
% 2.96/1.96 (! [X0 : $i,X1 : $i] : ((ksnd @ (kpair @ X1 @ X0)) = X0) | (ksndpairEq != $true)) & ((ksndpairEq = $true) | ((ksnd @ (kpair @ sK161 @ sK160)) != sK160))),
% 2.96/1.96 inference(skolemisation,[status(esa),new_symbols(skolem,[sK160,sK161])],[f1021,f1022])).
% 2.96/1.96 thf(f1022,plain,(
% 2.96/1.96 ? [X2 : $i,X3 : $i] : ((ksnd @ (kpair @ X3 @ X2)) != X2) => ((ksnd @ (kpair @ sK161 @ sK160)) != sK160)),
% 2.96/1.96 introduced(definition,[],[choice_axiom])).
% 2.96/1.96 thf(f1021,plain,(
% 2.96/1.96 (! [X0 : $i,X1 : $i] : ((ksnd @ (kpair @ X1 @ X0)) = X0) | (ksndpairEq != $true)) & ((ksndpairEq = $true) | ? [X2 : $i,X3 : $i] : ((ksnd @ (kpair @ X3 @ X2)) != X2))),
% 2.96/1.96 inference(rectify,[status(thm)],[f1020])).
% 2.96/1.96 thf(f1020,plain,(
% 2.96/1.96 (! [X0 : $i,X1 : $i] : ((ksnd @ (kpair @ X1 @ X0)) = X0) | (ksndpairEq != $true)) & ((ksndpairEq = $true) | ? [X0 : $i,X1 : $i] : ((ksnd @ (kpair @ X1 @ X0)) != X0))),
% 2.96/1.96 inference(nnf_transformation,[status(thm)],[f510])).
% 2.96/1.96 thf(f510,plain,(
% 2.96/1.96 ! [X0 : $i,X1 : $i] : ((ksnd @ (kpair @ X1 @ X0)) = X0) <=> (ksndpairEq = $true)),
% 2.96/1.96 inference(fool_elimination,[status(thm)],[f509])).
% 2.96/1.96 thf(f509,plain,(
% 2.96/1.96 (ksndpairEq = ! [X0 : $i,X1 : $i] : ((ksnd @ (kpair @ X1 @ X0)) = X0))),
% 2.96/1.96 inference(rectify,[status(thm)],[f177])).
% 2.96/1.96 thf(f177,axiom,(
% 2.96/1.96 (ksndpairEq = ! [X2 : $i,X1 : $i] : ((ksnd @ (kpair @ X1 @ X2)) = X2))),
% 2.96/1.96 file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',ksndpairEq)).
% 2.96/1.96 thf(f2578,plain,(
% 2.96/1.96 ((in @ (ksnd @ (kpair @ sK181 @ sK178)) @ sK180) = $true)),
% 2.96/1.96 inference(trivial_inequality_removal,[status(thm)],[f2577])).
% 2.96/1.96 thf(f2577,plain,(
% 2.96/1.96 ((in @ (ksnd @ (kpair @ sK181 @ sK178)) @ sK180) = $true) | ($true != $true)),
% 2.96/1.96 inference(forward_demodulation,[status(thm)],[f2570,f1853])).
% 2.96/1.96 thf(f1853,plain,(
% 2.96/1.96 (cartprodsndin = $true)),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f1054])).
% 2.96/1.96 thf(f2570,plain,(
% 2.96/1.96 (cartprodsndin != $true) | ((in @ (ksnd @ (kpair @ sK181 @ sK178)) @ sK180) = $true)),
% 2.96/1.96 inference(trivial_inequality_removal,[status(thm)],[f2566])).
% 2.96/1.96 thf(f2566,plain,(
% 2.96/1.96 ($true != $true) | (cartprodsndin != $true) | ((in @ (ksnd @ (kpair @ sK181 @ sK178)) @ sK180) = $true)),
% 2.96/1.96 inference(superposition,[status(thm)],[f2296,f1829])).
% 2.96/1.96 thf(f1829,plain,(
% 2.96/1.96 ($true = (in @ (kpair @ sK181 @ sK178) @ (cartprod @ sK179 @ sK180)))),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f1054])).
% 2.96/1.96 thf(f2296,plain,(
% 2.96/1.96 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ (cartprod @ X4 @ X3)) != $true) | (cartprodsndin != $true) | ($true = (in @ (ksnd @ X5) @ X3))) )),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f1423])).
% 2.96/1.96 thf(f1423,plain,(
% 2.96/1.96 ((cartprodsndin = $true) | (($true != (in @ (ksnd @ sK399) @ sK397)) & ($true = (in @ sK399 @ (cartprod @ sK398 @ sK397))))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ (ksnd @ X5) @ X3)) | ((in @ X5 @ (cartprod @ X4 @ X3)) != $true)) | (cartprodsndin != $true))),
% 2.96/1.96 inference(skolemisation,[status(esa),new_symbols(skolem,[sK397,sK398,sK399])],[f1421,f1422])).
% 2.96/1.96 thf(f1422,plain,(
% 2.96/1.96 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ (ksnd @ X2) @ X0) != $true) & ($true = (in @ X2 @ (cartprod @ X1 @ X0)))) => (($true != (in @ (ksnd @ sK399) @ sK397)) & ($true = (in @ sK399 @ (cartprod @ sK398 @ sK397))))),
% 2.96/1.96 introduced(definition,[],[choice_axiom])).
% 2.96/1.96 thf(f1421,plain,(
% 2.96/1.96 ((cartprodsndin = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ (ksnd @ X2) @ X0) != $true) & ($true = (in @ X2 @ (cartprod @ X1 @ X0))))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ (ksnd @ X5) @ X3)) | ((in @ X5 @ (cartprod @ X4 @ X3)) != $true)) | (cartprodsndin != $true))),
% 2.96/1.96 inference(rectify,[status(thm)],[f1420])).
% 2.96/1.96 thf(f1420,plain,(
% 2.96/1.96 ((cartprodsndin = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ (ksnd @ X2) @ X0) != $true) & ($true = (in @ X2 @ (cartprod @ X1 @ X0))))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ (ksnd @ X2) @ X0) = $true) | ($true != (in @ X2 @ (cartprod @ X1 @ X0)))) | (cartprodsndin != $true))),
% 2.96/1.96 inference(nnf_transformation,[status(thm)],[f602])).
% 2.96/1.96 thf(f602,plain,(
% 2.96/1.96 (cartprodsndin = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ (ksnd @ X2) @ X0) = $true) | ($true != (in @ X2 @ (cartprod @ X1 @ X0))))),
% 2.96/1.96 inference(ennf_transformation,[status(thm)],[f392])).
% 2.96/1.96 thf(f392,plain,(
% 2.96/1.96 (cartprodsndin = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (cartprod @ X1 @ X0))) => ((in @ (ksnd @ X2) @ X0) = $true))),
% 2.96/1.96 inference(fool_elimination,[status(thm)],[f391])).
% 2.96/1.96 thf(f391,plain,(
% 2.96/1.96 (cartprodsndin = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (cartprod @ X1 @ X0)) => (in @ (ksnd @ X2) @ X0)))),
% 2.96/1.96 inference(rectify,[status(thm)],[f179])).
% 2.96/1.96 thf(f179,axiom,(
% 2.96/1.96 (cartprodsndin = ! [X4 : $i,X3 : $i,X10 : $i] : ((in @ X10 @ (cartprod @ X3 @ X4)) => (in @ (ksnd @ X10) @ X4)))),
% 2.96/1.96 file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',cartprodsndin)).
% 2.96/1.96 thf(f3329,plain,(
% 2.96/1.96 ~spl494_15),
% 2.96/1.96 inference(avatar_split_clause,[status(thm)],[f2783,f3194])).
% 2.96/1.96 thf(f2783,plain,(
% 2.96/1.96 ($true != (in @ sK178 @ sK180))),
% 2.96/1.96 inference(trivial_inequality_removal,[status(thm)],[f2782])).
% 2.96/1.96 thf(f2782,plain,(
% 2.96/1.96 ($true != $true) | ($true != (in @ sK178 @ sK180))),
% 2.96/1.96 inference(forward_demodulation,[status(thm)],[f2752,f1887])).
% 2.96/1.96 thf(f1887,plain,(
% 2.96/1.96 (eqimpsubset1 = $true)),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f1054])).
% 2.96/1.96 thf(f2752,plain,(
% 2.96/1.96 ($true != (in @ sK178 @ sK180)) | (eqimpsubset1 != $true)),
% 2.96/1.96 inference(trivial_inequality_removal,[status(thm)],[f2735])).
% 2.96/1.96 thf(f2735,plain,(
% 2.96/1.96 ($true != (in @ sK178 @ sK180)) | ($true != $true) | (eqimpsubset1 != $true)),
% 2.96/1.96 inference(superposition,[status(thm)],[f2562,f2439])).
% 2.96/1.96 thf(f2439,plain,(
% 2.96/1.96 ( ! [X3 : $i] : (((subset @ X3 @ X3) = $true) | (eqimpsubset1 != $true)) )),
% 2.96/1.96 inference(equality_resolution,[status(thm)],[f1656])).
% 2.96/1.96 thf(f1656,plain,(
% 2.96/1.96 ( ! [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) | (X2 != X3) | (eqimpsubset1 != $true)) )),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f868])).
% 2.96/1.96 thf(f868,plain,(
% 2.96/1.96 ((eqimpsubset1 = $true) | (($true != (subset @ sK63 @ sK64)) & (sK64 = sK63))) & (! [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) | (X2 != X3)) | (eqimpsubset1 != $true))),
% 2.96/1.96 inference(skolemisation,[status(esa),new_symbols(skolem,[sK63,sK64])],[f866,f867])).
% 2.96/1.96 thf(f867,plain,(
% 2.96/1.96 ? [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) != $true) & (X0 = X1)) => (($true != (subset @ sK63 @ sK64)) & (sK64 = sK63))),
% 2.96/1.96 introduced(definition,[],[choice_axiom])).
% 2.96/1.96 thf(f866,plain,(
% 2.96/1.96 ((eqimpsubset1 = $true) | ? [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) != $true) & (X0 = X1))) & (! [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) | (X2 != X3)) | (eqimpsubset1 != $true))),
% 2.96/1.96 inference(rectify,[status(thm)],[f865])).
% 2.96/1.96 thf(f865,plain,(
% 2.96/1.96 ((eqimpsubset1 = $true) | ? [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) != $true) & (X0 = X1))) & (! [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) = $true) | (X0 != X1)) | (eqimpsubset1 != $true))),
% 2.96/1.96 inference(nnf_transformation,[status(thm)],[f638])).
% 2.96/1.96 thf(f638,plain,(
% 2.96/1.96 (eqimpsubset1 = $true) <=> ! [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) = $true) | (X0 != X1))),
% 2.96/1.96 inference(ennf_transformation,[status(thm)],[f490])).
% 2.96/1.96 thf(f490,plain,(
% 2.96/1.96 (eqimpsubset1 = $true) <=> ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((subset @ X0 @ X1) = $true))),
% 2.96/1.96 inference(fool_elimination,[status(thm)],[f489])).
% 2.96/1.96 thf(f489,plain,(
% 2.96/1.96 (! [X0 : $i,X1 : $i] : ((X0 = X1) => (subset @ X0 @ X1)) = eqimpsubset1)),
% 2.96/1.96 inference(rectify,[status(thm)],[f80])).
% 2.96/1.96 thf(f80,axiom,(
% 2.96/1.96 (! [X3 : $i,X4 : $i] : ((X3 = X4) => (subset @ X3 @ X4)) = eqimpsubset1)),
% 2.96/1.96 file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',eqimpsubset1)).
% 2.96/1.96 thf(f2562,plain,(
% 2.96/1.96 ( ! [X0 : $i] : (($true != (subset @ X0 @ sK180)) | ((in @ sK178 @ X0) != $true)) )),
% 2.96/1.96 inference(trivial_inequality_removal,[status(thm)],[f2561])).
% 2.96/1.96 thf(f2561,plain,(
% 2.96/1.96 ( ! [X0 : $i] : (($true != $true) | ($true != (subset @ X0 @ sK180)) | ((in @ sK178 @ X0) != $true)) )),
% 2.96/1.96 inference(forward_demodulation,[status(thm)],[f2554,f1975])).
% 2.96/1.96 thf(f1975,plain,(
% 2.96/1.96 (notsubsetI = $true)),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f1054])).
% 2.96/1.96 thf(f2554,plain,(
% 2.96/1.96 ( ! [X0 : $i] : (((in @ sK178 @ X0) != $true) | ($true != (subset @ X0 @ sK180)) | (notsubsetI != $true)) )),
% 2.96/1.96 inference(trivial_inequality_removal,[status(thm)],[f2552])).
% 2.96/1.96 thf(f2552,plain,(
% 2.96/1.96 ( ! [X0 : $i] : (($true != (subset @ X0 @ sK180)) | (notsubsetI != $true) | ((in @ sK178 @ X0) != $true) | ($true != $true)) )),
% 2.96/1.96 inference(superposition,[status(thm)],[f1830,f1703])).
% 2.96/1.96 thf(f1703,plain,(
% 2.96/1.96 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ X4) = $true) | ((subset @ X5 @ X4) != $true) | (notsubsetI != $true) | ((in @ X3 @ X5) != $true)) )),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f920])).
% 2.96/1.96 thf(f920,plain,(
% 2.96/1.96 ((notsubsetI = $true) | (((in @ sK95 @ sK97) = $true) & ($true != (in @ sK95 @ sK96)) & ((subset @ sK97 @ sK96) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ X5) != $true) | ((in @ X3 @ X4) = $true) | ((subset @ X5 @ X4) != $true)) | (notsubsetI != $true))),
% 2.96/1.96 inference(skolemisation,[status(esa),new_symbols(skolem,[sK95,sK96,sK97])],[f918,f919])).
% 2.96/1.96 thf(f919,plain,(
% 2.96/1.96 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X2)) & ($true != (in @ X0 @ X1)) & ((subset @ X2 @ X1) = $true)) => (((in @ sK95 @ sK97) = $true) & ($true != (in @ sK95 @ sK96)) & ((subset @ sK97 @ sK96) = $true))),
% 2.96/1.96 introduced(definition,[],[choice_axiom])).
% 2.96/1.96 thf(f918,plain,(
% 2.96/1.96 ((notsubsetI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X2)) & ($true != (in @ X0 @ X1)) & ((subset @ X2 @ X1) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ X5) != $true) | ((in @ X3 @ X4) = $true) | ((subset @ X5 @ X4) != $true)) | (notsubsetI != $true))),
% 2.96/1.96 inference(rectify,[status(thm)],[f917])).
% 2.96/1.96 thf(f917,plain,(
% 2.96/1.96 ((notsubsetI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X2)) & ($true != (in @ X0 @ X1)) & ((subset @ X2 @ X1) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X2)) | ($true = (in @ X0 @ X1)) | ((subset @ X2 @ X1) != $true)) | (notsubsetI != $true))),
% 2.96/1.96 inference(nnf_transformation,[status(thm)],[f713])).
% 2.96/1.96 thf(f713,plain,(
% 2.96/1.96 (notsubsetI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X2)) | ($true = (in @ X0 @ X1)) | ((subset @ X2 @ X1) != $true))),
% 2.96/1.96 inference(flattening,[status(thm)],[f712])).
% 2.96/1.96 thf(f712,plain,(
% 2.96/1.96 ! [X0 : $i,X1 : $i,X2 : $i] : ((((subset @ X2 @ X1) != $true) | ($true = (in @ X0 @ X1))) | ($true != (in @ X0 @ X2))) <=> (notsubsetI = $true)),
% 2.96/1.96 inference(ennf_transformation,[status(thm)],[f567])).
% 2.96/1.96 thf(f567,plain,(
% 2.96/1.96 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X2)) => (($true != (in @ X0 @ X1)) => ((subset @ X2 @ X1) != $true))) <=> (notsubsetI = $true)),
% 2.96/1.96 inference(flattening,[status(thm)],[f442])).
% 2.96/1.96 thf(f442,plain,(
% 2.96/1.96 (notsubsetI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X2)) => (~($true = (in @ X0 @ X1)) => ~((subset @ X2 @ X1) = $true)))),
% 2.96/1.96 inference(fool_elimination,[status(thm)],[f441])).
% 2.96/1.96 thf(f441,plain,(
% 2.96/1.96 (notsubsetI = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => (~(in @ X0 @ X1) => ~(subset @ X2 @ X1))))),
% 2.96/1.96 inference(rectify,[status(thm)],[f85])).
% 2.96/1.96 thf(f85,axiom,(
% 2.96/1.96 (notsubsetI = ! [X1 : $i,X4 : $i,X3 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ X4) => ~(subset @ X3 @ X4))))),
% 2.96/1.96 file('/export/starexec/sandbox/tmp/tmp.jryoenH2sp/DTF2THF_16138.p',notsubsetI)).
% 2.96/1.96 thf(f1830,plain,(
% 2.96/1.96 ($true != (in @ sK178 @ sK180))),
% 2.96/1.96 inference(cnf_transformation,[status(thm)],[f1054])).
% 2.96/1.96 % SZS output end Proof for DTF2THF_16138
% 2.96/1.96 % (16258)------------------------------
% 2.96/1.96 % (16258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.96 % (16258)Termination reason: Refutation
% 2.96/1.96
% 2.96/1.96 % (16258)Memory used [KB]: 8699
% 2.96/1.96 % (16258)Time elapsed: 0.147 s
% 2.96/1.96 % (16258)Instructions burned: 290 (million)
% 2.96/1.96 % (16258)------------------------------
% 2.96/1.96 % (16258)------------------------------
% 2.96/1.96 % (16235)Success in time 0.262 s
%------------------------------------------------------------------------------