%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU664^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:02 AM UTC 2026
% Result : Theorem 2.49s 1.60s
% Output : Refutation 2.49s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 6
% Syntax : Number of formulae : 40 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 3014 ( 824 equ; 0 cnn)
% Maximal formula atoms : 182 ( 75 avg)
% Number of connectives : 1598 ( 50 ~; 32 |; 561 &; 218 @)
% ( 4 <=>; 733 =>; 0 <=; 0 <~>)
% Maximal formula depth : 189 ( 36 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 330 ( 327 usr; 189 con; 0-4 aty)
% Number of variables : 65 ( 0 ^; 44 !; 21 ?; 65 :)
% 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_226,type,
sP2: $i > $i > $o ).
thf(func_def_227,type,
sP3: $i > $o ).
thf(func_def_228,type,
sP4: $i > $i > $o ).
thf(func_def_229,type,
sP5: $i > $i > $o ).
thf(func_def_230,type,
sP6: $i > $o > $i > $i > $o ).
thf(func_def_233,type,
sK9: $i > $o ).
thf(func_def_234,type,
sK10: ( $i > $o ) > $i > $i ).
thf(func_def_255,type,
sK31: $i > $o ).
thf(func_def_256,type,
sK32: $i > $o ).
thf(func_def_257,type,
sK33: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_258,type,
sK34: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_266,type,
sK42: $i > $i > $o ).
thf(func_def_268,type,
sK44: $i > $i ).
thf(func_def_269,type,
sK45: $i > $i ).
thf(func_def_270,type,
sK46: $i > ( $i > $i > $o ) > $i ).
thf(func_def_271,type,
sK47: $i > ( $i > $i > $o ) > $i ).
thf(func_def_272,type,
sK48: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_275,type,
sK51: $i > $i ).
thf(func_def_283,type,
sK59: $i > $i > $i ).
thf(func_def_290,type,
sK66: $i > $i > $i ).
thf(func_def_302,type,
sK78: ( $i > $o ) > $i > $i ).
thf(func_def_304,type,
sK80: $i > $o ).
thf(func_def_308,type,
sK84: $i > ( $i > $o ) > $i ).
thf(func_def_309,type,
sK85: $i > $o ).
thf(func_def_314,type,
sK90: $i > $o ).
thf(func_def_315,type,
sK91: $i > $i ).
thf(func_def_316,type,
sK92: ( $i > $o ) > $i ).
thf(func_def_319,type,
sK95: ( $i > $o ) > $i ).
thf(func_def_320,type,
sK96: $i > $o ).
thf(func_def_323,type,
sK99: $i > $o ).
thf(func_def_326,type,
sK102: $i > $o ).
thf(func_def_330,type,
sK106: $i > $i > $i ).
thf(func_def_331,type,
sK107: $i > $o ).
thf(func_def_334,type,
sK110: $i > ( $i > $o ) > $i ).
thf(func_def_335,type,
sK111: $i > ( $i > $o ) > $i ).
thf(func_def_336,type,
sK112: $i > $o ).
thf(func_def_341,type,
sK117: $i > $i ).
thf(func_def_348,type,
sK124: $i > $o ).
thf(func_def_350,type,
sK126: ( $i > $o ) > $i > $i ).
thf(func_def_351,type,
sK127: ( $i > $o ) > $i > $i ).
thf(func_def_379,type,
sK155: ( $i > $o ) > $i ).
thf(func_def_380,type,
sK156: $i > $o ).
thf(func_def_381,type,
sK157: $i > $i ).
thf(func_def_397,type,
sK173: $i > $o ).
thf(func_def_403,type,
sK179: $i > $i > $i ).
thf(func_def_406,type,
sK182: $i > $o ).
thf(func_def_408,type,
sK184: $i > ( $i > $o ) > $i ).
thf(func_def_414,type,
sK190: $i > $o ).
thf(func_def_415,type,
sK191: $i > $i ).
thf(func_def_418,type,
sK194: $i > $o ).
thf(func_def_420,type,
sK196: $i > ( $i > $o ) > $i > $i ).
thf(func_def_421,type,
sK197: $i > $i > $i ).
thf(func_def_422,type,
sK198: $i > $i > $i ).
thf(func_def_423,type,
sK199: $i > $i > $i ).
thf(func_def_424,type,
sK200: $i > $i > $i ).
thf(func_def_425,type,
sK201: $i > $i > $i ).
thf(func_def_426,type,
sK202: $i > $i > $i > $i ).
thf(func_def_427,type,
sK203: $i > $i ).
thf(func_def_428,type,
sK204: $i > $i ).
thf(func_def_429,type,
sK205: $i > $i ).
thf(func_def_430,type,
sK206: $i > $i ).
thf(func_def_431,type,
sK207: $i > $i > $i ).
thf(func_def_432,type,
sK208: $i > $i > $i > $i ).
thf(func_def_433,type,
sK209: $i > $i > $i > $i ).
thf(func_def_434,type,
sK210: $i > $i > $i ).
thf(func_def_435,type,
sK211: $i > $i > $i ).
thf(func_def_436,type,
sK212: $i > $i ).
thf(func_def_438,type,
sK214: $i > $i ).
thf(func_def_439,type,
sK215: $i > $i ).
thf(func_def_474,type,
sK250: $i > $i > $i ).
thf(func_def_484,type,
sK260: $i > $o ).
thf(func_def_485,type,
sK261: $i > $o ).
thf(func_def_486,type,
sK262: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_487,type,
sK263: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_506,type,
sK282: $i > $o ).
thf(func_def_508,type,
sK284: ( $i > $o ) > $i > $i ).
thf(func_def_513,type,
sK289: $i > $i > $i > $i ).
thf(func_def_514,type,
sK290: $i > $i > $i > $i ).
thf(func_def_521,type,
sK297: $i > $o ).
thf(func_def_522,type,
sK298: $i > $o ).
thf(func_def_523,type,
sK299: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_524,type,
sK300: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_545,type,
sK321: $i > $o ).
thf(func_def_558,type,
sK334: $i > $i > $i ).
thf(func_def_562,type,
sK338: $i > $o ).
thf(func_def_574,type,
sK350: $i > ( $i > $o ) > $i ).
thf(func_def_575,type,
sK351: $i > $o ).
thf(func_def_578,type,
sK354: $i > $i ).
thf(func_def_587,type,
sK363: $i > $i > $i ).
thf(func_def_589,type,
sK365: $i > $o ).
thf(func_def_597,type,
sK373: $o > $i > $i > $i ).
thf(func_def_600,type,
sK376: $i > $o ).
thf(func_def_615,type,
sK391: $i > $i > $i ).
thf(func_def_616,type,
sK392: $i > $i > $i ).
thf(func_def_619,type,
sK395: $i > $i > $i ).
thf(func_def_622,type,
sK398: $i > $o ).
thf(func_def_627,type,
sK403: $i > $o ).
thf(func_def_638,type,
sK414: $i > $o ).
thf(func_def_640,type,
sK416: ( $i > $o ) > $i > $i ).
thf(func_def_641,type,
sK417: ( $i > $o ) > $i > $i ).
thf(func_def_671,type,
sK447: $i > $o ).
thf(func_def_696,type,
sK472: $i > $o ).
thf(func_def_705,type,
sK481: ( $i > $o ) > $i > $i ).
thf(func_def_707,type,
sK483: $i > $o ).
thf(func_def_717,type,
sK493: ( $i > $o ) > $i ).
thf(func_def_718,type,
sK494: ( $i > $o ) > $i ).
thf(func_def_719,type,
sK495: $i > $o ).
thf(func_def_726,type,
sK502: $i > $i ).
thf(func_def_728,type,
sK504: $i > $i ).
thf(func_def_731,type,
sK507: $i > $i ).
thf(func_def_734,type,
ph510:
!>[X0: $tType] : X0 ).
thf(f2625,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2624]) ).
thf(f2624,plain,
$true != $true,
inference(superposition,[status(thm)],[f2607,f2623]) ).
thf(f2623,plain,
( $true
= ( iskpair @ sK407 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2622]) ).
thf(f2622,plain,
( ( $true != $true )
| ( $true
= ( iskpair @ sK407 ) ) ),
inference(forward_demodulation,[status(thm)],[f2612,f2340]) ).
thf(f2340,plain,
cartprodmempair = $true,
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f1443,plain,
( ( kpairp = $true )
& ( notinemptyset = $true )
& ( upairequniteq = $true )
& ( replAx = $true )
& ( binunionIL = $true )
& ( wellorderingAx = $true )
& ( notsubsetI = $true )
& ( ex1I = $true )
& ( binintersectSubset5 = $true )
& ( cartprodpairmemEL = $true )
& ( upairset2IR = $true )
& ( exuE3u = $true )
& ( noeltsimpempty = $true )
& ( ubforcartprodlem3 = $true )
& ( subsetI2 = $true )
& ( cartprodmempair = $true )
& ( setunionsingleton1 = $true )
& ( in__Cong = $true )
& ( prop2set2propI = $true )
& ( omega0Ax = $true )
& ( subsetE2 = $true )
& ( singletonprop = $true )
& ( quantDeMorgan3 = $true )
& ( cartprodpairin = $true )
& ( subsetRefl = $true )
& ( setukpairinjR11 = $true )
& ( binunionRsub = $true )
& ( dsetconstrER = $true )
& ( setminusILneg = $true )
& ( exuE2 = $true )
& ( setukpairinjL2 = $true )
& ( kfstsingleton = $true )
& ( secondinupair = $true )
& ( subsetI1 = $true )
& ( setunionsingleton2 = $true )
& ( setextAx = $true )
& ( ksndsingleton = $true )
& ( ksndpairEq = $true )
& ( omega__Cong = $true )
& ( symdiffI2 = $true )
& ( singletonsubset = $true )
& ( upairsubunion = $true )
& ( cartprodsndpairEq = $true )
& ( setukpairinjL1 = $true )
& ( setunionI = $true )
& ( setminusLsub = $true )
& ( emptyInPowerset = $true )
& ( setminusERneg = $true )
& ( ex1E1 = $true )
& ( singletonsswitch = $true )
& ( setadjoinIL = $true )
& ( symdiffE = $true )
& ( powersetI1 = $true )
& ( setukpairinjL = $true )
& ( setextsub = $true )
& ( subsetE = $true )
& ( prop2setE = $true )
& ( setext = $true )
& ( ubforcartprodlem2 = $true )
& ( nonemptyI1 = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( emptyE1 = $true )
& ( setunionsingleton = $true )
& ( theprop = $true )
& ( singletoninpowunion = $true )
& ( setminusSubset2 = $true )
& ( subset2powerset = $true )
& ( notequalI1 = $true )
& ( symdiffI1 = $true )
& ( setunionAx = $true )
& ( disjointsetsI1 = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectSubset1 = $true )
& ( binunionE = $true )
& ( setunionE = $true )
& ( setadjoinE = $true )
& ( omegaIndAx = $true )
& ( eqinunit = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinOr = $true )
& ( emptyinunitempty = $true )
& ( emptyinPowerset = $true )
& ( dsetconstr__Cong = $true )
& ( bs114d = $true )
& ( powersetAx = $true )
& ( binunionIR = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjR12 = $true )
& ( setminusER = $true )
& ( sepInPowerset = $true )
& ( powersetI = $true )
& ( ex1I2 = $true )
& ( setunionE2 = $true )
& ( setminusELneg = $true )
& ( setukpairIL = $true )
& ( setukpairinjR2 = $true )
& ( binintersectSubset3 = $true )
& ( vacuousDall = $true )
& ( nonemptyI = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinIR = $true )
& ( dsetconstrEL = $true )
& ( upairsetIL = $true )
& ( omegaSAx = $true )
& ( powersetE1 = $true )
& ( powersetsubset = $true )
& ( sK407
!= ( kpair @ ( kfst @ sK407 ) @ ( ksnd @ sK407 ) ) )
& ( $true
= ( in @ sK407 @ ( cartprod @ sK408 @ sK409 ) ) )
& ( notdallE = $true )
& ( cartprodfstpairEq = $true )
& ( emptysetAx = $true )
& ( setukpairinjR1 = $true )
& ( exuI1 = $true )
& ( binintersectI = $true )
& ( descr__Cong = $true )
& ( dsetconstrI = $true )
& ( setadjoinSub2 = $true )
& ( upairset2E = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairIR = $true )
& ( emptyset__Cong = $true )
& ( prop2setI = $true )
& ( setminusSubset1 = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( upairinpowunion = $true )
& ( kfstpairEq = $true )
& ( uniqinunit = $true )
& ( singletonsuniq = $true )
& ( setminusEL = $true )
& ( exuEu = $true )
& ( upairsetIR = $true )
& ( symdiffIneg2 = $true )
& ( subsetTrans = $true )
& ( upairsetE = $true )
& ( symdiffIneg1 = $true )
& ( cartprodmempair1 = $true )
& ( notdexE = $true )
& ( inPowerset = $true )
& ( eqimpsubset2 = $true )
& ( nonemptyImpWitness = $true )
& ( setbeta = $true )
& ( exuE1 = $true )
& ( setoftrueEq = $true )
& ( foundationAx = $true )
& ( cartprodfstin = $true )
& ( cartprodsndin = $true )
& ( emptyI = $true )
& ( binunionEcases = $true )
& ( exuE3e = $true )
& ( notequalI2 = $true )
& ( kpairiskpair = $true )
& ( binintersectLsub = $true )
& ( singletoninpowerset = $true )
& ( binunionLsub = $true )
& ( notinsingleton = $true )
& ( setadjoinSub = $true )
& ( sepSubset = $true )
& ( binintersectER = $true )
& ( descrp = $true )
& ( exuI3 = $true )
& ( setadjoinAx = $true )
& ( binintersectSubset4 = $true )
& ( setadjoin__Cong = $true )
& ( setminusIRneg = $true )
& ( powersetE = $true )
& ( binintersectEL = $true )
& ( subsetemptysetimpeq = $true )
& ( ubforcartprodlem1 = $true )
& ( exu__Cong = $true )
& ( setukpairinjR = $true )
& ( emptysetE = $true )
& ( binintersectRsub = $true )
& ( subPowSU = $true )
& ( emptysetsubset = $true )
& ( eqimpsubset1 = $true )
& ( nonemptyE1 = $true )
& ( kpairsurjEq = $true )
& ( setminusI = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK407,sK408,sK409])],[f764,f1442]) ).
thf(f1442,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
!= X0 )
& ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
= $true ) )
=> ( ( sK407
!= ( kpair @ ( kfst @ sK407 ) @ ( ksnd @ sK407 ) ) )
& ( $true
= ( in @ sK407 @ ( cartprod @ sK408 @ sK409 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f764,plain,
( ( kpairp = $true )
& ( notinemptyset = $true )
& ( upairequniteq = $true )
& ( replAx = $true )
& ( binunionIL = $true )
& ( wellorderingAx = $true )
& ( notsubsetI = $true )
& ( ex1I = $true )
& ( binintersectSubset5 = $true )
& ( cartprodpairmemEL = $true )
& ( upairset2IR = $true )
& ( exuE3u = $true )
& ( noeltsimpempty = $true )
& ( ubforcartprodlem3 = $true )
& ( subsetI2 = $true )
& ( cartprodmempair = $true )
& ( setunionsingleton1 = $true )
& ( in__Cong = $true )
& ( prop2set2propI = $true )
& ( omega0Ax = $true )
& ( subsetE2 = $true )
& ( singletonprop = $true )
& ( quantDeMorgan3 = $true )
& ( cartprodpairin = $true )
& ( subsetRefl = $true )
& ( setukpairinjR11 = $true )
& ( binunionRsub = $true )
& ( dsetconstrER = $true )
& ( setminusILneg = $true )
& ( exuE2 = $true )
& ( setukpairinjL2 = $true )
& ( kfstsingleton = $true )
& ( secondinupair = $true )
& ( subsetI1 = $true )
& ( setunionsingleton2 = $true )
& ( setextAx = $true )
& ( ksndsingleton = $true )
& ( ksndpairEq = $true )
& ( omega__Cong = $true )
& ( symdiffI2 = $true )
& ( singletonsubset = $true )
& ( upairsubunion = $true )
& ( cartprodsndpairEq = $true )
& ( setukpairinjL1 = $true )
& ( setunionI = $true )
& ( setminusLsub = $true )
& ( emptyInPowerset = $true )
& ( setminusERneg = $true )
& ( ex1E1 = $true )
& ( singletonsswitch = $true )
& ( setadjoinIL = $true )
& ( symdiffE = $true )
& ( powersetI1 = $true )
& ( setukpairinjL = $true )
& ( setextsub = $true )
& ( subsetE = $true )
& ( prop2setE = $true )
& ( setext = $true )
& ( ubforcartprodlem2 = $true )
& ( nonemptyI1 = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( emptyE1 = $true )
& ( setunionsingleton = $true )
& ( theprop = $true )
& ( singletoninpowunion = $true )
& ( setminusSubset2 = $true )
& ( subset2powerset = $true )
& ( notequalI1 = $true )
& ( symdiffI1 = $true )
& ( setunionAx = $true )
& ( disjointsetsI1 = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectSubset1 = $true )
& ( binunionE = $true )
& ( setunionE = $true )
& ( setadjoinE = $true )
& ( omegaIndAx = $true )
& ( eqinunit = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinOr = $true )
& ( emptyinunitempty = $true )
& ( emptyinPowerset = $true )
& ( dsetconstr__Cong = $true )
& ( bs114d = $true )
& ( powersetAx = $true )
& ( binunionIR = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjR12 = $true )
& ( setminusER = $true )
& ( sepInPowerset = $true )
& ( powersetI = $true )
& ( ex1I2 = $true )
& ( setunionE2 = $true )
& ( setminusELneg = $true )
& ( setukpairIL = $true )
& ( setukpairinjR2 = $true )
& ( binintersectSubset3 = $true )
& ( vacuousDall = $true )
& ( nonemptyI = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinIR = $true )
& ( dsetconstrEL = $true )
& ( upairsetIL = $true )
& ( omegaSAx = $true )
& ( powersetE1 = $true )
& ( powersetsubset = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
!= X0 )
& ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
= $true ) )
& ( notdallE = $true )
& ( cartprodfstpairEq = $true )
& ( emptysetAx = $true )
& ( setukpairinjR1 = $true )
& ( exuI1 = $true )
& ( binintersectI = $true )
& ( descr__Cong = $true )
& ( dsetconstrI = $true )
& ( setadjoinSub2 = $true )
& ( upairset2E = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairIR = $true )
& ( emptyset__Cong = $true )
& ( prop2setI = $true )
& ( setminusSubset1 = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( upairinpowunion = $true )
& ( kfstpairEq = $true )
& ( uniqinunit = $true )
& ( singletonsuniq = $true )
& ( setminusEL = $true )
& ( exuEu = $true )
& ( upairsetIR = $true )
& ( symdiffIneg2 = $true )
& ( subsetTrans = $true )
& ( upairsetE = $true )
& ( symdiffIneg1 = $true )
& ( cartprodmempair1 = $true )
& ( notdexE = $true )
& ( inPowerset = $true )
& ( eqimpsubset2 = $true )
& ( nonemptyImpWitness = $true )
& ( setbeta = $true )
& ( exuE1 = $true )
& ( setoftrueEq = $true )
& ( foundationAx = $true )
& ( cartprodfstin = $true )
& ( cartprodsndin = $true )
& ( emptyI = $true )
& ( binunionEcases = $true )
& ( exuE3e = $true )
& ( notequalI2 = $true )
& ( kpairiskpair = $true )
& ( binintersectLsub = $true )
& ( singletoninpowerset = $true )
& ( binunionLsub = $true )
& ( notinsingleton = $true )
& ( setadjoinSub = $true )
& ( sepSubset = $true )
& ( binintersectER = $true )
& ( descrp = $true )
& ( exuI3 = $true )
& ( setadjoinAx = $true )
& ( binintersectSubset4 = $true )
& ( setadjoin__Cong = $true )
& ( setminusIRneg = $true )
& ( powersetE = $true )
& ( binintersectEL = $true )
& ( subsetemptysetimpeq = $true )
& ( ubforcartprodlem1 = $true )
& ( exu__Cong = $true )
& ( setukpairinjR = $true )
& ( emptysetE = $true )
& ( binintersectRsub = $true )
& ( subPowSU = $true )
& ( emptysetsubset = $true )
& ( eqimpsubset1 = $true )
& ( nonemptyE1 = $true )
& ( kpairsurjEq = $true )
& ( setminusI = $true ) ),
inference(flattening,[status(thm)],[f763]) ).
thf(f763,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
!= X0 )
& ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
= $true ) )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f393]) ).
thf(f393,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
= $true )
=> ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
= X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f392]) ).
thf(f392,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
= X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f186]) ).
thf(f186,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ! [X10: $i,X3: $i,X4: $i] :
( ( in @ X10 @ ( cartprod @ X3 @ X4 ) )
=> ( ( kpair @ ( kfst @ X10 ) @ ( ksnd @ X10 ) )
= X10 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f185]) ).
thf(f185,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ! [X10: $i,X3: $i,X4: $i] :
( ( in @ X10 @ ( cartprod @ X3 @ X4 ) )
=> ( ( kpair @ ( kfst @ X10 ) @ ( ksnd @ X10 ) )
= X10 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.7RtdXlarj4/DTF2THF_4583.p',cartprodpairsurjEq) ).
thf(f2612,plain,
( ( cartprodmempair != $true )
| ( $true
= ( iskpair @ sK407 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2610]) ).
thf(f2610,plain,
( ( $true != $true )
| ( $true
= ( iskpair @ sK407 ) )
| ( cartprodmempair != $true ) ),
inference(superposition,[status(thm)],[f2069,f2245]) ).
thf(f2245,plain,
( $true
= ( in @ sK407 @ ( cartprod @ sK408 @ sK409 ) ) ),
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f2069,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( cartprod @ X2 @ X0 ) ) )
| ( ( iskpair @ X1 )
= $true )
| ( cartprodmempair != $true ) ),
inference(cnf_transformation,[status(thm)],[f1324]) ).
thf(f1324,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( iskpair @ X1 )
= $true )
| ( $true
!= ( in @ X1 @ ( cartprod @ X2 @ X0 ) ) ) )
| ( cartprodmempair != $true ) )
& ( ( cartprodmempair = $true )
| ( ( ( iskpair @ sK330 )
!= $true )
& ( $true
= ( in @ sK330 @ ( cartprod @ sK331 @ sK329 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK329,sK330,sK331])],[f1322,f1323]) ).
thf(f1323,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( iskpair @ X4 )
!= $true )
& ( ( in @ X4 @ ( cartprod @ X5 @ X3 ) )
= $true ) )
=> ( ( ( iskpair @ sK330 )
!= $true )
& ( $true
= ( in @ sK330 @ ( cartprod @ sK331 @ sK329 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1322,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( iskpair @ X1 )
= $true )
| ( $true
!= ( in @ X1 @ ( cartprod @ X2 @ X0 ) ) ) )
| ( cartprodmempair != $true ) )
& ( ( cartprodmempair = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( iskpair @ X4 )
!= $true )
& ( ( in @ X4 @ ( cartprod @ X5 @ X3 ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1321]) ).
thf(f1321,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( iskpair @ X1 )
= $true )
| ( $true
!= ( in @ X1 @ ( cartprod @ X2 @ X0 ) ) ) )
| ( cartprodmempair != $true ) )
& ( ( cartprodmempair = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( iskpair @ X1 )
!= $true )
& ( $true
= ( in @ X1 @ ( cartprod @ X2 @ X0 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f590]) ).
thf(f590,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( iskpair @ X1 )
= $true )
| ( $true
!= ( in @ X1 @ ( cartprod @ X2 @ X0 ) ) ) )
<=> ( cartprodmempair = $true ) ),
inference(ennf_transformation,[status(thm)],[f423]) ).
thf(f423,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( cartprod @ X2 @ X0 ) ) )
=> ( ( iskpair @ X1 )
= $true ) )
<=> ( cartprodmempair = $true ) ),
inference(fool_elimination,[status(thm)],[f422]) ).
thf(f422,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( cartprod @ X2 @ X0 ) )
=> ( iskpair @ X1 ) )
= cartprodmempair ),
inference(rectify,[status(thm)],[f151]) ).
thf(f151,axiom,
! [X4: $i,X10: $i,X3: $i] :
( ( ( in @ X10 @ ( cartprod @ X3 @ X4 ) )
=> ( iskpair @ X10 ) )
= cartprodmempair ),
file('/export/starexec/sandbox/tmp/tmp.7RtdXlarj4/DTF2THF_4583.p',cartprodmempair) ).
thf(f2607,plain,
( $true
!= ( iskpair @ sK407 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2606]) ).
thf(f2606,plain,
( ( $true
!= ( iskpair @ sK407 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2605,f2175]) ).
thf(f2175,plain,
kpairsurjEq = $true,
inference(cnf_transformation,[status(thm)],[f1443]) ).
thf(f2605,plain,
( ( kpairsurjEq != $true )
| ( $true
!= ( iskpair @ sK407 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2604]) ).
thf(f2604,plain,
( ( kpairsurjEq != $true )
| ( $true
!= ( iskpair @ sK407 ) )
| ( sK407 != sK407 ) ),
inference(superposition,[status(thm)],[f2246,f1745]) ).
thf(f1745,plain,
! [X1: $i] :
( ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 )
| ( kpairsurjEq != $true )
| ( ( iskpair @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f953]) ).
thf(f953,plain,
( ( ( kpairsurjEq = $true )
| ( ( sK103
!= ( kpair @ ( kfst @ sK103 ) @ ( ksnd @ sK103 ) ) )
& ( ( iskpair @ sK103 )
= $true ) ) )
& ( ! [X1: $i] :
( ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 )
| ( ( iskpair @ X1 )
!= $true ) )
| ( kpairsurjEq != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK103])],[f951,f952]) ).
thf(f952,plain,
( ? [X0: $i] :
( ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
!= X0 )
& ( ( iskpair @ X0 )
= $true ) )
=> ( ( sK103
!= ( kpair @ ( kfst @ sK103 ) @ ( ksnd @ sK103 ) ) )
& ( ( iskpair @ sK103 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f951,plain,
( ( ( kpairsurjEq = $true )
| ? [X0: $i] :
( ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
!= X0 )
& ( ( iskpair @ X0 )
= $true ) ) )
& ( ! [X1: $i] :
( ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 )
| ( ( iskpair @ X1 )
!= $true ) )
| ( kpairsurjEq != $true ) ) ),
inference(rectify,[status(thm)],[f950]) ).
thf(f950,plain,
( ( ( kpairsurjEq = $true )
| ? [X0: $i] :
( ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
!= X0 )
& ( ( iskpair @ X0 )
= $true ) ) )
& ( ! [X0: $i] :
( ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
= X0 )
| ( ( iskpair @ X0 )
!= $true ) )
| ( kpairsurjEq != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f677]) ).
thf(f677,plain,
( ( kpairsurjEq = $true )
<=> ! [X0: $i] :
( ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
= X0 )
| ( ( iskpair @ X0 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f459]) ).
thf(f459,plain,
( ! [X0: $i] :
( ( ( iskpair @ X0 )
= $true )
=> ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
= X0 ) )
<=> ( kpairsurjEq = $true ) ),
inference(fool_elimination,[status(thm)],[f458]) ).
thf(f458,plain,
( kpairsurjEq
= ( ! [X0: $i] :
( ( iskpair @ X0 )
=> ( ( kpair @ ( kfst @ X0 ) @ ( ksnd @ X0 ) )
= X0 ) ) ) ),
inference(rectify,[status(thm)],[f178]) ).
thf(f178,axiom,
( kpairsurjEq
= ( ! [X10: $i] :
( ( iskpair @ X10 )
=> ( ( kpair @ ( kfst @ X10 ) @ ( ksnd @ X10 ) )
= X10 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.7RtdXlarj4/DTF2THF_4583.p',kpairsurjEq) ).
thf(f2246,plain,
( sK407
!= ( kpair @ ( kfst @ sK407 ) @ ( ksnd @ sK407 ) ) ),
inference(cnf_transformation,[status(thm)],[f1443]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU664^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.12/0.33 % Computer : n007.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Feb 24 12:29:23 EST 2026
% 0.12/0.33 % CPUTime :
% 0.12/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.41 ---- Original DTF file ---
% 0.16/0.41 thf(spec,logic,$$dhol).
% 0.16/0.41 %------------------------------------------------------------------------------
% 0.16/0.41 % File : SEU664^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.41 % Domain : Set Theory
% 0.16/0.41 % Problem : Ordered Pairs - Properties of Pairs
% 0.16/0.41 % Version : Especial.
% 0.16/0.41 % English : (! A:i.! B:i.! u:i.in u (cartprod A B) -> kpair (kfst u)
% 0.16/0.41 % (ksnd u) = u)
% 0.16/0.41
% 0.16/0.41 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.41 % Source : [Bro08]
% 0.16/0.41 % Names : ZFC166g [Bro08]
% 0.16/0.41
% 0.16/0.41 % Status : Theorem
% 0.16/0.41 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 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 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.16/0.41 % Syntax : Number of formulae : 396 ( 184 unt; 211 typ; 184 def)
% 0.16/0.41 % Number of atoms : 979 ( 277 equ; 0 cnn)
% 0.16/0.41 % Maximal formula atoms : 182 ( 5 avg)
% 0.16/0.41 % Number of connectives : 1718 ( 47 ~; 7 |; 35 &;1182 @)
% 0.16/0.41 % ( 14 <=>; 433 =>; 0 <=; 0 <~>)
% 0.16/0.41 % Maximal formula depth : 188 ( 2 avg)
% 0.16/0.41 % Number of types : 2 ( 0 usr)
% 0.16/0.41 % Number of type conns : 84 ( 84 >; 0 *; 0 +; 0 <<)
% 0.16/0.41 % Number of symbols : 214 ( 211 usr; 184 con; 0-2 aty)
% 0.16/0.41 % Number of variables : 541 ( 40 ^; 465 !; 36 ?; 541 :)
% 0.16/0.41 % SPC : TH0_THM_EQU_NAR
% 0.16/0.41
% 0.16/0.41 % Comments : http://mathgate.info/detsetitem.php?id=218
% 0.16/0.41 %------------------------------------------------------------------------------
% 0.16/0.41 thf(in_type,type,
% 0.16/0.41 in: $i > $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(exu_type,type,
% 0.16/0.41 exu: ( $i > $o ) > $o ).
% 0.16/0.41
% 0.16/0.41 thf(exu,definition,
% 0.16/0.41 ( exu
% 0.16/0.41 = ( ^ [Xphi: $i > $o] :
% 0.16/0.41 ? [Xx: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 & ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setextAx_type,type,
% 0.16/0.41 setextAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setextAx,definition,
% 0.16/0.41 ( setextAx
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 <=> ( in @ Xx @ B ) )
% 0.16/0.41 => ( A = B ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyset_type,type,
% 0.16/0.41 emptyset: $i ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetAx_type,type,
% 0.16/0.41 emptysetAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetAx,definition,
% 0.16/0.41 ( emptysetAx
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoin_type,type,
% 0.16/0.41 setadjoin: $i > $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinAx_type,type,
% 0.16/0.41 setadjoinAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinAx,definition,
% 0.16/0.41 ( setadjoinAx
% 0.16/0.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.41 <=> ( ( Xy = Xx )
% 0.16/0.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(powerset_type,type,
% 0.16/0.41 powerset: $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(powersetAx_type,type,
% 0.16/0.41 powersetAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(powersetAx,definition,
% 0.16/0.41 ( powersetAx
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.41 <=> ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setunion_type,type,
% 0.16/0.41 setunion: $i > $i ).
% 0.16/0.41
% 0.16/0.41 thf(setunionAx_type,type,
% 0.16/0.41 setunionAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setunionAx,definition,
% 0.16/0.41 ( setunionAx
% 0.16/0.41 = ( ! [A: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.41 <=> ? [B: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omega_type,type,
% 0.16/0.41 omega: $i ).
% 0.16/0.41
% 0.16/0.41 thf(omega0Ax_type,type,
% 0.16/0.41 omega0Ax: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omega0Ax,definition,
% 0.16/0.41 ( omega0Ax
% 0.16/0.41 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omegaSAx_type,type,
% 0.16/0.41 omegaSAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omegaSAx,definition,
% 0.16/0.41 ( omegaSAx
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ omega )
% 0.16/0.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(omegaIndAx_type,type,
% 0.16/0.41 omegaIndAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(omegaIndAx,definition,
% 0.16/0.41 ( omegaIndAx
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( ( in @ emptyset @ A )
% 0.16/0.41 & ! [Xx: $i] :
% 0.16/0.41 ( ( ( in @ Xx @ omega )
% 0.16/0.41 & ( in @ Xx @ A ) )
% 0.16/0.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ omega )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(replAx_type,type,
% 0.16/0.41 replAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(replAx,definition,
% 0.16/0.41 ( replAx
% 0.16/0.41 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( exu
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.41 => ? [B: $i] :
% 0.16/0.41 ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 <=> ? [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ A )
% 0.16/0.41 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(foundationAx_type,type,
% 0.16/0.41 foundationAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(foundationAx,definition,
% 0.16/0.41 ( foundationAx
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.41 => ? [B: $i] :
% 0.16/0.41 ( ( in @ B @ A )
% 0.16/0.41 & ~ ? [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(wellorderingAx_type,type,
% 0.16/0.41 wellorderingAx: $o ).
% 0.16/0.41
% 0.16/0.41 thf(wellorderingAx,definition,
% 0.16/0.41 ( wellorderingAx
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ? [B: $i] :
% 0.16/0.41 ( ! [C: $i] :
% 0.16/0.41 ( ( in @ C @ B )
% 0.16/0.41 => ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ C )
% 0.16/0.41 => ( in @ Xx @ A ) ) )
% 0.16/0.41 & ! [Xx: $i,Xy: $i] :
% 0.16/0.41 ( ( ( in @ Xx @ A )
% 0.16/0.41 & ( in @ Xy @ A ) )
% 0.16/0.41 => ( ! [C: $i] :
% 0.16/0.41 ( ( in @ C @ B )
% 0.16/0.41 => ( ( in @ Xx @ C )
% 0.16/0.41 <=> ( in @ Xy @ C ) ) )
% 0.16/0.41 => ( Xx = Xy ) ) )
% 0.16/0.41 & ! [C: $i,D: $i] :
% 0.16/0.41 ( ( ( in @ C @ B )
% 0.16/0.41 & ( in @ D @ B ) )
% 0.16/0.41 => ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ C )
% 0.16/0.41 => ( in @ Xx @ D ) )
% 0.16/0.41 | ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ D )
% 0.16/0.41 => ( in @ Xx @ C ) ) ) )
% 0.16/0.41 & ! [C: $i] :
% 0.16/0.41 ( ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ C )
% 0.16/0.41 => ( in @ Xx @ A ) )
% 0.16/0.41 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.41 => ? [D: $i,Xx: $i] :
% 0.16/0.41 ( ( in @ D @ B )
% 0.16/0.41 & ( in @ Xx @ C )
% 0.16/0.41 & ~ ? [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ D )
% 0.16/0.41 & ( in @ Xy @ C ) )
% 0.16/0.41 & ! [E: $i] :
% 0.16/0.41 ( ( in @ E @ B )
% 0.16/0.41 => ( ! [Xy: $i] :
% 0.16/0.41 ( ( in @ Xy @ E )
% 0.16/0.41 => ( in @ Xy @ D ) )
% 0.16/0.41 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(descr_type,type,
% 0.16/0.41 descr: ( $i > $o ) > $i ).
% 0.16/0.41
% 0.16/0.41 thf(descrp_type,type,
% 0.16/0.41 descrp: $o ).
% 0.16/0.41
% 0.16/0.41 thf(descrp,definition,
% 0.16/0.41 ( descrp
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ( Xphi
% 0.16/0.41 @ ( descr
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstr_type,type,
% 0.16/0.41 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrI_type,type,
% 0.16/0.41 dsetconstrI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrI,definition,
% 0.16/0.41 ( dsetconstrI
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 => ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrEL_type,type,
% 0.16/0.41 dsetconstrEL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrEL,definition,
% 0.16/0.41 ( dsetconstrEL
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.41 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrER_type,type,
% 0.16/0.41 dsetconstrER: $o ).
% 0.16/0.41
% 0.16/0.41 thf(dsetconstrER,definition,
% 0.16/0.41 ( dsetconstrER
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.41 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE1_type,type,
% 0.16/0.41 exuE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE1,definition,
% 0.16/0.41 ( exuE1
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] :
% 0.16/0.41 ( ( Xphi @ Xx )
% 0.16/0.41 & ! [Xy: $i] :
% 0.16/0.41 ( ( Xphi @ Xy )
% 0.16/0.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(prop2set_type,type,
% 0.16/0.41 prop2set: $o > $i ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setE_type,type,
% 0.16/0.41 prop2setE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(prop2setE,definition,
% 0.16/0.41 ( prop2setE
% 0.16/0.41 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.41 => Xphi ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetE_type,type,
% 0.16/0.41 emptysetE: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetE,definition,
% 0.16/0.41 ( emptysetE
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ emptyset )
% 0.16/0.41 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetimpfalse_type,type,
% 0.16/0.41 emptysetimpfalse: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptysetimpfalse,definition,
% 0.16/0.41 ( emptysetimpfalse
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ emptyset )
% 0.16/0.41 => $false ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(notinemptyset_type,type,
% 0.16/0.41 notinemptyset: $o ).
% 0.16/0.41
% 0.16/0.41 thf(notinemptyset,definition,
% 0.16/0.41 ( notinemptyset
% 0.16/0.41 = ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3e_type,type,
% 0.16/0.41 exuE3e: $o ).
% 0.16/0.41
% 0.16/0.41 thf(exuE3e,definition,
% 0.16/0.41 ( exuE3e
% 0.16/0.41 = ( ! [Xphi: $i > $o] :
% 0.16/0.41 ( ( exu
% 0.16/0.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.41 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setext_type,type,
% 0.16/0.41 setext: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setext,definition,
% 0.16/0.41 ( setext
% 0.16/0.41 = ( ! [A: $i,B: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( in @ Xx @ B ) )
% 0.16/0.41 => ( ! [Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ B )
% 0.16/0.41 => ( in @ Xx @ A ) )
% 0.16/0.41 => ( A = B ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyI_type,type,
% 0.16/0.41 emptyI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyI,definition,
% 0.16/0.41 ( emptyI
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ A )
% 0.16/0.41 => ( A = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(noeltsimpempty_type,type,
% 0.16/0.41 noeltsimpempty: $o ).
% 0.16/0.41
% 0.16/0.41 thf(noeltsimpempty,definition,
% 0.16/0.41 ( noeltsimpempty
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ! [Xx: $i] :
% 0.16/0.41 ~ ( in @ Xx @ A )
% 0.16/0.41 => ( A = emptyset ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setbeta_type,type,
% 0.16/0.41 setbeta: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setbeta,definition,
% 0.16/0.41 ( setbeta
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( in @ Xx
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.41 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonempty_type,type,
% 0.16/0.41 nonempty: $i > $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonempty,definition,
% 0.16/0.41 ( nonempty
% 0.16/0.41 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyE1_type,type,
% 0.16/0.41 nonemptyE1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyE1,definition,
% 0.16/0.41 ( nonemptyE1
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ( nonempty @ A )
% 0.16/0.41 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI_type,type,
% 0.16/0.41 nonemptyI: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI,definition,
% 0.16/0.41 ( nonemptyI
% 0.16/0.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.41 ( ( in @ Xx @ A )
% 0.16/0.41 => ( ( Xphi @ Xx )
% 0.16/0.41 => ( nonempty
% 0.16/0.41 @ ( dsetconstr @ A
% 0.16/0.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI1_type,type,
% 0.16/0.41 nonemptyI1: $o ).
% 0.16/0.41
% 0.16/0.41 thf(nonemptyI1,definition,
% 0.16/0.41 ( nonemptyI1
% 0.16/0.41 = ( ! [A: $i] :
% 0.16/0.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.41 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIL_type,type,
% 0.16/0.41 setadjoinIL: $o ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIL,definition,
% 0.16/0.41 ( setadjoinIL
% 0.16/0.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinunitempty_type,type,
% 0.16/0.41 emptyinunitempty: $o ).
% 0.16/0.41
% 0.16/0.41 thf(emptyinunitempty,definition,
% 0.16/0.41 ( emptyinunitempty
% 0.16/0.41 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.41
% 0.16/0.41 thf(setadjoinIR_type,type,
% 0.16/0.41 setadjoinIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIR,definition,
% 0.16/0.42 ( setadjoinIR
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinE_type,type,
% 0.16/0.42 setadjoinE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinE,definition,
% 0.16/0.42 ( setadjoinE
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ( ( Xy = Xx )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => ( ( ( in @ Xy @ A )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinOr_type,type,
% 0.16/0.42 setadjoinOr: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinOr,definition,
% 0.16/0.42 ( setadjoinOr
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 => ( ( Xy = Xx )
% 0.16/0.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setoftrueEq_type,type,
% 0.16/0.42 setoftrueEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setoftrueEq,definition,
% 0.16/0.42 ( setoftrueEq
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : $true )
% 0.16/0.42 = A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI_type,type,
% 0.16/0.42 powersetI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI,definition,
% 0.16/0.42 ( powersetI
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinPowerset_type,type,
% 0.16/0.42 emptyinPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinPowerset,definition,
% 0.16/0.42 ( emptyinPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyInPowerset_type,type,
% 0.16/0.42 emptyInPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyInPowerset,definition,
% 0.16/0.42 ( emptyInPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE_type,type,
% 0.16/0.42 powersetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE,definition,
% 0.16/0.42 ( powersetE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionI_type,type,
% 0.16/0.42 setunionI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionI,definition,
% 0.16/0.42 ( setunionI
% 0.16/0.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( ( in @ B @ A )
% 0.16/0.42 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE_type,type,
% 0.16/0.42 setunionE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE,definition,
% 0.16/0.42 ( setunionE
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ! [B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( ( in @ B @ A )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subPowSU_type,type,
% 0.16/0.42 subPowSU: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subPowSU,definition,
% 0.16/0.42 ( subPowSU
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE2_type,type,
% 0.16/0.42 exuE2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE2,definition,
% 0.16/0.42 ( exuE2
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyImpWitness_type,type,
% 0.16/0.42 nonemptyImpWitness: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyImpWitness,definition,
% 0.16/0.42 ( nonemptyImpWitness
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( nonempty @ A )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & $true ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit_type,type,
% 0.16/0.42 uniqinunit: $o ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit,definition,
% 0.16/0.42 ( uniqinunit
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notinsingleton_type,type,
% 0.16/0.42 notinsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notinsingleton,definition,
% 0.16/0.42 ( notinsingleton
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx != Xy )
% 0.16/0.42 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqinunit_type,type,
% 0.16/0.42 eqinunit: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqinunit,definition,
% 0.16/0.42 ( eqinunit
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsswitch_type,type,
% 0.16/0.42 singletonsswitch: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsswitch,definition,
% 0.16/0.42 ( singletonsswitch
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetE_type,type,
% 0.16/0.42 upairsetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetE,definition,
% 0.16/0.42 ( upairsetE
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.42 => ( ( Xz = Xx )
% 0.16/0.42 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIL_type,type,
% 0.16/0.42 upairsetIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIL,definition,
% 0.16/0.42 ( upairsetIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIR_type,type,
% 0.16/0.42 upairsetIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIR,definition,
% 0.16/0.42 ( upairsetIR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyE1_type,type,
% 0.16/0.42 emptyE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyE1,definition,
% 0.16/0.42 ( emptyE1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ( ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = emptyset )
% 0.16/0.42 => $false ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(vacuousDall_type,type,
% 0.16/0.42 vacuousDall: $o ).
% 0.16/0.42
% 0.16/0.42 thf(vacuousDall,definition,
% 0.16/0.42 ( vacuousDall
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan1_type,type,
% 0.16/0.42 quantDeMorgan1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan1,definition,
% 0.16/0.42 ( quantDeMorgan1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan2_type,type,
% 0.16/0.42 quantDeMorgan2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan2,definition,
% 0.16/0.42 ( quantDeMorgan2
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) )
% 0.16/0.42 => ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan3_type,type,
% 0.16/0.42 quantDeMorgan3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan3,definition,
% 0.16/0.42 ( quantDeMorgan3
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan4_type,type,
% 0.16/0.42 quantDeMorgan4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan4,definition,
% 0.16/0.42 ( quantDeMorgan4
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) )
% 0.16/0.42 => ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setI_type,type,
% 0.16/0.42 prop2setI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setI,definition,
% 0.16/0.42 ( prop2setI
% 0.16/0.42 = ( ! [Xphi: $o] :
% 0.16/0.42 ( Xphi
% 0.16/0.42 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(set2prop_type,type,
% 0.16/0.42 set2prop: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set2propI_type,type,
% 0.16/0.42 prop2set2propI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set2propI,definition,
% 0.16/0.42 ( prop2set2propI
% 0.16/0.42 = ( ! [Xphi: $o] :
% 0.16/0.42 ( Xphi
% 0.16/0.42 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notdexE_type,type,
% 0.16/0.42 notdexE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notdexE,definition,
% 0.16/0.42 ( notdexE
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notdallE_type,type,
% 0.16/0.42 notdallE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notdallE,definition,
% 0.16/0.42 ( notdallE
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI1_type,type,
% 0.16/0.42 exuI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI1,definition,
% 0.16/0.42 ( exuI1
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI3_type,type,
% 0.16/0.42 exuI3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI3,definition,
% 0.16/0.42 ( exuI3
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.42 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI2_type,type,
% 0.16/0.42 exuI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI2,definition,
% 0.16/0.42 ( exuI2
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 <=> ( Xy = Xx ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inCongP_type,type,
% 0.16/0.42 inCongP: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inCongP,definition,
% 0.16/0.42 ( inCongP
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong_type,type,
% 0.16/0.42 in__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong,definition,
% 0.16/0.42 ( in__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3u_type,type,
% 0.16/0.42 exuE3u: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3u,definition,
% 0.16/0.42 ( exuE3u
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exu__Cong_type,type,
% 0.16/0.42 exu__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu__Cong,definition,
% 0.16/0.42 ( exu__Cong
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 <=> ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset__Cong_type,type,
% 0.16/0.42 emptyset__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset__Cong,definition,
% 0.16/0.42 ( emptyset__Cong
% 0.16/0.42 = ( emptyset = emptyset ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin__Cong_type,type,
% 0.16/0.42 setadjoin__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin__Cong,definition,
% 0.16/0.42 ( setadjoin__Cong
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ! [Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( Xz = Xu )
% 0.16/0.42 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.42 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powerset__Cong_type,type,
% 0.16/0.42 powerset__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powerset__Cong,definition,
% 0.16/0.42 ( powerset__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( powerset @ A )
% 0.16/0.42 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunion__Cong_type,type,
% 0.16/0.42 setunion__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunion__Cong,definition,
% 0.16/0.42 ( setunion__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( ( setunion @ A )
% 0.16/0.42 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omega__Cong_type,type,
% 0.16/0.42 omega__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omega__Cong,definition,
% 0.16/0.42 ( omega__Cong
% 0.16/0.42 = ( omega = omega ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuEu_type,type,
% 0.16/0.42 exuEu: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuEu,definition,
% 0.16/0.42 ( exuEu
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(descr__Cong_type,type,
% 0.16/0.42 descr__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(descr__Cong,definition,
% 0.16/0.42 ( descr__Cong
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.42 => ( ( descr
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = ( descr
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstr__Cong_type,type,
% 0.16/0.42 dsetconstr__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstr__Cong,definition,
% 0.16/0.42 ( dsetconstr__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.42 => ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = ( dsetconstr @ B
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subset_type,type,
% 0.16/0.42 subset: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(disjoint_type,type,
% 0.16/0.42 disjoint: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(setsmeet_type,type,
% 0.16/0.42 setsmeet: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI1_type,type,
% 0.16/0.42 subsetI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI1,definition,
% 0.16/0.42 ( subsetI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset2_type,type,
% 0.16/0.42 eqimpsubset2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset2,definition,
% 0.16/0.42 ( eqimpsubset2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset1_type,type,
% 0.16/0.42 eqimpsubset1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqimpsubset1,definition,
% 0.16/0.42 ( eqimpsubset1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI2_type,type,
% 0.16/0.42 subsetI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetI2,definition,
% 0.16/0.42 ( subsetI2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetsubset_type,type,
% 0.16/0.42 emptysetsubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetsubset,definition,
% 0.16/0.42 ( emptysetsubset
% 0.16/0.42 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE_type,type,
% 0.16/0.42 subsetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE,definition,
% 0.16/0.42 ( subsetE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE2_type,type,
% 0.16/0.42 subsetE2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetE2,definition,
% 0.16/0.42 ( subsetE2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notsubsetI_type,type,
% 0.16/0.42 notsubsetI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notsubsetI,definition,
% 0.16/0.42 ( notsubsetI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI1_type,type,
% 0.16/0.42 notequalI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI1,definition,
% 0.16/0.42 ( notequalI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ~ ( subset @ A @ B )
% 0.16/0.42 => ( A != B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI2_type,type,
% 0.16/0.42 notequalI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notequalI2,definition,
% 0.16/0.42 ( notequalI2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ( A != B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetRefl_type,type,
% 0.16/0.42 subsetRefl: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetRefl,definition,
% 0.16/0.42 ( subsetRefl
% 0.16/0.42 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetTrans_type,type,
% 0.16/0.42 subsetTrans: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetTrans,definition,
% 0.16/0.42 ( subsetTrans
% 0.16/0.42 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( subset @ B @ C )
% 0.16/0.42 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub_type,type,
% 0.16/0.42 setadjoinSub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub,definition,
% 0.16/0.42 ( setadjoinSub
% 0.16/0.42 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub2_type,type,
% 0.16/0.42 setadjoinSub2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinSub2,definition,
% 0.16/0.42 ( setadjoinSub2
% 0.16/0.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subset2powerset_type,type,
% 0.16/0.42 subset2powerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subset2powerset,definition,
% 0.16/0.42 ( subset2powerset
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setextsub_type,type,
% 0.16/0.42 setextsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setextsub,definition,
% 0.16/0.42 ( setextsub
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( subset @ B @ A )
% 0.16/0.42 => ( A = B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subsetemptysetimpeq_type,type,
% 0.16/0.42 subsetemptysetimpeq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subsetemptysetimpeq,definition,
% 0.16/0.42 ( subsetemptysetimpeq
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( subset @ A @ emptyset )
% 0.16/0.42 => ( A = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI1_type,type,
% 0.16/0.42 powersetI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI1,definition,
% 0.16/0.42 ( powersetI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ B @ A )
% 0.16/0.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE1_type,type,
% 0.16/0.42 powersetE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE1,definition,
% 0.16/0.42 ( powersetE1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inPowerset_type,type,
% 0.16/0.42 inPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inPowerset,definition,
% 0.16/0.42 ( inPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetsubset_type,type,
% 0.16/0.42 powersetsubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetsubset,definition,
% 0.16/0.42 ( powersetsubset
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(sepInPowerset_type,type,
% 0.16/0.42 sepInPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(sepInPowerset,definition,
% 0.16/0.42 ( sepInPowerset
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( in
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(sepSubset_type,type,
% 0.16/0.42 sepSubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(sepSubset,definition,
% 0.16/0.42 ( sepSubset
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( subset
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunion_type,type,
% 0.16/0.42 binunion: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIL_type,type,
% 0.16/0.42 binunionIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIL,definition,
% 0.16/0.42 ( binunionIL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2IR_type,type,
% 0.16/0.42 upairset2IR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2IR,definition,
% 0.16/0.42 ( upairset2IR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIR_type,type,
% 0.16/0.42 binunionIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionIR,definition,
% 0.16/0.42 ( binunionIR
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionEcases_type,type,
% 0.16/0.42 binunionEcases: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionEcases,definition,
% 0.16/0.42 ( binunionEcases
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.42 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.42 => ( ( ( in @ Xx @ A )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => ( ( ( in @ Xx @ B )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionE_type,type,
% 0.16/0.42 binunionE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionE,definition,
% 0.16/0.42 ( binunionE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionLsub_type,type,
% 0.16/0.42 binunionLsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionLsub,definition,
% 0.16/0.42 ( binunionLsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binunionRsub_type,type,
% 0.16/0.42 binunionRsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binunionRsub,definition,
% 0.16/0.42 ( binunionRsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersect_type,type,
% 0.16/0.42 binintersect: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectI_type,type,
% 0.16/0.42 binintersectI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectI,definition,
% 0.16/0.42 ( binintersectI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset5_type,type,
% 0.16/0.42 binintersectSubset5: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset5,definition,
% 0.16/0.42 ( binintersectSubset5
% 0.16/0.42 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.42 ( ( subset @ C @ A )
% 0.16/0.42 => ( ( subset @ C @ B )
% 0.16/0.42 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectEL_type,type,
% 0.16/0.42 binintersectEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectEL,definition,
% 0.16/0.42 ( binintersectEL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectLsub_type,type,
% 0.16/0.42 binintersectLsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectLsub,definition,
% 0.16/0.42 ( binintersectLsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset2_type,type,
% 0.16/0.42 binintersectSubset2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset2,definition,
% 0.16/0.42 ( binintersectSubset2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( binintersect @ A @ B )
% 0.16/0.42 = A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset3_type,type,
% 0.16/0.42 binintersectSubset3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset3,definition,
% 0.16/0.42 ( binintersectSubset3
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( ( binintersect @ A @ B )
% 0.16/0.42 = B )
% 0.16/0.42 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectER_type,type,
% 0.16/0.42 binintersectER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectER,definition,
% 0.16/0.42 ( binintersectER
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(disjointsetsI1_type,type,
% 0.16/0.42 disjointsetsI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(disjointsetsI1,definition,
% 0.16/0.42 ( disjointsetsI1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( in @ Xx @ B ) )
% 0.16/0.42 => ( ( binintersect @ A @ B )
% 0.16/0.42 = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectRsub_type,type,
% 0.16/0.42 binintersectRsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectRsub,definition,
% 0.16/0.42 ( binintersectRsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset4_type,type,
% 0.16/0.42 binintersectSubset4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset4,definition,
% 0.16/0.42 ( binintersectSubset4
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ B @ A )
% 0.16/0.42 => ( ( binintersect @ A @ B )
% 0.16/0.42 = B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset1_type,type,
% 0.16/0.42 binintersectSubset1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(binintersectSubset1,definition,
% 0.16/0.42 ( binintersectSubset1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( ( binintersect @ A @ B )
% 0.16/0.42 = A )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(bs114d_type,type,
% 0.16/0.42 bs114d: $o ).
% 0.16/0.42
% 0.16/0.42 thf(bs114d,definition,
% 0.16/0.42 ( bs114d
% 0.16/0.42 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.42 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.42 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(regular_type,type,
% 0.16/0.42 regular: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminus_type,type,
% 0.16/0.42 setminus: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(setminusI_type,type,
% 0.16/0.42 setminusI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusI,definition,
% 0.16/0.42 ( setminusI
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusEL_type,type,
% 0.16/0.42 setminusEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusEL,definition,
% 0.16/0.42 ( setminusEL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusER_type,type,
% 0.16/0.42 setminusER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusER,definition,
% 0.16/0.42 ( setminusER
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.42 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset2_type,type,
% 0.16/0.42 setminusSubset2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset2,definition,
% 0.16/0.42 ( setminusSubset2
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( subset @ A @ B )
% 0.16/0.42 => ( ( setminus @ A @ B )
% 0.16/0.42 = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusERneg_type,type,
% 0.16/0.42 setminusERneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusERneg,definition,
% 0.16/0.42 ( setminusERneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusELneg_type,type,
% 0.16/0.42 setminusELneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusELneg,definition,
% 0.16/0.42 ( setminusELneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusILneg_type,type,
% 0.16/0.42 setminusILneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusILneg,definition,
% 0.16/0.42 ( setminusILneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusIRneg_type,type,
% 0.16/0.42 setminusIRneg: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusIRneg,definition,
% 0.16/0.42 ( setminusIRneg
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusLsub_type,type,
% 0.16/0.42 setminusLsub: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusLsub,definition,
% 0.16/0.42 ( setminusLsub
% 0.16/0.42 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset1_type,type,
% 0.16/0.42 setminusSubset1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setminusSubset1,definition,
% 0.16/0.42 ( setminusSubset1
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( ( setminus @ A @ B )
% 0.16/0.42 = emptyset )
% 0.16/0.42 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiff_type,type,
% 0.16/0.42 symdiff: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffE_type,type,
% 0.16/0.42 symdiffE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffE,definition,
% 0.16/0.42 ( symdiffE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI1_type,type,
% 0.16/0.42 symdiffI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI1,definition,
% 0.16/0.42 ( symdiffI1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI2_type,type,
% 0.16/0.42 symdiffI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffI2,definition,
% 0.16/0.42 ( symdiffI2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg1_type,type,
% 0.16/0.42 symdiffIneg1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg1,definition,
% 0.16/0.42 ( symdiffIneg1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg2_type,type,
% 0.16/0.42 symdiffIneg2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(symdiffIneg2,definition,
% 0.16/0.42 ( symdiffIneg2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ~ ( in @ Xx @ A )
% 0.16/0.42 => ( ~ ( in @ Xx @ B )
% 0.16/0.42 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(iskpair_type,type,
% 0.16/0.42 iskpair: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(secondinupair_type,type,
% 0.16/0.42 secondinupair: $o ).
% 0.16/0.42
% 0.16/0.42 thf(secondinupair,definition,
% 0.16/0.42 ( secondinupair
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairIL_type,type,
% 0.16/0.42 setukpairIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairIL,definition,
% 0.16/0.42 ( setukpairIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairIR_type,type,
% 0.16/0.42 setukpairIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairIR,definition,
% 0.16/0.42 ( setukpairIR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kpairiskpair_type,type,
% 0.16/0.42 kpairiskpair: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kpairiskpair,definition,
% 0.16/0.42 ( kpairiskpair
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kpair_type,type,
% 0.16/0.42 kpair: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(kpairp_type,type,
% 0.16/0.42 kpairp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kpairp,definition,
% 0.16/0.42 ( kpairp
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprod_type,type,
% 0.16/0.42 cartprod: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsubset_type,type,
% 0.16/0.42 singletonsubset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsubset,definition,
% 0.16/0.42 ( singletonsubset
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletoninpowerset_type,type,
% 0.16/0.42 singletoninpowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletoninpowerset,definition,
% 0.16/0.42 ( singletoninpowerset
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletoninpowunion_type,type,
% 0.16/0.42 singletoninpowunion: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletoninpowunion,definition,
% 0.16/0.42 ( singletoninpowunion
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2E_type,type,
% 0.16/0.42 upairset2E: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairset2E,definition,
% 0.16/0.42 ( upairset2E
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.42 => ( ( Xz = Xx )
% 0.16/0.42 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsubunion_type,type,
% 0.16/0.42 upairsubunion: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsubunion,definition,
% 0.16/0.42 ( upairsubunion
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairinpowunion_type,type,
% 0.16/0.42 upairinpowunion: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairinpowunion,definition,
% 0.16/0.42 ( upairinpowunion
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem1_type,type,
% 0.16/0.42 ubforcartprodlem1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem1,definition,
% 0.16/0.42 ( ubforcartprodlem1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem2_type,type,
% 0.16/0.42 ubforcartprodlem2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem2,definition,
% 0.16/0.42 ( ubforcartprodlem2
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem3_type,type,
% 0.16/0.42 ubforcartprodlem3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ubforcartprodlem3,definition,
% 0.16/0.42 ( ubforcartprodlem3
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairin_type,type,
% 0.16/0.42 cartprodpairin: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairin,definition,
% 0.16/0.42 ( cartprodpairin
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempair1_type,type,
% 0.16/0.42 cartprodmempair1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempair1,definition,
% 0.16/0.42 ( cartprodmempair1
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 & ( Xu
% 0.16/0.42 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempair_type,type,
% 0.16/0.42 cartprodmempair: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempair,definition,
% 0.16/0.42 ( cartprodmempair
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( iskpair @ Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE2_type,type,
% 0.16/0.42 setunionE2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE2,definition,
% 0.16/0.42 ( setunionE2
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 => ? [X: $i] :
% 0.16/0.42 ( ( in @ X @ A )
% 0.16/0.42 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton1_type,type,
% 0.16/0.42 setunionsingleton1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton1,definition,
% 0.16/0.42 ( setunionsingleton1
% 0.16/0.42 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton2_type,type,
% 0.16/0.42 setunionsingleton2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton2,definition,
% 0.16/0.42 ( setunionsingleton2
% 0.16/0.42 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton_type,type,
% 0.16/0.42 setunionsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionsingleton,definition,
% 0.16/0.42 ( setunionsingleton
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.16/0.42 = Xx ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singleton_type,type,
% 0.16/0.42 singleton: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(singleton,definition,
% 0.16/0.42 ( singleton
% 0.16/0.42 = ( ^ [A: $i] :
% 0.16/0.42 ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( A
% 0.16/0.42 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletonprop_type,type,
% 0.16/0.42 singletonprop: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonprop,definition,
% 0.16/0.42 ( singletonprop
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) )
% 0.16/0.42 => ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1_type,type,
% 0.16/0.42 ex1: $i > ( $i > $o ) > $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1,definition,
% 0.16/0.42 ( ex1
% 0.16/0.42 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( singleton
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1E1_type,type,
% 0.16/0.42 ex1E1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1E1,definition,
% 0.16/0.42 ( ex1E1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ( ex1 @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1I_type,type,
% 0.16/0.42 ex1I: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1I,definition,
% 0.16/0.42 ( ex1I
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xy = Xx ) ) )
% 0.16/0.42 => ( ex1 @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ex1I2_type,type,
% 0.16/0.42 ex1I2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ex1I2,definition,
% 0.16/0.42 ( ex1I2
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) )
% 0.16/0.42 => ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ( ex1 @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsuniq_type,type,
% 0.16/0.42 singletonsuniq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsuniq,definition,
% 0.16/0.42 ( singletonsuniq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( ( setadjoin @ Xx @ emptyset )
% 0.16/0.42 = ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(atmost1p_type,type,
% 0.16/0.42 atmost1p: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(atleast2p_type,type,
% 0.16/0.42 atleast2p: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(atmost2p_type,type,
% 0.16/0.42 atmost2p: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetp_type,type,
% 0.16/0.42 upairsetp: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL1_type,type,
% 0.16/0.42 setukpairinjL1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL1,definition,
% 0.16/0.42 ( setukpairinjL1
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kfstsingleton_type,type,
% 0.16/0.42 kfstsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kfstsingleton,definition,
% 0.16/0.42 ( kfstsingleton
% 0.16/0.42 = ( ! [Xu: $i] :
% 0.16/0.42 ( ( iskpair @ Xu )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.42 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(theprop_type,type,
% 0.16/0.42 theprop: $o ).
% 0.16/0.42
% 0.16/0.42 thf(theprop,definition,
% 0.16/0.42 ( theprop
% 0.16/0.42 = ( ! [X: $i] :
% 0.16/0.42 ( ( singleton @ X )
% 0.16/0.42 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kfst_type,type,
% 0.16/0.42 kfst: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(kfstpairEq_type,type,
% 0.16/0.42 kfstpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kfstpairEq,definition,
% 0.16/0.42 ( kfstpairEq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xx ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstin_type,type,
% 0.16/0.42 cartprodfstin: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstin,definition,
% 0.16/0.42 ( cartprodfstin
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL2_type,type,
% 0.16/0.42 setukpairinjL2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL2,definition,
% 0.16/0.42 ( setukpairinjL2
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL_type,type,
% 0.16/0.42 setukpairinjL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjL,definition,
% 0.16/0.42 ( setukpairinjL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.42 = ( kpair @ Xz @ Xu ) )
% 0.16/0.42 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR11_type,type,
% 0.16/0.42 setukpairinjR11: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR11,definition,
% 0.16/0.42 ( setukpairinjR11
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR12_type,type,
% 0.16/0.42 setukpairinjR12: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR12,definition,
% 0.16/0.42 ( setukpairinjR12
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR1_type,type,
% 0.16/0.42 setukpairinjR1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR1,definition,
% 0.16/0.42 ( setukpairinjR1
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( ( Xz = Xu )
% 0.16/0.42 => ( Xy = Xu ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairequniteq_type,type,
% 0.16/0.42 upairequniteq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairequniteq,definition,
% 0.16/0.42 ( upairequniteq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ Xz @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR2_type,type,
% 0.16/0.42 setukpairinjR2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR2,definition,
% 0.16/0.42 ( setukpairinjR2
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.42 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.42 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR_type,type,
% 0.16/0.42 setukpairinjR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setukpairinjR,definition,
% 0.16/0.42 ( setukpairinjR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.42 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.42 = ( kpair @ Xz @ Xu ) )
% 0.16/0.42 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ksndsingleton_type,type,
% 0.16/0.42 ksndsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ksndsingleton,definition,
% 0.16/0.42 ( ksndsingleton
% 0.16/0.42 = ( ! [Xu: $i] :
% 0.16/0.42 ( ( iskpair @ Xu )
% 0.16/0.42 => ( singleton
% 0.16/0.42 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.42 @ ^ [Xx: $i] :
% 0.16/0.42 ( Xu
% 0.16/0.42 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(ksnd_type,type,
% 0.16/0.42 ksnd: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(ksndpairEq_type,type,
% 0.16/0.42 ksndpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(ksndpairEq,definition,
% 0.16/0.42 ( ksndpairEq
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xy ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(kpairsurjEq_type,type,
% 0.16/0.42 kpairsurjEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(kpairsurjEq,definition,
% 0.16/0.42 ( kpairsurjEq
% 0.16/0.42 = ( ! [Xu: $i] :
% 0.16/0.42 ( ( iskpair @ Xu )
% 0.16/0.42 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.42 = Xu ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndin_type,type,
% 0.16/0.42 cartprodsndin: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndin,definition,
% 0.16/0.42 ( cartprodsndin
% 0.16/0.42 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemEL_type,type,
% 0.16/0.42 cartprodpairmemEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemEL,definition,
% 0.16/0.42 ( cartprodpairmemEL
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemER_type,type,
% 0.16/0.42 cartprodpairmemER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairmemER,definition,
% 0.16/0.42 ( cartprodpairmemER
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempaircEq_type,type,
% 0.16/0.42 cartprodmempaircEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodmempaircEq,definition,
% 0.16/0.42 ( cartprodmempaircEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( kpair @ Xx @ Xy )
% 0.16/0.42 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstpairEq_type,type,
% 0.16/0.42 cartprodfstpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodfstpairEq,definition,
% 0.16/0.42 ( cartprodfstpairEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndpairEq_type,type,
% 0.16/0.42 cartprodsndpairEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodsndpairEq,definition,
% 0.16/0.42 ( cartprodsndpairEq
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ B )
% 0.16/0.42 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.42 = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(cartprodpairsurjEq,conjecture,
% 0.16/0.42 ( setextAx
% 0.16/0.42 => ( emptysetAx
% 0.16/0.42 => ( setadjoinAx
% 0.16/0.42 => ( powersetAx
% 0.16/0.42 => ( setunionAx
% 0.16/0.42 => ( omega0Ax
% 0.16/0.42 => ( omegaSAx
% 0.16/0.42 => ( omegaIndAx
% 0.16/0.42 => ( replAx
% 0.16/0.42 => ( foundationAx
% 0.16/0.42 => ( wellorderingAx
% 0.16/0.42 => ( descrp
% 0.16/0.42 => ( dsetconstrI
% 0.16/0.42 => ( dsetconstrEL
% 0.16/0.42 => ( dsetconstrER
% 0.16/0.42 => ( exuE1
% 0.16/0.42 => ( prop2setE
% 0.16/0.42 => ( emptysetE
% 0.16/0.42 => ( emptysetimpfalse
% 0.16/0.42 => ( notinemptyset
% 0.16/0.42 => ( exuE3e
% 0.16/0.42 => ( setext
% 0.16/0.42 => ( emptyI
% 0.16/0.42 => ( noeltsimpempty
% 0.16/0.42 => ( setbeta
% 0.16/0.42 => ( nonemptyE1
% 0.16/0.42 => ( nonemptyI
% 0.16/0.42 => ( nonemptyI1
% 0.16/0.42 => ( setadjoinIL
% 0.16/0.42 => ( emptyinunitempty
% 0.16/0.42 => ( setadjoinIR
% 0.16/0.42 => ( setadjoinE
% 0.16/0.42 => ( setadjoinOr
% 0.16/0.42 => ( setoftrueEq
% 0.16/0.42 => ( powersetI
% 0.16/0.42 => ( emptyinPowerset
% 0.16/0.42 => ( emptyInPowerset
% 0.16/0.42 => ( powersetE
% 0.16/0.42 => ( setunionI
% 0.16/0.42 => ( setunionE
% 0.16/0.42 => ( subPowSU
% 0.16/0.42 => ( exuE2
% 0.16/0.42 => ( nonemptyImpWitness
% 0.16/0.42 => ( uniqinunit
% 0.16/0.42 => ( notinsingleton
% 0.16/0.42 => ( eqinunit
% 0.16/0.42 => ( singletonsswitch
% 0.16/0.42 => ( upairsetE
% 0.16/0.42 => ( upairsetIL
% 0.16/0.42 => ( upairsetIR
% 0.16/0.42 => ( emptyE1
% 0.16/0.42 => ( vacuousDall
% 0.16/0.42 => ( quantDeMorgan1
% 0.16/0.42 => ( quantDeMorgan2
% 0.16/0.42 => ( quantDeMorgan3
% 0.16/0.42 => ( quantDeMorgan4
% 0.16/0.42 => ( prop2setI
% 0.16/0.42 => ( prop2set2propI
% 0.16/0.42 => ( notdexE
% 0.16/0.42 => ( notdallE
% 0.16/0.42 => ( exuI1
% 0.16/0.42 => ( exuI3
% 0.16/0.42 => ( exuI2
% 0.16/0.42 => ( inCongP
% 0.16/0.42 => ( in__Cong
% 0.16/0.42 => ( exuE3u
% 0.16/0.42 => ( exu__Cong
% 0.16/0.42 => ( emptyset__Cong
% 0.16/0.42 => ( setadjoin__Cong
% 0.16/0.42 => ( powerset__Cong
% 0.16/0.42 => ( setunion__Cong
% 0.16/0.42 => ( omega__Cong
% 0.16/0.42 => ( exuEu
% 0.16/0.42 => ( descr__Cong
% 0.16/0.42 => ( dsetconstr__Cong
% 0.16/0.42 => ( subsetI1
% 0.16/0.42 => ( eqimpsubset2
% 0.16/0.42 => ( eqimpsubset1
% 0.16/0.42 => ( subsetI2
% 0.16/0.42 => ( emptysetsubset
% 0.16/0.42 => ( subsetE
% 0.16/0.42 => ( subsetE2
% 0.16/0.42 => ( notsubsetI
% 0.16/0.42 => ( notequalI1
% 0.16/0.42 => ( notequalI2
% 0.16/0.42 => ( subsetRefl
% 0.16/0.42 => ( subsetTrans
% 0.16/0.42 => ( setadjoinSub
% 0.16/0.42 => ( setadjoinSub2
% 0.16/0.42 => ( subset2powerset
% 0.16/0.42 => ( setextsub
% 0.16/0.42 => ( subsetemptysetimpeq
% 0.16/0.42 => ( powersetI1
% 0.16/0.42 => ( powersetE1
% 0.16/0.42 => ( inPowerset
% 0.16/0.42 => ( powersetsubset
% 0.16/0.42 => ( sepInPowerset
% 0.16/0.42 => ( sepSubset
% 0.16/0.42 => ( binunionIL
% 0.16/0.42 => ( upairset2IR
% 0.16/0.42 => ( binunionIR
% 0.16/0.42 => ( binunionEcases
% 0.16/0.42 => ( binunionE
% 0.16/0.42 => ( binunionLsub
% 0.16/0.42 => ( binunionRsub
% 0.16/0.42 => ( binintersectI
% 0.16/0.42 => ( binintersectSubset5
% 0.16/0.42 => ( binintersectEL
% 0.16/0.42 => ( binintersectLsub
% 0.16/0.42 => ( binintersectSubset2
% 0.16/0.42 => ( binintersectSubset3
% 0.16/0.42 => ( binintersectER
% 0.16/0.42 => ( disjointsetsI1
% 0.16/0.42 => ( binintersectRsub
% 0.16/0.42 => ( binintersectSubset4
% 0.16/0.42 => ( binintersectSubset1
% 0.16/0.42 => ( bs114d
% 0.16/0.42 => ( setminusI
% 0.16/0.42 => ( setminusEL
% 0.16/0.42 => ( setminusER
% 0.16/0.42 => ( setminusSubset2
% 0.16/0.42 => ( setminusERneg
% 0.16/0.42 => ( setminusELneg
% 0.16/0.42 => ( setminusILneg
% 0.16/0.42 => ( setminusIRneg
% 0.16/0.42 => ( setminusLsub
% 0.16/0.42 => ( setminusSubset1
% 0.16/0.42 => ( symdiffE
% 0.16/0.42 => ( symdiffI1
% 0.16/0.42 => ( symdiffI2
% 0.16/0.42 => ( symdiffIneg1
% 0.16/0.42 => ( symdiffIneg2
% 0.16/0.42 => ( secondinupair
% 0.16/0.42 => ( setukpairIL
% 0.16/0.42 => ( setukpairIR
% 0.16/0.42 => ( kpairiskpair
% 0.16/0.42 => ( kpairp
% 0.16/0.42 => ( singletonsubset
% 0.16/0.42 => ( singletoninpowerset
% 0.16/0.42 => ( singletoninpowunion
% 0.16/0.42 => ( upairset2E
% 0.16/0.42 => ( upairsubunion
% 0.16/0.42 => ( upairinpowunion
% 0.16/0.42 => ( ubforcartprodlem1
% 0.16/0.42 => ( ubforcartprodlem2
% 0.16/0.42 => ( ubforcartprodlem3
% 0.16/0.42 => ( cartprodpairin
% 0.16/0.42 => ( cartprodmempair1
% 0.16/0.42 => ( cartprodmempair
% 0.16/0.42 => ( setunionE2
% 0.16/0.42 => ( setunionsingleton1
% 0.16/0.42 => ( setunionsingleton2
% 0.16/0.42 => ( setunionsingleton
% 0.16/0.42 => ( singletonprop
% 0.16/0.42 => ( ex1E1
% 0.16/0.42 => ( ex1I
% 0.16/0.42 => ( ex1I2
% 0.16/0.42 => ( singletonsuniq
% 0.16/0.42 => ( setukpairinjL1
% 0.16/0.42 => ( kfstsingleton
% 0.16/0.42 => ( theprop
% 0.16/0.42 => ( kfstpairEq
% 0.16/0.42 => ( cartprodfstin
% 0.16/0.42 => ( setukpairinjL2
% 0.16/0.42 => ( setukpairinjL
% 0.16/0.42 => ( setukpairinjR11
% 0.16/0.42 => ( setukpairinjR12
% 0.16/0.42 => ( setukpairinjR1
% 0.16/0.42 => ( upairequniteq
% 0.16/0.42 => ( setukpairinjR2
% 0.16/0.42 => ( setukpairinjR
% 0.16/0.42 => ( ksndsingleton
% 0.16/0.42 => ( ksndpairEq
% 0.16/0.42 => ( kpairsurjEq
% 0.16/0.42 => ( cartprodsndin
% 0.16/0.42 => ( cartprodpairmemEL
% 0.16/0.42 => ( cartprodpairmemER
% 0.16/0.42 => ( cartprodmempaircEq
% 0.16/0.42 => ( cartprodfstpairEq
% 0.16/0.42 => ( cartprodsndpairEq
% 0.16/0.42 => ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.42 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.42 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.42 = Xu ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 ------------------------
% 2.31/1.36 ---- Embedded in THF ---
% 2.31/1.36 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.31/1.36 %%% Generated on Tue Feb 24 12:29:24 EST 2026
% 2.31/1.36 %%% using '$$dhol' embedding, version 1.3.0.
% 2.31/1.36 %%% Logic specification used:
% 2.31/1.36 %%% thf(spec, logic, $$dhol).
% 2.31/1.36
% 2.31/1.36 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.7RtdXlarj4/DTF2DTF4583.p
% See solution above
% 2.31/1.36 ------------------------
% 2.46/1.40 ---- Cleaned THF ---
% 2.46/1.40 thf(in_type,type,
% 2.46/1.40 in: $i > $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(exu_type,type,
% 2.46/1.40 exu: ( $i > $o ) > $o ).
% 2.46/1.40
% 2.46/1.40 thf(exu,definition,
% 2.46/1.40 ( exu
% 2.46/1.40 = ( ^ [Xphi: $i > $o] :
% 2.46/1.40 ? [Xx: $i] :
% 2.46/1.40 ( ( Xphi @ Xx )
% 2.46/1.40 & ! [Xy: $i] :
% 2.46/1.40 ( ( Xphi @ Xy )
% 2.46/1.40 => ( Xx = Xy ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setextAx_type,type,
% 2.46/1.40 setextAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setextAx,definition,
% 2.46/1.40 ( setextAx
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 <=> ( in @ Xx @ B ) )
% 2.46/1.40 => ( A = B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptyset_type,type,
% 2.46/1.40 emptyset: $i ).
% 2.46/1.40
% 2.46/1.40 thf(emptysetAx_type,type,
% 2.46/1.40 emptysetAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptysetAx,definition,
% 2.46/1.40 ( emptysetAx
% 2.46/1.40 = ( ! [Xx: $i] :
% 2.46/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoin_type,type,
% 2.46/1.40 setadjoin: $i > $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinAx_type,type,
% 2.46/1.40 setadjoinAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinAx,definition,
% 2.46/1.40 ( setadjoinAx
% 2.46/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.46/1.40 <=> ( ( Xy = Xx )
% 2.46/1.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(powerset_type,type,
% 2.46/1.40 powerset: $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(powersetAx_type,type,
% 2.46/1.40 powersetAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(powersetAx,definition,
% 2.46/1.40 ( powersetAx
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.46/1.40 <=> ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setunion_type,type,
% 2.46/1.40 setunion: $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(setunionAx_type,type,
% 2.46/1.40 setunionAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setunionAx,definition,
% 2.46/1.40 ( setunionAx
% 2.46/1.40 = ( ! [A: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.46/1.40 <=> ? [B: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 & ( in @ B @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(omega_type,type,
% 2.46/1.40 omega: $i ).
% 2.46/1.40
% 2.46/1.40 thf(omega0Ax_type,type,
% 2.46/1.40 omega0Ax: $o ).
% 2.46/1.40
% 2.46/1.40 thf(omega0Ax,definition,
% 2.46/1.40 ( omega0Ax
% 2.46/1.40 = ( in @ emptyset @ omega ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(omegaSAx_type,type,
% 2.46/1.40 omegaSAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(omegaSAx,definition,
% 2.46/1.40 ( omegaSAx
% 2.46/1.40 = ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ omega )
% 2.46/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(omegaIndAx_type,type,
% 2.46/1.40 omegaIndAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(omegaIndAx,definition,
% 2.46/1.40 ( omegaIndAx
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ( ( ( in @ emptyset @ A )
% 2.46/1.40 & ! [Xx: $i] :
% 2.46/1.40 ( ( ( in @ Xx @ omega )
% 2.46/1.40 & ( in @ Xx @ A ) )
% 2.46/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.46/1.40 => ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ omega )
% 2.46/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(replAx_type,type,
% 2.46/1.40 replAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(replAx,definition,
% 2.46/1.40 ( replAx
% 2.46/1.40 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( exu
% 2.46/1.40 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.46/1.40 => ? [B: $i] :
% 2.46/1.40 ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 <=> ? [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ A )
% 2.46/1.40 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(foundationAx_type,type,
% 2.46/1.40 foundationAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(foundationAx,definition,
% 2.46/1.40 ( foundationAx
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.46/1.40 => ? [B: $i] :
% 2.46/1.40 ( ( in @ B @ A )
% 2.46/1.40 & ~ ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(wellorderingAx_type,type,
% 2.46/1.40 wellorderingAx: $o ).
% 2.46/1.40
% 2.46/1.40 thf(wellorderingAx,definition,
% 2.46/1.40 ( wellorderingAx
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ? [B: $i] :
% 2.46/1.40 ( ! [C: $i] :
% 2.46/1.40 ( ( in @ C @ B )
% 2.46/1.40 => ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ C )
% 2.46/1.40 => ( in @ Xx @ A ) ) )
% 2.46/1.40 & ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( ( in @ Xx @ A )
% 2.46/1.40 & ( in @ Xy @ A ) )
% 2.46/1.40 => ( ! [C: $i] :
% 2.46/1.40 ( ( in @ C @ B )
% 2.46/1.40 => ( ( in @ Xx @ C )
% 2.46/1.40 <=> ( in @ Xy @ C ) ) )
% 2.46/1.40 => ( Xx = Xy ) ) )
% 2.46/1.40 & ! [C: $i,D: $i] :
% 2.46/1.40 ( ( ( in @ C @ B )
% 2.46/1.40 & ( in @ D @ B ) )
% 2.46/1.40 => ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ C )
% 2.46/1.40 => ( in @ Xx @ D ) )
% 2.46/1.40 | ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ D )
% 2.46/1.40 => ( in @ Xx @ C ) ) ) )
% 2.46/1.40 & ! [C: $i] :
% 2.46/1.40 ( ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ C )
% 2.46/1.40 => ( in @ Xx @ A ) )
% 2.46/1.40 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.46/1.40 => ? [D: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ D @ B )
% 2.46/1.40 & ( in @ Xx @ C )
% 2.46/1.40 & ~ ? [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ D )
% 2.46/1.40 & ( in @ Xy @ C ) )
% 2.46/1.40 & ! [E: $i] :
% 2.46/1.40 ( ( in @ E @ B )
% 2.46/1.40 => ( ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ E )
% 2.46/1.40 => ( in @ Xy @ D ) )
% 2.46/1.40 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(descr_type,type,
% 2.46/1.40 descr: ( $i > $o ) > $i ).
% 2.46/1.40
% 2.46/1.40 thf(descrp_type,type,
% 2.46/1.40 descrp: $o ).
% 2.46/1.40
% 2.46/1.40 thf(descrp,definition,
% 2.46/1.40 ( descrp
% 2.46/1.40 = ( ! [Xphi: $i > $o] :
% 2.46/1.40 ( ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 => ( Xphi
% 2.46/1.40 @ ( descr
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(dsetconstr_type,type,
% 2.46/1.40 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.46/1.40
% 2.46/1.40 thf(dsetconstrI_type,type,
% 2.46/1.40 dsetconstrI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(dsetconstrI,definition,
% 2.46/1.40 ( dsetconstrI
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ( Xphi @ Xx )
% 2.46/1.40 => ( in @ Xx
% 2.46/1.40 @ ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(dsetconstrEL_type,type,
% 2.46/1.40 dsetconstrEL: $o ).
% 2.46/1.40
% 2.46/1.40 thf(dsetconstrEL,definition,
% 2.46/1.40 ( dsetconstrEL
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx
% 2.46/1.40 @ ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.46/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(dsetconstrER_type,type,
% 2.46/1.40 dsetconstrER: $o ).
% 2.46/1.40
% 2.46/1.40 thf(dsetconstrER,definition,
% 2.46/1.40 ( dsetconstrER
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx
% 2.46/1.40 @ ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.46/1.40 => ( Xphi @ Xx ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(exuE1_type,type,
% 2.46/1.40 exuE1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(exuE1,definition,
% 2.46/1.40 ( exuE1
% 2.46/1.40 = ( ! [Xphi: $i > $o] :
% 2.46/1.40 ( ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 => ? [Xx: $i] :
% 2.46/1.40 ( ( Xphi @ Xx )
% 2.46/1.40 & ! [Xy: $i] :
% 2.46/1.40 ( ( Xphi @ Xy )
% 2.46/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(prop2set_type,type,
% 2.46/1.40 prop2set: $o > $i ).
% 2.46/1.40
% 2.46/1.40 thf(prop2setE_type,type,
% 2.46/1.40 prop2setE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(prop2setE,definition,
% 2.46/1.40 ( prop2setE
% 2.46/1.40 = ( ! [Xphi: $o,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.46/1.40 => Xphi ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptysetE_type,type,
% 2.46/1.40 emptysetE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptysetE,definition,
% 2.46/1.40 ( emptysetE
% 2.46/1.40 = ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ emptyset )
% 2.46/1.40 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptysetimpfalse_type,type,
% 2.46/1.40 emptysetimpfalse: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptysetimpfalse,definition,
% 2.46/1.40 ( emptysetimpfalse
% 2.46/1.40 = ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ emptyset )
% 2.46/1.40 => $false ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(notinemptyset_type,type,
% 2.46/1.40 notinemptyset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(notinemptyset,definition,
% 2.46/1.40 ( notinemptyset
% 2.46/1.40 = ( ! [Xx: $i] :
% 2.46/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(exuE3e_type,type,
% 2.46/1.40 exuE3e: $o ).
% 2.46/1.40
% 2.46/1.40 thf(exuE3e,definition,
% 2.46/1.40 ( exuE3e
% 2.46/1.40 = ( ! [Xphi: $i > $o] :
% 2.46/1.40 ( ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setext_type,type,
% 2.46/1.40 setext: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setext,definition,
% 2.46/1.40 ( setext
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ Xx @ B ) )
% 2.46/1.40 => ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 => ( in @ Xx @ A ) )
% 2.46/1.40 => ( A = B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptyI_type,type,
% 2.46/1.40 emptyI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptyI,definition,
% 2.46/1.40 ( emptyI
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ~ ( in @ Xx @ A )
% 2.46/1.40 => ( A = emptyset ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(noeltsimpempty_type,type,
% 2.46/1.40 noeltsimpempty: $o ).
% 2.46/1.40
% 2.46/1.40 thf(noeltsimpempty,definition,
% 2.46/1.40 ( noeltsimpempty
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ~ ( in @ Xx @ A )
% 2.46/1.40 => ( A = emptyset ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setbeta_type,type,
% 2.46/1.40 setbeta: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setbeta,definition,
% 2.46/1.40 ( setbeta
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ( in @ Xx
% 2.46/1.40 @ ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.46/1.40 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(nonempty_type,type,
% 2.46/1.40 nonempty: $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(nonempty,definition,
% 2.46/1.40 ( nonempty
% 2.46/1.40 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(nonemptyE1_type,type,
% 2.46/1.40 nonemptyE1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(nonemptyE1,definition,
% 2.46/1.40 ( nonemptyE1
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ( ( nonempty @ A )
% 2.46/1.40 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(nonemptyI_type,type,
% 2.46/1.40 nonemptyI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(nonemptyI,definition,
% 2.46/1.40 ( nonemptyI
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ( Xphi @ Xx )
% 2.46/1.40 => ( nonempty
% 2.46/1.40 @ ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(nonemptyI1_type,type,
% 2.46/1.40 nonemptyI1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(nonemptyI1,definition,
% 2.46/1.40 ( nonemptyI1
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.46/1.40 => ( nonempty @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinIL_type,type,
% 2.46/1.40 setadjoinIL: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinIL,definition,
% 2.46/1.40 ( setadjoinIL
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptyinunitempty_type,type,
% 2.46/1.40 emptyinunitempty: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptyinunitempty,definition,
% 2.46/1.40 ( emptyinunitempty
% 2.46/1.40 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinIR_type,type,
% 2.46/1.40 setadjoinIR: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinIR,definition,
% 2.46/1.40 ( setadjoinIR
% 2.46/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ A )
% 2.46/1.40 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinE_type,type,
% 2.46/1.40 setadjoinE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinE,definition,
% 2.46/1.40 ( setadjoinE
% 2.46/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.46/1.40 => ! [Xphi: $o] :
% 2.46/1.40 ( ( ( Xy = Xx )
% 2.46/1.40 => Xphi )
% 2.46/1.40 => ( ( ( in @ Xy @ A )
% 2.46/1.40 => Xphi )
% 2.46/1.40 => Xphi ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinOr_type,type,
% 2.46/1.40 setadjoinOr: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinOr,definition,
% 2.46/1.40 ( setadjoinOr
% 2.46/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.46/1.40 => ( ( Xy = Xx )
% 2.46/1.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setoftrueEq_type,type,
% 2.46/1.40 setoftrueEq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setoftrueEq,definition,
% 2.46/1.40 ( setoftrueEq
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ( ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xx: $i] : $true )
% 2.46/1.40 = A ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(powersetI_type,type,
% 2.46/1.40 powersetI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(powersetI,definition,
% 2.46/1.40 ( powersetI
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 => ( in @ Xx @ A ) )
% 2.46/1.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptyinPowerset_type,type,
% 2.46/1.40 emptyinPowerset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptyinPowerset,definition,
% 2.46/1.40 ( emptyinPowerset
% 2.46/1.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptyInPowerset_type,type,
% 2.46/1.40 emptyInPowerset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptyInPowerset,definition,
% 2.46/1.40 ( emptyInPowerset
% 2.46/1.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(powersetE_type,type,
% 2.46/1.40 powersetE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(powersetE,definition,
% 2.46/1.40 ( powersetE
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.46/1.40 => ( ( in @ Xx @ B )
% 2.46/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setunionI_type,type,
% 2.46/1.40 setunionI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setunionI,definition,
% 2.46/1.40 ( setunionI
% 2.46/1.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 => ( ( in @ B @ A )
% 2.46/1.40 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setunionE_type,type,
% 2.46/1.40 setunionE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setunionE,definition,
% 2.46/1.40 ( setunionE
% 2.46/1.40 = ( ! [A: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.46/1.40 => ! [Xphi: $o] :
% 2.46/1.40 ( ! [B: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 => ( ( in @ B @ A )
% 2.46/1.40 => Xphi ) )
% 2.46/1.40 => Xphi ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(subPowSU_type,type,
% 2.46/1.40 subPowSU: $o ).
% 2.46/1.40
% 2.46/1.40 thf(subPowSU,definition,
% 2.46/1.40 ( subPowSU
% 2.46/1.40 = ( ! [A: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(exuE2_type,type,
% 2.46/1.40 exuE2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(exuE2,definition,
% 2.46/1.40 ( exuE2
% 2.46/1.40 = ( ! [Xphi: $i > $o] :
% 2.46/1.40 ( ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 => ? [Xx: $i] :
% 2.46/1.40 ! [Xy: $i] :
% 2.46/1.40 ( ( Xphi @ Xy )
% 2.46/1.40 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(nonemptyImpWitness_type,type,
% 2.46/1.40 nonemptyImpWitness: $o ).
% 2.46/1.40
% 2.46/1.40 thf(nonemptyImpWitness,definition,
% 2.46/1.40 ( nonemptyImpWitness
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ( ( nonempty @ A )
% 2.46/1.40 => ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & $true ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(uniqinunit_type,type,
% 2.46/1.40 uniqinunit: $o ).
% 2.46/1.40
% 2.46/1.40 thf(uniqinunit,definition,
% 2.46/1.40 ( uniqinunit
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.46/1.40 => ( Xx = Xy ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(notinsingleton_type,type,
% 2.46/1.40 notinsingleton: $o ).
% 2.46/1.40
% 2.46/1.40 thf(notinsingleton,definition,
% 2.46/1.40 ( notinsingleton
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xx != Xy )
% 2.46/1.40 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(eqinunit_type,type,
% 2.46/1.40 eqinunit: $o ).
% 2.46/1.40
% 2.46/1.40 thf(eqinunit,definition,
% 2.46/1.40 ( eqinunit
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xx = Xy )
% 2.46/1.40 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(singletonsswitch_type,type,
% 2.46/1.40 singletonsswitch: $o ).
% 2.46/1.40
% 2.46/1.40 thf(singletonsswitch,definition,
% 2.46/1.40 ( singletonsswitch
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.46/1.40 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(upairsetE_type,type,
% 2.46/1.40 upairsetE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(upairsetE,definition,
% 2.46/1.40 ( upairsetE
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.46/1.40 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.46/1.40 => ( ( Xz = Xx )
% 2.46/1.40 | ( Xz = Xy ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(upairsetIL_type,type,
% 2.46/1.40 upairsetIL: $o ).
% 2.46/1.40
% 2.46/1.40 thf(upairsetIL,definition,
% 2.46/1.40 ( upairsetIL
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(upairsetIR_type,type,
% 2.46/1.40 upairsetIR: $o ).
% 2.46/1.40
% 2.46/1.40 thf(upairsetIR,definition,
% 2.46/1.40 ( upairsetIR
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptyE1_type,type,
% 2.46/1.40 emptyE1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptyE1,definition,
% 2.46/1.40 ( emptyE1
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ( Xphi @ Xx ) )
% 2.46/1.40 => ( ( ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 = emptyset )
% 2.46/1.40 => $false ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(vacuousDall_type,type,
% 2.46/1.40 vacuousDall: $o ).
% 2.46/1.40
% 2.46/1.40 thf(vacuousDall,definition,
% 2.46/1.40 ( vacuousDall
% 2.46/1.40 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ emptyset )
% 2.46/1.40 => ( Xphi @ Xx ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(quantDeMorgan1_type,type,
% 2.46/1.40 quantDeMorgan1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(quantDeMorgan1,definition,
% 2.46/1.40 ( quantDeMorgan1
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ~ ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( Xphi @ Xx ) )
% 2.46/1.40 => ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(quantDeMorgan2_type,type,
% 2.46/1.40 quantDeMorgan2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(quantDeMorgan2,definition,
% 2.46/1.40 ( quantDeMorgan2
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ~ ( Xphi @ Xx ) )
% 2.46/1.40 => ~ ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(quantDeMorgan3_type,type,
% 2.46/1.40 quantDeMorgan3: $o ).
% 2.46/1.40
% 2.46/1.40 thf(quantDeMorgan3,definition,
% 2.46/1.40 ( quantDeMorgan3
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ~ ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ( Xphi @ Xx ) )
% 2.46/1.40 => ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(quantDeMorgan4_type,type,
% 2.46/1.40 quantDeMorgan4: $o ).
% 2.46/1.40
% 2.46/1.40 thf(quantDeMorgan4,definition,
% 2.46/1.40 ( quantDeMorgan4
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ~ ( Xphi @ Xx ) )
% 2.46/1.40 => ~ ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(prop2setI_type,type,
% 2.46/1.40 prop2setI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(prop2setI,definition,
% 2.46/1.40 ( prop2setI
% 2.46/1.40 = ( ! [Xphi: $o] :
% 2.46/1.40 ( Xphi
% 2.46/1.40 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(set2prop_type,type,
% 2.46/1.40 set2prop: $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(prop2set2propI_type,type,
% 2.46/1.40 prop2set2propI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(prop2set2propI,definition,
% 2.46/1.40 ( prop2set2propI
% 2.46/1.40 = ( ! [Xphi: $o] :
% 2.46/1.40 ( Xphi
% 2.46/1.40 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(notdexE_type,type,
% 2.46/1.40 notdexE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(notdexE,definition,
% 2.46/1.40 ( notdexE
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ~ ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ( Xphi @ Xx ) )
% 2.46/1.40 => ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(notdallE_type,type,
% 2.46/1.40 notdallE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(notdallE,definition,
% 2.46/1.40 ( notdallE
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ~ ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( Xphi @ Xx ) )
% 2.46/1.40 => ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(exuI1_type,type,
% 2.46/1.40 exuI1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(exuI1,definition,
% 2.46/1.40 ( exuI1
% 2.46/1.40 = ( ! [Xphi: $i > $o] :
% 2.46/1.40 ( ? [Xx: $i] :
% 2.46/1.40 ( ( Xphi @ Xx )
% 2.46/1.40 & ! [Xy: $i] :
% 2.46/1.40 ( ( Xphi @ Xy )
% 2.46/1.40 => ( Xx = Xy ) ) )
% 2.46/1.40 => ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(exuI3_type,type,
% 2.46/1.40 exuI3: $o ).
% 2.46/1.40
% 2.46/1.40 thf(exuI3,definition,
% 2.46/1.40 ( exuI3
% 2.46/1.40 = ( ! [Xphi: $i > $o] :
% 2.46/1.40 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.46/1.40 => ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xphi @ Xx )
% 2.46/1.40 => ( ( Xphi @ Xy )
% 2.46/1.40 => ( Xx = Xy ) ) )
% 2.46/1.40 => ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(exuI2_type,type,
% 2.46/1.40 exuI2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(exuI2,definition,
% 2.46/1.40 ( exuI2
% 2.46/1.40 = ( ! [Xphi: $i > $o] :
% 2.46/1.40 ( ? [Xx: $i] :
% 2.46/1.40 ! [Xy: $i] :
% 2.46/1.40 ( ( Xphi @ Xy )
% 2.46/1.40 <=> ( Xy = Xx ) )
% 2.46/1.40 => ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(inCongP_type,type,
% 2.46/1.40 inCongP: $o ).
% 2.46/1.40
% 2.46/1.40 thf(inCongP,definition,
% 2.46/1.40 ( inCongP
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( A = B )
% 2.46/1.40 => ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xx = Xy )
% 2.46/1.40 => ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(in__Cong_type,type,
% 2.46/1.40 in__Cong: $o ).
% 2.46/1.40
% 2.46/1.40 thf(in__Cong,definition,
% 2.46/1.40 ( in__Cong
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( A = B )
% 2.46/1.40 => ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xx = Xy )
% 2.46/1.40 => ( ( in @ Xx @ A )
% 2.46/1.40 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(exuE3u_type,type,
% 2.46/1.40 exuE3u: $o ).
% 2.46/1.40
% 2.46/1.40 thf(exuE3u,definition,
% 2.46/1.40 ( exuE3u
% 2.46/1.40 = ( ! [Xphi: $i > $o] :
% 2.46/1.40 ( ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 => ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xphi @ Xx )
% 2.46/1.40 => ( ( Xphi @ Xy )
% 2.46/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(exu__Cong_type,type,
% 2.46/1.40 exu__Cong: $o ).
% 2.46/1.40
% 2.46/1.40 thf(exu__Cong,definition,
% 2.46/1.40 ( exu__Cong
% 2.46/1.40 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.46/1.40 ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xx = Xy )
% 2.46/1.40 => ( ( Xphi @ Xx )
% 2.46/1.40 <=> ( Xpsi @ Xy ) ) )
% 2.46/1.40 => ( ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 <=> ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptyset__Cong_type,type,
% 2.46/1.40 emptyset__Cong: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptyset__Cong,definition,
% 2.46/1.40 ( emptyset__Cong
% 2.46/1.40 = ( emptyset = emptyset ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoin__Cong_type,type,
% 2.46/1.40 setadjoin__Cong: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoin__Cong,definition,
% 2.46/1.40 ( setadjoin__Cong
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xx = Xy )
% 2.46/1.40 => ! [Xz: $i,Xu: $i] :
% 2.46/1.40 ( ( Xz = Xu )
% 2.46/1.40 => ( ( setadjoin @ Xx @ Xz )
% 2.46/1.40 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(powerset__Cong_type,type,
% 2.46/1.40 powerset__Cong: $o ).
% 2.46/1.40
% 2.46/1.40 thf(powerset__Cong,definition,
% 2.46/1.40 ( powerset__Cong
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( A = B )
% 2.46/1.40 => ( ( powerset @ A )
% 2.46/1.40 = ( powerset @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setunion__Cong_type,type,
% 2.46/1.40 setunion__Cong: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setunion__Cong,definition,
% 2.46/1.40 ( setunion__Cong
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( A = B )
% 2.46/1.40 => ( ( setunion @ A )
% 2.46/1.40 = ( setunion @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(omega__Cong_type,type,
% 2.46/1.40 omega__Cong: $o ).
% 2.46/1.40
% 2.46/1.40 thf(omega__Cong,definition,
% 2.46/1.40 ( omega__Cong
% 2.46/1.40 = ( omega = omega ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(exuEu_type,type,
% 2.46/1.40 exuEu: $o ).
% 2.46/1.40
% 2.46/1.40 thf(exuEu,definition,
% 2.46/1.40 ( exuEu
% 2.46/1.40 = ( ! [Xphi: $i > $o] :
% 2.46/1.40 ( ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 => ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xphi @ Xx )
% 2.46/1.40 => ( ( Xphi @ Xy )
% 2.46/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(descr__Cong_type,type,
% 2.46/1.40 descr__Cong: $o ).
% 2.46/1.40
% 2.46/1.40 thf(descr__Cong,definition,
% 2.46/1.40 ( descr__Cong
% 2.46/1.40 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.46/1.40 ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xx = Xy )
% 2.46/1.40 => ( ( Xphi @ Xx )
% 2.46/1.40 <=> ( Xpsi @ Xy ) ) )
% 2.46/1.40 => ( ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 => ( ( exu
% 2.46/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.46/1.40 => ( ( descr
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 = ( descr
% 2.46/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(dsetconstr__Cong_type,type,
% 2.46/1.40 dsetconstr__Cong: $o ).
% 2.46/1.40
% 2.46/1.40 thf(dsetconstr__Cong,definition,
% 2.46/1.40 ( dsetconstr__Cong
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( A = B )
% 2.46/1.40 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( ( Xx = Xy )
% 2.46/1.40 => ( ( Xphi @ Xx )
% 2.46/1.40 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.46/1.40 => ( ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 = ( dsetconstr @ B
% 2.46/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(subset_type,type,
% 2.46/1.40 subset: $i > $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(disjoint_type,type,
% 2.46/1.40 disjoint: $i > $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(setsmeet_type,type,
% 2.46/1.40 setsmeet: $i > $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(subsetI1_type,type,
% 2.46/1.40 subsetI1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(subsetI1,definition,
% 2.46/1.40 ( subsetI1
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ Xx @ B ) )
% 2.46/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(eqimpsubset2_type,type,
% 2.46/1.40 eqimpsubset2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(eqimpsubset2,definition,
% 2.46/1.40 ( eqimpsubset2
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( A = B )
% 2.46/1.40 => ( subset @ B @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(eqimpsubset1_type,type,
% 2.46/1.40 eqimpsubset1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(eqimpsubset1,definition,
% 2.46/1.40 ( eqimpsubset1
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( A = B )
% 2.46/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(subsetI2_type,type,
% 2.46/1.40 subsetI2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(subsetI2,definition,
% 2.46/1.40 ( subsetI2
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ Xx @ B ) )
% 2.46/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(emptysetsubset_type,type,
% 2.46/1.40 emptysetsubset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(emptysetsubset,definition,
% 2.46/1.40 ( emptysetsubset
% 2.46/1.40 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(subsetE_type,type,
% 2.46/1.40 subsetE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(subsetE,definition,
% 2.46/1.40 ( subsetE
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( subset @ A @ B )
% 2.46/1.40 => ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(subsetE2_type,type,
% 2.46/1.40 subsetE2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(subsetE2,definition,
% 2.46/1.40 ( subsetE2
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( subset @ A @ B )
% 2.46/1.40 => ( ~ ( in @ Xx @ B )
% 2.46/1.40 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(notsubsetI_type,type,
% 2.46/1.40 notsubsetI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(notsubsetI,definition,
% 2.46/1.40 ( notsubsetI
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ~ ( in @ Xx @ B )
% 2.46/1.40 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(notequalI1_type,type,
% 2.46/1.40 notequalI1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(notequalI1,definition,
% 2.46/1.40 ( notequalI1
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ~ ( subset @ A @ B )
% 2.46/1.40 => ( A != B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(notequalI2_type,type,
% 2.46/1.40 notequalI2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(notequalI2,definition,
% 2.46/1.40 ( notequalI2
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ~ ( in @ Xx @ B )
% 2.46/1.40 => ( A != B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(subsetRefl_type,type,
% 2.46/1.40 subsetRefl: $o ).
% 2.46/1.40
% 2.46/1.40 thf(subsetRefl,definition,
% 2.46/1.40 ( subsetRefl
% 2.46/1.40 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(subsetTrans_type,type,
% 2.46/1.40 subsetTrans: $o ).
% 2.46/1.40
% 2.46/1.40 thf(subsetTrans,definition,
% 2.46/1.40 ( subsetTrans
% 2.46/1.40 = ( ! [A: $i,B: $i,C: $i] :
% 2.46/1.40 ( ( subset @ A @ B )
% 2.46/1.40 => ( ( subset @ B @ C )
% 2.46/1.40 => ( subset @ A @ C ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinSub_type,type,
% 2.46/1.40 setadjoinSub: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinSub,definition,
% 2.46/1.40 ( setadjoinSub
% 2.46/1.40 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinSub2_type,type,
% 2.46/1.40 setadjoinSub2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setadjoinSub2,definition,
% 2.46/1.40 ( setadjoinSub2
% 2.46/1.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.46/1.40 ( ( subset @ A @ B )
% 2.46/1.40 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(subset2powerset_type,type,
% 2.46/1.40 subset2powerset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(subset2powerset,definition,
% 2.46/1.40 ( subset2powerset
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( subset @ A @ B )
% 2.46/1.40 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setextsub_type,type,
% 2.46/1.40 setextsub: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setextsub,definition,
% 2.46/1.40 ( setextsub
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( subset @ A @ B )
% 2.46/1.40 => ( ( subset @ B @ A )
% 2.46/1.40 => ( A = B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(subsetemptysetimpeq_type,type,
% 2.46/1.40 subsetemptysetimpeq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(subsetemptysetimpeq,definition,
% 2.46/1.40 ( subsetemptysetimpeq
% 2.46/1.40 = ( ! [A: $i] :
% 2.46/1.40 ( ( subset @ A @ emptyset )
% 2.46/1.40 => ( A = emptyset ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(powersetI1_type,type,
% 2.46/1.40 powersetI1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(powersetI1,definition,
% 2.46/1.40 ( powersetI1
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( subset @ B @ A )
% 2.46/1.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(powersetE1_type,type,
% 2.46/1.40 powersetE1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(powersetE1,definition,
% 2.46/1.40 ( powersetE1
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.46/1.40 => ( subset @ B @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(inPowerset_type,type,
% 2.46/1.40 inPowerset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(inPowerset,definition,
% 2.46/1.40 ( inPowerset
% 2.46/1.40 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(powersetsubset_type,type,
% 2.46/1.40 powersetsubset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(powersetsubset,definition,
% 2.46/1.40 ( powersetsubset
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( subset @ A @ B )
% 2.46/1.40 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(sepInPowerset_type,type,
% 2.46/1.40 sepInPowerset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(sepInPowerset,definition,
% 2.46/1.40 ( sepInPowerset
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( in
% 2.46/1.40 @ ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 @ ( powerset @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(sepSubset_type,type,
% 2.46/1.40 sepSubset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(sepSubset,definition,
% 2.46/1.40 ( sepSubset
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( subset
% 2.46/1.40 @ ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 @ A ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binunion_type,type,
% 2.46/1.40 binunion: $i > $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(binunionIL_type,type,
% 2.46/1.40 binunionIL: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binunionIL,definition,
% 2.46/1.40 ( binunionIL
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(upairset2IR_type,type,
% 2.46/1.40 upairset2IR: $o ).
% 2.46/1.40
% 2.46/1.40 thf(upairset2IR,definition,
% 2.46/1.40 ( upairset2IR
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binunionIR_type,type,
% 2.46/1.40 binunionIR: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binunionIR,definition,
% 2.46/1.40 ( binunionIR
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binunionEcases_type,type,
% 2.46/1.40 binunionEcases: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binunionEcases,definition,
% 2.46/1.40 ( binunionEcases
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.46/1.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.46/1.40 => ( ( ( in @ Xx @ A )
% 2.46/1.40 => Xphi )
% 2.46/1.40 => ( ( ( in @ Xx @ B )
% 2.46/1.40 => Xphi )
% 2.46/1.40 => Xphi ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binunionE_type,type,
% 2.46/1.40 binunionE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binunionE,definition,
% 2.46/1.40 ( binunionE
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.46/1.40 => ( ( in @ Xx @ A )
% 2.46/1.40 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binunionLsub_type,type,
% 2.46/1.40 binunionLsub: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binunionLsub,definition,
% 2.46/1.40 ( binunionLsub
% 2.46/1.40 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binunionRsub_type,type,
% 2.46/1.40 binunionRsub: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binunionRsub,definition,
% 2.46/1.40 ( binunionRsub
% 2.46/1.40 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersect_type,type,
% 2.46/1.40 binintersect: $i > $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectI_type,type,
% 2.46/1.40 binintersectI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectI,definition,
% 2.46/1.40 ( binintersectI
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ( in @ Xx @ B )
% 2.46/1.40 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset5_type,type,
% 2.46/1.40 binintersectSubset5: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset5,definition,
% 2.46/1.40 ( binintersectSubset5
% 2.46/1.40 = ( ! [A: $i,B: $i,C: $i] :
% 2.46/1.40 ( ( subset @ C @ A )
% 2.46/1.40 => ( ( subset @ C @ B )
% 2.46/1.40 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectEL_type,type,
% 2.46/1.40 binintersectEL: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectEL,definition,
% 2.46/1.40 ( binintersectEL
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.46/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectLsub_type,type,
% 2.46/1.40 binintersectLsub: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectLsub,definition,
% 2.46/1.40 ( binintersectLsub
% 2.46/1.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset2_type,type,
% 2.46/1.40 binintersectSubset2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset2,definition,
% 2.46/1.40 ( binintersectSubset2
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( subset @ A @ B )
% 2.46/1.40 => ( ( binintersect @ A @ B )
% 2.46/1.40 = A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset3_type,type,
% 2.46/1.40 binintersectSubset3: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset3,definition,
% 2.46/1.40 ( binintersectSubset3
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( ( binintersect @ A @ B )
% 2.46/1.40 = B )
% 2.46/1.40 => ( subset @ B @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectER_type,type,
% 2.46/1.40 binintersectER: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectER,definition,
% 2.46/1.40 ( binintersectER
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.46/1.40 => ( in @ Xx @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(disjointsetsI1_type,type,
% 2.46/1.40 disjointsetsI1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(disjointsetsI1,definition,
% 2.46/1.40 ( disjointsetsI1
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ~ ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ( in @ Xx @ B ) )
% 2.46/1.40 => ( ( binintersect @ A @ B )
% 2.46/1.40 = emptyset ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectRsub_type,type,
% 2.46/1.40 binintersectRsub: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectRsub,definition,
% 2.46/1.40 ( binintersectRsub
% 2.46/1.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset4_type,type,
% 2.46/1.40 binintersectSubset4: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset4,definition,
% 2.46/1.40 ( binintersectSubset4
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( subset @ B @ A )
% 2.46/1.40 => ( ( binintersect @ A @ B )
% 2.46/1.40 = B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset1_type,type,
% 2.46/1.40 binintersectSubset1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(binintersectSubset1,definition,
% 2.46/1.40 ( binintersectSubset1
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( ( binintersect @ A @ B )
% 2.46/1.40 = A )
% 2.46/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(bs114d_type,type,
% 2.46/1.40 bs114d: $o ).
% 2.46/1.40
% 2.46/1.40 thf(bs114d,definition,
% 2.46/1.40 ( bs114d
% 2.46/1.40 = ( ! [A: $i,B: $i,C: $i] :
% 2.46/1.40 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.46/1.40 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(regular_type,type,
% 2.46/1.40 regular: $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminus_type,type,
% 2.46/1.40 setminus: $i > $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(setminusI_type,type,
% 2.46/1.40 setminusI: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusI,definition,
% 2.46/1.40 ( setminusI
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ~ ( in @ Xx @ B )
% 2.46/1.40 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setminusEL_type,type,
% 2.46/1.40 setminusEL: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusEL,definition,
% 2.46/1.40 ( setminusEL
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.46/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setminusER_type,type,
% 2.46/1.40 setminusER: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusER,definition,
% 2.46/1.40 ( setminusER
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.46/1.40 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setminusSubset2_type,type,
% 2.46/1.40 setminusSubset2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusSubset2,definition,
% 2.46/1.40 ( setminusSubset2
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( subset @ A @ B )
% 2.46/1.40 => ( ( setminus @ A @ B )
% 2.46/1.40 = emptyset ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setminusERneg_type,type,
% 2.46/1.40 setminusERneg: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusERneg,definition,
% 2.46/1.40 ( setminusERneg
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.46/1.40 => ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setminusELneg_type,type,
% 2.46/1.40 setminusELneg: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusELneg,definition,
% 2.46/1.40 ( setminusELneg
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.46/1.40 => ( ~ ( in @ Xx @ B )
% 2.46/1.40 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setminusILneg_type,type,
% 2.46/1.40 setminusILneg: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusILneg,definition,
% 2.46/1.40 ( setminusILneg
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ~ ( in @ Xx @ A )
% 2.46/1.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setminusIRneg_type,type,
% 2.46/1.40 setminusIRneg: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusIRneg,definition,
% 2.46/1.40 ( setminusIRneg
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ B )
% 2.46/1.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setminusLsub_type,type,
% 2.46/1.40 setminusLsub: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusLsub,definition,
% 2.46/1.40 ( setminusLsub
% 2.46/1.40 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setminusSubset1_type,type,
% 2.46/1.40 setminusSubset1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setminusSubset1,definition,
% 2.46/1.40 ( setminusSubset1
% 2.46/1.40 = ( ! [A: $i,B: $i] :
% 2.46/1.40 ( ( ( setminus @ A @ B )
% 2.46/1.40 = emptyset )
% 2.46/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(symdiff_type,type,
% 2.46/1.40 symdiff: $i > $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffE_type,type,
% 2.46/1.40 symdiffE: $o ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffE,definition,
% 2.46/1.40 ( symdiffE
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.46/1.40 => ! [Xphi: $o] :
% 2.46/1.40 ( ( ( in @ Xx @ A )
% 2.46/1.40 => ( ~ ( in @ Xx @ B )
% 2.46/1.40 => Xphi ) )
% 2.46/1.40 => ( ( ~ ( in @ Xx @ A )
% 2.46/1.40 => ( ( in @ Xx @ B )
% 2.46/1.40 => Xphi ) )
% 2.46/1.40 => Xphi ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffI1_type,type,
% 2.46/1.40 symdiffI1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffI1,definition,
% 2.46/1.40 ( symdiffI1
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ~ ( in @ Xx @ B )
% 2.46/1.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffI2_type,type,
% 2.46/1.40 symdiffI2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffI2,definition,
% 2.46/1.40 ( symdiffI2
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ~ ( in @ Xx @ A )
% 2.46/1.40 => ( ( in @ Xx @ B )
% 2.46/1.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffIneg1_type,type,
% 2.46/1.40 symdiffIneg1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffIneg1,definition,
% 2.46/1.40 ( symdiffIneg1
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ( in @ Xx @ B )
% 2.46/1.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffIneg2_type,type,
% 2.46/1.40 symdiffIneg2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(symdiffIneg2,definition,
% 2.46/1.40 ( symdiffIneg2
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ~ ( in @ Xx @ A )
% 2.46/1.40 => ( ~ ( in @ Xx @ B )
% 2.46/1.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(iskpair_type,type,
% 2.46/1.40 iskpair: $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(secondinupair_type,type,
% 2.46/1.40 secondinupair: $o ).
% 2.46/1.40
% 2.46/1.40 thf(secondinupair,definition,
% 2.46/1.40 ( secondinupair
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairIL_type,type,
% 2.46/1.40 setukpairIL: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairIL,definition,
% 2.46/1.40 ( setukpairIL
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairIR_type,type,
% 2.46/1.40 setukpairIR: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairIR,definition,
% 2.46/1.40 ( setukpairIR
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(kpairiskpair_type,type,
% 2.46/1.40 kpairiskpair: $o ).
% 2.46/1.40
% 2.46/1.40 thf(kpairiskpair,definition,
% 2.46/1.40 ( kpairiskpair
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(kpair_type,type,
% 2.46/1.40 kpair: $i > $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(kpairp_type,type,
% 2.46/1.40 kpairp: $o ).
% 2.46/1.40
% 2.46/1.40 thf(kpairp,definition,
% 2.46/1.40 ( kpairp
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprod_type,type,
% 2.46/1.40 cartprod: $i > $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(singletonsubset_type,type,
% 2.46/1.40 singletonsubset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(singletonsubset,definition,
% 2.46/1.40 ( singletonsubset
% 2.46/1.40 = ( ! [A: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(singletoninpowerset_type,type,
% 2.46/1.40 singletoninpowerset: $o ).
% 2.46/1.40
% 2.46/1.40 thf(singletoninpowerset,definition,
% 2.46/1.40 ( singletoninpowerset
% 2.46/1.40 = ( ! [A: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(singletoninpowunion_type,type,
% 2.46/1.40 singletoninpowunion: $o ).
% 2.46/1.40
% 2.46/1.40 thf(singletoninpowunion,definition,
% 2.46/1.40 ( singletoninpowunion
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(upairset2E_type,type,
% 2.46/1.40 upairset2E: $o ).
% 2.46/1.40
% 2.46/1.40 thf(upairset2E,definition,
% 2.46/1.40 ( upairset2E
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.46/1.40 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.46/1.40 => ( ( Xz = Xx )
% 2.46/1.40 | ( Xz = Xy ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(upairsubunion_type,type,
% 2.46/1.40 upairsubunion: $o ).
% 2.46/1.40
% 2.46/1.40 thf(upairsubunion,definition,
% 2.46/1.40 ( upairsubunion
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(upairinpowunion_type,type,
% 2.46/1.40 upairinpowunion: $o ).
% 2.46/1.40
% 2.46/1.40 thf(upairinpowunion,definition,
% 2.46/1.40 ( upairinpowunion
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(ubforcartprodlem1_type,type,
% 2.46/1.40 ubforcartprodlem1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(ubforcartprodlem1,definition,
% 2.46/1.40 ( ubforcartprodlem1
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(ubforcartprodlem2_type,type,
% 2.46/1.40 ubforcartprodlem2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(ubforcartprodlem2,definition,
% 2.46/1.40 ( ubforcartprodlem2
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(ubforcartprodlem3_type,type,
% 2.46/1.40 ubforcartprodlem3: $o ).
% 2.46/1.40
% 2.46/1.40 thf(ubforcartprodlem3,definition,
% 2.46/1.40 ( ubforcartprodlem3
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodpairin_type,type,
% 2.46/1.40 cartprodpairin: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodpairin,definition,
% 2.46/1.40 ( cartprodpairin
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodmempair1_type,type,
% 2.46/1.40 cartprodmempair1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodmempair1,definition,
% 2.46/1.40 ( cartprodmempair1
% 2.46/1.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.46/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.46/1.40 => ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ? [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 & ( Xu
% 2.46/1.40 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodmempair_type,type,
% 2.46/1.40 cartprodmempair: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodmempair,definition,
% 2.46/1.40 ( cartprodmempair
% 2.46/1.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.46/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.46/1.40 => ( iskpair @ Xu ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setunionE2_type,type,
% 2.46/1.40 setunionE2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setunionE2,definition,
% 2.46/1.40 ( setunionE2
% 2.46/1.40 = ( ! [A: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.46/1.40 => ? [X: $i] :
% 2.46/1.40 ( ( in @ X @ A )
% 2.46/1.40 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setunionsingleton1_type,type,
% 2.46/1.40 setunionsingleton1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setunionsingleton1,definition,
% 2.46/1.40 ( setunionsingleton1
% 2.46/1.40 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setunionsingleton2_type,type,
% 2.46/1.40 setunionsingleton2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setunionsingleton2,definition,
% 2.46/1.40 ( setunionsingleton2
% 2.46/1.40 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setunionsingleton_type,type,
% 2.46/1.40 setunionsingleton: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setunionsingleton,definition,
% 2.46/1.40 ( setunionsingleton
% 2.46/1.40 = ( ! [Xx: $i] :
% 2.46/1.40 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.46/1.40 = Xx ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(singleton_type,type,
% 2.46/1.40 singleton: $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(singleton,definition,
% 2.46/1.40 ( singleton
% 2.46/1.40 = ( ^ [A: $i] :
% 2.46/1.40 ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ( A
% 2.46/1.40 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(singletonprop_type,type,
% 2.46/1.40 singletonprop: $o ).
% 2.46/1.40
% 2.46/1.40 thf(singletonprop,definition,
% 2.46/1.40 ( singletonprop
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ A )
% 2.46/1.40 => ( ( Xphi @ Xx )
% 2.46/1.40 => ( ( Xphi @ Xy )
% 2.46/1.40 => ( Xx = Xy ) ) ) ) )
% 2.46/1.40 => ( ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ( Xphi @ Xx ) )
% 2.46/1.40 => ( singleton
% 2.46/1.40 @ ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(ex1_type,type,
% 2.46/1.40 ex1: $i > ( $i > $o ) > $o ).
% 2.46/1.40
% 2.46/1.40 thf(ex1,definition,
% 2.46/1.40 ( ex1
% 2.46/1.40 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( singleton
% 2.46/1.40 @ ( dsetconstr @ A
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(ex1E1_type,type,
% 2.46/1.40 ex1E1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(ex1E1,definition,
% 2.46/1.40 ( ex1E1
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ( ex1 @ A
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.46/1.40 => ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(ex1I_type,type,
% 2.46/1.40 ex1I: $o ).
% 2.46/1.40
% 2.46/1.40 thf(ex1I,definition,
% 2.46/1.40 ( ex1I
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ( ( Xphi @ Xx )
% 2.46/1.40 => ( ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ A )
% 2.46/1.40 => ( ( Xphi @ Xy )
% 2.46/1.40 => ( Xy = Xx ) ) )
% 2.46/1.40 => ( ex1 @ A
% 2.46/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(ex1I2_type,type,
% 2.46/1.40 ex1I2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(ex1I2,definition,
% 2.46/1.40 ( ex1I2
% 2.46/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.46/1.40 ( ! [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ A )
% 2.46/1.40 => ( ( Xphi @ Xx )
% 2.46/1.40 => ( ( Xphi @ Xy )
% 2.46/1.40 => ( Xx = Xy ) ) ) ) )
% 2.46/1.40 => ( ? [Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 & ( Xphi @ Xx ) )
% 2.46/1.40 => ( ex1 @ A
% 2.46/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(singletonsuniq_type,type,
% 2.46/1.40 singletonsuniq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(singletonsuniq,definition,
% 2.46/1.40 ( singletonsuniq
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( ( setadjoin @ Xx @ emptyset )
% 2.46/1.40 = ( setadjoin @ Xy @ emptyset ) )
% 2.46/1.40 => ( Xx = Xy ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(atmost1p_type,type,
% 2.46/1.40 atmost1p: $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(atleast2p_type,type,
% 2.46/1.40 atleast2p: $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(atmost2p_type,type,
% 2.46/1.40 atmost2p: $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(upairsetp_type,type,
% 2.46/1.40 upairsetp: $i > $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjL1_type,type,
% 2.46/1.40 setukpairinjL1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjL1,definition,
% 2.46/1.40 ( setukpairinjL1
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.46/1.40 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.46/1.40 => ( Xx = Xz ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(kfstsingleton_type,type,
% 2.46/1.40 kfstsingleton: $o ).
% 2.46/1.40
% 2.46/1.40 thf(kfstsingleton,definition,
% 2.46/1.40 ( kfstsingleton
% 2.46/1.40 = ( ! [Xu: $i] :
% 2.46/1.40 ( ( iskpair @ Xu )
% 2.46/1.40 => ( singleton
% 2.46/1.40 @ ( dsetconstr @ ( setunion @ Xu )
% 2.46/1.40 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(theprop_type,type,
% 2.46/1.40 theprop: $o ).
% 2.46/1.40
% 2.46/1.40 thf(theprop,definition,
% 2.46/1.40 ( theprop
% 2.46/1.40 = ( ! [X: $i] :
% 2.46/1.40 ( ( singleton @ X )
% 2.46/1.40 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(kfst_type,type,
% 2.46/1.40 kfst: $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(kfstpairEq_type,type,
% 2.46/1.40 kfstpairEq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(kfstpairEq,definition,
% 2.46/1.40 ( kfstpairEq
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.46/1.40 = Xx ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodfstin_type,type,
% 2.46/1.40 cartprodfstin: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodfstin,definition,
% 2.46/1.40 ( cartprodfstin
% 2.46/1.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.46/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.46/1.40 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjL2_type,type,
% 2.46/1.40 setukpairinjL2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjL2,definition,
% 2.46/1.40 ( setukpairinjL2
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.46/1.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.46/1.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.46/1.40 => ( Xx = Xz ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjL_type,type,
% 2.46/1.40 setukpairinjL: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjL,definition,
% 2.46/1.40 ( setukpairinjL
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.46/1.40 ( ( ( kpair @ Xx @ Xy )
% 2.46/1.40 = ( kpair @ Xz @ Xu ) )
% 2.46/1.40 => ( Xx = Xz ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR11_type,type,
% 2.46/1.40 setukpairinjR11: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR11,definition,
% 2.46/1.40 ( setukpairinjR11
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xx = Xy )
% 2.46/1.40 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.46/1.40 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR12_type,type,
% 2.46/1.40 setukpairinjR12: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR12,definition,
% 2.46/1.40 ( setukpairinjR12
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( Xx = Xy )
% 2.46/1.40 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.46/1.40 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR1_type,type,
% 2.46/1.40 setukpairinjR1: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR1,definition,
% 2.46/1.40 ( setukpairinjR1
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.46/1.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.46/1.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.46/1.40 => ( ( Xz = Xu )
% 2.46/1.40 => ( Xy = Xu ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(upairequniteq_type,type,
% 2.46/1.40 upairequniteq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(upairequniteq,definition,
% 2.46/1.40 ( upairequniteq
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.46/1.40 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.46/1.40 = ( setadjoin @ Xz @ emptyset ) )
% 2.46/1.40 => ( Xx = Xy ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR2_type,type,
% 2.46/1.40 setukpairinjR2: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR2,definition,
% 2.46/1.40 ( setukpairinjR2
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.46/1.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.46/1.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.46/1.40 => ( Xy = Xu ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR_type,type,
% 2.46/1.40 setukpairinjR: $o ).
% 2.46/1.40
% 2.46/1.40 thf(setukpairinjR,definition,
% 2.46/1.40 ( setukpairinjR
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.46/1.40 ( ( ( kpair @ Xx @ Xy )
% 2.46/1.40 = ( kpair @ Xz @ Xu ) )
% 2.46/1.40 => ( Xy = Xu ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(ksndsingleton_type,type,
% 2.46/1.40 ksndsingleton: $o ).
% 2.46/1.40
% 2.46/1.40 thf(ksndsingleton,definition,
% 2.46/1.40 ( ksndsingleton
% 2.46/1.40 = ( ! [Xu: $i] :
% 2.46/1.40 ( ( iskpair @ Xu )
% 2.46/1.40 => ( singleton
% 2.46/1.40 @ ( dsetconstr @ ( setunion @ Xu )
% 2.46/1.40 @ ^ [Xx: $i] :
% 2.46/1.40 ( Xu
% 2.46/1.40 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(ksnd_type,type,
% 2.46/1.40 ksnd: $i > $i ).
% 2.46/1.40
% 2.46/1.40 thf(ksndpairEq_type,type,
% 2.46/1.40 ksndpairEq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(ksndpairEq,definition,
% 2.46/1.40 ( ksndpairEq
% 2.46/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.46/1.40 = Xy ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(kpairsurjEq_type,type,
% 2.46/1.40 kpairsurjEq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(kpairsurjEq,definition,
% 2.46/1.40 ( kpairsurjEq
% 2.46/1.40 = ( ! [Xu: $i] :
% 2.46/1.40 ( ( iskpair @ Xu )
% 2.46/1.40 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.46/1.40 = Xu ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodsndin_type,type,
% 2.46/1.40 cartprodsndin: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodsndin,definition,
% 2.46/1.40 ( cartprodsndin
% 2.46/1.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.46/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.46/1.40 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodpairmemEL_type,type,
% 2.46/1.40 cartprodpairmemEL: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodpairmemEL,definition,
% 2.46/1.40 ( cartprodpairmemEL
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.46/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodpairmemER_type,type,
% 2.46/1.40 cartprodpairmemER: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodpairmemER,definition,
% 2.46/1.40 ( cartprodpairmemER
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.46/1.40 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.46/1.40 => ( in @ Xy @ B ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodmempaircEq_type,type,
% 2.46/1.40 cartprodmempaircEq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodmempaircEq,definition,
% 2.46/1.40 ( cartprodmempaircEq
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( ( kpair @ Xx @ Xy )
% 2.46/1.40 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodfstpairEq_type,type,
% 2.46/1.40 cartprodfstpairEq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodfstpairEq,definition,
% 2.46/1.40 ( cartprodfstpairEq
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.46/1.40 = Xx ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodsndpairEq_type,type,
% 2.46/1.40 cartprodsndpairEq: $o ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodsndpairEq,definition,
% 2.46/1.40 ( cartprodsndpairEq
% 2.46/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.46/1.40 ( ( in @ Xx @ A )
% 2.46/1.40 => ! [Xy: $i] :
% 2.46/1.40 ( ( in @ Xy @ B )
% 2.46/1.40 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.46/1.40 = Xy ) ) ) ) ) ).
% 2.46/1.40
% 2.46/1.40 thf(cartprodpairsurjEq,conjecture,
% 2.46/1.40 ( setextAx
% 2.46/1.40 => ( emptysetAx
% 2.46/1.40 => ( setadjoinAx
% 2.46/1.40 => ( powersetAx
% 2.46/1.40 => ( setunionAx
% 2.46/1.40 => ( omega0Ax
% 2.46/1.40 => ( omegaSAx
% 2.46/1.40 => ( omegaIndAx
% 2.46/1.40 => ( replAx
% 2.46/1.40 => ( foundationAx
% 2.46/1.40 => ( wellorderingAx
% 2.46/1.40 => ( descrp
% 2.46/1.40 => ( dsetconstrI
% 2.46/1.40 => ( dsetconstrEL
% 2.46/1.40 => ( dsetconstrER
% 2.46/1.40 => ( exuE1
% 2.46/1.40 => ( prop2setE
% 2.46/1.40 => ( emptysetE
% 2.46/1.40 => ( emptysetimpfalse
% 2.46/1.40 => ( notinemptyset
% 2.46/1.40 => ( exuE3e
% 2.46/1.40 => ( setext
% 2.46/1.40 => ( emptyI
% 2.46/1.40 => ( noeltsimpempty
% 2.46/1.40 => ( setbeta
% 2.46/1.40 => ( nonemptyE1
% 2.46/1.40 => ( nonemptyI
% 2.46/1.40 => ( nonemptyI1
% 2.46/1.40 => ( setadjoinIL
% 2.46/1.40 => ( emptyinunitempty
% 2.46/1.40 => ( setadjoinIR
% 2.46/1.40 => ( setadjoinE
% 2.46/1.40 => ( setadjoinOr
% 2.46/1.40 => ( setoftrueEq
% 2.46/1.40 => ( powersetI
% 2.46/1.40 => ( emptyinPowerset
% 2.46/1.40 => ( emptyInPowerset
% 2.46/1.40 => ( powersetE
% 2.46/1.40 => ( setunionI
% 2.46/1.40 => ( setunionE
% 2.46/1.40 => ( subPowSU
% 2.46/1.40 => ( exuE2
% 2.46/1.40 => ( nonemptyImpWitness
% 2.46/1.40 => ( uniqinunit
% 2.46/1.40 => ( notinsingleton
% 2.46/1.40 => ( eqinunit
% 2.46/1.40 => ( singletonsswitch
% 2.46/1.40 => ( upairsetE
% 2.46/1.40 => ( upairsetIL
% 2.46/1.40 => ( upairsetIR
% 2.46/1.40 => ( emptyE1
% 2.46/1.40 => ( vacuousDall
% 2.46/1.40 => ( quantDeMorgan1
% 2.46/1.40 => ( quantDeMorgan2
% 2.46/1.40 => ( quantDeMorgan3
% 2.46/1.40 => ( quantDeMorgan4
% 2.46/1.40 => ( prop2setI
% 2.46/1.40 => ( prop2set2propI
% 2.46/1.40 => ( notdexE
% 2.46/1.40 => ( notdallE
% 2.46/1.40 => ( exuI1
% 2.46/1.40 => ( exuI3
% 2.46/1.40 => ( exuI2
% 2.46/1.40 => ( inCongP
% 2.46/1.40 => ( in__Cong
% 2.46/1.40 => ( exuE3u
% 2.46/1.40 => ( exu__Cong
% 2.46/1.40 => ( emptyset__Cong
% 2.46/1.40 => ( setadjoin__Cong
% 2.46/1.40 => ( powerset__Cong
% 2.46/1.40 => ( setunion__Cong
% 2.46/1.40 => ( omega__Cong
% 2.46/1.40 => ( exuEu
% 2.46/1.40 => ( descr__Cong
% 2.46/1.40 => ( dsetconstr__Cong
% 2.46/1.40 => ( subsetI1
% 2.46/1.40 => ( eqimpsubset2
% 2.46/1.40 => ( eqimpsubset1
% 2.46/1.40 => ( subsetI2
% 2.46/1.40 => ( emptysetsubset
% 2.46/1.40 => ( subsetE
% 2.46/1.40 => ( subsetE2
% 2.46/1.40 => ( notsubsetI
% 2.46/1.40 => ( notequalI1
% 2.46/1.40 => ( notequalI2
% 2.46/1.40 => ( subsetRefl
% 2.46/1.40 => ( subsetTrans
% 2.46/1.40 => ( setadjoinSub
% 2.46/1.40 => ( setadjoinSub2
% 2.46/1.40 => ( subset2powerset
% 2.46/1.40 => ( setextsub
% 2.46/1.40 => ( subsetemptysetimpeq
% 2.46/1.40 => ( powersetI1
% 2.46/1.40 => ( powersetE1
% 2.46/1.40 => ( inPowerset
% 2.46/1.40 => ( powersetsubset
% 2.46/1.40 => ( sepInPowerset
% 2.46/1.40 => ( sepSubset
% 2.46/1.40 => ( binunionIL
% 2.46/1.40 => ( upairset2IR
% 2.46/1.40 => ( binunionIR
% 2.46/1.40 => ( binunionEcases
% 2.46/1.40 => ( binunionE
% 2.46/1.40 => ( binunionLsub
% 2.46/1.40 => ( binunionRsub
% 2.46/1.40 => ( binintersectI
% 2.46/1.40 => ( binintersectSubset5
% 2.46/1.40 => ( binintersectEL
% 2.46/1.40 => ( binintersectLsub
% 2.46/1.40 => ( binintersectSubset2
% 2.46/1.40 => ( binintersectSubset3
% 2.46/1.40 => ( binintersectER
% 2.46/1.40 => ( disjointsetsI1
% 2.46/1.40 => ( binintersectRsub
% 2.46/1.40 => ( binintersectSubset4
% 2.46/1.40 => ( binintersectSubset1
% 2.46/1.40 => ( bs114d
% 2.46/1.40 => ( setminusI
% 2.46/1.40 => ( setminusEL
% 2.46/1.40 => ( setminusER
% 2.46/1.40 => ( setminusSubset2
% 2.46/1.40 => ( setminusERneg
% 2.46/1.40 => ( setminusELneg
% 2.46/1.40 => ( setminusILneg
% 2.46/1.40 => ( setminusIRneg
% 2.46/1.40 => ( setminusLsub
% 2.46/1.40 => ( setminusSubset1
% 2.46/1.40 => ( symdiffE
% 2.46/1.40 => ( symdiffI1
% 2.46/1.40 => ( symdiffI2
% 2.46/1.40 => ( symdiffIneg1
% 2.46/1.40 => ( symdiffIneg2
% 2.46/1.40 => ( secondinupair
% 2.46/1.40 => ( setukpairIL
% 2.46/1.40 => ( setukpairIR
% 2.46/1.40 => ( kpairiskpair
% 2.46/1.40 => ( kpairp
% 2.46/1.40 => ( singletonsubset
% 2.46/1.40 => ( singletoninpowerset
% 2.46/1.40 => ( singletoninpowunion
% 2.46/1.40 => ( upairset2E
% 2.46/1.40 => ( upairsubunion
% 2.46/1.40 => ( upairinpowunion
% 2.46/1.40 => ( ubforcartprodlem1
% 2.46/1.40 => ( ubforcartprodlem2
% 2.46/1.40 => ( ubforcartprodlem3
% 2.46/1.40 => ( cartprodpairin
% 2.46/1.40 => ( cartprodmempair1
% 2.46/1.40 => ( cartprodmempair
% 2.46/1.40 => ( setunionE2
% 2.46/1.40 => ( setunionsingleton1
% 2.46/1.40 => ( setunionsingleton2
% 2.46/1.40 => ( setunionsingleton
% 2.46/1.40 => ( singletonprop
% 2.46/1.40 => ( ex1E1
% 2.46/1.40 => ( ex1I
% 2.46/1.40 => ( ex1I2
% 2.46/1.40 => ( singletonsuniq
% 2.46/1.40 => ( setukpairinjL1
% 2.46/1.40 => ( kfstsingleton
% 2.46/1.40 => ( theprop
% 2.46/1.40 => ( kfstpairEq
% 2.46/1.40 => ( cartprodfstin
% 2.46/1.40 => ( setukpairinjL2
% 2.46/1.40 => ( setukpairinjL
% 2.46/1.40 => ( setukpairinjR11
% 2.46/1.40 => ( setukpairinjR12
% 2.46/1.40 => ( setukpairinjR1
% 2.46/1.40 => ( upairequniteq
% 2.46/1.40 => ( setukpairinjR2
% 2.46/1.40 => ( setukpairinjR
% 2.46/1.40 => ( ksndsingleton
% 2.46/1.40 => ( ksndpairEq
% 2.46/1.40 => ( kpairsurjEq
% 2.46/1.40 => ( cartprodsndin
% 2.46/1.40 => ( cartprodpairmemEL
% 2.46/1.40 => ( cartprodpairmemER
% 2.46/1.40 => ( cartprodmempaircEq
% 2.46/1.40 => ( cartprodfstpairEq
% 2.46/1.40 => ( cartprodsndpairEq
% 2.46/1.40 => ! [A: $i,B: $i,Xu: $i] :
% 2.46/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.46/1.40 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.46/1.40 = Xu ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.46/1.40 ------------------------
% 2.46/1.42 % (4689)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/183Mi)
% 2.46/1.43 % (4690)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_4583 for (2999ds/4Mi)
% 2.46/1.43 % (4693)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_4583 for (2999ds/2Mi)
% 2.46/1.43 % (4691)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_4583 for (2999ds/27Mi)
% 2.46/1.43 % (4694)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_4583 for (2999ds/275Mi)
% 2.46/1.43 % (4692)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/2Mi)
% 2.46/1.43 % (4695)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/18Mi)
% 2.46/1.43 % (4692)Instruction limit reached!
% 2.46/1.43 % (4692)------------------------------
% 2.46/1.43 % (4692)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.43 % (4692)Termination reason: Unknown
% 2.46/1.43 % (4693)Instruction limit reached!
% 2.46/1.43 % (4693)------------------------------
% 2.46/1.43 % (4693)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.43 % (4693)Termination reason: Unknown
% 2.46/1.43 % (4693)Termination phase: shuffling
% 2.46/1.43
% 2.46/1.43 % (4693)Memory used [KB]: 1279
% 2.46/1.43 % (4693)Time elapsed: 0.003 s
% 2.46/1.43 % (4693)Instructions burned: 3 (million)
% 2.46/1.43 % (4693)------------------------------
% 2.46/1.43 % (4693)------------------------------
% 2.46/1.43 % (4692)Termination phase: shuffling
% 2.46/1.43
% 2.46/1.43 % (4692)Memory used [KB]: 1279
% 2.46/1.43 % (4692)Time elapsed: 0.003 s
% 2.46/1.43 % (4692)Instructions burned: 3 (million)
% 2.46/1.43 % (4692)------------------------------
% 2.46/1.43 % (4692)------------------------------
% 2.46/1.43 % (4690)Instruction limit reached!
% 2.46/1.43 % (4690)------------------------------
% 2.46/1.43 % (4690)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.43 % (4690)Termination reason: Unknown
% 2.46/1.43 % (4690)Termination phase: shuffling
% 2.46/1.43
% 2.46/1.43 % (4690)Memory used [KB]: 1407
% 2.46/1.43 % (4690)Time elapsed: 0.004 s
% 2.46/1.43 % (4690)Instructions burned: 6 (million)
% 2.46/1.43 % (4690)------------------------------
% 2.46/1.43 % (4690)------------------------------
% 2.46/1.44 % (4695)Instruction limit reached!
% 2.46/1.44 % (4695)------------------------------
% 2.46/1.44 % (4695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.44 % (4695)Termination reason: Unknown
% 2.46/1.44 % (4695)Termination phase: Property scanning
% 2.46/1.44
% 2.46/1.44 % (4695)Memory used [KB]: 1663
% 2.46/1.44 % (4695)Time elapsed: 0.011 s
% 2.46/1.44 % (4695)Instructions burned: 19 (million)
% 2.46/1.44 % (4695)------------------------------
% 2.46/1.44 % (4695)------------------------------
% 2.46/1.44 % (4691)Instruction limit reached!
% 2.46/1.44 % (4691)------------------------------
% 2.46/1.44 % (4691)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.44 % (4691)Termination reason: Unknown
% 2.46/1.44 % (4691)Termination phase: shuffling
% 2.46/1.44
% 2.46/1.44 % (4691)Memory used [KB]: 1918
% 2.46/1.44 % (4691)Time elapsed: 0.014 s
% 2.46/1.44 % (4691)Instructions burned: 27 (million)
% 2.46/1.44 % (4691)------------------------------
% 2.46/1.44 % (4691)------------------------------
% 2.46/1.44 % (4696)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/3Mi)
% 2.46/1.44 % (4697)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_4583 for (2999ds/37Mi)
% 2.46/1.44 % (4698)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_4583 for (2999ds/15Mi)
% 2.46/1.44 % (4696)Instruction limit reached!
% 2.46/1.44 % (4696)------------------------------
% 2.46/1.44 % (4696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.45 % (4696)Termination reason: Unknown
% 2.46/1.45 % (4696)Termination phase: shuffling
% 2.46/1.45
% 2.46/1.45 % (4696)Memory used [KB]: 1279
% 2.46/1.45 % (4696)Time elapsed: 0.003 s
% 2.46/1.45 % (4696)Instructions burned: 4 (million)
% 2.46/1.45 % (4696)------------------------------
% 2.46/1.45 % (4696)------------------------------
% 2.46/1.45 % (4699)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_4583 for (2999ds/3Mi)
% 2.46/1.45 % (4698)Instruction limit reached!
% 2.46/1.45 % (4698)------------------------------
% 2.46/1.45 % (4698)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.45 % (4698)Termination reason: Unknown
% 2.46/1.45 % (4698)Termination phase: shuffling
% 2.46/1.45
% 2.46/1.45 % (4698)Memory used [KB]: 1663
% 2.46/1.45 % (4698)Time elapsed: 0.009 s
% 2.46/1.45 % (4698)Instructions burned: 15 (million)
% 2.46/1.45 % (4698)------------------------------
% 2.46/1.45 % (4698)------------------------------
% 2.46/1.45 % (4699)Instruction limit reached!
% 2.46/1.45 % (4699)------------------------------
% 2.46/1.45 % (4699)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.45 % (4699)Termination reason: Unknown
% 2.46/1.45 % (4699)Termination phase: shuffling
% 2.46/1.45
% 2.46/1.45 % (4699)Memory used [KB]: 1407
% 2.46/1.45 % (4699)Time elapsed: 0.003 s
% 2.46/1.45 % (4699)Instructions burned: 3 (million)
% 2.46/1.45 % (4699)------------------------------
% 2.46/1.45 % (4699)------------------------------
% 2.46/1.46 % (4700)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_4583 for (2999ds/1041Mi)
% 2.46/1.46 % (4697)Instruction limit reached!
% 2.46/1.46 % (4697)------------------------------
% 2.46/1.46 % (4697)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.46 % (4697)Termination reason: Unknown
% 2.46/1.46 % (4697)Termination phase: Property scanning
% 2.46/1.46
% 2.46/1.46 % (4697)Memory used [KB]: 1918
% 2.46/1.46 % (4697)Time elapsed: 0.018 s
% 2.46/1.46 % (4697)Instructions burned: 37 (million)
% 2.46/1.46 % (4697)------------------------------
% 2.46/1.46 % (4697)------------------------------
% 2.46/1.46 % (4701)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/7Mi)
% 2.46/1.46 % (4701)Instruction limit reached!
% 2.46/1.46 % (4701)------------------------------
% 2.46/1.46 % (4701)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.46 % (4701)Termination reason: Unknown
% 2.46/1.46 % (4701)Termination phase: shuffling
% 2.46/1.46
% 2.46/1.46 % (4701)Memory used [KB]: 1407
% 2.46/1.46 % (4701)Time elapsed: 0.005 s
% 2.46/1.46 % (4701)Instructions burned: 8 (million)
% 2.46/1.46 % (4701)------------------------------
% 2.46/1.46 % (4701)------------------------------
% 2.46/1.47 % (4702)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/16Mi)
% 2.46/1.47 % (4689)Instruction limit reached!
% 2.46/1.47 % (4689)------------------------------
% 2.46/1.47 % (4689)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.47 % (4689)Termination reason: Unknown
% 2.46/1.47 % (4689)Termination phase: Saturation
% 2.46/1.47
% 2.46/1.47 % (4689)Memory used [KB]: 7547
% 2.46/1.47 % (4689)Time elapsed: 0.051 s
% 2.46/1.47 % (4689)Instructions burned: 186 (million)
% 2.46/1.47 % (4689)------------------------------
% 2.46/1.47 % (4689)------------------------------
% 2.46/1.47 % (4703)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_4583 for (2999ds/3Mi)
% 2.46/1.47 % (4703)Instruction limit reached!
% 2.46/1.47 % (4703)------------------------------
% 2.46/1.47 % (4703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.47 % (4703)Termination reason: Unknown
% 2.46/1.47 % (4703)Termination phase: shuffling
% 2.46/1.47
% 2.46/1.47 % (4703)Memory used [KB]: 1279
% 2.46/1.47 % (4703)Time elapsed: 0.003 s
% 2.46/1.47 % (4703)Instructions burned: 4 (million)
% 2.46/1.47 % (4703)------------------------------
% 2.46/1.47 % (4703)------------------------------
% 2.46/1.48 % (4702)Instruction limit reached!
% 2.46/1.48 % (4702)------------------------------
% 2.46/1.48 % (4702)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.48 % (4702)Termination reason: Unknown
% 2.46/1.48 % (4702)Termination phase: shuffling
% 2.46/1.48
% 2.46/1.48 % (4702)Memory used [KB]: 1663
% 2.46/1.48 % (4702)Time elapsed: 0.009 s
% 2.46/1.48 % (4702)Instructions burned: 17 (million)
% 2.46/1.48 % (4702)------------------------------
% 2.46/1.48 % (4702)------------------------------
% 2.46/1.48 % (4704)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_4583 for (2999ds/3Mi)
% 2.46/1.48 % (4704)Instruction limit reached!
% 2.46/1.48 % (4704)------------------------------
% 2.46/1.48 % (4704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.48 % (4704)Termination reason: Unknown
% 2.46/1.48 % (4704)Termination phase: shuffling
% 2.46/1.48
% 2.46/1.48 % (4704)Memory used [KB]: 1279
% 2.46/1.48 % (4704)Time elapsed: 0.003 s
% 2.46/1.48 % (4704)Instructions burned: 3 (million)
% 2.46/1.48 % (4704)------------------------------
% 2.46/1.48 % (4704)------------------------------
% 2.46/1.48 % (4706)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/3Mi)
% 2.46/1.48 % (4706)Instruction limit reached!
% 2.46/1.48 % (4706)------------------------------
% 2.46/1.48 % (4706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.48 % (4706)Termination reason: Unknown
% 2.46/1.48 % (4706)Termination phase: shuffling
% 2.46/1.48
% 2.46/1.48 % (4706)Memory used [KB]: 1279
% 2.46/1.48 % (4706)Time elapsed: 0.002 s
% 2.46/1.48 % (4706)Instructions burned: 4 (million)
% 2.46/1.48 % (4706)------------------------------
% 2.46/1.48 % (4706)------------------------------
% 2.46/1.48 % (4705)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/7Mi)
% 2.46/1.48 % (4705)Instruction limit reached!
% 2.46/1.48 % (4705)------------------------------
% 2.46/1.48 % (4705)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.48 % (4705)Termination reason: Unknown
% 2.46/1.48 % (4705)Termination phase: shuffling
% 2.46/1.48
% 2.46/1.48 % (4705)Memory used [KB]: 1407
% 2.46/1.48 % (4705)Time elapsed: 0.005 s
% 2.46/1.48 % (4705)Instructions burned: 8 (million)
% 2.46/1.48 % (4705)------------------------------
% 2.46/1.48 % (4705)------------------------------
% 2.46/1.49 % (4710)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/6Mi)
% 2.46/1.49 % (4708)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/18Mi)
% 2.46/1.49 % (4710)Instruction limit reached!
% 2.46/1.49 % (4710)------------------------------
% 2.46/1.49 % (4710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.49 % (4710)Termination reason: Unknown
% 2.46/1.49 % (4710)Termination phase: shuffling
% 2.46/1.49
% 2.46/1.49 % (4710)Memory used [KB]: 1407
% 2.46/1.49 % (4710)Time elapsed: 0.003 s
% 2.46/1.49 % (4710)Instructions burned: 7 (million)
% 2.46/1.49 % (4710)------------------------------
% 2.46/1.49 % (4710)------------------------------
% 2.46/1.49 % (4707)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/4Mi)
% 2.46/1.49 % (4709)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/710Mi)
% 2.46/1.50 % (4707)Instruction limit reached!
% 2.46/1.50 % (4707)------------------------------
% 2.46/1.50 % (4707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.46/1.50 % (4707)Termination reason: Unknown
% 2.46/1.50 % (4707)Termination phase: shuffling
% 2.46/1.50
% 2.46/1.50 % (4707)Memory used [KB]: 1407
% 2.46/1.50 % (4707)Time elapsed: 0.003 s
% 2.46/1.50 % (4707)Instructions burned: 4 (million)
% 2.46/1.50 % (4707)------------------------------
% 2.46/1.50 % (4707)------------------------------
% 2.49/1.50 % (4708)Instruction limit reached!
% 2.49/1.50 % (4708)------------------------------
% 2.49/1.50 % (4708)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.50 % (4708)Termination reason: Unknown
% 2.49/1.50 % (4708)Termination phase: shuffling
% 2.49/1.50
% 2.49/1.50 % (4708)Memory used [KB]: 1663
% 2.49/1.50 % (4708)Time elapsed: 0.010 s
% 2.49/1.50 % (4708)Instructions burned: 19 (million)
% 2.49/1.50 % (4708)------------------------------
% 2.49/1.50 % (4708)------------------------------
% 2.49/1.50 % (4711)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_4583 for (2999ds/902Mi)
% 2.49/1.50 % (4712)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_4583 for (2999ds/21Mi)
% 2.49/1.51 % (4712)Instruction limit reached!
% 2.49/1.51 % (4712)------------------------------
% 2.49/1.51 % (4712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.51 % (4712)Termination reason: Unknown
% 2.49/1.51 % (4712)Termination phase: shuffling
% 2.49/1.51
% 2.49/1.51 % (4712)Memory used [KB]: 1791
% 2.49/1.51 % (4712)Time elapsed: 0.007 s
% 2.49/1.51 % (4712)Instructions burned: 22 (million)
% 2.49/1.51 % (4712)------------------------------
% 2.49/1.51 % (4712)------------------------------
% 2.49/1.51 % (4713)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/5Mi)
% 2.49/1.51 % (4713)Instruction limit reached!
% 2.49/1.51 % (4713)------------------------------
% 2.49/1.51 % (4713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.51 % (4713)Termination reason: Unknown
% 2.49/1.51 % (4713)Termination phase: shuffling
% 2.49/1.51
% 2.49/1.51 % (4713)Memory used [KB]: 1407
% 2.49/1.51 % (4713)Time elapsed: 0.004 s
% 2.49/1.51 % (4713)Instructions burned: 6 (million)
% 2.49/1.51 % (4713)------------------------------
% 2.49/1.51 % (4713)------------------------------
% 2.49/1.52 % (4715)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_4583 for (2999ds/6Mi)
% 2.49/1.52 % (4715)Instruction limit reached!
% 2.49/1.52 % (4715)------------------------------
% 2.49/1.52 % (4715)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.52 % (4715)Termination reason: Unknown
% 2.49/1.52 % (4715)Termination phase: shuffling
% 2.49/1.52
% 2.49/1.52 % (4715)Memory used [KB]: 1407
% 2.49/1.52 % (4715)Time elapsed: 0.003 s
% 2.49/1.52 % (4715)Instructions burned: 7 (million)
% 2.49/1.52 % (4715)------------------------------
% 2.49/1.52 % (4715)------------------------------
% 2.49/1.52 % (4716)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_4583 for (2999ds/377Mi)
% 2.49/1.53 % (4718)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_4583 for (2998ds/19Mi)
% 2.49/1.53 % (4717)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_4583 for (2999ds/779Mi)
% 2.49/1.53 % (4718)Instruction limit reached!
% 2.49/1.53 % (4718)------------------------------
% 2.49/1.53 % (4718)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.53 % (4718)Termination reason: Unknown
% 2.49/1.53 % (4718)Termination phase: shuffling
% 2.49/1.53
% 2.49/1.53 % (4718)Memory used [KB]: 1791
% 2.49/1.53 % (4718)Time elapsed: 0.007 s
% 2.49/1.53 % (4718)Instructions burned: 22 (million)
% 2.49/1.53 % (4718)------------------------------
% 2.49/1.53 % (4718)------------------------------
% 2.49/1.54 % (4719)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_4583 for (2998ds/879Mi)
% 2.49/1.56 % (4694)Instruction limit reached!
% 2.49/1.56 % (4694)------------------------------
% 2.49/1.56 % (4694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.56 % (4694)Termination reason: Unknown
% 2.49/1.56 % (4694)Termination phase: Saturation
% 2.49/1.56
% 2.49/1.56 % (4694)Memory used [KB]: 8315
% 2.49/1.56 % (4694)Time elapsed: 0.136 s
% 2.49/1.56 % (4694)Instructions burned: 276 (million)
% 2.49/1.56 % (4694)------------------------------
% 2.49/1.56 % (4694)------------------------------
% 2.49/1.58 % (4720)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_4583 for (2998ds/17Mi)
% 2.49/1.58 % (4720)Instruction limit reached!
% 2.49/1.58 % (4720)------------------------------
% 2.49/1.58 % (4720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.58 % (4720)Termination reason: Unknown
% 2.49/1.58 % (4720)Termination phase: shuffling
% 2.49/1.58
% 2.49/1.58 % (4720)Memory used [KB]: 1791
% 2.49/1.58 % (4720)Time elapsed: 0.008 s
% 2.49/1.58 % (4720)Instructions burned: 19 (million)
% 2.49/1.58 % (4720)------------------------------
% 2.49/1.58 % (4720)------------------------------
% 2.49/1.59 % (4721)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_4583 for (2998ds/3Mi)
% 2.49/1.59 % (4721)Instruction limit reached!
% 2.49/1.59 % (4721)------------------------------
% 2.49/1.59 % (4721)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.59 % (4721)Termination reason: Unknown
% 2.49/1.59 % (4721)Termination phase: shuffling
% 2.49/1.59
% 2.49/1.59 % (4721)Memory used [KB]: 1279
% 2.49/1.59 % (4721)Time elapsed: 0.002 s
% 2.49/1.59 % (4721)Instructions burned: 3 (million)
% 2.49/1.59 % (4721)------------------------------
% 2.49/1.59 % (4721)------------------------------
% 2.49/1.60 % (4711)First to succeed.
% 2.49/1.60 % (4722)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_4583 for (2998ds/30Mi)
% 2.49/1.60 % (4711)Refutation found. Thanks to Tanya!
% 2.49/1.60 % SZS status Theorem for DTF2THF_4583
% 2.49/1.60 % SZS output start Proof for DTF2THF_4583
% 2.49/1.60 thf(func_def_0, type, in: $i > $i > $o).
% 2.49/1.60 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.49/1.60 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.49/1.60 thf(func_def_8, type, powerset: $i > $i).
% 2.49/1.60 thf(func_def_10, type, setunion: $i > $i).
% 2.49/1.60 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.49/1.60 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_26, type, prop2set: $o > $i).
% 2.49/1.60 thf(func_def_36, type, nonempty: $i > $o).
% 2.49/1.60 thf(func_def_69, type, set2prop: $i > $o).
% 2.49/1.60 thf(func_def_88, type, subset: $i > $i > $o).
% 2.49/1.60 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.49/1.60 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.49/1.60 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.49/1.60 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.49/1.60 thf(func_def_135, type, regular: $i > $o).
% 2.49/1.60 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.49/1.60 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.49/1.60 thf(func_def_153, type, iskpair: $i > $o).
% 2.49/1.60 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.49/1.60 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.49/1.60 thf(func_def_177, type, singleton: $i > $o).
% 2.49/1.60 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.49/1.60 thf(func_def_184, type, atmost1p: $i > $o).
% 2.49/1.60 thf(func_def_185, type, atleast2p: $i > $o).
% 2.49/1.60 thf(func_def_186, type, atmost2p: $i > $o).
% 2.49/1.60 thf(func_def_187, type, upairsetp: $i > $o).
% 2.49/1.60 thf(func_def_191, type, kfst: $i > $i).
% 2.49/1.60 thf(func_def_203, type, ksnd: $i > $i).
% 2.49/1.60 thf(func_def_226, type, sP2: $i > $i > $o).
% 2.49/1.60 thf(func_def_227, type, sP3: $i > $o).
% 2.49/1.60 thf(func_def_228, type, sP4: $i > $i > $o).
% 2.49/1.60 thf(func_def_229, type, sP5: $i > $i > $o).
% 2.49/1.60 thf(func_def_230, type, sP6: $i > $o > $i > $i > $o).
% 2.49/1.60 thf(func_def_233, type, sK9: $i > $o).
% 2.49/1.60 thf(func_def_234, type, sK10: ($i > $o) > $i > $i).
% 2.49/1.60 thf(func_def_255, type, sK31: $i > $o).
% 2.49/1.60 thf(func_def_256, type, sK32: $i > $o).
% 2.49/1.60 thf(func_def_257, type, sK33: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.49/1.60 thf(func_def_258, type, sK34: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.49/1.60 thf(func_def_266, type, sK42: $i > $i > $o).
% 2.49/1.60 thf(func_def_268, type, sK44: $i > $i).
% 2.49/1.60 thf(func_def_269, type, sK45: $i > $i).
% 2.49/1.60 thf(func_def_270, type, sK46: $i > ($i > $i > $o) > $i).
% 2.49/1.60 thf(func_def_271, type, sK47: $i > ($i > $i > $o) > $i).
% 2.49/1.60 thf(func_def_272, type, sK48: $i > $i > ($i > $i > $o) > $i).
% 2.49/1.60 thf(func_def_275, type, sK51: $i > $i).
% 2.49/1.60 thf(func_def_283, type, sK59: $i > $i > $i).
% 2.49/1.60 thf(func_def_290, type, sK66: $i > $i > $i).
% 2.49/1.60 thf(func_def_302, type, sK78: ($i > $o) > $i > $i).
% 2.49/1.60 thf(func_def_304, type, sK80: $i > $o).
% 2.49/1.60 thf(func_def_308, type, sK84: $i > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_309, type, sK85: $i > $o).
% 2.49/1.60 thf(func_def_314, type, sK90: $i > $o).
% 2.49/1.60 thf(func_def_315, type, sK91: $i > $i).
% 2.49/1.60 thf(func_def_316, type, sK92: ($i > $o) > $i).
% 2.49/1.60 thf(func_def_319, type, sK95: ($i > $o) > $i).
% 2.49/1.60 thf(func_def_320, type, sK96: $i > $o).
% 2.49/1.60 thf(func_def_323, type, sK99: $i > $o).
% 2.49/1.60 thf(func_def_326, type, sK102: $i > $o).
% 2.49/1.60 thf(func_def_330, type, sK106: $i > $i > $i).
% 2.49/1.60 thf(func_def_331, type, sK107: $i > $o).
% 2.49/1.60 thf(func_def_334, type, sK110: $i > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_335, type, sK111: $i > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_336, type, sK112: $i > $o).
% 2.49/1.60 thf(func_def_341, type, sK117: $i > $i).
% 2.49/1.60 thf(func_def_348, type, sK124: $i > $o).
% 2.49/1.60 thf(func_def_350, type, sK126: ($i > $o) > $i > $i).
% 2.49/1.60 thf(func_def_351, type, sK127: ($i > $o) > $i > $i).
% 2.49/1.60 thf(func_def_379, type, sK155: ($i > $o) > $i).
% 2.49/1.60 thf(func_def_380, type, sK156: $i > $o).
% 2.49/1.60 thf(func_def_381, type, sK157: $i > $i).
% 2.49/1.60 thf(func_def_397, type, sK173: $i > $o).
% 2.49/1.60 thf(func_def_403, type, sK179: $i > $i > $i).
% 2.49/1.60 thf(func_def_406, type, sK182: $i > $o).
% 2.49/1.60 thf(func_def_408, type, sK184: $i > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_414, type, sK190: $i > $o).
% 2.49/1.60 thf(func_def_415, type, sK191: $i > $i).
% 2.49/1.60 thf(func_def_418, type, sK194: $i > $o).
% 2.49/1.60 thf(func_def_420, type, sK196: $i > ($i > $o) > $i > $i).
% 2.49/1.60 thf(func_def_421, type, sK197: $i > $i > $i).
% 2.49/1.60 thf(func_def_422, type, sK198: $i > $i > $i).
% 2.49/1.60 thf(func_def_423, type, sK199: $i > $i > $i).
% 2.49/1.60 thf(func_def_424, type, sK200: $i > $i > $i).
% 2.49/1.60 thf(func_def_425, type, sK201: $i > $i > $i).
% 2.49/1.60 thf(func_def_426, type, sK202: $i > $i > $i > $i).
% 2.49/1.60 thf(func_def_427, type, sK203: $i > $i).
% 2.49/1.60 thf(func_def_428, type, sK204: $i > $i).
% 2.49/1.60 thf(func_def_429, type, sK205: $i > $i).
% 2.49/1.60 thf(func_def_430, type, sK206: $i > $i).
% 2.49/1.60 thf(func_def_431, type, sK207: $i > $i > $i).
% 2.49/1.60 thf(func_def_432, type, sK208: $i > $i > $i > $i).
% 2.49/1.60 thf(func_def_433, type, sK209: $i > $i > $i > $i).
% 2.49/1.60 thf(func_def_434, type, sK210: $i > $i > $i).
% 2.49/1.60 thf(func_def_435, type, sK211: $i > $i > $i).
% 2.49/1.60 thf(func_def_436, type, sK212: $i > $i).
% 2.49/1.60 thf(func_def_438, type, sK214: $i > $i).
% 2.49/1.60 thf(func_def_439, type, sK215: $i > $i).
% 2.49/1.60 thf(func_def_474, type, sK250: $i > $i > $i).
% 2.49/1.60 thf(func_def_484, type, sK260: $i > $o).
% 2.49/1.60 thf(func_def_485, type, sK261: $i > $o).
% 2.49/1.60 thf(func_def_486, type, sK262: ($i > $o) > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_487, type, sK263: ($i > $o) > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_506, type, sK282: $i > $o).
% 2.49/1.60 thf(func_def_508, type, sK284: ($i > $o) > $i > $i).
% 2.49/1.60 thf(func_def_513, type, sK289: $i > $i > $i > $i).
% 2.49/1.60 thf(func_def_514, type, sK290: $i > $i > $i > $i).
% 2.49/1.60 thf(func_def_521, type, sK297: $i > $o).
% 2.49/1.60 thf(func_def_522, type, sK298: $i > $o).
% 2.49/1.60 thf(func_def_523, type, sK299: ($i > $o) > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_524, type, sK300: ($i > $o) > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_545, type, sK321: $i > $o).
% 2.49/1.60 thf(func_def_558, type, sK334: $i > $i > $i).
% 2.49/1.60 thf(func_def_562, type, sK338: $i > $o).
% 2.49/1.60 thf(func_def_574, type, sK350: $i > ($i > $o) > $i).
% 2.49/1.60 thf(func_def_575, type, sK351: $i > $o).
% 2.49/1.60 thf(func_def_578, type, sK354: $i > $i).
% 2.49/1.60 thf(func_def_587, type, sK363: $i > $i > $i).
% 2.49/1.60 thf(func_def_589, type, sK365: $i > $o).
% 2.49/1.60 thf(func_def_597, type, sK373: $o > $i > $i > $i).
% 2.49/1.60 thf(func_def_600, type, sK376: $i > $o).
% 2.49/1.60 thf(func_def_615, type, sK391: $i > $i > $i).
% 2.49/1.60 thf(func_def_616, type, sK392: $i > $i > $i).
% 2.49/1.60 thf(func_def_619, type, sK395: $i > $i > $i).
% 2.49/1.60 thf(func_def_622, type, sK398: $i > $o).
% 2.49/1.60 thf(func_def_627, type, sK403: $i > $o).
% 2.49/1.60 thf(func_def_638, type, sK414: $i > $o).
% 2.49/1.60 thf(func_def_640, type, sK416: ($i > $o) > $i > $i).
% 2.49/1.60 thf(func_def_641, type, sK417: ($i > $o) > $i > $i).
% 2.49/1.60 thf(func_def_671, type, sK447: $i > $o).
% 2.49/1.60 thf(func_def_696, type, sK472: $i > $o).
% 2.49/1.60 thf(func_def_705, type, sK481: ($i > $o) > $i > $i).
% 2.49/1.60 thf(func_def_707, type, sK483: $i > $o).
% 2.49/1.60 thf(func_def_717, type, sK493: ($i > $o) > $i).
% 2.49/1.60 thf(func_def_718, type, sK494: ($i > $o) > $i).
% 2.49/1.60 thf(func_def_719, type, sK495: $i > $o).
% 2.49/1.60 thf(func_def_726, type, sK502: $i > $i).
% 2.49/1.60 thf(func_def_728, type, sK504: $i > $i).
% 2.49/1.60 thf(func_def_731, type, sK507: $i > $i).
% 2.49/1.60 thf(func_def_734, type, ph510: !>[X0: $tType]:(X0)).
% 2.49/1.60 thf(f2625,plain,(
% 2.49/1.60 $false),
% 2.49/1.60 inference(trivial_inequality_removal,[status(thm)],[f2624])).
% 2.49/1.60 thf(f2624,plain,(
% 2.49/1.60 ($true != $true)),
% 2.49/1.60 inference(superposition,[status(thm)],[f2607,f2623])).
% 2.49/1.60 thf(f2623,plain,(
% 2.49/1.60 ($true = (iskpair @ sK407))),
% 2.49/1.60 inference(trivial_inequality_removal,[status(thm)],[f2622])).
% 2.49/1.60 thf(f2622,plain,(
% 2.49/1.60 ($true != $true) | ($true = (iskpair @ sK407))),
% 2.49/1.60 inference(forward_demodulation,[status(thm)],[f2612,f2340])).
% 2.49/1.60 thf(f2340,plain,(
% 2.49/1.60 (cartprodmempair = $true)),
% 2.49/1.60 inference(cnf_transformation,[status(thm)],[f1443])).
% 2.49/1.60 thf(f1443,plain,(
% 2.49/1.60 (kpairp = $true) & (notinemptyset = $true) & (upairequniteq = $true) & (replAx = $true) & (binunionIL = $true) & (wellorderingAx = $true) & (notsubsetI = $true) & (ex1I = $true) & (binintersectSubset5 = $true) & (cartprodpairmemEL = $true) & (upairset2IR = $true) & (exuE3u = $true) & (noeltsimpempty = $true) & (ubforcartprodlem3 = $true) & (subsetI2 = $true) & (cartprodmempair = $true) & (setunionsingleton1 = $true) & (in__Cong = $true) & (prop2set2propI = $true) & (omega0Ax = $true) & (subsetE2 = $true) & (singletonprop = $true) & (quantDeMorgan3 = $true) & (cartprodpairin = $true) & (subsetRefl = $true) & (setukpairinjR11 = $true) & (binunionRsub = $true) & (dsetconstrER = $true) & (setminusILneg = $true) & (exuE2 = $true) & (setukpairinjL2 = $true) & (kfstsingleton = $true) & (secondinupair = $true) & (subsetI1 = $true) & (setunionsingleton2 = $true) & (setextAx = $true) & (ksndsingleton = $true) & (ksndpairEq = $true) & (omega__Cong = $true) & (symdiffI2 = $true) & (singletonsubset = $true) & (upairsubunion = $true) & (cartprodsndpairEq = $true) & (setukpairinjL1 = $true) & (setunionI = $true) & (setminusLsub = $true) & (emptyInPowerset = $true) & (setminusERneg = $true) & (ex1E1 = $true) & (singletonsswitch = $true) & (setadjoinIL = $true) & (symdiffE = $true) & (powersetI1 = $true) & (setukpairinjL = $true) & (setextsub = $true) & (subsetE = $true) & (prop2setE = $true) & (setext = $true) & (ubforcartprodlem2 = $true) & (nonemptyI1 = $true) & (setunion__Cong = $true) & (powerset__Cong = $true) & (quantDeMorgan2 = $true) & (quantDeMorgan1 = $true) & (emptyE1 = $true) & (setunionsingleton = $true) & (theprop = $true) & (singletoninpowunion = $true) & (setminusSubset2 = $true) & (subset2powerset = $true) & (notequalI1 = $true) & (symdiffI1 = $true) & (setunionAx = $true) & (disjointsetsI1 = $true) & (quantDeMorgan4 = $true) & (binintersectSubset1 = $true) & (binunionE = $true) & (setunionE = $true) & (setadjoinE = $true) & (omegaIndAx = $true) & (eqinunit = $true) & (binintersectSubset2 = $true) & (setadjoinOr = $true) & (emptyinunitempty = $true) & (emptyinPowerset = $true) & (dsetconstr__Cong = $true) & (bs114d = $true) & (powersetAx = $true) & (binunionIR = $true) & (cartprodpairmemER = $true) & (setukpairinjR12 = $true) & (setminusER = $true) & (sepInPowerset = $true) & (powersetI = $true) & (ex1I2 = $true) & (setunionE2 = $true) & (setminusELneg = $true) & (setukpairIL = $true) & (setukpairinjR2 = $true) & (binintersectSubset3 = $true) & (vacuousDall = $true) & (nonemptyI = $true) & (emptysetimpfalse = $true) & (setadjoinIR = $true) & (dsetconstrEL = $true) & (upairsetIL = $true) & (omegaSAx = $true) & (powersetE1 = $true) & (powersetsubset = $true) & ((sK407 != (kpair @ (kfst @ sK407) @ (ksnd @ sK407))) & ($true = (in @ sK407 @ (cartprod @ sK408 @ sK409)))) & (notdallE = $true) & (cartprodfstpairEq = $true) & (emptysetAx = $true) & (setukpairinjR1 = $true) & (exuI1 = $true) & (binintersectI = $true) & (descr__Cong = $true) & (dsetconstrI = $true) & (setadjoinSub2 = $true) & (upairset2E = $true) & (cartprodmempaircEq = $true) & (setukpairIR = $true) & (emptyset__Cong = $true) & (prop2setI = $true) & (setminusSubset1 = $true) & (inCongP = $true) & (exuI2 = $true) & (upairinpowunion = $true) & (kfstpairEq = $true) & (uniqinunit = $true) & (singletonsuniq = $true) & (setminusEL = $true) & (exuEu = $true) & (upairsetIR = $true) & (symdiffIneg2 = $true) & (subsetTrans = $true) & (upairsetE = $true) & (symdiffIneg1 = $true) & (cartprodmempair1 = $true) & (notdexE = $true) & (inPowerset = $true) & (eqimpsubset2 = $true) & (nonemptyImpWitness = $true) & (setbeta = $true) & (exuE1 = $true) & (setoftrueEq = $true) & (foundationAx = $true) & (cartprodfstin = $true) & (cartprodsndin = $true) & (emptyI = $true) & (binunionEcases = $true) & (exuE3e = $true) & (notequalI2 = $true) & (kpairiskpair = $true) & (binintersectLsub = $true) & (singletoninpowerset = $true) & (binunionLsub = $true) & (notinsingleton = $true) & (setadjoinSub = $true) & (sepSubset = $true) & (binintersectER = $true) & (descrp = $true) & (exuI3 = $true) & (setadjoinAx = $true) & (binintersectSubset4 = $true) & (setadjoin__Cong = $true) & (setminusIRneg = $true) & (powersetE = $true) & (binintersectEL = $true) & (subsetemptysetimpeq = $true) & (ubforcartprodlem1 = $true) & (exu__Cong = $true) & (setukpairinjR = $true) & (emptysetE = $true) & (binintersectRsub = $true) & (subPowSU = $true) & (emptysetsubset = $true) & (eqimpsubset1 = $true) & (nonemptyE1 = $true) & (kpairsurjEq = $true) & (setminusI = $true)),
% 2.49/1.60 inference(skolemisation,[status(esa),new_symbols(skolem,[sK407,sK408,sK409])],[f764,f1442])).
% 2.49/1.60 thf(f1442,plain,(
% 2.49/1.60 ? [X0 : $i,X1 : $i,X2 : $i] : (((kpair @ (kfst @ X0) @ (ksnd @ X0)) != X0) & ((in @ X0 @ (cartprod @ X1 @ X2)) = $true)) => ((sK407 != (kpair @ (kfst @ sK407) @ (ksnd @ sK407))) & ($true = (in @ sK407 @ (cartprod @ sK408 @ sK409))))),
% 2.49/1.60 introduced(definition,[],[choice_axiom])).
% 2.49/1.60 thf(f764,plain,(
% 2.49/1.60 (kpairp = $true) & (notinemptyset = $true) & (upairequniteq = $true) & (replAx = $true) & (binunionIL = $true) & (wellorderingAx = $true) & (notsubsetI = $true) & (ex1I = $true) & (binintersectSubset5 = $true) & (cartprodpairmemEL = $true) & (upairset2IR = $true) & (exuE3u = $true) & (noeltsimpempty = $true) & (ubforcartprodlem3 = $true) & (subsetI2 = $true) & (cartprodmempair = $true) & (setunionsingleton1 = $true) & (in__Cong = $true) & (prop2set2propI = $true) & (omega0Ax = $true) & (subsetE2 = $true) & (singletonprop = $true) & (quantDeMorgan3 = $true) & (cartprodpairin = $true) & (subsetRefl = $true) & (setukpairinjR11 = $true) & (binunionRsub = $true) & (dsetconstrER = $true) & (setminusILneg = $true) & (exuE2 = $true) & (setukpairinjL2 = $true) & (kfstsingleton = $true) & (secondinupair = $true) & (subsetI1 = $true) & (setunionsingleton2 = $true) & (setextAx = $true) & (ksndsingleton = $true) & (ksndpairEq = $true) & (omega__Cong = $true) & (symdiffI2 = $true) & (singletonsubset = $true) & (upairsubunion = $true) & (cartprodsndpairEq = $true) & (setukpairinjL1 = $true) & (setunionI = $true) & (setminusLsub = $true) & (emptyInPowerset = $true) & (setminusERneg = $true) & (ex1E1 = $true) & (singletonsswitch = $true) & (setadjoinIL = $true) & (symdiffE = $true) & (powersetI1 = $true) & (setukpairinjL = $true) & (setextsub = $true) & (subsetE = $true) & (prop2setE = $true) & (setext = $true) & (ubforcartprodlem2 = $true) & (nonemptyI1 = $true) & (setunion__Cong = $true) & (powerset__Cong = $true) & (quantDeMorgan2 = $true) & (quantDeMorgan1 = $true) & (emptyE1 = $true) & (setunionsingleton = $true) & (theprop = $true) & (singletoninpowunion = $true) & (setminusSubset2 = $true) & (subset2powerset = $true) & (notequalI1 = $true) & (symdiffI1 = $true) & (setunionAx = $true) & (disjointsetsI1 = $true) & (quantDeMorgan4 = $true) & (binintersectSubset1 = $true) & (binunionE = $true) & (setunionE = $true) & (setadjoinE = $true) & (omegaIndAx = $true) & (eqinunit = $true) & (binintersectSubset2 = $true) & (setadjoinOr = $true) & (emptyinunitempty = $true) & (emptyinPowerset = $true) & (dsetconstr__Cong = $true) & (bs114d = $true) & (powersetAx = $true) & (binunionIR = $true) & (cartprodpairmemER = $true) & (setukpairinjR12 = $true) & (setminusER = $true) & (sepInPowerset = $true) & (powersetI = $true) & (ex1I2 = $true) & (setunionE2 = $true) & (setminusELneg = $true) & (setukpairIL = $true) & (setukpairinjR2 = $true) & (binintersectSubset3 = $true) & (vacuousDall = $true) & (nonemptyI = $true) & (emptysetimpfalse = $true) & (setadjoinIR = $true) & (dsetconstrEL = $true) & (upairsetIL = $true) & (omegaSAx = $true) & (powersetE1 = $true) & (powersetsubset = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((kpair @ (kfst @ X0) @ (ksnd @ X0)) != X0) & ((in @ X0 @ (cartprod @ X1 @ X2)) = $true)) & (notdallE = $true) & (cartprodfstpairEq = $true) & (emptysetAx = $true) & (setukpairinjR1 = $true) & (exuI1 = $true) & (binintersectI = $true) & (descr__Cong = $true) & (dsetconstrI = $true) & (setadjoinSub2 = $true) & (upairset2E = $true) & (cartprodmempaircEq = $true) & (setukpairIR = $true) & (emptyset__Cong = $true) & (prop2setI = $true) & (setminusSubset1 = $true) & (inCongP = $true) & (exuI2 = $true) & (upairinpowunion = $true) & (kfstpairEq = $true) & (uniqinunit = $true) & (singletonsuniq = $true) & (setminusEL = $true) & (exuEu = $true) & (upairsetIR = $true) & (symdiffIneg2 = $true) & (subsetTrans = $true) & (upairsetE = $true) & (symdiffIneg1 = $true) & (cartprodmempair1 = $true) & (notdexE = $true) & (inPowerset = $true) & (eqimpsubset2 = $true) & (nonemptyImpWitness = $true) & (setbeta = $true) & (exuE1 = $true) & (setoftrueEq = $true) & (foundationAx = $true) & (cartprodfstin = $true) & (cartprodsndin = $true) & (emptyI = $true) & (binunionEcases = $true) & (exuE3e = $true) & (notequalI2 = $true) & (kpairiskpair = $true) & (binintersectLsub = $true) & (singletoninpowerset = $true) & (binunionLsub = $true) & (notinsingleton = $true) & (setadjoinSub = $true) & (sepSubset = $true) & (binintersectER = $true) & (descrp = $true) & (exuI3 = $true) & (setadjoinAx = $true) & (binintersectSubset4 = $true) & (setadjoin__Cong = $true) & (setminusIRneg = $true) & (powersetE = $true) & (binintersectEL = $true) & (subsetemptysetimpeq = $true) & (ubforcartprodlem1 = $true) & (exu__Cong = $true) & (setukpairinjR = $true) & (emptysetE = $true) & (binintersectRsub = $true) & (subPowSU = $true) & (emptysetsubset = $true) & (eqimpsubset1 = $true) & (nonemptyE1 = $true) & (kpairsurjEq = $true) & (setminusI = $true)),
% 2.49/1.60 inference(flattening,[status(thm)],[f763])).
% 2.49/1.60 thf(f763,plain,(
% 2.49/1.60 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (((kpair @ (kfst @ X0) @ (ksnd @ X0)) != X0) & ((in @ X0 @ (cartprod @ X1 @ X2)) = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.49/1.60 inference(ennf_transformation,[status(thm)],[f393])).
% 2.49/1.60 thf(f393,plain,(
% 2.49/1.60 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (cartprod @ X1 @ X2)) = $true) => ((kpair @ (kfst @ X0) @ (ksnd @ X0)) = X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.49/1.60 inference(fool_elimination,[status(thm)],[f392])).
% 2.49/1.60 thf(f392,plain,(
% 2.49/1.60 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (cartprod @ X1 @ X2)) => ((kpair @ (kfst @ X0) @ (ksnd @ X0)) = X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.49/1.60 inference(rectify,[status(thm)],[f186])).
% 2.49/1.60 thf(f186,negated_conjecture,(
% 2.49/1.60 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => ! [X10 : $i,X3 : $i,X4 : $i] : ((in @ X10 @ (cartprod @ X3 @ X4)) => ((kpair @ (kfst @ X10) @ (ksnd @ X10)) = X10))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.49/1.60 inference(negated_conjecture,[status(cth)],[f185])).
% 2.49/1.60 thf(f185,conjecture,(
% 2.49/1.60 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => ! [X10 : $i,X3 : $i,X4 : $i] : ((in @ X10 @ (cartprod @ X3 @ X4)) => ((kpair @ (kfst @ X10) @ (ksnd @ X10)) = X10)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.49/1.60 file('/export/starexec/sandbox/tmp/tmp.7RtdXlarj4/DTF2THF_4583.p',cartprodpairsurjEq)).
% 2.49/1.60 thf(f2612,plain,(
% 2.49/1.60 (cartprodmempair != $true) | ($true = (iskpair @ sK407))),
% 2.49/1.60 inference(trivial_inequality_removal,[status(thm)],[f2610])).
% 2.49/1.60 thf(f2610,plain,(
% 2.49/1.60 ($true != $true) | ($true = (iskpair @ sK407)) | (cartprodmempair != $true)),
% 2.49/1.60 inference(superposition,[status(thm)],[f2069,f2245])).
% 2.49/1.60 thf(f2245,plain,(
% 2.49/1.60 ($true = (in @ sK407 @ (cartprod @ sK408 @ sK409)))),
% 2.49/1.60 inference(cnf_transformation,[status(thm)],[f1443])).
% 2.49/1.60 thf(f2069,plain,(
% 2.49/1.60 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X1 @ (cartprod @ X2 @ X0))) | ((iskpair @ X1) = $true) | (cartprodmempair != $true)) )),
% 2.49/1.60 inference(cnf_transformation,[status(thm)],[f1324])).
% 2.49/1.60 thf(f1324,plain,(
% 2.49/1.60 (! [X0 : $i,X1 : $i,X2 : $i] : (((iskpair @ X1) = $true) | ($true != (in @ X1 @ (cartprod @ X2 @ X0)))) | (cartprodmempair != $true)) & ((cartprodmempair = $true) | (((iskpair @ sK330) != $true) & ($true = (in @ sK330 @ (cartprod @ sK331 @ sK329)))))),
% 2.49/1.60 inference(skolemisation,[status(esa),new_symbols(skolem,[sK329,sK330,sK331])],[f1322,f1323])).
% 2.49/1.60 thf(f1323,plain,(
% 2.49/1.60 ? [X3 : $i,X4 : $i,X5 : $i] : (((iskpair @ X4) != $true) & ((in @ X4 @ (cartprod @ X5 @ X3)) = $true)) => (((iskpair @ sK330) != $true) & ($true = (in @ sK330 @ (cartprod @ sK331 @ sK329))))),
% 2.49/1.60 introduced(definition,[],[choice_axiom])).
% 2.49/1.60 thf(f1322,plain,(
% 2.49/1.60 (! [X0 : $i,X1 : $i,X2 : $i] : (((iskpair @ X1) = $true) | ($true != (in @ X1 @ (cartprod @ X2 @ X0)))) | (cartprodmempair != $true)) & ((cartprodmempair = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((iskpair @ X4) != $true) & ((in @ X4 @ (cartprod @ X5 @ X3)) = $true)))),
% 2.49/1.60 inference(rectify,[status(thm)],[f1321])).
% 2.49/1.60 thf(f1321,plain,(
% 2.49/1.60 (! [X0 : $i,X1 : $i,X2 : $i] : (((iskpair @ X1) = $true) | ($true != (in @ X1 @ (cartprod @ X2 @ X0)))) | (cartprodmempair != $true)) & ((cartprodmempair = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((iskpair @ X1) != $true) & ($true = (in @ X1 @ (cartprod @ X2 @ X0)))))),
% 2.49/1.60 inference(nnf_transformation,[status(thm)],[f590])).
% 2.49/1.60 thf(f590,plain,(
% 2.49/1.60 ! [X0 : $i,X1 : $i,X2 : $i] : (((iskpair @ X1) = $true) | ($true != (in @ X1 @ (cartprod @ X2 @ X0)))) <=> (cartprodmempair = $true)),
% 2.49/1.60 inference(ennf_transformation,[status(thm)],[f423])).
% 2.49/1.60 thf(f423,plain,(
% 2.49/1.60 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (cartprod @ X2 @ X0))) => ((iskpair @ X1) = $true)) <=> (cartprodmempair = $true)),
% 2.49/1.60 inference(fool_elimination,[status(thm)],[f422])).
% 2.49/1.60 thf(f422,plain,(
% 2.49/1.60 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (cartprod @ X2 @ X0)) => (iskpair @ X1)) = cartprodmempair)),
% 2.49/1.60 inference(rectify,[status(thm)],[f151])).
% 2.49/1.60 thf(f151,axiom,(
% 2.49/1.60 (! [X4 : $i,X10 : $i,X3 : $i] : ((in @ X10 @ (cartprod @ X3 @ X4)) => (iskpair @ X10)) = cartprodmempair)),
% 2.49/1.60 file('/export/starexec/sandbox/tmp/tmp.7RtdXlarj4/DTF2THF_4583.p',cartprodmempair)).
% 2.49/1.60 thf(f2607,plain,(
% 2.49/1.60 ($true != (iskpair @ sK407))),
% 2.49/1.60 inference(trivial_inequality_removal,[status(thm)],[f2606])).
% 2.49/1.60 thf(f2606,plain,(
% 2.49/1.60 ($true != (iskpair @ sK407)) | ($true != $true)),
% 2.49/1.60 inference(forward_demodulation,[status(thm)],[f2605,f2175])).
% 2.49/1.60 thf(f2175,plain,(
% 2.49/1.60 (kpairsurjEq = $true)),
% 2.49/1.60 inference(cnf_transformation,[status(thm)],[f1443])).
% 2.49/1.60 thf(f2605,plain,(
% 2.49/1.60 (kpairsurjEq != $true) | ($true != (iskpair @ sK407))),
% 2.49/1.60 inference(trivial_inequality_removal,[status(thm)],[f2604])).
% 2.49/1.60 thf(f2604,plain,(
% 2.49/1.60 (kpairsurjEq != $true) | ($true != (iskpair @ sK407)) | (sK407 != sK407)),
% 2.49/1.60 inference(superposition,[status(thm)],[f2246,f1745])).
% 2.49/1.60 thf(f1745,plain,(
% 2.49/1.60 ( ! [X1 : $i] : (((kpair @ (kfst @ X1) @ (ksnd @ X1)) = X1) | (kpairsurjEq != $true) | ((iskpair @ X1) != $true)) )),
% 2.49/1.60 inference(cnf_transformation,[status(thm)],[f953])).
% 2.49/1.60 thf(f953,plain,(
% 2.49/1.60 ((kpairsurjEq = $true) | ((sK103 != (kpair @ (kfst @ sK103) @ (ksnd @ sK103))) & ((iskpair @ sK103) = $true))) & (! [X1 : $i] : (((kpair @ (kfst @ X1) @ (ksnd @ X1)) = X1) | ((iskpair @ X1) != $true)) | (kpairsurjEq != $true))),
% 2.49/1.60 inference(skolemisation,[status(esa),new_symbols(skolem,[sK103])],[f951,f952])).
% 2.49/1.60 thf(f952,plain,(
% 2.49/1.60 ? [X0 : $i] : (((kpair @ (kfst @ X0) @ (ksnd @ X0)) != X0) & ((iskpair @ X0) = $true)) => ((sK103 != (kpair @ (kfst @ sK103) @ (ksnd @ sK103))) & ((iskpair @ sK103) = $true))),
% 2.49/1.60 introduced(definition,[],[choice_axiom])).
% 2.49/1.60 thf(f951,plain,(
% 2.49/1.60 ((kpairsurjEq = $true) | ? [X0 : $i] : (((kpair @ (kfst @ X0) @ (ksnd @ X0)) != X0) & ((iskpair @ X0) = $true))) & (! [X1 : $i] : (((kpair @ (kfst @ X1) @ (ksnd @ X1)) = X1) | ((iskpair @ X1) != $true)) | (kpairsurjEq != $true))),
% 2.49/1.60 inference(rectify,[status(thm)],[f950])).
% 2.49/1.60 thf(f950,plain,(
% 2.49/1.60 ((kpairsurjEq = $true) | ? [X0 : $i] : (((kpair @ (kfst @ X0) @ (ksnd @ X0)) != X0) & ((iskpair @ X0) = $true))) & (! [X0 : $i] : (((kpair @ (kfst @ X0) @ (ksnd @ X0)) = X0) | ((iskpair @ X0) != $true)) | (kpairsurjEq != $true))),
% 2.49/1.60 inference(nnf_transformation,[status(thm)],[f677])).
% 2.49/1.60 thf(f677,plain,(
% 2.49/1.60 (kpairsurjEq = $true) <=> ! [X0 : $i] : (((kpair @ (kfst @ X0) @ (ksnd @ X0)) = X0) | ((iskpair @ X0) != $true))),
% 2.49/1.60 inference(ennf_transformation,[status(thm)],[f459])).
% 2.49/1.60 thf(f459,plain,(
% 2.49/1.60 ! [X0 : $i] : (((iskpair @ X0) = $true) => ((kpair @ (kfst @ X0) @ (ksnd @ X0)) = X0)) <=> (kpairsurjEq = $true)),
% 2.49/1.60 inference(fool_elimination,[status(thm)],[f458])).
% 2.49/1.60 thf(f458,plain,(
% 2.49/1.60 (kpairsurjEq = ! [X0 : $i] : ((iskpair @ X0) => ((kpair @ (kfst @ X0) @ (ksnd @ X0)) = X0)))),
% 2.49/1.60 inference(rectify,[status(thm)],[f178])).
% 2.49/1.60 thf(f178,axiom,(
% 2.49/1.60 (kpairsurjEq = ! [X10 : $i] : ((iskpair @ X10) => ((kpair @ (kfst @ X10) @ (ksnd @ X10)) = X10)))),
% 2.49/1.60 file('/export/starexec/sandbox/tmp/tmp.7RtdXlarj4/DTF2THF_4583.p',kpairsurjEq)).
% 2.49/1.60 thf(f2246,plain,(
% 2.49/1.60 (sK407 != (kpair @ (kfst @ sK407) @ (ksnd @ sK407)))),
% 2.49/1.60 inference(cnf_transformation,[status(thm)],[f1443])).
% 2.49/1.60 % SZS output end Proof for DTF2THF_4583
% 2.49/1.60 % (4711)------------------------------
% 2.49/1.60 % (4711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.49/1.60 % (4711)Termination reason: Refutation
% 2.49/1.60
% 2.49/1.60 % (4711)Memory used [KB]: 8443
% 2.49/1.60 % (4711)Time elapsed: 0.103 s
% 2.49/1.60 % (4711)Instructions burned: 203 (million)
% 2.49/1.60 % (4711)------------------------------
% 2.49/1.60 % (4711)------------------------------
% 2.49/1.60 % (4688)Success in time 0.185 s
%------------------------------------------------------------------------------