%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU708^1 : 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 : n022.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 09:36:33 AM UTC 2026
% Result : Theorem 9.05s 2.87s
% Output : Refutation 9.05s
% Verified :
% SZS Type : Refutation
% Derivation depth : 29
% Number of leaves : 14
% Syntax : Number of formulae : 71 ( 38 unt; 0 typ; 6 def)
% Number of atoms : 4573 (1309 equ; 0 cnn)
% Maximal formula atoms : 228 ( 64 avg)
% Number of connectives : 2495 ( 84 ~; 66 |; 743 &; 391 @)
% ( 6 <=>;1177 =>; 0 <=; 0 <~>)
% Maximal formula depth : 234 ( 29 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 391 ( 387 usr; 238 con; 0-4 aty)
% ( 28 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 165 ( 79 ^; 72 !; 14 ?; 165 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_286,type,
sK5: $i > ( $i > $o ) > $i ).
thf(func_def_287,type,
sK6: ( $i > $o ) > $i > $i ).
thf(func_def_288,type,
sK7: $i > $i > $i ).
thf(func_def_289,type,
sK8: $i > $i > $i ).
thf(func_def_290,type,
sK9: $i > $i > $i ).
thf(func_def_291,type,
sK10: ( $i > $i > $o ) > $i > $i ).
thf(func_def_292,type,
sK11: ( $i > $i > $o ) > $i > $i ).
thf(func_def_293,type,
sK12: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_294,type,
sK13: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_295,type,
sK14: ( $i > $i > $o ) > $i > $i ).
thf(func_def_296,type,
sK15: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_297,type,
sK16: ( $i > $o ) > $i ).
thf(func_def_298,type,
sK17: ( $i > $o ) > $i > $i ).
thf(func_def_299,type,
sK18: ( $i > $o ) > $i > $i ).
thf(func_def_300,type,
sK19: $i > $i > $i > $i ).
thf(func_def_301,type,
sK20: $i > $i > $i > $i ).
thf(func_def_302,type,
sK21: ( $i > $o ) > $i ).
thf(func_def_303,type,
sK22: ( $i > $o ) > $i > $i ).
thf(func_def_304,type,
sK23: ( $i > $o ) > $i > $i ).
thf(func_def_305,type,
sK24: $i > $i ).
thf(func_def_306,type,
sK25: $i > $i > $i ).
thf(func_def_307,type,
sK26: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_308,type,
sK27: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_309,type,
sK28: $i > $i > $i > $i > $i ).
thf(func_def_310,type,
sK29: $i > $i > $i > $i > $i ).
thf(func_def_311,type,
sK30: ( $i > $i ) > $i > $i > $i ).
thf(func_def_312,type,
sK31: $i > $i > ( $i > $i ) > $i ).
thf(func_def_313,type,
sK32: $i > $i ).
thf(func_def_314,type,
sK33: $i > $i > $i ).
thf(func_def_315,type,
sK34: $i > $i > $i ).
thf(func_def_316,type,
sK35: $i > $i > $i ).
thf(func_def_317,type,
sK36: $i > $i > $i > $i ).
thf(func_def_318,type,
sK37: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_319,type,
sK38: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_320,type,
sK39: ( $i > $o ) > $i > $i ).
thf(func_def_321,type,
sK40: ( $i > $o ) > $i > $i ).
thf(func_def_322,type,
sK41: ( $i > $o ) > $i > $i ).
thf(func_def_323,type,
sK42: ( $i > $o ) > $i > $i ).
thf(func_def_324,type,
sK43: ( $i > $o ) > $i > $i ).
thf(func_def_325,type,
sK44: ( $i > $o ) > $i > $i ).
thf(func_def_326,type,
sK45: ( $i > $o ) > $i > $i ).
thf(func_def_327,type,
sK46: ( $i > $o ) > $i > $i ).
thf(func_def_328,type,
sK47: $i > $o > $i > $i ).
thf(func_def_329,type,
sK48: $i > $i > $i ).
thf(func_def_330,type,
sK49: ( $i > $i ) > $i > $i > $i ).
thf(func_def_331,type,
sK50: ( $i > $o ) > $i ).
thf(func_def_332,type,
sK51: ( $i > $o ) > $i > $i ).
thf(func_def_333,type,
sK52: ( $i > $o ) > $i > $i ).
thf(func_def_334,type,
sK53: ( $i > $o ) > $i ).
thf(func_def_335,type,
sK54: ( $i > $o ) > $i > $i ).
thf(func_def_336,type,
sK55: ( $i > $o ) > $i > $i ).
thf(func_def_337,type,
sK56: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_338,type,
sK57: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_339,type,
sK58: $i > $i > $i ).
thf(func_def_340,type,
sK59: ( $i > $o ) > $i > $i ).
thf(func_def_341,type,
sK60: ( $i > $o ) > $i > $i ).
thf(func_def_342,type,
sK61: $i > $i > $i ).
thf(func_def_343,type,
sK62: ( $i > $o ) > $i ).
thf(func_def_344,type,
sK63: ( $i > $o ) > $i > $i ).
thf(func_def_345,type,
sK64: $i > ( $i > $o ) > $i ).
thf(func_def_346,type,
sK65: ( $i > $o ) > $i > $i ).
thf(func_def_347,type,
sK66: ( $i > $o ) > $i > $i ).
thf(func_def_348,type,
sK67: $i > ( $i > $i ) > $i > $i ).
thf(func_def_349,type,
sK68: $i > $i > ( $i > $o ) > $i ).
thf(func_def_350,type,
sK69: $i > ( $i > $i ) > $i > $i ).
thf(func_def_351,type,
sK70: $i > $i ).
thf(func_def_352,type,
sK71: $i > ( $i > $o ) > $i ).
thf(func_def_353,type,
sK72: $i > ( $i > $o ) > $i ).
thf(func_def_354,type,
sK73: $i > $i > $i > $i > $i ).
thf(func_def_355,type,
sK74: $i > $i > $i > $i > $i ).
thf(func_def_356,type,
sK75: $i > $i > $i > $i > $i ).
thf(func_def_357,type,
sK76: $i > $i > $i > $i > $i ).
thf(func_def_358,type,
sK77: $i > $i > $i > $i > $i ).
thf(func_def_359,type,
sK78: $i > $i > $i > $i > $i ).
thf(func_def_360,type,
sK79: $i > $i > $i > $i > $i ).
thf(func_def_361,type,
sK80: $i > $i > $i > $i > $i ).
thf(func_def_362,type,
sK81: $i > $i > $i > $i > $i ).
thf(func_def_363,type,
sK82: $i > $i > $i ).
thf(func_def_364,type,
sK83: ( $i > $o ) > $i > $i ).
thf(func_def_365,type,
sK84: $i > $i ).
thf(func_def_366,type,
sK85: $i > $i > $i > $i > $i ).
thf(func_def_367,type,
sK86: ( $i > $o ) > $i > $i ).
thf(func_def_368,type,
sK87: $i > $i > $i > $i > $i ).
thf(func_def_369,type,
sK88: $i > ( $i > $o ) > $i ).
thf(func_def_370,type,
sK89: $i > $i ).
thf(func_def_371,type,
sK90: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_372,type,
sK91: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_373,type,
sK92: ( $i > $o ) > $i ).
thf(func_def_374,type,
sK93: ( $i > $o ) > $i > $i ).
thf(func_def_375,type,
sK94: ( $i > $o ) > $i > $i ).
thf(func_def_376,type,
sK95: ( $i > $o ) > $i ).
thf(func_def_377,type,
sK96: ( $i > $o ) > $i > $i ).
thf(func_def_378,type,
sK97: ( $i > $o ) > $i > $i ).
thf(func_def_379,type,
sK98: ( $i > $o ) > $i ).
thf(func_def_380,type,
sK99: ( $i > $o ) > $i > $i ).
thf(func_def_381,type,
sK100: ( $i > $o ) > $i > $i ).
thf(func_def_382,type,
sK101: ( $i > $o ) > $i > $i ).
thf(func_def_383,type,
sK102: ( $i > $o ) > $i ).
thf(func_def_384,type,
sK103: ( $i > $o ) > $i ).
thf(func_def_385,type,
sK104: ( $i > $o ) > $i > $i ).
thf(func_def_386,type,
sK105: ( $i > $o ) > $i > $i ).
thf(func_def_387,type,
sK106: ( $i > $o ) > $i > $i ).
thf(func_def_388,type,
sK107: ( $i > $o ) > $i ).
thf(func_def_389,type,
sK108: $i > $i > $i > $i > $i ).
thf(func_def_390,type,
sK109: $i > $i > $i ).
thf(func_def_391,type,
sK110: ( $i > $o ) > $i > $i ).
thf(func_def_392,type,
sK111: ( $i > $o ) > $i ).
thf(func_def_393,type,
sK112: ( $i > $o ) > $i ).
thf(func_def_394,type,
sK113: ( $i > $o ) > $i ).
thf(func_def_395,type,
sK114: $i > $i > $i ).
thf(func_def_396,type,
sK115: $i > $i ).
thf(func_def_397,type,
sK116: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_398,type,
sK117: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_399,type,
sK118: $i > $i ).
thf(func_def_400,type,
sK119: $i > $i > $i ).
thf(func_def_401,type,
ph120:
!>[X0: $tType] : X0 ).
thf(f5029,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3985,f3995,f4055,f4061,f4076,f4999,f5028]) ).
thf(f5028,plain,
( ( $false
!= ( in @ sK3 @ sK2 ) )
| ( $true
!= ( in @ sK3 @ sK2 ) )
| ( $false = $true ) ),
introduced(definition,[],[theory_tautology_sat_conflict]) ).
thf(f4999,plain,
( spl4_37
| spl4_1
| ~ spl4_19
| spl4_22 ),
inference(avatar_split_clause,[status(thm)],[f4998,f4073,f4058,f3982,f4902]) ).
thf(f4902,definition,
( spl4_37
<=> ( $false
= ( in @ sK3 @ sK2 ) ) ),
introduced(definition,[new_symbols(naming,[spl4_37])],[avatar_definition]) ).
thf(f3982,definition,
( spl4_1
<=> ( ( setunion
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] :
( ( ( sK1 = Y0 )
& sK0 )
| ( ~ sK0
& ( sK3 = Y0 ) ) ) ) )
= sK3 ) ),
introduced(definition,[new_symbols(naming,[spl4_1])],[avatar_definition]) ).
thf(f4058,definition,
( spl4_19
<=> ( $true
= ( in @ sK1 @ sK2 ) ) ),
introduced(definition,[new_symbols(naming,[spl4_19])],[avatar_definition]) ).
thf(f4073,definition,
( spl4_22
<=> ( $true = sK0 ) ),
introduced(definition,[new_symbols(naming,[spl4_22])],[avatar_definition]) ).
thf(f4998,plain,
( ( $false
= ( in @ sK3 @ sK2 ) )
| spl4_1
| ~ spl4_19
| spl4_22 ),
inference(trivial_inequality_removal,[status(thm)],[f4997]) ).
thf(f4997,plain,
( ( $false
= ( in @ sK3 @ sK2 ) )
| ( $false = $true )
| spl4_1
| ~ spl4_19
| spl4_22 ),
inference(forward_demodulation,[status(thm)],[f4996,f4060]) ).
thf(f4060,plain,
( ( $true
= ( in @ sK1 @ sK2 ) )
| ~ spl4_19 ),
inference(avatar_component_clause,[status(thm)],[f4058]) ).
thf(f4996,plain,
( ( $false
= ( in @ sK1 @ sK2 ) )
| ( $false
= ( in @ sK3 @ sK2 ) )
| spl4_1
| spl4_22 ),
inference(subsumption_resolution,[status(thm)],[f4995,f3288]) ).
thf(f3288,plain,
! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ),
inference(equality_proxy_clausification,[status(thm)],[f3287]) ).
thf(f3287,plain,
! [X1: $i] :
( ( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f3286]) ).
thf(f3286,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( ( setunion @ ( setadjoin @ Y0 @ emptyset ) )
= Y0 )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1238]) ).
thf(f1238,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( setunion @ ( setadjoin @ Y0 @ emptyset ) )
= Y0 ) ) ),
inference(definition_unfolding,[status(thm)],[f1006,f768]) ).
thf(f768,plain,
setunionsingleton = $true,
inference(cnf_transformation,[status(thm)],[f692]) ).
thf(f692,plain,
( ( eqinunit = $true )
& ( setunionE = $true )
& ( cartprodpairmemER = $true )
& ( powersetI1 = $true )
& ( binunionE = $true )
& ( powersetsubset = $true )
& ( kfstpairEq = $true )
& ( powersetAx = $true )
& ( emptyinPowerset = $true )
& ( upairinpowunion = $true )
& ( dpsetconstrEL1 = $true )
& ( emptyI = $true )
& ( nonemptyImpWitness = $true )
& ( exu__Cong = $true )
& ( ex1E1 = $true )
& ( iffalseProp2 = $true )
& ( notequalI2 = $true )
& ( eqimpsubset1 = $true )
& ( subset2powerset = $true )
& ( setunion__Cong = $true )
& ( binunionIR = $true )
& ( binintersectER = $true )
& ( upairsetIL = $true )
& ( upairsetIR = $true )
& ( funcGraphProp1 = $true )
& ( dsetconstrI = $true )
& ( foundationAx = $true )
& ( singletoninpowunion = $true )
& ( omega__Cong = $true )
& ( $true
= ( in @ sK1 @ sK2 ) )
& ( sK3
!= ( if @ sK2 @ sK0 @ sK1 @ sK3 ) )
& ( $true
= ( in @ sK3 @ sK2 ) )
& ( $true != sK0 )
& ( setukpairIL = $true )
& ( emptyset__Cong = $true )
& ( upairset2E = $true )
& ( app = $true )
& ( funcinfuncset = $true )
& ( lam2p = $true )
& ( brelnall2 = $true )
& ( setadjoin__Cong = $true )
& ( subbreln = $true )
& ( setadjoinIR = $true )
& ( singletoninpowerset = $true )
& ( setadjoinIL = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairIR = $true )
& ( singletonsuniq = $true )
& ( inCongP = $true )
& ( symdiffI1 = $true )
& ( powersetE = $true )
& ( emptyE1 = $true )
& ( prop2set2propI = $true )
& ( setOfPairsIsBReln = $true )
& ( exuI3 = $true )
& ( eta2 = $true )
& ( omega0Ax = $true )
& ( binunionRsub = $true )
& ( binunionIL = $true )
& ( setadjoinSub2 = $true )
& ( ex1E2 = $true )
& ( uniqinunit = $true )
& ( upairsetE = $true )
& ( ifSingleton = $true )
& ( prop2setE = $true )
& ( ubforcartprodlem2 = $true )
& ( cartprodmempaircEq = $true )
& ( iftrue = $true )
& ( in__Cong = $true )
& ( powersetE1 = $true )
& ( setunionI = $true )
& ( binintersectSubset3 = $true )
& ( quantDeMorgan1 = $true )
& ( subsetE2 = $true )
& ( setminusELneg = $true )
& ( exuI2 = $true )
& ( binintersectSubset1 = $true )
& ( ex1I = $true )
& ( lamProp = $true )
& ( replAx = $true )
& ( binintersectI = $true )
& ( emptysetE = $true )
& ( beta2 = $true )
& ( powerset__Cong = $true )
& ( theeq = $true )
& ( exuEu = $true )
& ( ksndsingleton = $true )
& ( notdexE = $true )
& ( cartprodpairsurjEq = $true )
& ( kpairsurjEq = $true )
& ( setext = $true )
& ( exuI1 = $true )
& ( eqimpsubset2 = $true )
& ( binintersectLsub = $true )
& ( exuE3e = $true )
& ( subsetI2 = $true )
& ( setukpairinjL2 = $true )
& ( dpsetconstrEL2 = $true )
& ( exuE1 = $true )
& ( cartprodfstin = $true )
& ( descr__Cong = $true )
& ( emptysetsubset = $true )
& ( setminusI = $true )
& ( symdiffIneg1 = $true )
& ( setunionsingleton2 = $true )
& ( omegaSAx = $true )
& ( setukpairinjL = $true )
& ( setukpairinjR2 = $true )
& ( inPowerset = $true )
& ( notinsingleton = $true )
& ( setminusIRneg = $true )
& ( noeltsimpempty = $true )
& ( emptysetimpfalse = $true )
& ( setukpairinjR1 = $true )
& ( cartprodfstpairEq = $true )
& ( exuE3u = $true )
& ( subsetI1 = $true )
& ( emptyinunitempty = $true )
& ( funcext2 = $true )
& ( funcGraphProp2 = $true )
& ( notinemptyset = $true )
& ( dsetconstrER = $true )
& ( dpsetconstrSub = $true )
& ( binintersectSubset2 = $true )
& ( dpsetconstrER = $true )
& ( kpairiskpair = $true )
& ( ksndpairEq = $true )
& ( cartprodmempair1 = $true )
& ( prop2setI = $true )
& ( setbeta = $true )
& ( eqbreln = $true )
& ( funcGraphProp4 = $true )
& ( subPowSU = $true )
& ( cartprodpairin = $true )
& ( subsetemptysetimpeq = $true )
& ( iffalseProp1 = $true )
& ( omegaIndAx = $true )
& ( dpsetconstrERa = $true )
& ( symdiffI2 = $true )
& ( cartprodsndpairEq = $true )
& ( setadjoinE = $true )
& ( setminusILneg = $true )
& ( upairset2IR = $true )
& ( sepInPowerset = $true )
& ( beta1 = $true )
& ( emptyInPowerset = $true )
& ( funcGraphProp3 = $true )
& ( setunionE2 = $true )
& ( descrp = $true )
& ( setminusSubset2 = $true )
& ( setminusSubset1 = $true )
& ( singletonsubset = $true )
& ( theprop = $true )
& ( eta1 = $true )
& ( notequalI1 = $true )
& ( kfstsingleton = $true )
& ( ifp = $true )
& ( funcext = $true )
& ( ap2apEq2 = $true )
& ( quantDeMorgan4 = $true )
& ( ap2apEq1 = $true )
& ( subsetTrans = $true )
& ( notdallE = $true )
& ( cartprodpairmemEL = $true )
& ( dsetconstr__Cong = $true )
& ( lam2lamEq = $true )
& ( nonemptyI = $true )
& ( setukpairinjL1 = $true )
& ( notsubsetI = $true )
& ( ubforcartprodlem3 = $true )
& ( dsetconstrEL = $true )
& ( binunionLsub = $true )
& ( symdiffIneg2 = $true )
& ( upairsubunion = $true )
& ( singletonprop = $true )
& ( setadjoinAx = $true )
& ( nonemptyI1 = $true )
& ( powersetI = $true )
& ( binintersectEL = $true )
& ( ap2p = $true )
& ( quantDeMorgan3 = $true )
& ( ex1I2 = $true )
& ( subsetRefl = $true )
& ( exuE2 = $true )
& ( setadjoinSub = $true )
& ( apProp = $true )
& ( lamp = $true )
& ( subsetE = $true )
& ( funcImageSingleton = $true )
& ( setunionsingleton = $true )
& ( setukpairinjR = $true )
& ( setminusER = $true )
& ( binintersectRsub = $true )
& ( iftrueProp1 = $true )
& ( setextsub = $true )
& ( upairequniteq = $true )
& ( wellorderingAx = $true )
& ( brelnall1 = $true )
& ( funcextLem = $true )
& ( setminusERneg = $true )
& ( setukpairinjR12 = $true )
& ( quantDeMorgan2 = $true )
& ( binunionEcases = $true )
& ( symdiffE = $true )
& ( secondinupair = $true )
& ( vacuousDall = $true )
& ( setoftrueEq = $true )
& ( setunionsingleton1 = $true )
& ( singletonsswitch = $true )
& ( disjointsetsI1 = $true )
& ( binintersectSubset5 = $true )
& ( sepSubset = $true )
& ( dpsetconstrI = $true )
& ( cartprodsndin = $true )
& ( kpairp = $true )
& ( iftrueProp2 = $true )
& ( binintersectSubset4 = $true )
& ( setunionAx = $true )
& ( infuncsetfunc = $true )
& ( setminusEL = $true )
& ( setminusLsub = $true )
& ( bs114d = $true )
& ( setextAx = $true )
& ( cartprodmempair = $true )
& ( emptysetAx = $true )
& ( nonemptyE1 = $true )
& ( setadjoinOr = $true )
& ( setukpairinjR11 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f689,f691,f690]) ).
thf(f690,plain,
( ? [X0: $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
& ? [X3: $i] :
( ( ( if @ X2 @ X0 @ X1 @ X3 )
!= X3 )
& ( ( in @ X3 @ X2 )
= $true )
& ( $true != X0 ) ) )
=> ( ( $true
= ( in @ sK1 @ sK2 ) )
& ? [X3: $i] :
( ( ( if @ sK2 @ sK0 @ sK1 @ X3 )
!= X3 )
& ( ( in @ X3 @ sK2 )
= $true )
& ( $true != sK0 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f691,plain,
( ? [X3: $i] :
( ( ( if @ sK2 @ sK0 @ sK1 @ X3 )
!= X3 )
& ( ( in @ X3 @ sK2 )
= $true )
& ( $true != sK0 ) )
=> ( ( sK3
!= ( if @ sK2 @ sK0 @ sK1 @ sK3 ) )
& ( $true
= ( in @ sK3 @ sK2 ) )
& ( $true != sK0 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f689,plain,
( ( eqinunit = $true )
& ( setunionE = $true )
& ( cartprodpairmemER = $true )
& ( powersetI1 = $true )
& ( binunionE = $true )
& ( powersetsubset = $true )
& ( kfstpairEq = $true )
& ( powersetAx = $true )
& ( emptyinPowerset = $true )
& ( upairinpowunion = $true )
& ( dpsetconstrEL1 = $true )
& ( emptyI = $true )
& ( nonemptyImpWitness = $true )
& ( exu__Cong = $true )
& ( ex1E1 = $true )
& ( iffalseProp2 = $true )
& ( notequalI2 = $true )
& ( eqimpsubset1 = $true )
& ( subset2powerset = $true )
& ( setunion__Cong = $true )
& ( binunionIR = $true )
& ( binintersectER = $true )
& ( upairsetIL = $true )
& ( upairsetIR = $true )
& ( funcGraphProp1 = $true )
& ( dsetconstrI = $true )
& ( foundationAx = $true )
& ( singletoninpowunion = $true )
& ( omega__Cong = $true )
& ? [X0: $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
& ? [X3: $i] :
( ( ( if @ X2 @ X0 @ X1 @ X3 )
!= X3 )
& ( ( in @ X3 @ X2 )
= $true )
& ( $true != X0 ) ) )
& ( setukpairIL = $true )
& ( emptyset__Cong = $true )
& ( upairset2E = $true )
& ( app = $true )
& ( funcinfuncset = $true )
& ( lam2p = $true )
& ( brelnall2 = $true )
& ( setadjoin__Cong = $true )
& ( subbreln = $true )
& ( setadjoinIR = $true )
& ( singletoninpowerset = $true )
& ( setadjoinIL = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairIR = $true )
& ( singletonsuniq = $true )
& ( inCongP = $true )
& ( symdiffI1 = $true )
& ( powersetE = $true )
& ( emptyE1 = $true )
& ( prop2set2propI = $true )
& ( setOfPairsIsBReln = $true )
& ( exuI3 = $true )
& ( eta2 = $true )
& ( omega0Ax = $true )
& ( binunionRsub = $true )
& ( binunionIL = $true )
& ( setadjoinSub2 = $true )
& ( ex1E2 = $true )
& ( uniqinunit = $true )
& ( upairsetE = $true )
& ( ifSingleton = $true )
& ( prop2setE = $true )
& ( ubforcartprodlem2 = $true )
& ( cartprodmempaircEq = $true )
& ( iftrue = $true )
& ( in__Cong = $true )
& ( powersetE1 = $true )
& ( setunionI = $true )
& ( binintersectSubset3 = $true )
& ( quantDeMorgan1 = $true )
& ( subsetE2 = $true )
& ( setminusELneg = $true )
& ( exuI2 = $true )
& ( binintersectSubset1 = $true )
& ( ex1I = $true )
& ( lamProp = $true )
& ( replAx = $true )
& ( binintersectI = $true )
& ( emptysetE = $true )
& ( beta2 = $true )
& ( powerset__Cong = $true )
& ( theeq = $true )
& ( exuEu = $true )
& ( ksndsingleton = $true )
& ( notdexE = $true )
& ( cartprodpairsurjEq = $true )
& ( kpairsurjEq = $true )
& ( setext = $true )
& ( exuI1 = $true )
& ( eqimpsubset2 = $true )
& ( binintersectLsub = $true )
& ( exuE3e = $true )
& ( subsetI2 = $true )
& ( setukpairinjL2 = $true )
& ( dpsetconstrEL2 = $true )
& ( exuE1 = $true )
& ( cartprodfstin = $true )
& ( descr__Cong = $true )
& ( emptysetsubset = $true )
& ( setminusI = $true )
& ( symdiffIneg1 = $true )
& ( setunionsingleton2 = $true )
& ( omegaSAx = $true )
& ( setukpairinjL = $true )
& ( setukpairinjR2 = $true )
& ( inPowerset = $true )
& ( notinsingleton = $true )
& ( setminusIRneg = $true )
& ( noeltsimpempty = $true )
& ( emptysetimpfalse = $true )
& ( setukpairinjR1 = $true )
& ( cartprodfstpairEq = $true )
& ( exuE3u = $true )
& ( subsetI1 = $true )
& ( emptyinunitempty = $true )
& ( funcext2 = $true )
& ( funcGraphProp2 = $true )
& ( notinemptyset = $true )
& ( dsetconstrER = $true )
& ( dpsetconstrSub = $true )
& ( binintersectSubset2 = $true )
& ( dpsetconstrER = $true )
& ( kpairiskpair = $true )
& ( ksndpairEq = $true )
& ( cartprodmempair1 = $true )
& ( prop2setI = $true )
& ( setbeta = $true )
& ( eqbreln = $true )
& ( funcGraphProp4 = $true )
& ( subPowSU = $true )
& ( cartprodpairin = $true )
& ( subsetemptysetimpeq = $true )
& ( iffalseProp1 = $true )
& ( omegaIndAx = $true )
& ( dpsetconstrERa = $true )
& ( symdiffI2 = $true )
& ( cartprodsndpairEq = $true )
& ( setadjoinE = $true )
& ( setminusILneg = $true )
& ( upairset2IR = $true )
& ( sepInPowerset = $true )
& ( beta1 = $true )
& ( emptyInPowerset = $true )
& ( funcGraphProp3 = $true )
& ( setunionE2 = $true )
& ( descrp = $true )
& ( setminusSubset2 = $true )
& ( setminusSubset1 = $true )
& ( singletonsubset = $true )
& ( theprop = $true )
& ( eta1 = $true )
& ( notequalI1 = $true )
& ( kfstsingleton = $true )
& ( ifp = $true )
& ( funcext = $true )
& ( ap2apEq2 = $true )
& ( quantDeMorgan4 = $true )
& ( ap2apEq1 = $true )
& ( subsetTrans = $true )
& ( notdallE = $true )
& ( cartprodpairmemEL = $true )
& ( dsetconstr__Cong = $true )
& ( lam2lamEq = $true )
& ( nonemptyI = $true )
& ( setukpairinjL1 = $true )
& ( notsubsetI = $true )
& ( ubforcartprodlem3 = $true )
& ( dsetconstrEL = $true )
& ( binunionLsub = $true )
& ( symdiffIneg2 = $true )
& ( upairsubunion = $true )
& ( singletonprop = $true )
& ( setadjoinAx = $true )
& ( nonemptyI1 = $true )
& ( powersetI = $true )
& ( binintersectEL = $true )
& ( ap2p = $true )
& ( quantDeMorgan3 = $true )
& ( ex1I2 = $true )
& ( subsetRefl = $true )
& ( exuE2 = $true )
& ( setadjoinSub = $true )
& ( apProp = $true )
& ( lamp = $true )
& ( subsetE = $true )
& ( funcImageSingleton = $true )
& ( setunionsingleton = $true )
& ( setukpairinjR = $true )
& ( setminusER = $true )
& ( binintersectRsub = $true )
& ( iftrueProp1 = $true )
& ( setextsub = $true )
& ( upairequniteq = $true )
& ( wellorderingAx = $true )
& ( brelnall1 = $true )
& ( funcextLem = $true )
& ( setminusERneg = $true )
& ( setukpairinjR12 = $true )
& ( quantDeMorgan2 = $true )
& ( binunionEcases = $true )
& ( symdiffE = $true )
& ( secondinupair = $true )
& ( vacuousDall = $true )
& ( setoftrueEq = $true )
& ( setunionsingleton1 = $true )
& ( singletonsswitch = $true )
& ( disjointsetsI1 = $true )
& ( binintersectSubset5 = $true )
& ( sepSubset = $true )
& ( dpsetconstrI = $true )
& ( cartprodsndin = $true )
& ( kpairp = $true )
& ( iftrueProp2 = $true )
& ( binintersectSubset4 = $true )
& ( setunionAx = $true )
& ( infuncsetfunc = $true )
& ( setminusEL = $true )
& ( setminusLsub = $true )
& ( bs114d = $true )
& ( setextAx = $true )
& ( cartprodmempair = $true )
& ( emptysetAx = $true )
& ( nonemptyE1 = $true )
& ( setadjoinOr = $true )
& ( setukpairinjR11 = $true ) ),
inference(flattening,[status(thm)],[f688]) ).
thf(f688,plain,
( ? [X0: $o,X1: $i,X2: $i] :
( ? [X3: $i] :
( ( ( if @ X2 @ X0 @ X1 @ X3 )
!= X3 )
& ( $true != X0 )
& ( ( in @ X3 @ X2 )
= $true ) )
& ( ( in @ X1 @ X2 )
= $true ) )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f687]) ).
thf(f687,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ! [X0: $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ( ( $true != X0 )
=> ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f471]) ).
thf(f471,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ! [X0: $o,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X2 )
= $true )
=> ( ( $true != X0 )
=> ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f470]) ).
thf(f470,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ! [X0: $o,X1: $i,X2: $i] :
( ( in @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( ~ X0
=> ( ( if @ X2 @ X0 @ X1 @ X3 )
= X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f231]) ).
thf(f231,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ! [X0: $o,X1: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ~ X0
=> ( ( if @ X3 @ X0 @ X1 @ X2 )
= X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f230]) ).
thf(f230,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ! [X0: $o,X1: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ~ X0
=> ( ( if @ X3 @ X0 @ X1 @ X2 )
= X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.K93CCo4jzK/DTF2THF_21305.p',iffalse) ).
thf(f1006,plain,
( setunionsingleton
= ( !! @ $i
@ ^ [Y0: $i] :
( ( setunion @ ( setadjoin @ Y0 @ emptyset ) )
= Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f469]) ).
thf(f469,plain,
( setunionsingleton
= ( !! @ $i
@ ^ [Y0: $i] :
( ( setunion @ ( setadjoin @ Y0 @ emptyset ) )
= Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f468]) ).
thf(f468,plain,
( setunionsingleton
= ( ! [X0: $i] :
( ( setunion @ ( setadjoin @ X0 @ emptyset ) )
= X0 ) ) ),
inference(rectify,[status(thm)],[f155]) ).
thf(f155,axiom,
( setunionsingleton
= ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.K93CCo4jzK/DTF2THF_21305.p',setunionsingleton) ).
thf(f4995,plain,
( ( $false
= ( in @ sK1 @ sK2 ) )
| ( $false
= ( in @ sK3 @ sK2 ) )
| ( ( setunion @ ( setadjoin @ sK3 @ emptyset ) )
!= sK3 )
| spl4_1
| spl4_22 ),
inference(subsumption_resolution,[status(thm)],[f4951,f4075]) ).
thf(f4075,plain,
( ( $true != sK0 )
| spl4_22 ),
inference(avatar_component_clause,[status(thm)],[f4073]) ).
thf(f4951,plain,
( ( $false
= ( in @ sK1 @ sK2 ) )
| ( $true = sK0 )
| ( ( setunion @ ( setadjoin @ sK3 @ emptyset ) )
!= sK3 )
| ( $false
= ( in @ sK3 @ sK2 ) )
| spl4_1 ),
inference(superposition,[status(thm)],[f3984,f2695]) ).
thf(f2695,plain,
! [X2: $i,X3: $o,X1: $i,X4: $i] :
( ( ( dsetconstr @ X2
@ ^ [Y0: $i] :
( ( ( X1 = Y0 )
& X3 )
| ( ~ X3
& ( X4 = Y0 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) )
| ( $false
= ( in @ X4 @ X2 ) )
| ( $true = X3 )
| ( ( in @ X1 @ X2 )
= $false ) ),
inference(not_proxy_clausification,[status(thm)],[f2694]) ).
thf(f2694,plain,
! [X2: $i,X3: $o,X1: $i,X4: $i] :
( ( $false = ( ~ X3 ) )
| ( $false
= ( in @ X4 @ X2 ) )
| ( ( dsetconstr @ X2
@ ^ [Y0: $i] :
( ( ( X1 = Y0 )
& X3 )
| ( ~ X3
& ( X4 = Y0 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) )
| ( ( in @ X1 @ X2 )
= $false ) ),
inference(equality_proxy_clausification,[status(thm)],[f2693]) ).
thf(f2693,plain,
! [X2: $i,X3: $o,X1: $i,X4: $i] :
( ( $true
= ( ( setadjoin @ X4 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y0: $i] :
( ( ( X1 = Y0 )
& X3 )
| ( ~ X3
& ( X4 = Y0 ) ) ) ) ) )
| ( $false = ( ~ X3 ) )
| ( ( in @ X1 @ X2 )
= $false )
| ( $false
= ( in @ X4 @ X2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f2692]) ).
thf(f2692,plain,
! [X2: $i,X3: $o,X1: $i,X4: $i] :
( ( ( in @ X1 @ X2 )
= $false )
| ( $true
= ( ~ X3
=> ( ( setadjoin @ X4 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y0: $i] :
( ( ( X1 = Y0 )
& X3 )
| ( ~ X3
& ( X4 = Y0 ) ) ) ) ) ) )
| ( $false
= ( in @ X4 @ X2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f2691]) ).
thf(f2691,plain,
! [X2: $i,X3: $o,X1: $i,X4: $i] :
( ( ( ( in @ X4 @ X2 )
=> ( ~ X3
=> ( ( setadjoin @ X4 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y0: $i] :
( ( ( X1 = Y0 )
& X3 )
| ( ~ X3
& ( X4 = Y0 ) ) ) ) ) ) )
= $true )
| ( ( in @ X1 @ X2 )
= $false ) ),
inference(beta_eta_normalization,[status(thm)],[f2690]) ).
thf(f2690,plain,
! [X2: $i,X3: $o,X1: $i,X4: $i] :
( ( ( in @ X1 @ X2 )
= $false )
| ( ( ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ~ X3
=> ( ( setadjoin @ Y0 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y1: $i] :
( ( ( X1 = Y1 )
& X3 )
| ( ~ X3
& ( Y0 = Y1 ) ) ) ) ) ) )
@ X4 )
= $true ) ),
inference(pi_clausification,[status(thm)],[f2689]) ).
thf(f2689,plain,
! [X2: $i,X3: $o,X1: $i] :
( ( ( in @ X1 @ X2 )
= $false )
| ( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ~ X3
=> ( ( setadjoin @ Y0 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y1: $i] :
( ( ( X1 = Y1 )
& X3 )
| ( ~ X3
& ( Y0 = Y1 ) ) ) ) ) ) ) )
= $true ) ),
inference(binary_proxy_clausification,[status(thm)],[f2688]) ).
thf(f2688,plain,
! [X2: $i,X3: $o,X1: $i] :
( ( ( in @ X1 @ X2 )
=> ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ X2 )
=> ( ~ X3
=> ( ( setadjoin @ Y0 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y1: $i] :
( ( ( X1 = Y1 )
& X3 )
| ( ~ X3
& ( Y0 = Y1 ) ) ) ) ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f2687]) ).
thf(f2687,plain,
! [X2: $i,X3: $o,X1: $i] :
( ( ^ [Y0: $o] :
( ( in @ X1 @ X2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ~ Y0
=> ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y2: $i] :
( ( ( X1 = Y2 )
& Y0 )
| ( ~ Y0
& ( Y1 = Y2 ) ) ) ) ) ) ) ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f2686]) ).
thf(f2686,plain,
! [X2: $i,X1: $i] :
( $true
= ( !! @ $o
@ ^ [Y0: $o] :
( ( in @ X1 @ X2 )
=> ( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ X2 )
=> ( ~ Y0
=> ( ( setadjoin @ Y1 @ emptyset )
= ( dsetconstr @ X2
@ ^ [Y2: $i] :
( ( ( X1 = Y2 )
& Y0 )
| ( ~ Y0
& ( Y1 = Y2 ) ) ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2685]) ).
thf(f2685,plain,
! [X2: $i,X1: $i] :
( ( ^ [Y0: $i] :
( !! @ $o
@ ^ [Y1: $o] :
( ( in @ X1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ~ Y1
=> ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y3: $i] :
( ( ( X1 = Y3 )
& Y1 )
| ( ~ Y1
& ( Y2 = Y3 ) ) ) ) ) ) ) ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f2684]) ).
thf(f2684,plain,
! [X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $o
@ ^ [Y1: $o] :
( ( in @ X1 @ Y0 )
=> ( !! @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ~ Y1
=> ( ( setadjoin @ Y2 @ emptyset )
= ( dsetconstr @ Y0
@ ^ [Y3: $i] :
( ( ( X1 = Y3 )
& Y1 )
| ( ~ Y1
& ( Y2 = Y3 ) ) ) ) ) ) ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2683]) ).
thf(f2683,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $o
@ ^ [Y2: $o] :
( ( in @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ~ Y2
=> ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y4: $i] :
( ( ( Y0 = Y4 )
& Y2 )
| ( ~ Y2
& ( Y3 = Y4 ) ) ) ) ) ) ) ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1160]) ).
thf(f1160,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $o
@ ^ [Y2: $o] :
( ( in @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ~ Y2
=> ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y4: $i] :
( ( ( Y0 = Y4 )
& Y2 )
| ( ~ Y2
& ( Y3 = Y4 ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f942,f711]) ).
thf(f711,plain,
( iffalseProp2
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $o
@ ^ [Y2: $o] :
( ( in @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ~ Y2
=> ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y4: $i] :
( ( ( Y0 = Y4 )
& Y2 )
| ( ~ Y2
& ( Y3 = Y4 ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f621]) ).
thf(f621,plain,
( iffalseProp2
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( !! @ $o
@ ^ [Y2: $o] :
( ( in @ Y0 @ Y1 )
=> ( !! @ $i
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ~ Y2
=> ( ( setadjoin @ Y3 @ emptyset )
= ( dsetconstr @ Y1
@ ^ [Y4: $i] :
( ( ( Y0 = Y4 )
& Y2 )
| ( ~ Y2
& ( Y3 = Y4 ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f620]) ).
thf(f620,plain,
( iffalseProp2
= ( ! [X0: $o,X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ X0
=> ( ( setadjoin @ X3 @ emptyset )
= ( dsetconstr @ X1
@ ^ [X4: $i] :
( ( ( X3 = X4 )
& ~ X0 )
| ( X0
& ( X2 = X4 ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f222]) ).
thf(f222,axiom,
( iffalseProp2
= ( ! [X0: $o,X3: $i,X1: $i] :
( ( in @ X1 @ X3 )
=> ! [X2: $i] :
( ( in @ X2 @ X3 )
=> ( ~ X0
=> ( ( setadjoin @ X2 @ emptyset )
= ( dsetconstr @ X3
@ ^ [X8: $i] :
( ( ( X2 = X8 )
& ~ X0 )
| ( X0
& ( X1 = X8 ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.K93CCo4jzK/DTF2THF_21305.p',iffalseProp2) ).
thf(f942,plain,
iffalseProp2 = $true,
inference(cnf_transformation,[status(thm)],[f692]) ).
thf(f3984,plain,
( ( ( setunion
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] :
( ( ( sK1 = Y0 )
& sK0 )
| ( ~ sK0
& ( sK3 = Y0 ) ) ) ) )
!= sK3 )
| spl4_1 ),
inference(avatar_component_clause,[status(thm)],[f3982]) ).
thf(f4076,plain,
~ spl4_22,
inference(avatar_split_clause,[status(thm)],[f925,f4073]) ).
thf(f925,plain,
$true != sK0,
inference(cnf_transformation,[status(thm)],[f692]) ).
thf(f4061,plain,
spl4_19,
inference(avatar_split_clause,[status(thm)],[f928,f4058]) ).
thf(f928,plain,
( $true
= ( in @ sK1 @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f692]) ).
thf(f4055,plain,
spl4_18,
inference(avatar_split_clause,[status(thm)],[f926,f4052]) ).
thf(f4052,definition,
( spl4_18
<=> ( $true
= ( in @ sK3 @ sK2 ) ) ),
introduced(definition,[new_symbols(naming,[spl4_18])],[avatar_definition]) ).
thf(f926,plain,
( $true
= ( in @ sK3 @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f692]) ).
thf(f3995,plain,
~ spl4_3,
inference(avatar_split_clause,[status(thm)],[f232,f3992]) ).
thf(f3992,definition,
( spl4_3
<=> ( $false = $true ) ),
introduced(definition,[new_symbols(naming,[spl4_3])],[avatar_definition]) ).
thf(f232,plain,
$false != $true,
introduced(definition,[],[fool_axiom]) ).
thf(f3985,plain,
~ spl4_1,
inference(avatar_split_clause,[status(thm)],[f3802,f3982]) ).
thf(f3802,plain,
( ( setunion
@ ( dsetconstr @ sK2
@ ^ [Y0: $i] :
( ( ( sK1 = Y0 )
& sK0 )
| ( ~ sK0
& ( sK3 = Y0 ) ) ) ) )
!= sK3 ),
inference(beta_eta_normalization,[status(thm)],[f1165]) ).
thf(f1165,plain,
( sK3
!= ( ^ [Y0: $i,Y1: $o,Y2: $i,Y3: $i] :
( setunion
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] :
( ( ( Y2 = Y4 )
& Y1 )
| ( ~ Y1
& ( Y3 = Y4 ) ) ) ) )
@ sK2
@ sK0
@ sK1
@ sK3 ) ),
inference(definition_unfolding,[status(thm)],[f927,f965]) ).
thf(f965,plain,
( if
= ( ^ [Y0: $i,Y1: $o,Y2: $i,Y3: $i] :
( setunion
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] :
( ( ( Y2 = Y4 )
& Y1 )
| ( ~ Y1
& ( Y3 = Y4 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f272]) ).
thf(f272,plain,
( if
= ( ^ [Y0: $i,Y1: $o,Y2: $i,Y3: $i] :
( setunion
@ ( dsetconstr @ Y0
@ ^ [Y4: $i] :
( ( ( Y2 = Y4 )
& Y1 )
| ( ~ Y1
& ( Y3 = Y4 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f271]) ).
thf(f271,plain,
( ( ^ [X0: $i,X1: $o,X2: $i,X3: $i] :
( setunion
@ ( dsetconstr @ X0
@ ^ [X4: $i] :
( ( ( X3 = X4 )
& ~ X1 )
| ( X1
& ( X2 = X4 ) ) ) ) ) )
= if ),
inference(rectify,[status(thm)],[f226]) ).
thf(f226,axiom,
( ( ^ [X3: $i,X0: $o,X1: $i,X2: $i] :
( setunion
@ ( dsetconstr @ X3
@ ^ [X8: $i] :
( ( ( X2 = X8 )
& ~ X0 )
| ( X0
& ( X1 = X8 ) ) ) ) ) )
= if ),
file('/export/starexec/sandbox/tmp/tmp.K93CCo4jzK/DTF2THF_21305.p',if) ).
thf(f927,plain,
( sK3
!= ( if @ sK2 @ sK0 @ sK1 @ sK3 ) ),
inference(cnf_transformation,[status(thm)],[f692]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU708^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n022.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:31:36 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : SEU708^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.46 % Domain : Set Theory
% 0.19/0.46 % Problem : Conditionals
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English : (! A:i.! phi:o.! x:i.in x A -> (! y:i.in y A -> ~phi ->
% 0.19/0.46 % if A phi x y = y))
% 0.19/0.46
% 0.19/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.46 % Source : [Bro08]
% 0.19/0.46 % Names : ZFC210g [Bro08]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.50 v8.2.0, 0.46 v8.1.0, 0.45 v7.5.0, 0.71 v7.4.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.86 v6.0.0, 0.71 v5.5.0, 0.67 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.19/0.46 % Syntax : Number of formulae : 492 ( 229 unt; 262 typ; 229 def)
% 0.19/0.46 % Number of atoms : 1278 ( 355 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 227 ( 5 avg)
% 0.19/0.46 % Number of connectives : 2469 ( 56 ~; 13 |; 46 &;1740 @)
% 0.19/0.46 % ( 14 <=>; 600 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 233 ( 2 avg)
% 0.19/0.46 % Number of types : 2 ( 0 usr)
% 0.19/0.46 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 265 ( 262 usr; 228 con; 0-4 aty)
% 0.19/0.46 % Number of variables : 773 ( 79 ^; 659 !; 35 ?; 773 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : http://mathgate.info/detsetitem.php?id=264
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 thf(in_type,type,
% 0.19/0.46 in: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu_type,type,
% 0.19/0.46 exu: ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu,definition,
% 0.19/0.46 ( exu
% 0.19/0.46 = ( ^ [Xphi: $i > $o] :
% 0.19/0.46 ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextAx_type,type,
% 0.19/0.46 setextAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setextAx,definition,
% 0.19/0.46 ( setextAx
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 <=> ( in @ Xx @ B ) )
% 0.19/0.46 => ( A = B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset_type,type,
% 0.19/0.46 emptyset: $i ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetAx_type,type,
% 0.19/0.46 emptysetAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetAx,definition,
% 0.19/0.46 ( emptysetAx
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin_type,type,
% 0.19/0.46 setadjoin: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinAx_type,type,
% 0.19/0.46 setadjoinAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinAx,definition,
% 0.19/0.46 ( setadjoinAx
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 <=> ( ( Xy = Xx )
% 0.19/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powerset_type,type,
% 0.19/0.46 powerset: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(powersetAx_type,type,
% 0.19/0.46 powersetAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetAx,definition,
% 0.19/0.46 ( powersetAx
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 <=> ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunion_type,type,
% 0.19/0.46 setunion: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setunionAx_type,type,
% 0.19/0.46 setunionAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionAx,definition,
% 0.19/0.46 ( setunionAx
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 <=> ? [B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omega_type,type,
% 0.19/0.46 omega: $i ).
% 0.19/0.46
% 0.19/0.46 thf(omega0Ax_type,type,
% 0.19/0.46 omega0Ax: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omega0Ax,definition,
% 0.19/0.46 ( omega0Ax
% 0.19/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omegaSAx_type,type,
% 0.19/0.46 omegaSAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omegaSAx,definition,
% 0.19/0.46 ( omegaSAx
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ omega )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omegaIndAx_type,type,
% 0.19/0.46 omegaIndAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omegaIndAx,definition,
% 0.19/0.46 ( omegaIndAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( ( in @ emptyset @ A )
% 0.19/0.46 & ! [Xx: $i] :
% 0.19/0.46 ( ( ( in @ Xx @ omega )
% 0.19/0.46 & ( in @ Xx @ A ) )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ omega )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(replAx_type,type,
% 0.19/0.46 replAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(replAx,definition,
% 0.19/0.46 ( replAx
% 0.19/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.46 => ? [B: $i] :
% 0.19/0.46 ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 <=> ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(foundationAx_type,type,
% 0.19/0.46 foundationAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(foundationAx,definition,
% 0.19/0.46 ( foundationAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.46 => ? [B: $i] :
% 0.19/0.46 ( ( in @ B @ A )
% 0.19/0.46 & ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(wellorderingAx_type,type,
% 0.19/0.46 wellorderingAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(wellorderingAx,definition,
% 0.19/0.46 ( wellorderingAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ? [B: $i] :
% 0.19/0.46 ( ! [C: $i] :
% 0.19/0.46 ( ( in @ C @ B )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ A ) ) )
% 0.19/0.46 & ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ( in @ Xx @ A )
% 0.19/0.46 & ( in @ Xy @ A ) )
% 0.19/0.46 => ( ! [C: $i] :
% 0.19/0.46 ( ( in @ C @ B )
% 0.19/0.46 => ( ( in @ Xx @ C )
% 0.19/0.46 <=> ( in @ Xy @ C ) ) )
% 0.19/0.46 => ( Xx = Xy ) ) )
% 0.19/0.46 & ! [C: $i,D: $i] :
% 0.19/0.46 ( ( ( in @ C @ B )
% 0.19/0.46 & ( in @ D @ B ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ D ) )
% 0.19/0.46 | ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ D )
% 0.19/0.46 => ( in @ Xx @ C ) ) ) )
% 0.19/0.46 & ! [C: $i] :
% 0.19/0.46 ( ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.46 => ? [D: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ D @ B )
% 0.19/0.46 & ( in @ Xx @ C )
% 0.19/0.46 & ~ ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ D )
% 0.19/0.46 & ( in @ Xy @ C ) )
% 0.19/0.46 & ! [E: $i] :
% 0.19/0.46 ( ( in @ E @ B )
% 0.19/0.46 => ( ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ E )
% 0.19/0.46 => ( in @ Xy @ D ) )
% 0.19/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(descr_type,type,
% 0.19/0.46 descr: ( $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(descrp_type,type,
% 0.19/0.46 descrp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(descrp,definition,
% 0.19/0.46 ( descrp
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 @ ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr_type,type,
% 0.19/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrI_type,type,
% 0.19/0.46 dsetconstrI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrI,definition,
% 0.19/0.46 ( dsetconstrI
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrEL_type,type,
% 0.19/0.46 dsetconstrEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrEL,definition,
% 0.19/0.46 ( dsetconstrEL
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrER_type,type,
% 0.19/0.46 dsetconstrER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrER,definition,
% 0.19/0.46 ( dsetconstrER
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE1_type,type,
% 0.19/0.46 exuE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE1,definition,
% 0.19/0.46 ( exuE1
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(prop2set_type,type,
% 0.19/0.46 prop2set: $o > $i ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setE_type,type,
% 0.19/0.46 prop2setE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setE,definition,
% 0.19/0.46 ( prop2setE
% 0.19/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetE_type,type,
% 0.19/0.47 emptysetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetE,definition,
% 0.19/0.47 ( emptysetE
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetimpfalse_type,type,
% 0.19/0.47 emptysetimpfalse: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetimpfalse,definition,
% 0.19/0.47 ( emptysetimpfalse
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => $false ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notinemptyset_type,type,
% 0.19/0.47 notinemptyset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notinemptyset,definition,
% 0.19/0.47 ( notinemptyset
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3e_type,type,
% 0.19/0.47 exuE3e: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3e,definition,
% 0.19/0.47 ( exuE3e
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setext_type,type,
% 0.19/0.47 setext: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setext,definition,
% 0.19/0.47 ( setext
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ B ) )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) )
% 0.19/0.47 => ( A = B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyI_type,type,
% 0.19/0.47 emptyI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyI,definition,
% 0.19/0.47 ( emptyI
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ A )
% 0.19/0.47 => ( A = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(noeltsimpempty_type,type,
% 0.19/0.47 noeltsimpempty: $o ).
% 0.19/0.47
% 0.19/0.47 thf(noeltsimpempty,definition,
% 0.19/0.47 ( noeltsimpempty
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ A )
% 0.19/0.47 => ( A = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setbeta_type,type,
% 0.19/0.47 setbeta: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setbeta,definition,
% 0.19/0.47 ( setbeta
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonempty_type,type,
% 0.19/0.47 nonempty: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonempty,definition,
% 0.19/0.47 ( nonempty
% 0.19/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyE1_type,type,
% 0.19/0.47 nonemptyE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyE1,definition,
% 0.19/0.47 ( nonemptyE1
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( nonempty @ A )
% 0.19/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI_type,type,
% 0.19/0.47 nonemptyI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI,definition,
% 0.19/0.47 ( nonemptyI
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 => ( nonempty
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI1_type,type,
% 0.19/0.47 nonemptyI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI1,definition,
% 0.19/0.47 ( nonemptyI1
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIL_type,type,
% 0.19/0.47 setadjoinIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIL,definition,
% 0.19/0.47 ( setadjoinIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinunitempty_type,type,
% 0.19/0.47 emptyinunitempty: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinunitempty,definition,
% 0.19/0.47 ( emptyinunitempty
% 0.19/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIR_type,type,
% 0.19/0.47 setadjoinIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIR,definition,
% 0.19/0.47 ( setadjoinIR
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinE_type,type,
% 0.19/0.47 setadjoinE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinE,definition,
% 0.19/0.47 ( setadjoinE
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.47 => ! [Xphi: $o] :
% 0.19/0.47 ( ( ( Xy = Xx )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => ( ( ( in @ Xy @ A )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinOr_type,type,
% 0.19/0.47 setadjoinOr: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinOr,definition,
% 0.19/0.47 ( setadjoinOr
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.47 => ( ( Xy = Xx )
% 0.19/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setoftrueEq_type,type,
% 0.19/0.47 setoftrueEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setoftrueEq,definition,
% 0.19/0.47 ( setoftrueEq
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : $true )
% 0.19/0.47 = A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI_type,type,
% 0.19/0.47 powersetI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI,definition,
% 0.19/0.47 ( powersetI
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) )
% 0.19/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinPowerset_type,type,
% 0.19/0.47 emptyinPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinPowerset,definition,
% 0.19/0.47 ( emptyinPowerset
% 0.19/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyInPowerset_type,type,
% 0.19/0.47 emptyInPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyInPowerset,definition,
% 0.19/0.47 ( emptyInPowerset
% 0.19/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE_type,type,
% 0.19/0.47 powersetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE,definition,
% 0.19/0.47 ( powersetE
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionI_type,type,
% 0.19/0.47 setunionI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionI,definition,
% 0.19/0.47 ( setunionI
% 0.19/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( ( in @ B @ A )
% 0.19/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionE_type,type,
% 0.19/0.47 setunionE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionE,definition,
% 0.19/0.47 ( setunionE
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.47 => ! [Xphi: $o] :
% 0.19/0.47 ( ! [B: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( ( in @ B @ A )
% 0.19/0.47 => Xphi ) )
% 0.19/0.47 => Xphi ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subPowSU_type,type,
% 0.19/0.47 subPowSU: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subPowSU,definition,
% 0.19/0.47 ( subPowSU
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE2_type,type,
% 0.19/0.47 exuE2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE2,definition,
% 0.19/0.47 ( exuE2
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyImpWitness_type,type,
% 0.19/0.47 nonemptyImpWitness: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyImpWitness,definition,
% 0.19/0.47 ( nonemptyImpWitness
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( nonempty @ A )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & $true ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(uniqinunit_type,type,
% 0.19/0.47 uniqinunit: $o ).
% 0.19/0.47
% 0.19/0.47 thf(uniqinunit,definition,
% 0.19/0.47 ( uniqinunit
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notinsingleton_type,type,
% 0.19/0.47 notinsingleton: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notinsingleton,definition,
% 0.19/0.47 ( notinsingleton
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx != Xy )
% 0.19/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eqinunit_type,type,
% 0.19/0.47 eqinunit: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eqinunit,definition,
% 0.19/0.47 ( eqinunit
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsswitch_type,type,
% 0.19/0.47 singletonsswitch: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsswitch,definition,
% 0.19/0.47 ( singletonsswitch
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetE_type,type,
% 0.19/0.47 upairsetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetE,definition,
% 0.19/0.47 ( upairsetE
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.47 => ( ( Xz = Xx )
% 0.19/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIL_type,type,
% 0.19/0.47 upairsetIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIL,definition,
% 0.19/0.47 ( upairsetIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIR_type,type,
% 0.19/0.47 upairsetIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIR,definition,
% 0.19/0.47 ( upairsetIR
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyE1_type,type,
% 0.19/0.47 emptyE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyE1,definition,
% 0.19/0.47 ( emptyE1
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ( ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 = emptyset )
% 0.19/0.47 => $false ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(vacuousDall_type,type,
% 0.19/0.47 vacuousDall: $o ).
% 0.19/0.47
% 0.19/0.47 thf(vacuousDall,definition,
% 0.19/0.47 ( vacuousDall
% 0.19/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan1_type,type,
% 0.19/0.47 quantDeMorgan1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan1,definition,
% 0.19/0.47 ( quantDeMorgan1
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan2_type,type,
% 0.19/0.47 quantDeMorgan2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan2,definition,
% 0.19/0.47 ( quantDeMorgan2
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ~ ( Xphi @ Xx ) )
% 0.19/0.47 => ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan3_type,type,
% 0.19/0.47 quantDeMorgan3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan3,definition,
% 0.19/0.47 ( quantDeMorgan3
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan4_type,type,
% 0.19/0.47 quantDeMorgan4: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan4,definition,
% 0.19/0.47 ( quantDeMorgan4
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ~ ( Xphi @ Xx ) )
% 0.19/0.47 => ~ ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setI_type,type,
% 0.19/0.47 prop2setI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setI,definition,
% 0.19/0.47 ( prop2setI
% 0.19/0.47 = ( ! [Xphi: $o] :
% 0.19/0.47 ( Xphi
% 0.19/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(set2prop_type,type,
% 0.19/0.47 set2prop: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2set2propI_type,type,
% 0.19/0.47 prop2set2propI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2set2propI,definition,
% 0.19/0.47 ( prop2set2propI
% 0.19/0.47 = ( ! [Xphi: $o] :
% 0.19/0.47 ( Xphi
% 0.19/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notdexE_type,type,
% 0.19/0.47 notdexE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notdexE,definition,
% 0.19/0.47 ( notdexE
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notdallE_type,type,
% 0.19/0.47 notdallE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notdallE,definition,
% 0.19/0.47 ( notdallE
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuI1_type,type,
% 0.19/0.47 exuI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuI1,definition,
% 0.19/0.47 ( exuI1
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 & ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuI3_type,type,
% 0.19/0.47 exuI3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuI3,definition,
% 0.19/0.47 ( exuI3
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuI2_type,type,
% 0.19/0.47 exuI2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuI2,definition,
% 0.19/0.47 ( exuI2
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 <=> ( Xy = Xx ) )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(inCongP_type,type,
% 0.19/0.47 inCongP: $o ).
% 0.19/0.47
% 0.19/0.47 thf(inCongP,definition,
% 0.19/0.47 ( inCongP
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(in__Cong_type,type,
% 0.19/0.47 in__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(in__Cong,definition,
% 0.19/0.47 ( in__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3u_type,type,
% 0.19/0.47 exuE3u: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3u,definition,
% 0.19/0.47 ( exuE3u
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exu__Cong_type,type,
% 0.19/0.47 exu__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exu__Cong,definition,
% 0.19/0.47 ( exu__Cong
% 0.19/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.47 => ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 <=> ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyset__Cong_type,type,
% 0.19/0.47 emptyset__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyset__Cong,definition,
% 0.19/0.47 ( emptyset__Cong
% 0.19/0.47 = ( emptyset = emptyset ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoin__Cong_type,type,
% 0.19/0.47 setadjoin__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoin__Cong,definition,
% 0.19/0.47 ( setadjoin__Cong
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ! [Xz: $i,Xu: $i] :
% 0.19/0.47 ( ( Xz = Xu )
% 0.19/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powerset__Cong_type,type,
% 0.19/0.47 powerset__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powerset__Cong,definition,
% 0.19/0.47 ( powerset__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( ( powerset @ A )
% 0.19/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunion__Cong_type,type,
% 0.19/0.47 setunion__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunion__Cong,definition,
% 0.19/0.47 ( setunion__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( ( setunion @ A )
% 0.19/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(omega__Cong_type,type,
% 0.19/0.47 omega__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(omega__Cong,definition,
% 0.19/0.47 ( omega__Cong
% 0.19/0.47 = ( omega = omega ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuEu_type,type,
% 0.19/0.47 exuEu: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuEu,definition,
% 0.19/0.47 ( exuEu
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(descr__Cong_type,type,
% 0.19/0.47 descr__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(descr__Cong,definition,
% 0.19/0.47 ( descr__Cong
% 0.19/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.47 => ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.19/0.47 => ( ( descr
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 = ( descr
% 0.19/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstr__Cong_type,type,
% 0.19/0.47 dsetconstr__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstr__Cong,definition,
% 0.19/0.47 ( dsetconstr__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( Xx = Xy )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.47 => ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 = ( dsetconstr @ B
% 0.19/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subset_type,type,
% 0.19/0.47 subset: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(disjoint_type,type,
% 0.19/0.47 disjoint: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(setsmeet_type,type,
% 0.19/0.47 setsmeet: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetI1_type,type,
% 0.19/0.47 subsetI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetI1,definition,
% 0.19/0.47 ( subsetI1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ B ) )
% 0.19/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eqimpsubset2_type,type,
% 0.19/0.47 eqimpsubset2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eqimpsubset2,definition,
% 0.19/0.47 ( eqimpsubset2
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eqimpsubset1_type,type,
% 0.19/0.47 eqimpsubset1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eqimpsubset1,definition,
% 0.19/0.47 ( eqimpsubset1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetI2_type,type,
% 0.19/0.47 subsetI2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetI2,definition,
% 0.19/0.47 ( subsetI2
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ B ) )
% 0.19/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetsubset_type,type,
% 0.19/0.47 emptysetsubset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetsubset,definition,
% 0.19/0.47 ( emptysetsubset
% 0.19/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetE_type,type,
% 0.19/0.47 subsetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetE,definition,
% 0.19/0.47 ( subsetE
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetE2_type,type,
% 0.19/0.47 subsetE2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetE2,definition,
% 0.19/0.47 ( subsetE2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notsubsetI_type,type,
% 0.19/0.47 notsubsetI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notsubsetI,definition,
% 0.19/0.47 ( notsubsetI
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notequalI1_type,type,
% 0.19/0.47 notequalI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notequalI1,definition,
% 0.19/0.47 ( notequalI1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ~ ( subset @ A @ B )
% 0.19/0.47 => ( A != B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notequalI2_type,type,
% 0.19/0.47 notequalI2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notequalI2,definition,
% 0.19/0.47 ( notequalI2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ( A != B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetRefl_type,type,
% 0.19/0.47 subsetRefl: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetRefl,definition,
% 0.19/0.47 ( subsetRefl
% 0.19/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetTrans_type,type,
% 0.19/0.47 subsetTrans: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetTrans,definition,
% 0.19/0.47 ( subsetTrans
% 0.19/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( subset @ B @ C )
% 0.19/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinSub_type,type,
% 0.19/0.47 setadjoinSub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinSub,definition,
% 0.19/0.47 ( setadjoinSub
% 0.19/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinSub2_type,type,
% 0.19/0.47 setadjoinSub2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinSub2,definition,
% 0.19/0.47 ( setadjoinSub2
% 0.19/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subset2powerset_type,type,
% 0.19/0.47 subset2powerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subset2powerset,definition,
% 0.19/0.47 ( subset2powerset
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setextsub_type,type,
% 0.19/0.47 setextsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setextsub,definition,
% 0.19/0.47 ( setextsub
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( subset @ B @ A )
% 0.19/0.47 => ( A = B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetemptysetimpeq_type,type,
% 0.19/0.47 subsetemptysetimpeq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetemptysetimpeq,definition,
% 0.19/0.47 ( subsetemptysetimpeq
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( subset @ A @ emptyset )
% 0.19/0.47 => ( A = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI1_type,type,
% 0.19/0.47 powersetI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI1,definition,
% 0.19/0.47 ( powersetI1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ B @ A )
% 0.19/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE1_type,type,
% 0.19/0.47 powersetE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE1,definition,
% 0.19/0.47 ( powersetE1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(inPowerset_type,type,
% 0.19/0.47 inPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(inPowerset,definition,
% 0.19/0.47 ( inPowerset
% 0.19/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetsubset_type,type,
% 0.19/0.47 powersetsubset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetsubset,definition,
% 0.19/0.47 ( powersetsubset
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sepInPowerset_type,type,
% 0.19/0.47 sepInPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(sepInPowerset,definition,
% 0.19/0.47 ( sepInPowerset
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( in
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sepSubset_type,type,
% 0.19/0.47 sepSubset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(sepSubset,definition,
% 0.19/0.47 ( sepSubset
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( subset
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunion_type,type,
% 0.19/0.47 binunion: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(binunionIL_type,type,
% 0.19/0.47 binunionIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionIL,definition,
% 0.19/0.47 ( binunionIL
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairset2IR_type,type,
% 0.19/0.47 upairset2IR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairset2IR,definition,
% 0.19/0.47 ( upairset2IR
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionIR_type,type,
% 0.19/0.47 binunionIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionIR,definition,
% 0.19/0.47 ( binunionIR
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionEcases_type,type,
% 0.19/0.47 binunionEcases: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionEcases,definition,
% 0.19/0.47 ( binunionEcases
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.19/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.47 => ( ( ( in @ Xx @ A )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => ( ( ( in @ Xx @ B )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionE_type,type,
% 0.19/0.47 binunionE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionE,definition,
% 0.19/0.47 ( binunionE
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionLsub_type,type,
% 0.19/0.47 binunionLsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionLsub,definition,
% 0.19/0.47 ( binunionLsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionRsub_type,type,
% 0.19/0.47 binunionRsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionRsub,definition,
% 0.19/0.47 ( binunionRsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersect_type,type,
% 0.19/0.47 binintersect: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectI_type,type,
% 0.19/0.47 binintersectI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectI,definition,
% 0.19/0.47 ( binintersectI
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset5_type,type,
% 0.19/0.47 binintersectSubset5: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset5,definition,
% 0.19/0.47 ( binintersectSubset5
% 0.19/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.47 ( ( subset @ C @ A )
% 0.19/0.47 => ( ( subset @ C @ B )
% 0.19/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectEL_type,type,
% 0.19/0.47 binintersectEL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectEL,definition,
% 0.19/0.47 ( binintersectEL
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectLsub_type,type,
% 0.19/0.47 binintersectLsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectLsub,definition,
% 0.19/0.47 ( binintersectLsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset2_type,type,
% 0.19/0.47 binintersectSubset2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset2,definition,
% 0.19/0.47 ( binintersectSubset2
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( binintersect @ A @ B )
% 0.19/0.47 = A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset3_type,type,
% 0.19/0.47 binintersectSubset3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset3,definition,
% 0.19/0.47 ( binintersectSubset3
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( ( binintersect @ A @ B )
% 0.19/0.47 = B )
% 0.19/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectER_type,type,
% 0.19/0.47 binintersectER: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectER,definition,
% 0.19/0.47 ( binintersectER
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(disjointsetsI1_type,type,
% 0.19/0.47 disjointsetsI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(disjointsetsI1,definition,
% 0.19/0.47 ( disjointsetsI1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( in @ Xx @ B ) )
% 0.19/0.47 => ( ( binintersect @ A @ B )
% 0.19/0.47 = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectRsub_type,type,
% 0.19/0.47 binintersectRsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectRsub,definition,
% 0.19/0.47 ( binintersectRsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset4_type,type,
% 0.19/0.47 binintersectSubset4: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset4,definition,
% 0.19/0.47 ( binintersectSubset4
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ B @ A )
% 0.19/0.47 => ( ( binintersect @ A @ B )
% 0.19/0.47 = B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset1_type,type,
% 0.19/0.47 binintersectSubset1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset1,definition,
% 0.19/0.47 ( binintersectSubset1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( ( binintersect @ A @ B )
% 0.19/0.47 = A )
% 0.19/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(bs114d_type,type,
% 0.19/0.47 bs114d: $o ).
% 0.19/0.47
% 0.19/0.47 thf(bs114d,definition,
% 0.19/0.47 ( bs114d
% 0.19/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.19/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(regular_type,type,
% 0.19/0.47 regular: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminus_type,type,
% 0.19/0.47 setminus: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(setminusI_type,type,
% 0.19/0.47 setminusI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusI,definition,
% 0.19/0.47 ( setminusI
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusEL_type,type,
% 0.19/0.47 setminusEL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusEL,definition,
% 0.19/0.47 ( setminusEL
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusER_type,type,
% 0.19/0.47 setminusER: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusER,definition,
% 0.19/0.47 ( setminusER
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusSubset2_type,type,
% 0.19/0.47 setminusSubset2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusSubset2,definition,
% 0.19/0.47 ( setminusSubset2
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( setminus @ A @ B )
% 0.19/0.47 = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusERneg_type,type,
% 0.19/0.47 setminusERneg: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusERneg,definition,
% 0.19/0.47 ( setminusERneg
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusELneg_type,type,
% 0.19/0.47 setminusELneg: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusELneg,definition,
% 0.19/0.47 ( setminusELneg
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusILneg_type,type,
% 0.19/0.47 setminusILneg: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusILneg,definition,
% 0.19/0.47 ( setminusILneg
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ A )
% 0.19/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusIRneg_type,type,
% 0.19/0.47 setminusIRneg: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusIRneg,definition,
% 0.19/0.47 ( setminusIRneg
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusLsub_type,type,
% 0.19/0.47 setminusLsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusLsub,definition,
% 0.19/0.47 ( setminusLsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusSubset1_type,type,
% 0.19/0.47 setminusSubset1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusSubset1,definition,
% 0.19/0.47 ( setminusSubset1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( ( setminus @ A @ B )
% 0.19/0.47 = emptyset )
% 0.19/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiff_type,type,
% 0.19/0.47 symdiff: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffE_type,type,
% 0.19/0.47 symdiffE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffE,definition,
% 0.19/0.47 ( symdiffE
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.19/0.47 => ! [Xphi: $o] :
% 0.19/0.47 ( ( ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => Xphi ) )
% 0.19/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => Xphi ) )
% 0.19/0.47 => Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffI1_type,type,
% 0.19/0.47 symdiffI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffI1,definition,
% 0.19/0.47 ( symdiffI1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffI2_type,type,
% 0.19/0.47 symdiffI2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffI2,definition,
% 0.19/0.47 ( symdiffI2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffIneg1_type,type,
% 0.19/0.47 symdiffIneg1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffIneg1,definition,
% 0.19/0.47 ( symdiffIneg1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffIneg2_type,type,
% 0.19/0.47 symdiffIneg2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffIneg2,definition,
% 0.19/0.47 ( symdiffIneg2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(iskpair_type,type,
% 0.19/0.47 iskpair: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(secondinupair_type,type,
% 0.19/0.47 secondinupair: $o ).
% 0.19/0.47
% 0.19/0.47 thf(secondinupair,definition,
% 0.19/0.47 ( secondinupair
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIL_type,type,
% 0.19/0.47 setukpairIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIL,definition,
% 0.19/0.47 ( setukpairIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIR_type,type,
% 0.19/0.47 setukpairIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIR,definition,
% 0.19/0.47 ( setukpairIR
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kpairiskpair_type,type,
% 0.19/0.47 kpairiskpair: $o ).
% 0.19/0.47
% 0.19/0.47 thf(kpairiskpair,definition,
% 0.19/0.47 ( kpairiskpair
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kpair_type,type,
% 0.19/0.47 kpair: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(kpairp_type,type,
% 0.19/0.47 kpairp: $o ).
% 0.19/0.47
% 0.19/0.47 thf(kpairp,definition,
% 0.19/0.47 ( kpairp
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprod_type,type,
% 0.19/0.47 cartprod: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsubset_type,type,
% 0.19/0.47 singletonsubset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsubset,definition,
% 0.19/0.47 ( singletonsubset
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(singletoninpowerset_type,type,
% 0.19/0.47 singletoninpowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletoninpowerset,definition,
% 0.19/0.47 ( singletoninpowerset
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(singletoninpowunion_type,type,
% 0.19/0.47 singletoninpowunion: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletoninpowunion,definition,
% 0.19/0.47 ( singletoninpowunion
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairset2E_type,type,
% 0.19/0.47 upairset2E: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairset2E,definition,
% 0.19/0.47 ( upairset2E
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.47 => ( ( Xz = Xx )
% 0.19/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsubunion_type,type,
% 0.19/0.47 upairsubunion: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsubunion,definition,
% 0.19/0.47 ( upairsubunion
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairinpowunion_type,type,
% 0.19/0.47 upairinpowunion: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairinpowunion,definition,
% 0.19/0.47 ( upairinpowunion
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ubforcartprodlem1_type,type,
% 0.19/0.47 ubforcartprodlem1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ubforcartprodlem1,definition,
% 0.19/0.47 ( ubforcartprodlem1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ubforcartprodlem2_type,type,
% 0.19/0.47 ubforcartprodlem2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ubforcartprodlem2,definition,
% 0.19/0.47 ( ubforcartprodlem2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ubforcartprodlem3_type,type,
% 0.19/0.47 ubforcartprodlem3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ubforcartprodlem3,definition,
% 0.19/0.47 ( ubforcartprodlem3
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodpairin_type,type,
% 0.19/0.47 cartprodpairin: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodpairin,definition,
% 0.19/0.47 ( cartprodpairin
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodmempair1_type,type,
% 0.19/0.47 cartprodmempair1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodmempair1,definition,
% 0.19/0.47 ( cartprodmempair1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ? [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 & ( Xu
% 0.19/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodmempair_type,type,
% 0.19/0.47 cartprodmempair: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodmempair,definition,
% 0.19/0.47 ( cartprodmempair
% 0.19/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionE2_type,type,
% 0.19/0.47 setunionE2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionE2,definition,
% 0.19/0.47 ( setunionE2
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.47 => ? [X: $i] :
% 0.19/0.47 ( ( in @ X @ A )
% 0.19/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionsingleton1_type,type,
% 0.19/0.47 setunionsingleton1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionsingleton1,definition,
% 0.19/0.47 ( setunionsingleton1
% 0.19/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionsingleton2_type,type,
% 0.19/0.47 setunionsingleton2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionsingleton2,definition,
% 0.19/0.47 ( setunionsingleton2
% 0.19/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionsingleton_type,type,
% 0.19/0.47 setunionsingleton: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionsingleton,definition,
% 0.19/0.47 ( setunionsingleton
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.19/0.47 = Xx ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(singleton_type,type,
% 0.19/0.47 singleton: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletonprop_type,type,
% 0.19/0.47 singletonprop: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletonprop,definition,
% 0.19/0.47 ( singletonprop
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) )
% 0.19/0.47 => ( ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ( singleton
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ex1_type,type,
% 0.19/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.19/0.47
% 0.19/0.47 thf(ex1,definition,
% 0.19/0.47 ( ex1
% 0.19/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( singleton
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ex1E1_type,type,
% 0.19/0.47 ex1E1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ex1E1,definition,
% 0.19/0.47 ( ex1E1
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ( ex1 @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ex1I_type,type,
% 0.19/0.47 ex1I: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ex1I,definition,
% 0.19/0.47 ( ex1I
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 => ( ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xy = Xx ) ) )
% 0.19/0.47 => ( ex1 @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ex1I2_type,type,
% 0.19/0.47 ex1I2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ex1I2,definition,
% 0.19/0.47 ( ex1I2
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) )
% 0.19/0.47 => ( ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ( ex1 @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsuniq_type,type,
% 0.19/0.47 singletonsuniq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsuniq,definition,
% 0.19/0.47 ( singletonsuniq
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.19/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(atmost1p_type,type,
% 0.19/0.47 atmost1p: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(atleast2p_type,type,
% 0.19/0.47 atleast2p: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(atmost2p_type,type,
% 0.19/0.47 atmost2p: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetp_type,type,
% 0.19/0.47 upairsetp: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjL1_type,type,
% 0.19/0.47 setukpairinjL1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjL1,definition,
% 0.19/0.47 ( setukpairinjL1
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kfstsingleton_type,type,
% 0.19/0.47 kfstsingleton: $o ).
% 0.19/0.47
% 0.19/0.47 thf(kfstsingleton,definition,
% 0.19/0.47 ( kfstsingleton
% 0.19/0.47 = ( ! [Xu: $i] :
% 0.19/0.47 ( ( iskpair @ Xu )
% 0.19/0.47 => ( singleton
% 0.19/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(theprop_type,type,
% 0.19/0.47 theprop: $o ).
% 0.19/0.47
% 0.19/0.47 thf(theprop,definition,
% 0.19/0.47 ( theprop
% 0.19/0.47 = ( ! [X: $i] :
% 0.19/0.47 ( ( singleton @ X )
% 0.19/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kfst_type,type,
% 0.19/0.47 kfst: $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(kfstpairEq_type,type,
% 0.19/0.47 kfstpairEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(kfstpairEq,definition,
% 0.19/0.47 ( kfstpairEq
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.47 = Xx ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodfstin_type,type,
% 0.19/0.47 cartprodfstin: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodfstin,definition,
% 0.19/0.47 ( cartprodfstin
% 0.19/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjL2_type,type,
% 0.19/0.47 setukpairinjL2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjL2,definition,
% 0.19/0.47 ( setukpairinjL2
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjL_type,type,
% 0.19/0.47 setukpairinjL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjL,definition,
% 0.19/0.47 ( setukpairinjL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.47 = ( kpair @ Xz @ Xu ) )
% 0.19/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR11_type,type,
% 0.19/0.47 setukpairinjR11: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR11,definition,
% 0.19/0.47 ( setukpairinjR11
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR12_type,type,
% 0.19/0.47 setukpairinjR12: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR12,definition,
% 0.19/0.47 ( setukpairinjR12
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR1_type,type,
% 0.19/0.47 setukpairinjR1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR1,definition,
% 0.19/0.47 ( setukpairinjR1
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.47 => ( ( Xz = Xu )
% 0.19/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairequniteq_type,type,
% 0.19/0.47 upairequniteq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairequniteq,definition,
% 0.19/0.47 ( upairequniteq
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR2_type,type,
% 0.19/0.47 setukpairinjR2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR2,definition,
% 0.19/0.47 ( setukpairinjR2
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR_type,type,
% 0.19/0.47 setukpairinjR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairinjR,definition,
% 0.19/0.47 ( setukpairinjR
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.47 = ( kpair @ Xz @ Xu ) )
% 0.19/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ksndsingleton_type,type,
% 0.19/0.47 ksndsingleton: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ksndsingleton,definition,
% 0.19/0.47 ( ksndsingleton
% 0.19/0.47 = ( ! [Xu: $i] :
% 0.19/0.47 ( ( iskpair @ Xu )
% 0.19/0.47 => ( singleton
% 0.19/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.47 @ ^ [Xx: $i] :
% 0.19/0.47 ( Xu
% 0.19/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ksnd_type,type,
% 0.19/0.47 ksnd: $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(ksndpairEq_type,type,
% 0.19/0.47 ksndpairEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ksndpairEq,definition,
% 0.19/0.47 ( ksndpairEq
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.47 = Xy ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kpairsurjEq_type,type,
% 0.19/0.47 kpairsurjEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(kpairsurjEq,definition,
% 0.19/0.47 ( kpairsurjEq
% 0.19/0.47 = ( ! [Xu: $i] :
% 0.19/0.47 ( ( iskpair @ Xu )
% 0.19/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.47 = Xu ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodsndin_type,type,
% 0.19/0.47 cartprodsndin: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodsndin,definition,
% 0.19/0.47 ( cartprodsndin
% 0.19/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodpairmemEL_type,type,
% 0.19/0.47 cartprodpairmemEL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodpairmemEL,definition,
% 0.19/0.47 ( cartprodpairmemEL
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodpairmemER_type,type,
% 0.19/0.47 cartprodpairmemER: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodpairmemER,definition,
% 0.19/0.47 ( cartprodpairmemER
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodmempaircEq_type,type,
% 0.19/0.47 cartprodmempaircEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodmempaircEq,definition,
% 0.19/0.47 ( cartprodmempaircEq
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( kpair @ Xx @ Xy )
% 0.19/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodfstpairEq_type,type,
% 0.19/0.47 cartprodfstpairEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodfstpairEq,definition,
% 0.19/0.47 ( cartprodfstpairEq
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.47 = Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodsndpairEq_type,type,
% 0.19/0.47 cartprodsndpairEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodsndpairEq,definition,
% 0.19/0.47 ( cartprodsndpairEq
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.47 = Xy ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodpairsurjEq_type,type,
% 0.19/0.47 cartprodpairsurjEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(cartprodpairsurjEq,definition,
% 0.19/0.47 ( cartprodpairsurjEq
% 0.19/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.47 = Xu ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(breln_type,type,
% 0.19/0.47 breln: $i > $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(breln,definition,
% 0.19/0.47 ( breln
% 0.19/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstr_type,type,
% 0.19/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrI_type,type,
% 0.19/0.47 dpsetconstrI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrI,definition,
% 0.19/0.47 ( dpsetconstrI
% 0.19/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.19/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.47 @ ( dpsetconstr @ A @ B
% 0.19/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrSub_type,type,
% 0.19/0.47 dpsetconstrSub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrSub,definition,
% 0.19/0.47 ( dpsetconstrSub
% 0.19/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.47 ( subset
% 0.19/0.47 @ ( dpsetconstr @ A @ B
% 0.19/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.19/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setOfPairsIsBReln_type,type,
% 0.19/0.47 setOfPairsIsBReln: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setOfPairsIsBReln,definition,
% 0.19/0.47 ( setOfPairsIsBReln
% 0.19/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.47 ( breln @ A @ B
% 0.19/0.47 @ ( dpsetconstr @ A @ B
% 0.19/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrERa_type,type,
% 0.19/0.47 dpsetconstrERa: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrERa,definition,
% 0.19/0.47 ( dpsetconstrERa
% 0.19/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.47 @ ( dpsetconstr @ A @ B
% 0.19/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrEL1_type,type,
% 0.19/0.47 dpsetconstrEL1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrEL1,definition,
% 0.19/0.47 ( dpsetconstrEL1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.47 @ ( dpsetconstr @ A @ B
% 0.19/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrEL2_type,type,
% 0.19/0.47 dpsetconstrEL2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrEL2,definition,
% 0.19/0.47 ( dpsetconstrEL2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.47 @ ( dpsetconstr @ A @ B
% 0.19/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrER_type,type,
% 0.19/0.47 dpsetconstrER: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dpsetconstrER,definition,
% 0.19/0.47 ( dpsetconstrER
% 0.19/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.47 @ ( dpsetconstr @ A @ B
% 0.19/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(func_type,type,
% 0.19/0.47 func: $i > $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcSet_type,type,
% 0.19/0.47 funcSet: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(funcImageSingleton_type,type,
% 0.19/0.47 funcImageSingleton: $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcImageSingleton,definition,
% 0.19/0.47 ( funcImageSingleton
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( singleton
% 0.19/0.47 @ ( dsetconstr @ B
% 0.19/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(apProp_type,type,
% 0.19/0.47 apProp: $o ).
% 0.19/0.47
% 0.19/0.47 thf(apProp,definition,
% 0.19/0.47 ( apProp
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in
% 0.19/0.47 @ ( setunion
% 0.19/0.47 @ ( dsetconstr @ B
% 0.19/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.19/0.47 @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ap_type,type,
% 0.19/0.47 ap: $i > $i > $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(app_type,type,
% 0.19/0.47 app: $o ).
% 0.19/0.47
% 0.19/0.47 thf(app,definition,
% 0.19/0.47 ( app
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(infuncsetfunc_type,type,
% 0.19/0.47 infuncsetfunc: $o ).
% 0.19/0.47
% 0.19/0.47 thf(infuncsetfunc,definition,
% 0.19/0.47 ( infuncsetfunc
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ap2p_type,type,
% 0.19/0.47 ap2p: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ap2p,definition,
% 0.19/0.47 ( ap2p
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(funcinfuncset_type,type,
% 0.19/0.47 funcinfuncset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcinfuncset,definition,
% 0.19/0.47 ( funcinfuncset
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(lamProp_type,type,
% 0.19/0.47 lamProp: $o ).
% 0.19/0.47
% 0.19/0.47 thf(lamProp,definition,
% 0.19/0.47 ( lamProp
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.47 => ( func @ A @ B
% 0.19/0.47 @ ( dpsetconstr @ A @ B
% 0.19/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xf @ Xx )
% 0.19/0.47 = Xy ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(lam_type,type,
% 0.19/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.19/0.47
% 0.19/0.47 thf(lamp_type,type,
% 0.19/0.47 lamp: $o ).
% 0.19/0.47
% 0.19/0.47 thf(lamp,definition,
% 0.19/0.47 ( lamp
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.47 => ( func @ A @ B
% 0.19/0.47 @ ( lam @ A @ B
% 0.19/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(lam2p_type,type,
% 0.19/0.47 lam2p: $o ).
% 0.19/0.47
% 0.19/0.47 thf(lam2p,definition,
% 0.19/0.47 ( lam2p
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.47 => ( in
% 0.19/0.47 @ ( lam @ A @ B
% 0.19/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(brelnall1_type,type,
% 0.19/0.47 brelnall1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(brelnall1,definition,
% 0.19/0.47 ( brelnall1
% 0.19/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.47 ( ( breln @ A @ B @ R )
% 0.19/0.47 => ! [Xphi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ R )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(brelnall2_type,type,
% 0.19/0.47 brelnall2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(brelnall2,definition,
% 0.19/0.47 ( brelnall2
% 0.19/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.47 ( ( breln @ A @ B @ R )
% 0.19/0.47 => ! [Xphi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ R )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ex1E2_type,type,
% 0.19/0.47 ex1E2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ex1E2,definition,
% 0.19/0.47 ( ex1E2
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ( ex1 @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(funcGraphProp1_type,type,
% 0.19/0.47 funcGraphProp1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcGraphProp1,definition,
% 0.19/0.47 ( funcGraphProp1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(funcGraphProp3_type,type,
% 0.19/0.47 funcGraphProp3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcGraphProp3,definition,
% 0.19/0.47 ( funcGraphProp3
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(funcGraphProp2_type,type,
% 0.19/0.47 funcGraphProp2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcGraphProp2,definition,
% 0.19/0.47 ( funcGraphProp2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(funcextLem_type,type,
% 0.19/0.47 funcextLem: $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcextLem,definition,
% 0.19/0.47 ( funcextLem
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ! [Xg: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xg )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.19/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(funcGraphProp4_type,type,
% 0.19/0.47 funcGraphProp4: $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcGraphProp4,definition,
% 0.19/0.47 ( funcGraphProp4
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subbreln_type,type,
% 0.19/0.47 subbreln: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subbreln,definition,
% 0.19/0.47 ( subbreln
% 0.19/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.47 ( ( breln @ A @ B @ R )
% 0.19/0.47 => ! [S: $i] :
% 0.19/0.47 ( ( breln @ A @ B @ S )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eqbreln_type,type,
% 0.19/0.47 eqbreln: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eqbreln,definition,
% 0.19/0.47 ( eqbreln
% 0.19/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.47 ( ( breln @ A @ B @ R )
% 0.19/0.47 => ! [S: $i] :
% 0.19/0.47 ( ( breln @ A @ B @ S )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.19/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.19/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(funcext_type,type,
% 0.19/0.47 funcext: $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcext,definition,
% 0.19/0.47 ( funcext
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ! [Xg: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xg )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(funcext2_type,type,
% 0.19/0.47 funcext2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(funcext2,definition,
% 0.19/0.47 ( funcext2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.47 => ! [Xg: $i] :
% 0.19/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ap2apEq1_type,type,
% 0.19/0.47 ap2apEq1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ap2apEq1,definition,
% 0.19/0.47 ( ap2apEq1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ap2apEq2_type,type,
% 0.19/0.47 ap2apEq2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ap2apEq2,definition,
% 0.19/0.47 ( ap2apEq2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(beta1_type,type,
% 0.19/0.47 beta1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(beta1,definition,
% 0.19/0.47 ( beta1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( ap @ A @ B
% 0.19/0.47 @ ( lam @ A @ B
% 0.19/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.47 @ Xx )
% 0.19/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eta1_type,type,
% 0.19/0.47 eta1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eta1,definition,
% 0.19/0.47 ( eta1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( func @ A @ B @ Xf )
% 0.19/0.47 => ( ( lam @ A @ B
% 0.19/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.47 = Xf ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(lam2lamEq_type,type,
% 0.19/0.47 lam2lamEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(lam2lamEq,definition,
% 0.19/0.47 ( lam2lamEq
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.47 => ( ( lam @ A @ B
% 0.19/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.47 = ( lam @ A @ B
% 0.19/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(beta2_type,type,
% 0.19/0.47 beta2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(beta2,definition,
% 0.19/0.47 ( beta2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( ap @ A @ B
% 0.19/0.47 @ ( lam @ A @ B
% 0.19/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.47 @ Xx )
% 0.19/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eta2_type,type,
% 0.19/0.47 eta2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eta2,definition,
% 0.19/0.47 ( eta2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.47 => ( ( lam @ A @ B
% 0.19/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.47 = Xf ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(iffalseProp1_type,type,
% 0.19/0.47 iffalseProp1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(iffalseProp1,definition,
% 0.19/0.47 ( iffalseProp1
% 0.19/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( ~ Xphi
% 0.19/0.47 => ( in @ Xy
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xz: $i] :
% 0.19/0.47 ( ( Xphi
% 0.19/0.47 & ( Xz = Xx ) )
% 0.19/0.47 | ( ~ Xphi
% 0.19/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(iffalseProp2_type,type,
% 0.19/0.47 iffalseProp2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(iffalseProp2,definition,
% 0.19/0.47 ( iffalseProp2
% 0.19/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( ~ Xphi
% 0.19/0.47 => ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xz: $i] :
% 0.19/0.47 ( ( Xphi
% 0.19/0.47 & ( Xz = Xx ) )
% 0.19/0.47 | ( ~ Xphi
% 0.19/0.47 & ( Xz = Xy ) ) ) )
% 0.19/0.47 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(iftrueProp1_type,type,
% 0.19/0.47 iftrueProp1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(iftrueProp1,definition,
% 0.19/0.47 ( iftrueProp1
% 0.19/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( Xphi
% 0.19/0.47 => ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xz: $i] :
% 0.19/0.47 ( ( Xphi
% 0.19/0.47 & ( Xz = Xx ) )
% 0.19/0.47 | ( ~ Xphi
% 0.19/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(iftrueProp2_type,type,
% 0.19/0.47 iftrueProp2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(iftrueProp2,definition,
% 0.19/0.47 ( iftrueProp2
% 0.19/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( Xphi
% 0.19/0.47 => ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xz: $i] :
% 0.19/0.47 ( ( Xphi
% 0.19/0.47 & ( Xz = Xx ) )
% 0.19/0.47 | ( ~ Xphi
% 0.19/0.47 & ( Xz = Xy ) ) ) )
% 0.19/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ifSingleton_type,type,
% 0.19/0.47 ifSingleton: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ifSingleton,definition,
% 0.19/0.47 ( ifSingleton
% 0.19/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( singleton
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xz: $i] :
% 0.19/0.47 ( ( Xphi
% 0.19/0.47 & ( Xz = Xx ) )
% 0.19/0.47 | ( ~ Xphi
% 0.19/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(if_type,type,
% 0.19/0.47 if: $i > $o > $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(if,definition,
% 0.19/0.47 ( if
% 0.19/0.47 = ( ^ [A: $i,Xphi: $o,Xx: $i,Xy: $i] :
% 0.19/0.47 ( setunion
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xz: $i] :
% 0.19/0.47 ( ( Xphi
% 0.19/0.47 & ( Xz = Xx ) )
% 0.19/0.47 | ( ~ Xphi
% 0.19/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ifp_type,type,
% 0.19/0.47 ifp: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ifp,definition,
% 0.19/0.47 ( ifp
% 0.19/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(theeq_type,type,
% 0.19/0.47 theeq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(theeq,definition,
% 0.19/0.47 ( theeq
% 0.19/0.47 = ( ! [X: $i] :
% 0.19/0.47 ( ( singleton @ X )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ X )
% 0.19/0.47 => ( ( setunion @ X )
% 0.19/0.47 = Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(iftrue_type,type,
% 0.19/0.47 iftrue: $o ).
% 0.19/0.47
% 0.19/0.47 thf(iftrue,definition,
% 0.19/0.47 ( iftrue
% 0.19/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( Xphi
% 0.19/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.47 = Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(iffalse,conjecture,
% 0.19/0.47 ( setextAx
% 0.19/0.47 => ( emptysetAx
% 0.19/0.47 => ( setadjoinAx
% 0.19/0.47 => ( powersetAx
% 0.19/0.47 => ( setunionAx
% 0.19/0.47 => ( omega0Ax
% 0.19/0.47 => ( omegaSAx
% 0.19/0.47 => ( omegaIndAx
% 0.19/0.47 => ( replAx
% 0.19/0.47 => ( foundationAx
% 0.19/0.47 => ( wellorderingAx
% 0.19/0.47 => ( descrp
% 0.19/0.47 => ( dsetconstrI
% 0.19/0.47 => ( dsetconstrEL
% 0.19/0.47 => ( dsetconstrER
% 0.19/0.47 => ( exuE1
% 0.19/0.47 => ( prop2setE
% 0.19/0.47 => ( emptysetE
% 0.19/0.47 => ( emptysetimpfalse
% 0.19/0.47 => ( notinemptyset
% 0.19/0.47 => ( exuE3e
% 0.19/0.47 => ( setext
% 0.19/0.47 => ( emptyI
% 0.19/0.47 => ( noeltsimpempty
% 0.19/0.47 => ( setbeta
% 0.19/0.47 => ( nonemptyE1
% 0.19/0.47 => ( nonemptyI
% 0.19/0.47 => ( nonemptyI1
% 0.19/0.47 => ( setadjoinIL
% 0.19/0.47 => ( emptyinunitempty
% 0.19/0.47 => ( setadjoinIR
% 0.19/0.47 => ( setadjoinE
% 0.19/0.47 => ( setadjoinOr
% 0.19/0.47 => ( setoftrueEq
% 0.19/0.47 => ( powersetI
% 0.19/0.47 => ( emptyinPowerset
% 0.19/0.47 => ( emptyInPowerset
% 0.19/0.47 => ( powersetE
% 0.19/0.47 => ( setunionI
% 0.19/0.47 => ( setunionE
% 0.19/0.47 => ( subPowSU
% 0.19/0.47 => ( exuE2
% 0.19/0.47 => ( nonemptyImpWitness
% 0.19/0.47 => ( uniqinunit
% 0.19/0.47 => ( notinsingleton
% 0.19/0.47 => ( eqinunit
% 0.19/0.47 => ( singletonsswitch
% 0.19/0.47 => ( upairsetE
% 0.19/0.47 => ( upairsetIL
% 0.19/0.47 => ( upairsetIR
% 0.19/0.47 => ( emptyE1
% 0.19/0.47 => ( vacuousDall
% 0.19/0.47 => ( quantDeMorgan1
% 0.19/0.47 => ( quantDeMorgan2
% 0.19/0.47 => ( quantDeMorgan3
% 0.19/0.47 => ( quantDeMorgan4
% 0.19/0.47 => ( prop2setI
% 0.19/0.47 => ( prop2set2propI
% 0.19/0.47 => ( notdexE
% 0.19/0.47 => ( notdallE
% 0.19/0.47 => ( exuI1
% 0.19/0.47 => ( exuI3
% 0.19/0.47 => ( exuI2
% 0.19/0.47 => ( inCongP
% 0.19/0.47 => ( in__Cong
% 0.19/0.47 => ( exuE3u
% 0.19/0.47 => ( exu__Cong
% 0.19/0.47 => ( emptyset__Cong
% 0.19/0.47 => ( setadjoin__Cong
% 0.19/0.47 => ( powerset__Cong
% 0.19/0.47 => ( setunion__Cong
% 0.19/0.47 => ( omega__Cong
% 0.19/0.47 => ( exuEu
% 0.19/0.47 => ( descr__Cong
% 0.19/0.47 => ( dsetconstr__Cong
% 0.19/0.47 => ( subsetI1
% 0.19/0.47 => ( eqimpsubset2
% 0.19/0.47 => ( eqimpsubset1
% 0.19/0.47 => ( subsetI2
% 0.19/0.47 => ( emptysetsubset
% 0.19/0.47 => ( subsetE
% 0.19/0.47 => ( subsetE2
% 0.19/0.47 => ( notsubsetI
% 0.19/0.47 => ( notequalI1
% 0.19/0.47 => ( notequalI2
% 0.19/0.47 => ( subsetRefl
% 0.19/0.47 => ( subsetTrans
% 0.19/0.47 => ( setadjoinSub
% 0.19/0.47 => ( setadjoinSub2
% 0.19/0.47 => ( subset2powerset
% 0.19/0.47 => ( setextsub
% 0.19/0.47 => ( subsetemptysetimpeq
% 0.19/0.47 => ( powersetI1
% 0.19/0.47 => ( powersetE1
% 0.19/0.47 => ( inPowerset
% 0.19/0.47 => ( powersetsubset
% 0.19/0.47 => ( sepInPowerset
% 0.19/0.47 => ( sepSubset
% 0.19/0.47 => ( binunionIL
% 0.19/0.47 => ( upairset2IR
% 0.19/0.47 => ( binunionIR
% 0.19/0.47 => ( binunionEcases
% 0.19/0.47 => ( binunionE
% 0.19/0.47 => ( binunionLsub
% 0.19/0.47 => ( binunionRsub
% 0.19/0.47 => ( binintersectI
% 0.19/0.47 => ( binintersectSubset5
% 0.19/0.47 => ( binintersectEL
% 0.19/0.47 => ( binintersectLsub
% 0.19/0.47 => ( binintersectSubset2
% 0.19/0.47 => ( binintersectSubset3
% 0.19/0.47 => ( binintersectER
% 0.19/0.47 => ( disjointsetsI1
% 0.19/0.47 => ( binintersectRsub
% 0.19/0.47 => ( binintersectSubset4
% 0.19/0.47 => ( binintersectSubset1
% 0.19/0.47 => ( bs114d
% 0.19/0.47 => ( setminusI
% 0.19/0.47 => ( setminusEL
% 0.19/0.47 => ( setminusER
% 0.19/0.47 => ( setminusSubset2
% 0.19/0.47 => ( setminusERneg
% 0.19/0.47 => ( setminusELneg
% 0.19/0.47 => ( setminusILneg
% 0.19/0.47 => ( setminusIRneg
% 0.19/0.47 => ( setminusLsub
% 0.19/0.47 => ( setminusSubset1
% 0.19/0.47 => ( symdiffE
% 0.19/0.47 => ( symdiffI1
% 0.19/0.47 => ( symdiffI2
% 0.19/0.47 => ( symdiffIneg1
% 0.19/0.47 => ( symdiffIneg2
% 0.19/0.47 => ( secondinupair
% 0.19/0.47 => ( setukpairIL
% 0.19/0.47 => ( setukpairIR
% 0.19/0.47 => ( kpairiskpair
% 0.19/0.47 => ( kpairp
% 0.19/0.47 => ( singletonsubset
% 0.19/0.47 => ( singletoninpowerset
% 0.19/0.47 => ( singletoninpowunion
% 0.19/0.47 => ( upairset2E
% 0.19/0.47 => ( upairsubunion
% 0.19/0.47 => ( upairinpowunion
% 0.19/0.47 => ( ubforcartprodlem1
% 0.19/0.47 => ( ubforcartprodlem2
% 0.19/0.47 => ( ubforcartprodlem3
% 0.19/0.47 => ( cartprodpairin
% 0.19/0.47 => ( cartprodmempair1
% 0.19/0.47 => ( cartprodmempair
% 0.19/0.47 => ( setunionE2
% 0.19/0.47 => ( setunionsingleton1
% 0.19/0.47 => ( setunionsingleton2
% 0.19/0.47 => ( setunionsingleton
% 0.19/0.47 => ( singletonprop
% 0.19/0.47 => ( ex1E1
% 0.19/0.47 => ( ex1I
% 0.19/0.47 => ( ex1I2
% 0.19/0.47 => ( singletonsuniq
% 0.19/0.47 => ( setukpairinjL1
% 0.19/0.47 => ( kfstsingleton
% 0.19/0.47 => ( theprop
% 0.19/0.47 => ( kfstpairEq
% 0.19/0.47 => ( cartprodfstin
% 0.19/0.47 => ( setukpairinjL2
% 0.19/0.47 => ( setukpairinjL
% 0.19/0.47 => ( setukpairinjR11
% 0.19/0.47 => ( setukpairinjR12
% 0.19/0.47 => ( setukpairinjR1
% 0.19/0.47 => ( upairequniteq
% 0.19/0.47 => ( setukpairinjR2
% 0.19/0.47 => ( setukpairinjR
% 0.19/0.47 => ( ksndsingleton
% 0.19/0.47 => ( ksndpairEq
% 0.19/0.47 => ( kpairsurjEq
% 0.19/0.47 => ( cartprodsndin
% 0.19/0.47 => ( cartprodpairmemEL
% 0.19/0.47 => ( cartprodpairmemER
% 0.19/0.47 => ( cartprodmempaircEq
% 0.19/0.47 => ( cartprodfstpairEq
% 0.19/0.47 => ( cartprodsndpairEq
% 0.19/0.47 => ( cartprodpairsurjEq
% 0.19/0.47 => ( dpsetconstrI
% 0.19/0.47 => ( dpsetconstrSub
% 0.19/0.48 => ( setOfPairsIsBReln
% 0.19/0.48 => ( dpsetconstrERa
% 0.19/0.48 => ( dpsetconstrEL1
% 0.19/0.48 => ( dpsetconstrEL2
% 0.19/0.48 => ( dpsetconstrER
% 0.19/0.48 => ( funcImageSingleton
% 0.19/0.48 => ( apProp
% 0.19/0.48 => ( app
% 0.19/0.48 => ( infuncsetfunc
% 0.19/0.48 => ( ap2p
% 0.19/0.48 => ( funcinfuncset
% 0.19/0.48 => ( lamProp
% 0.19/0.48 => ( lamp
% 0.19/0.48 => ( lam2p
% 0.19/0.48 => ( brelnall1
% 0.19/0.48 => ( brelnall2
% 0.19/0.48 => ( ex1E2
% 0.19/0.48 => ( funcGraphProp1
% 0.19/0.48 => ( funcGraphProp3
% 0.19/0.48 => ( funcGraphProp2
% 0.19/0.48 => ( funcextLem
% 0.19/0.48 => ( funcGraphProp4
% 0.19/0.48 => ( subbreln
% 0.19/0.48 => ( eqbreln
% 0.19/0.48 => ( funcext
% 0.19/0.48 => ( funcext2
% 0.19/0.48 => ( ap2apEq1
% 0.19/0.48 => ( ap2apEq2
% 0.19/0.48 => ( beta1
% 0.19/0.48 => ( eta1
% 0.19/0.48 => ( lam2lamEq
% 0.19/0.48 => ( beta2
% 0.19/0.48 => ( eta2
% 0.19/0.48 => ( iffalseProp1
% 0.19/0.48 => ( iffalseProp2
% 0.19/0.48 => ( iftrueProp1
% 0.19/0.48 => ( iftrueProp2
% 0.19/0.48 => ( ifSingleton
% 0.19/0.48 => ( ifp
% 0.19/0.48 => ( theeq
% 0.19/0.48 => ( iftrue
% 0.19/0.48 => ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( ~ Xphi
% 0.19/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.48 = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 %------------------------------------------------------------------------------
% 0.19/0.48 ------------------------
% 3.16/1.78 ---- Embedded in THF ---
% 3.16/1.78 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.16/1.78 %%% Generated on Tue Feb 24 12:31:37 EST 2026
% 3.16/1.78 %%% using '$$dhol' embedding, version 1.3.0.
% 3.16/1.78 %%% Logic specification used:
% 3.16/1.78 %%% thf(spec, logic, $$dhol).
% 3.16/1.78
% 3.16/1.78 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.K93CCo4jzK/DTF2DTF21305.p
% See solution above
% 3.16/1.78 ------------------------
% 3.43/1.83 ---- Cleaned THF ---
% 3.43/1.83 thf(in_type,type,
% 3.43/1.83 in: $i > $i > $o ).
% 3.43/1.83
% 3.43/1.83 thf(exu_type,type,
% 3.43/1.83 exu: ( $i > $o ) > $o ).
% 3.43/1.83
% 3.43/1.83 thf(exu,definition,
% 3.43/1.83 ( exu
% 3.43/1.83 = ( ^ [Xphi: $i > $o] :
% 3.43/1.83 ? [Xx: $i] :
% 3.43/1.83 ( ( Xphi @ Xx )
% 3.43/1.83 & ! [Xy: $i] :
% 3.43/1.83 ( ( Xphi @ Xy )
% 3.43/1.83 => ( Xx = Xy ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setextAx_type,type,
% 3.43/1.83 setextAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setextAx,definition,
% 3.43/1.83 ( setextAx
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 <=> ( in @ Xx @ B ) )
% 3.43/1.83 => ( A = B ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptyset_type,type,
% 3.43/1.83 emptyset: $i ).
% 3.43/1.83
% 3.43/1.83 thf(emptysetAx_type,type,
% 3.43/1.83 emptysetAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptysetAx,definition,
% 3.43/1.83 ( emptysetAx
% 3.43/1.83 = ( ! [Xx: $i] :
% 3.43/1.83 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoin_type,type,
% 3.43/1.83 setadjoin: $i > $i > $i ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinAx_type,type,
% 3.43/1.83 setadjoinAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinAx,definition,
% 3.43/1.83 ( setadjoinAx
% 3.43/1.83 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.43/1.83 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.43/1.83 <=> ( ( Xy = Xx )
% 3.43/1.83 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(powerset_type,type,
% 3.43/1.83 powerset: $i > $i ).
% 3.43/1.83
% 3.43/1.83 thf(powersetAx_type,type,
% 3.43/1.83 powersetAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(powersetAx,definition,
% 3.43/1.83 ( powersetAx
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( in @ B @ ( powerset @ A ) )
% 3.43/1.83 <=> ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ B )
% 3.43/1.83 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setunion_type,type,
% 3.43/1.83 setunion: $i > $i ).
% 3.43/1.83
% 3.43/1.83 thf(setunionAx_type,type,
% 3.43/1.83 setunionAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setunionAx,definition,
% 3.43/1.83 ( setunionAx
% 3.43/1.83 = ( ! [A: $i,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ ( setunion @ A ) )
% 3.43/1.83 <=> ? [B: $i] :
% 3.43/1.83 ( ( in @ Xx @ B )
% 3.43/1.83 & ( in @ B @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(omega_type,type,
% 3.43/1.83 omega: $i ).
% 3.43/1.83
% 3.43/1.83 thf(omega0Ax_type,type,
% 3.43/1.83 omega0Ax: $o ).
% 3.43/1.83
% 3.43/1.83 thf(omega0Ax,definition,
% 3.43/1.83 ( omega0Ax
% 3.43/1.83 = ( in @ emptyset @ omega ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(omegaSAx_type,type,
% 3.43/1.83 omegaSAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(omegaSAx,definition,
% 3.43/1.83 ( omegaSAx
% 3.43/1.83 = ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ omega )
% 3.43/1.83 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(omegaIndAx_type,type,
% 3.43/1.83 omegaIndAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(omegaIndAx,definition,
% 3.43/1.83 ( omegaIndAx
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ( ( ( in @ emptyset @ A )
% 3.43/1.83 & ! [Xx: $i] :
% 3.43/1.83 ( ( ( in @ Xx @ omega )
% 3.43/1.83 & ( in @ Xx @ A ) )
% 3.43/1.83 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.43/1.83 => ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ omega )
% 3.43/1.83 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(replAx_type,type,
% 3.43/1.83 replAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(replAx,definition,
% 3.43/1.83 ( replAx
% 3.43/1.83 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( exu
% 3.43/1.83 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.43/1.83 => ? [B: $i] :
% 3.43/1.83 ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ B )
% 3.43/1.83 <=> ? [Xy: $i] :
% 3.43/1.83 ( ( in @ Xy @ A )
% 3.43/1.83 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(foundationAx_type,type,
% 3.43/1.83 foundationAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(foundationAx,definition,
% 3.43/1.83 ( foundationAx
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.43/1.83 => ? [B: $i] :
% 3.43/1.83 ( ( in @ B @ A )
% 3.43/1.83 & ~ ? [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ B )
% 3.43/1.83 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(wellorderingAx_type,type,
% 3.43/1.83 wellorderingAx: $o ).
% 3.43/1.83
% 3.43/1.83 thf(wellorderingAx,definition,
% 3.43/1.83 ( wellorderingAx
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ? [B: $i] :
% 3.43/1.83 ( ! [C: $i] :
% 3.43/1.83 ( ( in @ C @ B )
% 3.43/1.83 => ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ C )
% 3.43/1.83 => ( in @ Xx @ A ) ) )
% 3.43/1.83 & ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( ( in @ Xx @ A )
% 3.43/1.83 & ( in @ Xy @ A ) )
% 3.43/1.83 => ( ! [C: $i] :
% 3.43/1.83 ( ( in @ C @ B )
% 3.43/1.83 => ( ( in @ Xx @ C )
% 3.43/1.83 <=> ( in @ Xy @ C ) ) )
% 3.43/1.83 => ( Xx = Xy ) ) )
% 3.43/1.83 & ! [C: $i,D: $i] :
% 3.43/1.83 ( ( ( in @ C @ B )
% 3.43/1.83 & ( in @ D @ B ) )
% 3.43/1.83 => ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ C )
% 3.43/1.83 => ( in @ Xx @ D ) )
% 3.43/1.83 | ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ D )
% 3.43/1.83 => ( in @ Xx @ C ) ) ) )
% 3.43/1.83 & ! [C: $i] :
% 3.43/1.83 ( ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ C )
% 3.43/1.83 => ( in @ Xx @ A ) )
% 3.43/1.83 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.43/1.83 => ? [D: $i,Xx: $i] :
% 3.43/1.83 ( ( in @ D @ B )
% 3.43/1.83 & ( in @ Xx @ C )
% 3.43/1.83 & ~ ? [Xy: $i] :
% 3.43/1.83 ( ( in @ Xy @ D )
% 3.43/1.83 & ( in @ Xy @ C ) )
% 3.43/1.83 & ! [E: $i] :
% 3.43/1.83 ( ( in @ E @ B )
% 3.43/1.83 => ( ! [Xy: $i] :
% 3.43/1.83 ( ( in @ Xy @ E )
% 3.43/1.83 => ( in @ Xy @ D ) )
% 3.43/1.83 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(descr_type,type,
% 3.43/1.83 descr: ( $i > $o ) > $i ).
% 3.43/1.83
% 3.43/1.83 thf(descrp_type,type,
% 3.43/1.83 descrp: $o ).
% 3.43/1.83
% 3.43/1.83 thf(descrp,definition,
% 3.43/1.83 ( descrp
% 3.43/1.83 = ( ! [Xphi: $i > $o] :
% 3.43/1.83 ( ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 => ( Xphi
% 3.43/1.83 @ ( descr
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(dsetconstr_type,type,
% 3.43/1.83 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.43/1.83
% 3.43/1.83 thf(dsetconstrI_type,type,
% 3.43/1.83 dsetconstrI: $o ).
% 3.43/1.83
% 3.43/1.83 thf(dsetconstrI,definition,
% 3.43/1.83 ( dsetconstrI
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( ( Xphi @ Xx )
% 3.43/1.83 => ( in @ Xx
% 3.43/1.83 @ ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(dsetconstrEL_type,type,
% 3.43/1.83 dsetconstrEL: $o ).
% 3.43/1.83
% 3.43/1.83 thf(dsetconstrEL,definition,
% 3.43/1.83 ( dsetconstrEL
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx
% 3.43/1.83 @ ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.43/1.83 => ( in @ Xx @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(dsetconstrER_type,type,
% 3.43/1.83 dsetconstrER: $o ).
% 3.43/1.83
% 3.43/1.83 thf(dsetconstrER,definition,
% 3.43/1.83 ( dsetconstrER
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx
% 3.43/1.83 @ ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.43/1.83 => ( Xphi @ Xx ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(exuE1_type,type,
% 3.43/1.83 exuE1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(exuE1,definition,
% 3.43/1.83 ( exuE1
% 3.43/1.83 = ( ! [Xphi: $i > $o] :
% 3.43/1.83 ( ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 => ? [Xx: $i] :
% 3.43/1.83 ( ( Xphi @ Xx )
% 3.43/1.83 & ! [Xy: $i] :
% 3.43/1.83 ( ( Xphi @ Xy )
% 3.43/1.83 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(prop2set_type,type,
% 3.43/1.83 prop2set: $o > $i ).
% 3.43/1.83
% 3.43/1.83 thf(prop2setE_type,type,
% 3.43/1.83 prop2setE: $o ).
% 3.43/1.83
% 3.43/1.83 thf(prop2setE,definition,
% 3.43/1.83 ( prop2setE
% 3.43/1.83 = ( ! [Xphi: $o,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.43/1.83 => Xphi ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptysetE_type,type,
% 3.43/1.83 emptysetE: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptysetE,definition,
% 3.43/1.83 ( emptysetE
% 3.43/1.83 = ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ emptyset )
% 3.43/1.83 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptysetimpfalse_type,type,
% 3.43/1.83 emptysetimpfalse: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptysetimpfalse,definition,
% 3.43/1.83 ( emptysetimpfalse
% 3.43/1.83 = ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ emptyset )
% 3.43/1.83 => $false ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(notinemptyset_type,type,
% 3.43/1.83 notinemptyset: $o ).
% 3.43/1.83
% 3.43/1.83 thf(notinemptyset,definition,
% 3.43/1.83 ( notinemptyset
% 3.43/1.83 = ( ! [Xx: $i] :
% 3.43/1.83 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(exuE3e_type,type,
% 3.43/1.83 exuE3e: $o ).
% 3.43/1.83
% 3.43/1.83 thf(exuE3e,definition,
% 3.43/1.83 ( exuE3e
% 3.43/1.83 = ( ! [Xphi: $i > $o] :
% 3.43/1.83 ( ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setext_type,type,
% 3.43/1.83 setext: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setext,definition,
% 3.43/1.83 ( setext
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( in @ Xx @ B ) )
% 3.43/1.83 => ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ B )
% 3.43/1.83 => ( in @ Xx @ A ) )
% 3.43/1.83 => ( A = B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptyI_type,type,
% 3.43/1.83 emptyI: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptyI,definition,
% 3.43/1.83 ( emptyI
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ~ ( in @ Xx @ A )
% 3.43/1.83 => ( A = emptyset ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(noeltsimpempty_type,type,
% 3.43/1.83 noeltsimpempty: $o ).
% 3.43/1.83
% 3.43/1.83 thf(noeltsimpempty,definition,
% 3.43/1.83 ( noeltsimpempty
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ~ ( in @ Xx @ A )
% 3.43/1.83 => ( A = emptyset ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setbeta_type,type,
% 3.43/1.83 setbeta: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setbeta,definition,
% 3.43/1.83 ( setbeta
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( ( in @ Xx
% 3.43/1.83 @ ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.43/1.83 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(nonempty_type,type,
% 3.43/1.83 nonempty: $i > $o ).
% 3.43/1.83
% 3.43/1.83 thf(nonempty,definition,
% 3.43/1.83 ( nonempty
% 3.43/1.83 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(nonemptyE1_type,type,
% 3.43/1.83 nonemptyE1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(nonemptyE1,definition,
% 3.43/1.83 ( nonemptyE1
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ( ( nonempty @ A )
% 3.43/1.83 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(nonemptyI_type,type,
% 3.43/1.83 nonemptyI: $o ).
% 3.43/1.83
% 3.43/1.83 thf(nonemptyI,definition,
% 3.43/1.83 ( nonemptyI
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( ( Xphi @ Xx )
% 3.43/1.83 => ( nonempty
% 3.43/1.83 @ ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(nonemptyI1_type,type,
% 3.43/1.83 nonemptyI1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(nonemptyI1,definition,
% 3.43/1.83 ( nonemptyI1
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.43/1.83 => ( nonempty @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinIL_type,type,
% 3.43/1.83 setadjoinIL: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinIL,definition,
% 3.43/1.83 ( setadjoinIL
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptyinunitempty_type,type,
% 3.43/1.83 emptyinunitempty: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptyinunitempty,definition,
% 3.43/1.83 ( emptyinunitempty
% 3.43/1.83 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinIR_type,type,
% 3.43/1.83 setadjoinIR: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinIR,definition,
% 3.43/1.83 ( setadjoinIR
% 3.43/1.83 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.43/1.83 ( ( in @ Xy @ A )
% 3.43/1.83 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinE_type,type,
% 3.43/1.83 setadjoinE: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinE,definition,
% 3.43/1.83 ( setadjoinE
% 3.43/1.83 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.43/1.83 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.43/1.83 => ! [Xphi: $o] :
% 3.43/1.83 ( ( ( Xy = Xx )
% 3.43/1.83 => Xphi )
% 3.43/1.83 => ( ( ( in @ Xy @ A )
% 3.43/1.83 => Xphi )
% 3.43/1.83 => Xphi ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinOr_type,type,
% 3.43/1.83 setadjoinOr: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinOr,definition,
% 3.43/1.83 ( setadjoinOr
% 3.43/1.83 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.43/1.83 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.43/1.83 => ( ( Xy = Xx )
% 3.43/1.83 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setoftrueEq_type,type,
% 3.43/1.83 setoftrueEq: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setoftrueEq,definition,
% 3.43/1.83 ( setoftrueEq
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ( ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xx: $i] : $true )
% 3.43/1.83 = A ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(powersetI_type,type,
% 3.43/1.83 powersetI: $o ).
% 3.43/1.83
% 3.43/1.83 thf(powersetI,definition,
% 3.43/1.83 ( powersetI
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ B )
% 3.43/1.83 => ( in @ Xx @ A ) )
% 3.43/1.83 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptyinPowerset_type,type,
% 3.43/1.83 emptyinPowerset: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptyinPowerset,definition,
% 3.43/1.83 ( emptyinPowerset
% 3.43/1.83 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptyInPowerset_type,type,
% 3.43/1.83 emptyInPowerset: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptyInPowerset,definition,
% 3.43/1.83 ( emptyInPowerset
% 3.43/1.83 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(powersetE_type,type,
% 3.43/1.83 powersetE: $o ).
% 3.43/1.83
% 3.43/1.83 thf(powersetE,definition,
% 3.43/1.83 ( powersetE
% 3.43/1.83 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.83 ( ( in @ B @ ( powerset @ A ) )
% 3.43/1.83 => ( ( in @ Xx @ B )
% 3.43/1.83 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setunionI_type,type,
% 3.43/1.83 setunionI: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setunionI,definition,
% 3.43/1.83 ( setunionI
% 3.43/1.83 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.43/1.83 ( ( in @ Xx @ B )
% 3.43/1.83 => ( ( in @ B @ A )
% 3.43/1.83 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setunionE_type,type,
% 3.43/1.83 setunionE: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setunionE,definition,
% 3.43/1.83 ( setunionE
% 3.43/1.83 = ( ! [A: $i,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ ( setunion @ A ) )
% 3.43/1.83 => ! [Xphi: $o] :
% 3.43/1.83 ( ! [B: $i] :
% 3.43/1.83 ( ( in @ Xx @ B )
% 3.43/1.83 => ( ( in @ B @ A )
% 3.43/1.83 => Xphi ) )
% 3.43/1.83 => Xphi ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(subPowSU_type,type,
% 3.43/1.83 subPowSU: $o ).
% 3.43/1.83
% 3.43/1.83 thf(subPowSU,definition,
% 3.43/1.83 ( subPowSU
% 3.43/1.83 = ( ! [A: $i,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(exuE2_type,type,
% 3.43/1.83 exuE2: $o ).
% 3.43/1.83
% 3.43/1.83 thf(exuE2,definition,
% 3.43/1.83 ( exuE2
% 3.43/1.83 = ( ! [Xphi: $i > $o] :
% 3.43/1.83 ( ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 => ? [Xx: $i] :
% 3.43/1.83 ! [Xy: $i] :
% 3.43/1.83 ( ( Xphi @ Xy )
% 3.43/1.83 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(nonemptyImpWitness_type,type,
% 3.43/1.83 nonemptyImpWitness: $o ).
% 3.43/1.83
% 3.43/1.83 thf(nonemptyImpWitness,definition,
% 3.43/1.83 ( nonemptyImpWitness
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ( ( nonempty @ A )
% 3.43/1.83 => ? [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 & $true ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(uniqinunit_type,type,
% 3.43/1.83 uniqinunit: $o ).
% 3.43/1.83
% 3.43/1.83 thf(uniqinunit,definition,
% 3.43/1.83 ( uniqinunit
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.43/1.83 => ( Xx = Xy ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(notinsingleton_type,type,
% 3.43/1.83 notinsingleton: $o ).
% 3.43/1.83
% 3.43/1.83 thf(notinsingleton,definition,
% 3.43/1.83 ( notinsingleton
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xx != Xy )
% 3.43/1.83 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(eqinunit_type,type,
% 3.43/1.83 eqinunit: $o ).
% 3.43/1.83
% 3.43/1.83 thf(eqinunit,definition,
% 3.43/1.83 ( eqinunit
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xx = Xy )
% 3.43/1.83 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(singletonsswitch_type,type,
% 3.43/1.83 singletonsswitch: $o ).
% 3.43/1.83
% 3.43/1.83 thf(singletonsswitch,definition,
% 3.43/1.83 ( singletonsswitch
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.43/1.83 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(upairsetE_type,type,
% 3.43/1.83 upairsetE: $o ).
% 3.43/1.83
% 3.43/1.83 thf(upairsetE,definition,
% 3.43/1.83 ( upairsetE
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.43/1.83 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.43/1.83 => ( ( Xz = Xx )
% 3.43/1.83 | ( Xz = Xy ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(upairsetIL_type,type,
% 3.43/1.83 upairsetIL: $o ).
% 3.43/1.83
% 3.43/1.83 thf(upairsetIL,definition,
% 3.43/1.83 ( upairsetIL
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(upairsetIR_type,type,
% 3.43/1.83 upairsetIR: $o ).
% 3.43/1.83
% 3.43/1.83 thf(upairsetIR,definition,
% 3.43/1.83 ( upairsetIR
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptyE1_type,type,
% 3.43/1.83 emptyE1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptyE1,definition,
% 3.43/1.83 ( emptyE1
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.83 ( ? [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 & ( Xphi @ Xx ) )
% 3.43/1.83 => ( ( ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 = emptyset )
% 3.43/1.83 => $false ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(vacuousDall_type,type,
% 3.43/1.83 vacuousDall: $o ).
% 3.43/1.83
% 3.43/1.83 thf(vacuousDall,definition,
% 3.43/1.83 ( vacuousDall
% 3.43/1.83 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ emptyset )
% 3.43/1.83 => ( Xphi @ Xx ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(quantDeMorgan1_type,type,
% 3.43/1.83 quantDeMorgan1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(quantDeMorgan1,definition,
% 3.43/1.83 ( quantDeMorgan1
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.83 ( ~ ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( Xphi @ Xx ) )
% 3.43/1.83 => ? [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(quantDeMorgan2_type,type,
% 3.43/1.83 quantDeMorgan2: $o ).
% 3.43/1.83
% 3.43/1.83 thf(quantDeMorgan2,definition,
% 3.43/1.83 ( quantDeMorgan2
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ~ ( Xphi @ Xx ) )
% 3.43/1.83 => ~ ? [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(quantDeMorgan3_type,type,
% 3.43/1.83 quantDeMorgan3: $o ).
% 3.43/1.83
% 3.43/1.83 thf(quantDeMorgan3,definition,
% 3.43/1.83 ( quantDeMorgan3
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.83 ( ~ ? [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 & ( Xphi @ Xx ) )
% 3.43/1.83 => ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(quantDeMorgan4_type,type,
% 3.43/1.83 quantDeMorgan4: $o ).
% 3.43/1.83
% 3.43/1.83 thf(quantDeMorgan4,definition,
% 3.43/1.83 ( quantDeMorgan4
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.83 ( ? [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 & ~ ( Xphi @ Xx ) )
% 3.43/1.83 => ~ ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(prop2setI_type,type,
% 3.43/1.83 prop2setI: $o ).
% 3.43/1.83
% 3.43/1.83 thf(prop2setI,definition,
% 3.43/1.83 ( prop2setI
% 3.43/1.83 = ( ! [Xphi: $o] :
% 3.43/1.83 ( Xphi
% 3.43/1.83 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(set2prop_type,type,
% 3.43/1.83 set2prop: $i > $o ).
% 3.43/1.83
% 3.43/1.83 thf(prop2set2propI_type,type,
% 3.43/1.83 prop2set2propI: $o ).
% 3.43/1.83
% 3.43/1.83 thf(prop2set2propI,definition,
% 3.43/1.83 ( prop2set2propI
% 3.43/1.83 = ( ! [Xphi: $o] :
% 3.43/1.83 ( Xphi
% 3.43/1.83 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(notdexE_type,type,
% 3.43/1.83 notdexE: $o ).
% 3.43/1.83
% 3.43/1.83 thf(notdexE,definition,
% 3.43/1.83 ( notdexE
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.83 ( ~ ? [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 & ( Xphi @ Xx ) )
% 3.43/1.83 => ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(notdallE_type,type,
% 3.43/1.83 notdallE: $o ).
% 3.43/1.83
% 3.43/1.83 thf(notdallE,definition,
% 3.43/1.83 ( notdallE
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.83 ( ~ ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( Xphi @ Xx ) )
% 3.43/1.83 => ? [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(exuI1_type,type,
% 3.43/1.83 exuI1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(exuI1,definition,
% 3.43/1.83 ( exuI1
% 3.43/1.83 = ( ! [Xphi: $i > $o] :
% 3.43/1.83 ( ? [Xx: $i] :
% 3.43/1.83 ( ( Xphi @ Xx )
% 3.43/1.83 & ! [Xy: $i] :
% 3.43/1.83 ( ( Xphi @ Xy )
% 3.43/1.83 => ( Xx = Xy ) ) )
% 3.43/1.83 => ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(exuI3_type,type,
% 3.43/1.83 exuI3: $o ).
% 3.43/1.83
% 3.43/1.83 thf(exuI3,definition,
% 3.43/1.83 ( exuI3
% 3.43/1.83 = ( ! [Xphi: $i > $o] :
% 3.43/1.83 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.43/1.83 => ( ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xphi @ Xx )
% 3.43/1.83 => ( ( Xphi @ Xy )
% 3.43/1.83 => ( Xx = Xy ) ) )
% 3.43/1.83 => ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(exuI2_type,type,
% 3.43/1.83 exuI2: $o ).
% 3.43/1.83
% 3.43/1.83 thf(exuI2,definition,
% 3.43/1.83 ( exuI2
% 3.43/1.83 = ( ! [Xphi: $i > $o] :
% 3.43/1.83 ( ? [Xx: $i] :
% 3.43/1.83 ! [Xy: $i] :
% 3.43/1.83 ( ( Xphi @ Xy )
% 3.43/1.83 <=> ( Xy = Xx ) )
% 3.43/1.83 => ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(inCongP_type,type,
% 3.43/1.83 inCongP: $o ).
% 3.43/1.83
% 3.43/1.83 thf(inCongP,definition,
% 3.43/1.83 ( inCongP
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( A = B )
% 3.43/1.83 => ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xx = Xy )
% 3.43/1.83 => ( ( in @ Xx @ A )
% 3.43/1.83 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(in__Cong_type,type,
% 3.43/1.83 in__Cong: $o ).
% 3.43/1.83
% 3.43/1.83 thf(in__Cong,definition,
% 3.43/1.83 ( in__Cong
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( A = B )
% 3.43/1.83 => ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xx = Xy )
% 3.43/1.83 => ( ( in @ Xx @ A )
% 3.43/1.83 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(exuE3u_type,type,
% 3.43/1.83 exuE3u: $o ).
% 3.43/1.83
% 3.43/1.83 thf(exuE3u,definition,
% 3.43/1.83 ( exuE3u
% 3.43/1.83 = ( ! [Xphi: $i > $o] :
% 3.43/1.83 ( ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 => ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xphi @ Xx )
% 3.43/1.83 => ( ( Xphi @ Xy )
% 3.43/1.83 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(exu__Cong_type,type,
% 3.43/1.83 exu__Cong: $o ).
% 3.43/1.83
% 3.43/1.83 thf(exu__Cong,definition,
% 3.43/1.83 ( exu__Cong
% 3.43/1.83 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.43/1.83 ( ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xx = Xy )
% 3.43/1.83 => ( ( Xphi @ Xx )
% 3.43/1.83 <=> ( Xpsi @ Xy ) ) )
% 3.43/1.83 => ( ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 <=> ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptyset__Cong_type,type,
% 3.43/1.83 emptyset__Cong: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptyset__Cong,definition,
% 3.43/1.83 ( emptyset__Cong
% 3.43/1.83 = ( emptyset = emptyset ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoin__Cong_type,type,
% 3.43/1.83 setadjoin__Cong: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoin__Cong,definition,
% 3.43/1.83 ( setadjoin__Cong
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xx = Xy )
% 3.43/1.83 => ! [Xz: $i,Xu: $i] :
% 3.43/1.83 ( ( Xz = Xu )
% 3.43/1.83 => ( ( setadjoin @ Xx @ Xz )
% 3.43/1.83 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(powerset__Cong_type,type,
% 3.43/1.83 powerset__Cong: $o ).
% 3.43/1.83
% 3.43/1.83 thf(powerset__Cong,definition,
% 3.43/1.83 ( powerset__Cong
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( A = B )
% 3.43/1.83 => ( ( powerset @ A )
% 3.43/1.83 = ( powerset @ B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setunion__Cong_type,type,
% 3.43/1.83 setunion__Cong: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setunion__Cong,definition,
% 3.43/1.83 ( setunion__Cong
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( A = B )
% 3.43/1.83 => ( ( setunion @ A )
% 3.43/1.83 = ( setunion @ B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(omega__Cong_type,type,
% 3.43/1.83 omega__Cong: $o ).
% 3.43/1.83
% 3.43/1.83 thf(omega__Cong,definition,
% 3.43/1.83 ( omega__Cong
% 3.43/1.83 = ( omega = omega ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(exuEu_type,type,
% 3.43/1.83 exuEu: $o ).
% 3.43/1.83
% 3.43/1.83 thf(exuEu,definition,
% 3.43/1.83 ( exuEu
% 3.43/1.83 = ( ! [Xphi: $i > $o] :
% 3.43/1.83 ( ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 => ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xphi @ Xx )
% 3.43/1.83 => ( ( Xphi @ Xy )
% 3.43/1.83 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(descr__Cong_type,type,
% 3.43/1.83 descr__Cong: $o ).
% 3.43/1.83
% 3.43/1.83 thf(descr__Cong,definition,
% 3.43/1.83 ( descr__Cong
% 3.43/1.83 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.43/1.83 ( ! [Xx: $i,Xy: $i] :
% 3.43/1.83 ( ( Xx = Xy )
% 3.43/1.83 => ( ( Xphi @ Xx )
% 3.43/1.83 <=> ( Xpsi @ Xy ) ) )
% 3.43/1.83 => ( ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 => ( ( exu
% 3.43/1.83 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.43/1.83 => ( ( descr
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 = ( descr
% 3.43/1.83 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(dsetconstr__Cong_type,type,
% 3.43/1.83 dsetconstr__Cong: $o ).
% 3.43/1.83
% 3.43/1.83 thf(dsetconstr__Cong,definition,
% 3.43/1.83 ( dsetconstr__Cong
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( A = B )
% 3.43/1.83 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ! [Xy: $i] :
% 3.43/1.83 ( ( in @ Xy @ B )
% 3.43/1.83 => ( ( Xx = Xy )
% 3.43/1.83 => ( ( Xphi @ Xx )
% 3.43/1.83 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.43/1.83 => ( ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 = ( dsetconstr @ B
% 3.43/1.83 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(subset_type,type,
% 3.43/1.83 subset: $i > $i > $o ).
% 3.43/1.83
% 3.43/1.83 thf(disjoint_type,type,
% 3.43/1.83 disjoint: $i > $i > $o ).
% 3.43/1.83
% 3.43/1.83 thf(setsmeet_type,type,
% 3.43/1.83 setsmeet: $i > $i > $o ).
% 3.43/1.83
% 3.43/1.83 thf(subsetI1_type,type,
% 3.43/1.83 subsetI1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(subsetI1,definition,
% 3.43/1.83 ( subsetI1
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( in @ Xx @ B ) )
% 3.43/1.83 => ( subset @ A @ B ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(eqimpsubset2_type,type,
% 3.43/1.83 eqimpsubset2: $o ).
% 3.43/1.83
% 3.43/1.83 thf(eqimpsubset2,definition,
% 3.43/1.83 ( eqimpsubset2
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( A = B )
% 3.43/1.83 => ( subset @ B @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(eqimpsubset1_type,type,
% 3.43/1.83 eqimpsubset1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(eqimpsubset1,definition,
% 3.43/1.83 ( eqimpsubset1
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( A = B )
% 3.43/1.83 => ( subset @ A @ B ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(subsetI2_type,type,
% 3.43/1.83 subsetI2: $o ).
% 3.43/1.83
% 3.43/1.83 thf(subsetI2,definition,
% 3.43/1.83 ( subsetI2
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ! [Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( in @ Xx @ B ) )
% 3.43/1.83 => ( subset @ A @ B ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(emptysetsubset_type,type,
% 3.43/1.83 emptysetsubset: $o ).
% 3.43/1.83
% 3.43/1.83 thf(emptysetsubset,definition,
% 3.43/1.83 ( emptysetsubset
% 3.43/1.83 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(subsetE_type,type,
% 3.43/1.83 subsetE: $o ).
% 3.43/1.83
% 3.43/1.83 thf(subsetE,definition,
% 3.43/1.83 ( subsetE
% 3.43/1.83 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.83 ( ( subset @ A @ B )
% 3.43/1.83 => ( ( in @ Xx @ A )
% 3.43/1.83 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(subsetE2_type,type,
% 3.43/1.83 subsetE2: $o ).
% 3.43/1.83
% 3.43/1.83 thf(subsetE2,definition,
% 3.43/1.83 ( subsetE2
% 3.43/1.83 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.83 ( ( subset @ A @ B )
% 3.43/1.83 => ( ~ ( in @ Xx @ B )
% 3.43/1.83 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(notsubsetI_type,type,
% 3.43/1.83 notsubsetI: $o ).
% 3.43/1.83
% 3.43/1.83 thf(notsubsetI,definition,
% 3.43/1.83 ( notsubsetI
% 3.43/1.83 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( ~ ( in @ Xx @ B )
% 3.43/1.83 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(notequalI1_type,type,
% 3.43/1.83 notequalI1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(notequalI1,definition,
% 3.43/1.83 ( notequalI1
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ~ ( subset @ A @ B )
% 3.43/1.83 => ( A != B ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(notequalI2_type,type,
% 3.43/1.83 notequalI2: $o ).
% 3.43/1.83
% 3.43/1.83 thf(notequalI2,definition,
% 3.43/1.83 ( notequalI2
% 3.43/1.83 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( ~ ( in @ Xx @ B )
% 3.43/1.83 => ( A != B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(subsetRefl_type,type,
% 3.43/1.83 subsetRefl: $o ).
% 3.43/1.83
% 3.43/1.83 thf(subsetRefl,definition,
% 3.43/1.83 ( subsetRefl
% 3.43/1.83 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(subsetTrans_type,type,
% 3.43/1.83 subsetTrans: $o ).
% 3.43/1.83
% 3.43/1.83 thf(subsetTrans,definition,
% 3.43/1.83 ( subsetTrans
% 3.43/1.83 = ( ! [A: $i,B: $i,C: $i] :
% 3.43/1.83 ( ( subset @ A @ B )
% 3.43/1.83 => ( ( subset @ B @ C )
% 3.43/1.83 => ( subset @ A @ C ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinSub_type,type,
% 3.43/1.83 setadjoinSub: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinSub,definition,
% 3.43/1.83 ( setadjoinSub
% 3.43/1.83 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinSub2_type,type,
% 3.43/1.83 setadjoinSub2: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setadjoinSub2,definition,
% 3.43/1.83 ( setadjoinSub2
% 3.43/1.83 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.43/1.83 ( ( subset @ A @ B )
% 3.43/1.83 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(subset2powerset_type,type,
% 3.43/1.83 subset2powerset: $o ).
% 3.43/1.83
% 3.43/1.83 thf(subset2powerset,definition,
% 3.43/1.83 ( subset2powerset
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( subset @ A @ B )
% 3.43/1.83 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(setextsub_type,type,
% 3.43/1.83 setextsub: $o ).
% 3.43/1.83
% 3.43/1.83 thf(setextsub,definition,
% 3.43/1.83 ( setextsub
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( subset @ A @ B )
% 3.43/1.83 => ( ( subset @ B @ A )
% 3.43/1.83 => ( A = B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(subsetemptysetimpeq_type,type,
% 3.43/1.83 subsetemptysetimpeq: $o ).
% 3.43/1.83
% 3.43/1.83 thf(subsetemptysetimpeq,definition,
% 3.43/1.83 ( subsetemptysetimpeq
% 3.43/1.83 = ( ! [A: $i] :
% 3.43/1.83 ( ( subset @ A @ emptyset )
% 3.43/1.83 => ( A = emptyset ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(powersetI1_type,type,
% 3.43/1.83 powersetI1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(powersetI1,definition,
% 3.43/1.83 ( powersetI1
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( subset @ B @ A )
% 3.43/1.83 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(powersetE1_type,type,
% 3.43/1.83 powersetE1: $o ).
% 3.43/1.83
% 3.43/1.83 thf(powersetE1,definition,
% 3.43/1.83 ( powersetE1
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( in @ B @ ( powerset @ A ) )
% 3.43/1.83 => ( subset @ B @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(inPowerset_type,type,
% 3.43/1.83 inPowerset: $o ).
% 3.43/1.83
% 3.43/1.83 thf(inPowerset,definition,
% 3.43/1.83 ( inPowerset
% 3.43/1.83 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(powersetsubset_type,type,
% 3.43/1.83 powersetsubset: $o ).
% 3.43/1.83
% 3.43/1.83 thf(powersetsubset,definition,
% 3.43/1.83 ( powersetsubset
% 3.43/1.83 = ( ! [A: $i,B: $i] :
% 3.43/1.83 ( ( subset @ A @ B )
% 3.43/1.83 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(sepInPowerset_type,type,
% 3.43/1.83 sepInPowerset: $o ).
% 3.43/1.83
% 3.43/1.83 thf(sepInPowerset,definition,
% 3.43/1.83 ( sepInPowerset
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.83 ( in
% 3.43/1.83 @ ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 @ ( powerset @ A ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(sepSubset_type,type,
% 3.43/1.83 sepSubset: $o ).
% 3.43/1.83
% 3.43/1.83 thf(sepSubset,definition,
% 3.43/1.83 ( sepSubset
% 3.43/1.83 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.83 ( subset
% 3.43/1.83 @ ( dsetconstr @ A
% 3.43/1.83 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.83 @ A ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(binunion_type,type,
% 3.43/1.83 binunion: $i > $i > $i ).
% 3.43/1.83
% 3.43/1.83 thf(binunionIL_type,type,
% 3.43/1.83 binunionIL: $o ).
% 3.43/1.83
% 3.43/1.83 thf(binunionIL,definition,
% 3.43/1.83 ( binunionIL
% 3.43/1.83 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.83 ( ( in @ Xx @ A )
% 3.43/1.83 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.43/1.83
% 3.43/1.83 thf(upairset2IR_type,type,
% 3.43/1.83 upairset2IR: $o ).
% 3.43/1.83
% 3.43/1.83 thf(upairset2IR,definition,
% 3.43/1.83 ( upairset2IR
% 3.43/1.83 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binunionIR_type,type,
% 3.43/1.84 binunionIR: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binunionIR,definition,
% 3.43/1.84 ( binunionIR
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ B )
% 3.43/1.84 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binunionEcases_type,type,
% 3.43/1.84 binunionEcases: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binunionEcases,definition,
% 3.43/1.84 ( binunionEcases
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.43/1.84 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.43/1.84 => ( ( ( in @ Xx @ A )
% 3.43/1.84 => Xphi )
% 3.43/1.84 => ( ( ( in @ Xx @ B )
% 3.43/1.84 => Xphi )
% 3.43/1.84 => Xphi ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binunionE_type,type,
% 3.43/1.84 binunionE: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binunionE,definition,
% 3.43/1.84 ( binunionE
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.43/1.84 => ( ( in @ Xx @ A )
% 3.43/1.84 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binunionLsub_type,type,
% 3.43/1.84 binunionLsub: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binunionLsub,definition,
% 3.43/1.84 ( binunionLsub
% 3.43/1.84 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binunionRsub_type,type,
% 3.43/1.84 binunionRsub: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binunionRsub,definition,
% 3.43/1.84 ( binunionRsub
% 3.43/1.84 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersect_type,type,
% 3.43/1.84 binintersect: $i > $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectI_type,type,
% 3.43/1.84 binintersectI: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectI,definition,
% 3.43/1.84 ( binintersectI
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( in @ Xx @ B )
% 3.43/1.84 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset5_type,type,
% 3.43/1.84 binintersectSubset5: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset5,definition,
% 3.43/1.84 ( binintersectSubset5
% 3.43/1.84 = ( ! [A: $i,B: $i,C: $i] :
% 3.43/1.84 ( ( subset @ C @ A )
% 3.43/1.84 => ( ( subset @ C @ B )
% 3.43/1.84 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectEL_type,type,
% 3.43/1.84 binintersectEL: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectEL,definition,
% 3.43/1.84 ( binintersectEL
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.43/1.84 => ( in @ Xx @ A ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectLsub_type,type,
% 3.43/1.84 binintersectLsub: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectLsub,definition,
% 3.43/1.84 ( binintersectLsub
% 3.43/1.84 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset2_type,type,
% 3.43/1.84 binintersectSubset2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset2,definition,
% 3.43/1.84 ( binintersectSubset2
% 3.43/1.84 = ( ! [A: $i,B: $i] :
% 3.43/1.84 ( ( subset @ A @ B )
% 3.43/1.84 => ( ( binintersect @ A @ B )
% 3.43/1.84 = A ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset3_type,type,
% 3.43/1.84 binintersectSubset3: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset3,definition,
% 3.43/1.84 ( binintersectSubset3
% 3.43/1.84 = ( ! [A: $i,B: $i] :
% 3.43/1.84 ( ( ( binintersect @ A @ B )
% 3.43/1.84 = B )
% 3.43/1.84 => ( subset @ B @ A ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectER_type,type,
% 3.43/1.84 binintersectER: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectER,definition,
% 3.43/1.84 ( binintersectER
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.43/1.84 => ( in @ Xx @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(disjointsetsI1_type,type,
% 3.43/1.84 disjointsetsI1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(disjointsetsI1,definition,
% 3.43/1.84 ( disjointsetsI1
% 3.43/1.84 = ( ! [A: $i,B: $i] :
% 3.43/1.84 ( ~ ? [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 & ( in @ Xx @ B ) )
% 3.43/1.84 => ( ( binintersect @ A @ B )
% 3.43/1.84 = emptyset ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectRsub_type,type,
% 3.43/1.84 binintersectRsub: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectRsub,definition,
% 3.43/1.84 ( binintersectRsub
% 3.43/1.84 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset4_type,type,
% 3.43/1.84 binintersectSubset4: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset4,definition,
% 3.43/1.84 ( binintersectSubset4
% 3.43/1.84 = ( ! [A: $i,B: $i] :
% 3.43/1.84 ( ( subset @ B @ A )
% 3.43/1.84 => ( ( binintersect @ A @ B )
% 3.43/1.84 = B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset1_type,type,
% 3.43/1.84 binintersectSubset1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(binintersectSubset1,definition,
% 3.43/1.84 ( binintersectSubset1
% 3.43/1.84 = ( ! [A: $i,B: $i] :
% 3.43/1.84 ( ( ( binintersect @ A @ B )
% 3.43/1.84 = A )
% 3.43/1.84 => ( subset @ A @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(bs114d_type,type,
% 3.43/1.84 bs114d: $o ).
% 3.43/1.84
% 3.43/1.84 thf(bs114d,definition,
% 3.43/1.84 ( bs114d
% 3.43/1.84 = ( ! [A: $i,B: $i,C: $i] :
% 3.43/1.84 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.43/1.84 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(regular_type,type,
% 3.43/1.84 regular: $i > $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminus_type,type,
% 3.43/1.84 setminus: $i > $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(setminusI_type,type,
% 3.43/1.84 setminusI: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusI,definition,
% 3.43/1.84 ( setminusI
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ~ ( in @ Xx @ B )
% 3.43/1.84 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setminusEL_type,type,
% 3.43/1.84 setminusEL: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusEL,definition,
% 3.43/1.84 ( setminusEL
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.43/1.84 => ( in @ Xx @ A ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setminusER_type,type,
% 3.43/1.84 setminusER: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusER,definition,
% 3.43/1.84 ( setminusER
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.43/1.84 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setminusSubset2_type,type,
% 3.43/1.84 setminusSubset2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusSubset2,definition,
% 3.43/1.84 ( setminusSubset2
% 3.43/1.84 = ( ! [A: $i,B: $i] :
% 3.43/1.84 ( ( subset @ A @ B )
% 3.43/1.84 => ( ( setminus @ A @ B )
% 3.43/1.84 = emptyset ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setminusERneg_type,type,
% 3.43/1.84 setminusERneg: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusERneg,definition,
% 3.43/1.84 ( setminusERneg
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.43/1.84 => ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setminusELneg_type,type,
% 3.43/1.84 setminusELneg: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusELneg,definition,
% 3.43/1.84 ( setminusELneg
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.43/1.84 => ( ~ ( in @ Xx @ B )
% 3.43/1.84 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setminusILneg_type,type,
% 3.43/1.84 setminusILneg: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusILneg,definition,
% 3.43/1.84 ( setminusILneg
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ~ ( in @ Xx @ A )
% 3.43/1.84 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setminusIRneg_type,type,
% 3.43/1.84 setminusIRneg: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusIRneg,definition,
% 3.43/1.84 ( setminusIRneg
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ B )
% 3.43/1.84 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setminusLsub_type,type,
% 3.43/1.84 setminusLsub: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusLsub,definition,
% 3.43/1.84 ( setminusLsub
% 3.43/1.84 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setminusSubset1_type,type,
% 3.43/1.84 setminusSubset1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setminusSubset1,definition,
% 3.43/1.84 ( setminusSubset1
% 3.43/1.84 = ( ! [A: $i,B: $i] :
% 3.43/1.84 ( ( ( setminus @ A @ B )
% 3.43/1.84 = emptyset )
% 3.43/1.84 => ( subset @ A @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(symdiff_type,type,
% 3.43/1.84 symdiff: $i > $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffE_type,type,
% 3.43/1.84 symdiffE: $o ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffE,definition,
% 3.43/1.84 ( symdiffE
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.43/1.84 => ! [Xphi: $o] :
% 3.43/1.84 ( ( ( in @ Xx @ A )
% 3.43/1.84 => ( ~ ( in @ Xx @ B )
% 3.43/1.84 => Xphi ) )
% 3.43/1.84 => ( ( ~ ( in @ Xx @ A )
% 3.43/1.84 => ( ( in @ Xx @ B )
% 3.43/1.84 => Xphi ) )
% 3.43/1.84 => Xphi ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffI1_type,type,
% 3.43/1.84 symdiffI1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffI1,definition,
% 3.43/1.84 ( symdiffI1
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ~ ( in @ Xx @ B )
% 3.43/1.84 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffI2_type,type,
% 3.43/1.84 symdiffI2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffI2,definition,
% 3.43/1.84 ( symdiffI2
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ~ ( in @ Xx @ A )
% 3.43/1.84 => ( ( in @ Xx @ B )
% 3.43/1.84 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffIneg1_type,type,
% 3.43/1.84 symdiffIneg1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffIneg1,definition,
% 3.43/1.84 ( symdiffIneg1
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( in @ Xx @ B )
% 3.43/1.84 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffIneg2_type,type,
% 3.43/1.84 symdiffIneg2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(symdiffIneg2,definition,
% 3.43/1.84 ( symdiffIneg2
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ~ ( in @ Xx @ A )
% 3.43/1.84 => ( ~ ( in @ Xx @ B )
% 3.43/1.84 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(iskpair_type,type,
% 3.43/1.84 iskpair: $i > $o ).
% 3.43/1.84
% 3.43/1.84 thf(secondinupair_type,type,
% 3.43/1.84 secondinupair: $o ).
% 3.43/1.84
% 3.43/1.84 thf(secondinupair,definition,
% 3.43/1.84 ( secondinupair
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairIL_type,type,
% 3.43/1.84 setukpairIL: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairIL,definition,
% 3.43/1.84 ( setukpairIL
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairIR_type,type,
% 3.43/1.84 setukpairIR: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairIR,definition,
% 3.43/1.84 ( setukpairIR
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(kpairiskpair_type,type,
% 3.43/1.84 kpairiskpair: $o ).
% 3.43/1.84
% 3.43/1.84 thf(kpairiskpair,definition,
% 3.43/1.84 ( kpairiskpair
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(kpair_type,type,
% 3.43/1.84 kpair: $i > $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(kpairp_type,type,
% 3.43/1.84 kpairp: $o ).
% 3.43/1.84
% 3.43/1.84 thf(kpairp,definition,
% 3.43/1.84 ( kpairp
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprod_type,type,
% 3.43/1.84 cartprod: $i > $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(singletonsubset_type,type,
% 3.43/1.84 singletonsubset: $o ).
% 3.43/1.84
% 3.43/1.84 thf(singletonsubset,definition,
% 3.43/1.84 ( singletonsubset
% 3.43/1.84 = ( ! [A: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(singletoninpowerset_type,type,
% 3.43/1.84 singletoninpowerset: $o ).
% 3.43/1.84
% 3.43/1.84 thf(singletoninpowerset,definition,
% 3.43/1.84 ( singletoninpowerset
% 3.43/1.84 = ( ! [A: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(singletoninpowunion_type,type,
% 3.43/1.84 singletoninpowunion: $o ).
% 3.43/1.84
% 3.43/1.84 thf(singletoninpowunion,definition,
% 3.43/1.84 ( singletoninpowunion
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(upairset2E_type,type,
% 3.43/1.84 upairset2E: $o ).
% 3.43/1.84
% 3.43/1.84 thf(upairset2E,definition,
% 3.43/1.84 ( upairset2E
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.43/1.84 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.43/1.84 => ( ( Xz = Xx )
% 3.43/1.84 | ( Xz = Xy ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(upairsubunion_type,type,
% 3.43/1.84 upairsubunion: $o ).
% 3.43/1.84
% 3.43/1.84 thf(upairsubunion,definition,
% 3.43/1.84 ( upairsubunion
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(upairinpowunion_type,type,
% 3.43/1.84 upairinpowunion: $o ).
% 3.43/1.84
% 3.43/1.84 thf(upairinpowunion,definition,
% 3.43/1.84 ( upairinpowunion
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ubforcartprodlem1_type,type,
% 3.43/1.84 ubforcartprodlem1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ubforcartprodlem1,definition,
% 3.43/1.84 ( ubforcartprodlem1
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ubforcartprodlem2_type,type,
% 3.43/1.84 ubforcartprodlem2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ubforcartprodlem2,definition,
% 3.43/1.84 ( ubforcartprodlem2
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ubforcartprodlem3_type,type,
% 3.43/1.84 ubforcartprodlem3: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ubforcartprodlem3,definition,
% 3.43/1.84 ( ubforcartprodlem3
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodpairin_type,type,
% 3.43/1.84 cartprodpairin: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodpairin,definition,
% 3.43/1.84 ( cartprodpairin
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodmempair1_type,type,
% 3.43/1.84 cartprodmempair1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodmempair1,definition,
% 3.43/1.84 ( cartprodmempair1
% 3.43/1.84 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.43/1.84 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.43/1.84 => ? [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 & ? [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 & ( Xu
% 3.43/1.84 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodmempair_type,type,
% 3.43/1.84 cartprodmempair: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodmempair,definition,
% 3.43/1.84 ( cartprodmempair
% 3.43/1.84 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.43/1.84 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.43/1.84 => ( iskpair @ Xu ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setunionE2_type,type,
% 3.43/1.84 setunionE2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setunionE2,definition,
% 3.43/1.84 ( setunionE2
% 3.43/1.84 = ( ! [A: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ ( setunion @ A ) )
% 3.43/1.84 => ? [X: $i] :
% 3.43/1.84 ( ( in @ X @ A )
% 3.43/1.84 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setunionsingleton1_type,type,
% 3.43/1.84 setunionsingleton1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setunionsingleton1,definition,
% 3.43/1.84 ( setunionsingleton1
% 3.43/1.84 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setunionsingleton2_type,type,
% 3.43/1.84 setunionsingleton2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setunionsingleton2,definition,
% 3.43/1.84 ( setunionsingleton2
% 3.43/1.84 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setunionsingleton_type,type,
% 3.43/1.84 setunionsingleton: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setunionsingleton,definition,
% 3.43/1.84 ( setunionsingleton
% 3.43/1.84 = ( ! [Xx: $i] :
% 3.43/1.84 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.43/1.84 = Xx ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(singleton_type,type,
% 3.43/1.84 singleton: $i > $o ).
% 3.43/1.84
% 3.43/1.84 thf(singletonprop_type,type,
% 3.43/1.84 singletonprop: $o ).
% 3.43/1.84
% 3.43/1.84 thf(singletonprop,definition,
% 3.43/1.84 ( singletonprop
% 3.43/1.84 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( ( Xphi @ Xx )
% 3.43/1.84 => ( ( Xphi @ Xy )
% 3.43/1.84 => ( Xx = Xy ) ) ) ) )
% 3.43/1.84 => ( ? [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 & ( Xphi @ Xx ) )
% 3.43/1.84 => ( singleton
% 3.43/1.84 @ ( dsetconstr @ A
% 3.43/1.84 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ex1_type,type,
% 3.43/1.84 ex1: $i > ( $i > $o ) > $o ).
% 3.43/1.84
% 3.43/1.84 thf(ex1,definition,
% 3.43/1.84 ( ex1
% 3.43/1.84 = ( ^ [A: $i,Xphi: $i > $o] :
% 3.43/1.84 ( singleton
% 3.43/1.84 @ ( dsetconstr @ A
% 3.43/1.84 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ex1E1_type,type,
% 3.43/1.84 ex1E1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ex1E1,definition,
% 3.43/1.84 ( ex1E1
% 3.43/1.84 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.84 ( ( ex1 @ A
% 3.43/1.84 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.84 => ? [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ex1I_type,type,
% 3.43/1.84 ex1I: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ex1I,definition,
% 3.43/1.84 ( ex1I
% 3.43/1.84 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( Xphi @ Xx )
% 3.43/1.84 => ( ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( ( Xphi @ Xy )
% 3.43/1.84 => ( Xy = Xx ) ) )
% 3.43/1.84 => ( ex1 @ A
% 3.43/1.84 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ex1I2_type,type,
% 3.43/1.84 ex1I2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ex1I2,definition,
% 3.43/1.84 ( ex1I2
% 3.43/1.84 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( ( Xphi @ Xx )
% 3.43/1.84 => ( ( Xphi @ Xy )
% 3.43/1.84 => ( Xx = Xy ) ) ) ) )
% 3.43/1.84 => ( ? [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 & ( Xphi @ Xx ) )
% 3.43/1.84 => ( ex1 @ A
% 3.43/1.84 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(singletonsuniq_type,type,
% 3.43/1.84 singletonsuniq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(singletonsuniq,definition,
% 3.43/1.84 ( singletonsuniq
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( ( setadjoin @ Xx @ emptyset )
% 3.43/1.84 = ( setadjoin @ Xy @ emptyset ) )
% 3.43/1.84 => ( Xx = Xy ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(atmost1p_type,type,
% 3.43/1.84 atmost1p: $i > $o ).
% 3.43/1.84
% 3.43/1.84 thf(atleast2p_type,type,
% 3.43/1.84 atleast2p: $i > $o ).
% 3.43/1.84
% 3.43/1.84 thf(atmost2p_type,type,
% 3.43/1.84 atmost2p: $i > $o ).
% 3.43/1.84
% 3.43/1.84 thf(upairsetp_type,type,
% 3.43/1.84 upairsetp: $i > $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjL1_type,type,
% 3.43/1.84 setukpairinjL1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjL1,definition,
% 3.43/1.84 ( setukpairinjL1
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.43/1.84 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.43/1.84 => ( Xx = Xz ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(kfstsingleton_type,type,
% 3.43/1.84 kfstsingleton: $o ).
% 3.43/1.84
% 3.43/1.84 thf(kfstsingleton,definition,
% 3.43/1.84 ( kfstsingleton
% 3.43/1.84 = ( ! [Xu: $i] :
% 3.43/1.84 ( ( iskpair @ Xu )
% 3.43/1.84 => ( singleton
% 3.43/1.84 @ ( dsetconstr @ ( setunion @ Xu )
% 3.43/1.84 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(theprop_type,type,
% 3.43/1.84 theprop: $o ).
% 3.43/1.84
% 3.43/1.84 thf(theprop,definition,
% 3.43/1.84 ( theprop
% 3.43/1.84 = ( ! [X: $i] :
% 3.43/1.84 ( ( singleton @ X )
% 3.43/1.84 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(kfst_type,type,
% 3.43/1.84 kfst: $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(kfstpairEq_type,type,
% 3.43/1.84 kfstpairEq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(kfstpairEq,definition,
% 3.43/1.84 ( kfstpairEq
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.43/1.84 = Xx ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodfstin_type,type,
% 3.43/1.84 cartprodfstin: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodfstin,definition,
% 3.43/1.84 ( cartprodfstin
% 3.43/1.84 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.43/1.84 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.43/1.84 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjL2_type,type,
% 3.43/1.84 setukpairinjL2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjL2,definition,
% 3.43/1.84 ( setukpairinjL2
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.43/1.84 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.43/1.84 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.43/1.84 => ( Xx = Xz ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjL_type,type,
% 3.43/1.84 setukpairinjL: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjL,definition,
% 3.43/1.84 ( setukpairinjL
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.43/1.84 ( ( ( kpair @ Xx @ Xy )
% 3.43/1.84 = ( kpair @ Xz @ Xu ) )
% 3.43/1.84 => ( Xx = Xz ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR11_type,type,
% 3.43/1.84 setukpairinjR11: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR11,definition,
% 3.43/1.84 ( setukpairinjR11
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( Xx = Xy )
% 3.43/1.84 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.43/1.84 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR12_type,type,
% 3.43/1.84 setukpairinjR12: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR12,definition,
% 3.43/1.84 ( setukpairinjR12
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( Xx = Xy )
% 3.43/1.84 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.43/1.84 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR1_type,type,
% 3.43/1.84 setukpairinjR1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR1,definition,
% 3.43/1.84 ( setukpairinjR1
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.43/1.84 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.43/1.84 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.43/1.84 => ( ( Xz = Xu )
% 3.43/1.84 => ( Xy = Xu ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(upairequniteq_type,type,
% 3.43/1.84 upairequniteq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(upairequniteq,definition,
% 3.43/1.84 ( upairequniteq
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.43/1.84 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.43/1.84 = ( setadjoin @ Xz @ emptyset ) )
% 3.43/1.84 => ( Xx = Xy ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR2_type,type,
% 3.43/1.84 setukpairinjR2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR2,definition,
% 3.43/1.84 ( setukpairinjR2
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.43/1.84 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.43/1.84 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.43/1.84 => ( Xy = Xu ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR_type,type,
% 3.43/1.84 setukpairinjR: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setukpairinjR,definition,
% 3.43/1.84 ( setukpairinjR
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.43/1.84 ( ( ( kpair @ Xx @ Xy )
% 3.43/1.84 = ( kpair @ Xz @ Xu ) )
% 3.43/1.84 => ( Xy = Xu ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ksndsingleton_type,type,
% 3.43/1.84 ksndsingleton: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ksndsingleton,definition,
% 3.43/1.84 ( ksndsingleton
% 3.43/1.84 = ( ! [Xu: $i] :
% 3.43/1.84 ( ( iskpair @ Xu )
% 3.43/1.84 => ( singleton
% 3.43/1.84 @ ( dsetconstr @ ( setunion @ Xu )
% 3.43/1.84 @ ^ [Xx: $i] :
% 3.43/1.84 ( Xu
% 3.43/1.84 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ksnd_type,type,
% 3.43/1.84 ksnd: $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(ksndpairEq_type,type,
% 3.43/1.84 ksndpairEq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ksndpairEq,definition,
% 3.43/1.84 ( ksndpairEq
% 3.43/1.84 = ( ! [Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.43/1.84 = Xy ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(kpairsurjEq_type,type,
% 3.43/1.84 kpairsurjEq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(kpairsurjEq,definition,
% 3.43/1.84 ( kpairsurjEq
% 3.43/1.84 = ( ! [Xu: $i] :
% 3.43/1.84 ( ( iskpair @ Xu )
% 3.43/1.84 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.43/1.84 = Xu ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodsndin_type,type,
% 3.43/1.84 cartprodsndin: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodsndin,definition,
% 3.43/1.84 ( cartprodsndin
% 3.43/1.84 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.43/1.84 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.43/1.84 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodpairmemEL_type,type,
% 3.43/1.84 cartprodpairmemEL: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodpairmemEL,definition,
% 3.43/1.84 ( cartprodpairmemEL
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.43/1.84 => ( in @ Xx @ A ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodpairmemER_type,type,
% 3.43/1.84 cartprodpairmemER: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodpairmemER,definition,
% 3.43/1.84 ( cartprodpairmemER
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.43/1.84 => ( in @ Xy @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodmempaircEq_type,type,
% 3.43/1.84 cartprodmempaircEq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodmempaircEq,definition,
% 3.43/1.84 ( cartprodmempaircEq
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( kpair @ Xx @ Xy )
% 3.43/1.84 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodfstpairEq_type,type,
% 3.43/1.84 cartprodfstpairEq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodfstpairEq,definition,
% 3.43/1.84 ( cartprodfstpairEq
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.43/1.84 = Xx ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodsndpairEq_type,type,
% 3.43/1.84 cartprodsndpairEq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodsndpairEq,definition,
% 3.43/1.84 ( cartprodsndpairEq
% 3.43/1.84 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.43/1.84 = Xy ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodpairsurjEq_type,type,
% 3.43/1.84 cartprodpairsurjEq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(cartprodpairsurjEq,definition,
% 3.43/1.84 ( cartprodpairsurjEq
% 3.43/1.84 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.43/1.84 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.43/1.84 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.43/1.84 = Xu ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(breln_type,type,
% 3.43/1.84 breln: $i > $i > $i > $o ).
% 3.43/1.84
% 3.43/1.84 thf(breln,definition,
% 3.43/1.84 ( breln
% 3.43/1.84 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstr_type,type,
% 3.43/1.84 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrI_type,type,
% 3.43/1.84 dpsetconstrI: $o ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrI,definition,
% 3.43/1.84 ( dpsetconstrI
% 3.43/1.84 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( Xphi @ Xx @ Xy )
% 3.43/1.84 => ( in @ ( kpair @ Xx @ Xy )
% 3.43/1.84 @ ( dpsetconstr @ A @ B
% 3.43/1.84 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrSub_type,type,
% 3.43/1.84 dpsetconstrSub: $o ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrSub,definition,
% 3.43/1.84 ( dpsetconstrSub
% 3.43/1.84 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.43/1.84 ( subset
% 3.43/1.84 @ ( dpsetconstr @ A @ B
% 3.43/1.84 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.43/1.84 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(setOfPairsIsBReln_type,type,
% 3.43/1.84 setOfPairsIsBReln: $o ).
% 3.43/1.84
% 3.43/1.84 thf(setOfPairsIsBReln,definition,
% 3.43/1.84 ( setOfPairsIsBReln
% 3.43/1.84 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.43/1.84 ( breln @ A @ B
% 3.43/1.84 @ ( dpsetconstr @ A @ B
% 3.43/1.84 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrERa_type,type,
% 3.43/1.84 dpsetconstrERa: $o ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrERa,definition,
% 3.43/1.84 ( dpsetconstrERa
% 3.43/1.84 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.43/1.84 @ ( dpsetconstr @ A @ B
% 3.43/1.84 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.43/1.84 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrEL1_type,type,
% 3.43/1.84 dpsetconstrEL1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrEL1,definition,
% 3.43/1.84 ( dpsetconstrEL1
% 3.43/1.84 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( in @ ( kpair @ Xx @ Xy )
% 3.43/1.84 @ ( dpsetconstr @ A @ B
% 3.43/1.84 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.43/1.84 => ( in @ Xx @ A ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrEL2_type,type,
% 3.43/1.84 dpsetconstrEL2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrEL2,definition,
% 3.43/1.84 ( dpsetconstrEL2
% 3.43/1.84 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( in @ ( kpair @ Xx @ Xy )
% 3.43/1.84 @ ( dpsetconstr @ A @ B
% 3.43/1.84 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.43/1.84 => ( in @ Xy @ B ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrER_type,type,
% 3.43/1.84 dpsetconstrER: $o ).
% 3.43/1.84
% 3.43/1.84 thf(dpsetconstrER,definition,
% 3.43/1.84 ( dpsetconstrER
% 3.43/1.84 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( in @ ( kpair @ Xx @ Xy )
% 3.43/1.84 @ ( dpsetconstr @ A @ B
% 3.43/1.84 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.43/1.84 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(func_type,type,
% 3.43/1.84 func: $i > $i > $i > $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcSet_type,type,
% 3.43/1.84 funcSet: $i > $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(funcImageSingleton_type,type,
% 3.43/1.84 funcImageSingleton: $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcImageSingleton,definition,
% 3.43/1.84 ( funcImageSingleton
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( singleton
% 3.43/1.84 @ ( dsetconstr @ B
% 3.43/1.84 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(apProp_type,type,
% 3.43/1.84 apProp: $o ).
% 3.43/1.84
% 3.43/1.84 thf(apProp,definition,
% 3.43/1.84 ( apProp
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in
% 3.43/1.84 @ ( setunion
% 3.43/1.84 @ ( dsetconstr @ B
% 3.43/1.84 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.43/1.84 @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ap_type,type,
% 3.43/1.84 ap: $i > $i > $i > $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(app_type,type,
% 3.43/1.84 app: $o ).
% 3.43/1.84
% 3.43/1.84 thf(app,definition,
% 3.43/1.84 ( app
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(infuncsetfunc_type,type,
% 3.43/1.84 infuncsetfunc: $o ).
% 3.43/1.84
% 3.43/1.84 thf(infuncsetfunc,definition,
% 3.43/1.84 ( infuncsetfunc
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.43/1.84 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ap2p_type,type,
% 3.43/1.84 ap2p: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ap2p,definition,
% 3.43/1.84 ( ap2p
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(funcinfuncset_type,type,
% 3.43/1.84 funcinfuncset: $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcinfuncset,definition,
% 3.43/1.84 ( funcinfuncset
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(lamProp_type,type,
% 3.43/1.84 lamProp: $o ).
% 3.43/1.84
% 3.43/1.84 thf(lamProp,definition,
% 3.43/1.84 ( lamProp
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.43/1.84 => ( func @ A @ B
% 3.43/1.84 @ ( dpsetconstr @ A @ B
% 3.43/1.84 @ ^ [Xx: $i,Xy: $i] :
% 3.43/1.84 ( ( Xf @ Xx )
% 3.43/1.84 = Xy ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(lam_type,type,
% 3.43/1.84 lam: $i > $i > ( $i > $i ) > $i ).
% 3.43/1.84
% 3.43/1.84 thf(lamp_type,type,
% 3.43/1.84 lamp: $o ).
% 3.43/1.84
% 3.43/1.84 thf(lamp,definition,
% 3.43/1.84 ( lamp
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.43/1.84 => ( func @ A @ B
% 3.43/1.84 @ ( lam @ A @ B
% 3.43/1.84 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(lam2p_type,type,
% 3.43/1.84 lam2p: $o ).
% 3.43/1.84
% 3.43/1.84 thf(lam2p,definition,
% 3.43/1.84 ( lam2p
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.43/1.84 => ( in
% 3.43/1.84 @ ( lam @ A @ B
% 3.43/1.84 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.43/1.84 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(brelnall1_type,type,
% 3.43/1.84 brelnall1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(brelnall1,definition,
% 3.43/1.84 ( brelnall1
% 3.43/1.84 = ( ! [A: $i,B: $i,R: $i] :
% 3.43/1.84 ( ( breln @ A @ B @ R )
% 3.43/1.84 => ! [Xphi: $i > $o] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.43/1.84 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ R )
% 3.43/1.84 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(brelnall2_type,type,
% 3.43/1.84 brelnall2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(brelnall2,definition,
% 3.43/1.84 ( brelnall2
% 3.43/1.84 = ( ! [A: $i,B: $i,R: $i] :
% 3.43/1.84 ( ( breln @ A @ B @ R )
% 3.43/1.84 => ! [Xphi: $i > $o] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.43/1.84 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ R )
% 3.43/1.84 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ex1E2_type,type,
% 3.43/1.84 ex1E2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ex1E2,definition,
% 3.43/1.84 ( ex1E2
% 3.43/1.84 = ( ! [A: $i,Xphi: $i > $o] :
% 3.43/1.84 ( ( ex1 @ A
% 3.43/1.84 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( ( Xphi @ Xx )
% 3.43/1.84 => ( ( Xphi @ Xy )
% 3.43/1.84 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(funcGraphProp1_type,type,
% 3.43/1.84 funcGraphProp1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcGraphProp1,definition,
% 3.43/1.84 ( funcGraphProp1
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(funcGraphProp3_type,type,
% 3.43/1.84 funcGraphProp3: $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcGraphProp3,definition,
% 3.43/1.84 ( funcGraphProp3
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(funcGraphProp2_type,type,
% 3.43/1.84 funcGraphProp2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcGraphProp2,definition,
% 3.43/1.84 ( funcGraphProp2
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.43/1.84 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.43/1.84 = Xy ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(funcextLem_type,type,
% 3.43/1.84 funcextLem: $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcextLem,definition,
% 3.43/1.84 ( funcextLem
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ! [Xg: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xg )
% 3.43/1.84 => ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.43/1.84 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.43/1.84 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(funcGraphProp4_type,type,
% 3.43/1.84 funcGraphProp4: $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcGraphProp4,definition,
% 3.43/1.84 ( funcGraphProp4
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.43/1.84 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.43/1.84 = Xy ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(subbreln_type,type,
% 3.43/1.84 subbreln: $o ).
% 3.43/1.84
% 3.43/1.84 thf(subbreln,definition,
% 3.43/1.84 ( subbreln
% 3.43/1.84 = ( ! [A: $i,B: $i,R: $i] :
% 3.43/1.84 ( ( breln @ A @ B @ R )
% 3.43/1.84 => ! [S: $i] :
% 3.43/1.84 ( ( breln @ A @ B @ S )
% 3.43/1.84 => ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.43/1.84 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.43/1.84 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(eqbreln_type,type,
% 3.43/1.84 eqbreln: $o ).
% 3.43/1.84
% 3.43/1.84 thf(eqbreln,definition,
% 3.43/1.84 ( eqbreln
% 3.43/1.84 = ( ! [A: $i,B: $i,R: $i] :
% 3.43/1.84 ( ( breln @ A @ B @ R )
% 3.43/1.84 => ! [S: $i] :
% 3.43/1.84 ( ( breln @ A @ B @ S )
% 3.43/1.84 => ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.43/1.84 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.43/1.84 => ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ B )
% 3.43/1.84 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.43/1.84 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.43/1.84 => ( R = S ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(funcext_type,type,
% 3.43/1.84 funcext: $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcext,definition,
% 3.43/1.84 ( funcext
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ! [Xg: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xg )
% 3.43/1.84 => ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.43/1.84 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.43/1.84 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(funcext2_type,type,
% 3.43/1.84 funcext2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(funcext2,definition,
% 3.43/1.84 ( funcext2
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.43/1.84 => ! [Xg: $i] :
% 3.43/1.84 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.43/1.84 => ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.43/1.84 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.43/1.84 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ap2apEq1_type,type,
% 3.43/1.84 ap2apEq1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ap2apEq1,definition,
% 3.43/1.84 ( ap2apEq1
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.43/1.84 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ap2apEq2_type,type,
% 3.43/1.84 ap2apEq2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ap2apEq2,definition,
% 3.43/1.84 ( ap2apEq2
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.43/1.84 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(beta1_type,type,
% 3.43/1.84 beta1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(beta1,definition,
% 3.43/1.84 ( beta1
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( ap @ A @ B
% 3.43/1.84 @ ( lam @ A @ B
% 3.43/1.84 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.43/1.84 @ Xx )
% 3.43/1.84 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(eta1_type,type,
% 3.43/1.84 eta1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(eta1,definition,
% 3.43/1.84 ( eta1
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( func @ A @ B @ Xf )
% 3.43/1.84 => ( ( lam @ A @ B
% 3.43/1.84 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.43/1.84 = Xf ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(lam2lamEq_type,type,
% 3.43/1.84 lam2lamEq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(lam2lamEq,definition,
% 3.43/1.84 ( lam2lamEq
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.43/1.84 => ( ( lam @ A @ B
% 3.43/1.84 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.43/1.84 = ( lam @ A @ B
% 3.43/1.84 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(beta2_type,type,
% 3.43/1.84 beta2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(beta2,definition,
% 3.43/1.84 ( beta2
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.43/1.84 ( ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ( ( ap @ A @ B
% 3.43/1.84 @ ( lam @ A @ B
% 3.43/1.84 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.43/1.84 @ Xx )
% 3.43/1.84 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(eta2_type,type,
% 3.43/1.84 eta2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(eta2,definition,
% 3.43/1.84 ( eta2
% 3.43/1.84 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.43/1.84 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.43/1.84 => ( ( lam @ A @ B
% 3.43/1.84 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.43/1.84 = Xf ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(iffalseProp1_type,type,
% 3.43/1.84 iffalseProp1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(iffalseProp1,definition,
% 3.43/1.84 ( iffalseProp1
% 3.43/1.84 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( ~ Xphi
% 3.43/1.84 => ( in @ Xy
% 3.43/1.84 @ ( dsetconstr @ A
% 3.43/1.84 @ ^ [Xz: $i] :
% 3.43/1.84 ( ( Xphi
% 3.43/1.84 & ( Xz = Xx ) )
% 3.43/1.84 | ( ~ Xphi
% 3.43/1.84 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(iffalseProp2_type,type,
% 3.43/1.84 iffalseProp2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(iffalseProp2,definition,
% 3.43/1.84 ( iffalseProp2
% 3.43/1.84 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( ~ Xphi
% 3.43/1.84 => ( ( dsetconstr @ A
% 3.43/1.84 @ ^ [Xz: $i] :
% 3.43/1.84 ( ( Xphi
% 3.43/1.84 & ( Xz = Xx ) )
% 3.43/1.84 | ( ~ Xphi
% 3.43/1.84 & ( Xz = Xy ) ) ) )
% 3.43/1.84 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(iftrueProp1_type,type,
% 3.43/1.84 iftrueProp1: $o ).
% 3.43/1.84
% 3.43/1.84 thf(iftrueProp1,definition,
% 3.43/1.84 ( iftrueProp1
% 3.43/1.84 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( Xphi
% 3.43/1.84 => ( in @ Xx
% 3.43/1.84 @ ( dsetconstr @ A
% 3.43/1.84 @ ^ [Xz: $i] :
% 3.43/1.84 ( ( Xphi
% 3.43/1.84 & ( Xz = Xx ) )
% 3.43/1.84 | ( ~ Xphi
% 3.43/1.84 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(iftrueProp2_type,type,
% 3.43/1.84 iftrueProp2: $o ).
% 3.43/1.84
% 3.43/1.84 thf(iftrueProp2,definition,
% 3.43/1.84 ( iftrueProp2
% 3.43/1.84 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( Xphi
% 3.43/1.84 => ( ( dsetconstr @ A
% 3.43/1.84 @ ^ [Xz: $i] :
% 3.43/1.84 ( ( Xphi
% 3.43/1.84 & ( Xz = Xx ) )
% 3.43/1.84 | ( ~ Xphi
% 3.43/1.84 & ( Xz = Xy ) ) ) )
% 3.43/1.84 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ifSingleton_type,type,
% 3.43/1.84 ifSingleton: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ifSingleton,definition,
% 3.43/1.84 ( ifSingleton
% 3.43/1.84 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( singleton
% 3.43/1.84 @ ( dsetconstr @ A
% 3.43/1.84 @ ^ [Xz: $i] :
% 3.43/1.84 ( ( Xphi
% 3.43/1.84 & ( Xz = Xx ) )
% 3.43/1.84 | ( ~ Xphi
% 3.43/1.84 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(if_type,type,
% 3.43/1.84 if: $i > $o > $i > $i > $i ).
% 3.43/1.84
% 3.43/1.84 thf(if,definition,
% 3.43/1.84 ( if
% 3.43/1.84 = ( ^ [A: $i,Xphi: $o,Xx: $i,Xy: $i] :
% 3.43/1.84 ( setunion
% 3.43/1.84 @ ( dsetconstr @ A
% 3.43/1.84 @ ^ [Xz: $i] :
% 3.43/1.84 ( ( Xphi
% 3.43/1.84 & ( Xz = Xx ) )
% 3.43/1.84 | ( ~ Xphi
% 3.43/1.84 & ( Xz = Xy ) ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(ifp_type,type,
% 3.43/1.84 ifp: $o ).
% 3.43/1.84
% 3.43/1.84 thf(ifp,definition,
% 3.43/1.84 ( ifp
% 3.43/1.84 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(theeq_type,type,
% 3.43/1.84 theeq: $o ).
% 3.43/1.84
% 3.43/1.84 thf(theeq,definition,
% 3.43/1.84 ( theeq
% 3.43/1.84 = ( ! [X: $i] :
% 3.43/1.84 ( ( singleton @ X )
% 3.43/1.84 => ! [Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ X )
% 3.43/1.84 => ( ( setunion @ X )
% 3.43/1.84 = Xx ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(iftrue_type,type,
% 3.43/1.84 iftrue: $o ).
% 3.43/1.84
% 3.43/1.84 thf(iftrue,definition,
% 3.43/1.84 ( iftrue
% 3.43/1.84 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( Xphi
% 3.43/1.84 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.43/1.84 = Xx ) ) ) ) ) ) ).
% 3.43/1.84
% 3.43/1.84 thf(iffalse,conjecture,
% 3.43/1.84 ( setextAx
% 3.43/1.84 => ( emptysetAx
% 3.43/1.84 => ( setadjoinAx
% 3.43/1.84 => ( powersetAx
% 3.43/1.84 => ( setunionAx
% 3.43/1.84 => ( omega0Ax
% 3.43/1.84 => ( omegaSAx
% 3.43/1.84 => ( omegaIndAx
% 3.43/1.84 => ( replAx
% 3.43/1.84 => ( foundationAx
% 3.43/1.84 => ( wellorderingAx
% 3.43/1.84 => ( descrp
% 3.43/1.84 => ( dsetconstrI
% 3.43/1.84 => ( dsetconstrEL
% 3.43/1.84 => ( dsetconstrER
% 3.43/1.84 => ( exuE1
% 3.43/1.84 => ( prop2setE
% 3.43/1.84 => ( emptysetE
% 3.43/1.84 => ( emptysetimpfalse
% 3.43/1.84 => ( notinemptyset
% 3.43/1.84 => ( exuE3e
% 3.43/1.84 => ( setext
% 3.43/1.84 => ( emptyI
% 3.43/1.84 => ( noeltsimpempty
% 3.43/1.84 => ( setbeta
% 3.43/1.84 => ( nonemptyE1
% 3.43/1.84 => ( nonemptyI
% 3.43/1.84 => ( nonemptyI1
% 3.43/1.84 => ( setadjoinIL
% 3.43/1.84 => ( emptyinunitempty
% 3.43/1.84 => ( setadjoinIR
% 3.43/1.84 => ( setadjoinE
% 3.43/1.84 => ( setadjoinOr
% 3.43/1.84 => ( setoftrueEq
% 3.43/1.84 => ( powersetI
% 3.43/1.84 => ( emptyinPowerset
% 3.43/1.84 => ( emptyInPowerset
% 3.43/1.84 => ( powersetE
% 3.43/1.84 => ( setunionI
% 3.43/1.84 => ( setunionE
% 3.43/1.84 => ( subPowSU
% 3.43/1.84 => ( exuE2
% 3.43/1.84 => ( nonemptyImpWitness
% 3.43/1.84 => ( uniqinunit
% 3.43/1.84 => ( notinsingleton
% 3.43/1.84 => ( eqinunit
% 3.43/1.84 => ( singletonsswitch
% 3.43/1.84 => ( upairsetE
% 3.43/1.84 => ( upairsetIL
% 3.43/1.84 => ( upairsetIR
% 3.43/1.84 => ( emptyE1
% 3.43/1.84 => ( vacuousDall
% 3.43/1.84 => ( quantDeMorgan1
% 3.43/1.84 => ( quantDeMorgan2
% 3.43/1.84 => ( quantDeMorgan3
% 3.43/1.84 => ( quantDeMorgan4
% 3.43/1.84 => ( prop2setI
% 3.43/1.84 => ( prop2set2propI
% 3.43/1.84 => ( notdexE
% 3.43/1.84 => ( notdallE
% 3.43/1.84 => ( exuI1
% 3.43/1.84 => ( exuI3
% 3.43/1.84 => ( exuI2
% 3.43/1.84 => ( inCongP
% 3.43/1.84 => ( in__Cong
% 3.43/1.84 => ( exuE3u
% 3.43/1.84 => ( exu__Cong
% 3.43/1.84 => ( emptyset__Cong
% 3.43/1.84 => ( setadjoin__Cong
% 3.43/1.84 => ( powerset__Cong
% 3.43/1.84 => ( setunion__Cong
% 3.43/1.84 => ( omega__Cong
% 3.43/1.84 => ( exuEu
% 3.43/1.84 => ( descr__Cong
% 3.43/1.84 => ( dsetconstr__Cong
% 3.43/1.84 => ( subsetI1
% 3.43/1.84 => ( eqimpsubset2
% 3.43/1.84 => ( eqimpsubset1
% 3.43/1.84 => ( subsetI2
% 3.43/1.84 => ( emptysetsubset
% 3.43/1.84 => ( subsetE
% 3.43/1.84 => ( subsetE2
% 3.43/1.84 => ( notsubsetI
% 3.43/1.84 => ( notequalI1
% 3.43/1.84 => ( notequalI2
% 3.43/1.84 => ( subsetRefl
% 3.43/1.84 => ( subsetTrans
% 3.43/1.84 => ( setadjoinSub
% 3.43/1.84 => ( setadjoinSub2
% 3.43/1.84 => ( subset2powerset
% 3.43/1.84 => ( setextsub
% 3.43/1.84 => ( subsetemptysetimpeq
% 3.43/1.84 => ( powersetI1
% 3.43/1.84 => ( powersetE1
% 3.43/1.84 => ( inPowerset
% 3.43/1.84 => ( powersetsubset
% 3.43/1.84 => ( sepInPowerset
% 3.43/1.84 => ( sepSubset
% 3.43/1.84 => ( binunionIL
% 3.43/1.84 => ( upairset2IR
% 3.43/1.84 => ( binunionIR
% 3.43/1.84 => ( binunionEcases
% 3.43/1.84 => ( binunionE
% 3.43/1.84 => ( binunionLsub
% 3.43/1.84 => ( binunionRsub
% 3.43/1.84 => ( binintersectI
% 3.43/1.84 => ( binintersectSubset5
% 3.43/1.84 => ( binintersectEL
% 3.43/1.84 => ( binintersectLsub
% 3.43/1.84 => ( binintersectSubset2
% 3.43/1.84 => ( binintersectSubset3
% 3.43/1.84 => ( binintersectER
% 3.43/1.84 => ( disjointsetsI1
% 3.43/1.84 => ( binintersectRsub
% 3.43/1.84 => ( binintersectSubset4
% 3.43/1.84 => ( binintersectSubset1
% 3.43/1.84 => ( bs114d
% 3.43/1.84 => ( setminusI
% 3.43/1.84 => ( setminusEL
% 3.43/1.84 => ( setminusER
% 3.43/1.84 => ( setminusSubset2
% 3.43/1.84 => ( setminusERneg
% 3.43/1.84 => ( setminusELneg
% 3.43/1.84 => ( setminusILneg
% 3.43/1.84 => ( setminusIRneg
% 3.43/1.84 => ( setminusLsub
% 3.43/1.84 => ( setminusSubset1
% 3.43/1.84 => ( symdiffE
% 3.43/1.84 => ( symdiffI1
% 3.43/1.84 => ( symdiffI2
% 3.43/1.84 => ( symdiffIneg1
% 3.43/1.84 => ( symdiffIneg2
% 3.43/1.84 => ( secondinupair
% 3.43/1.84 => ( setukpairIL
% 3.43/1.84 => ( setukpairIR
% 3.43/1.84 => ( kpairiskpair
% 3.43/1.84 => ( kpairp
% 3.43/1.84 => ( singletonsubset
% 3.43/1.84 => ( singletoninpowerset
% 3.43/1.84 => ( singletoninpowunion
% 3.43/1.84 => ( upairset2E
% 3.43/1.84 => ( upairsubunion
% 3.43/1.84 => ( upairinpowunion
% 3.43/1.84 => ( ubforcartprodlem1
% 3.43/1.84 => ( ubforcartprodlem2
% 3.43/1.84 => ( ubforcartprodlem3
% 3.43/1.84 => ( cartprodpairin
% 3.43/1.84 => ( cartprodmempair1
% 3.43/1.84 => ( cartprodmempair
% 3.43/1.84 => ( setunionE2
% 3.43/1.84 => ( setunionsingleton1
% 3.43/1.84 => ( setunionsingleton2
% 3.43/1.84 => ( setunionsingleton
% 3.43/1.84 => ( singletonprop
% 3.43/1.84 => ( ex1E1
% 3.43/1.84 => ( ex1I
% 3.43/1.84 => ( ex1I2
% 3.43/1.84 => ( singletonsuniq
% 3.43/1.84 => ( setukpairinjL1
% 3.43/1.84 => ( kfstsingleton
% 3.43/1.84 => ( theprop
% 3.43/1.84 => ( kfstpairEq
% 3.43/1.84 => ( cartprodfstin
% 3.43/1.84 => ( setukpairinjL2
% 3.43/1.84 => ( setukpairinjL
% 3.43/1.84 => ( setukpairinjR11
% 3.43/1.84 => ( setukpairinjR12
% 3.43/1.84 => ( setukpairinjR1
% 3.43/1.84 => ( upairequniteq
% 3.43/1.84 => ( setukpairinjR2
% 3.43/1.84 => ( setukpairinjR
% 3.43/1.84 => ( ksndsingleton
% 3.43/1.84 => ( ksndpairEq
% 3.43/1.84 => ( kpairsurjEq
% 3.43/1.84 => ( cartprodsndin
% 3.43/1.84 => ( cartprodpairmemEL
% 3.43/1.84 => ( cartprodpairmemER
% 3.43/1.84 => ( cartprodmempaircEq
% 3.43/1.84 => ( cartprodfstpairEq
% 3.43/1.84 => ( cartprodsndpairEq
% 3.43/1.84 => ( cartprodpairsurjEq
% 3.43/1.84 => ( dpsetconstrI
% 3.43/1.84 => ( dpsetconstrSub
% 3.43/1.84 => ( setOfPairsIsBReln
% 3.43/1.84 => ( dpsetconstrERa
% 3.43/1.84 => ( dpsetconstrEL1
% 3.43/1.84 => ( dpsetconstrEL2
% 3.43/1.84 => ( dpsetconstrER
% 3.43/1.84 => ( funcImageSingleton
% 3.43/1.84 => ( apProp
% 3.43/1.84 => ( app
% 3.43/1.84 => ( infuncsetfunc
% 3.43/1.84 => ( ap2p
% 3.43/1.84 => ( funcinfuncset
% 3.43/1.84 => ( lamProp
% 3.43/1.84 => ( lamp
% 3.43/1.84 => ( lam2p
% 3.43/1.84 => ( brelnall1
% 3.43/1.84 => ( brelnall2
% 3.43/1.84 => ( ex1E2
% 3.43/1.84 => ( funcGraphProp1
% 3.43/1.84 => ( funcGraphProp3
% 3.43/1.84 => ( funcGraphProp2
% 3.43/1.84 => ( funcextLem
% 3.43/1.84 => ( funcGraphProp4
% 3.43/1.84 => ( subbreln
% 3.43/1.84 => ( eqbreln
% 3.43/1.84 => ( funcext
% 3.43/1.84 => ( funcext2
% 3.43/1.84 => ( ap2apEq1
% 3.43/1.84 => ( ap2apEq2
% 3.43/1.84 => ( beta1
% 3.43/1.84 => ( eta1
% 3.43/1.84 => ( lam2lamEq
% 3.43/1.84 => ( beta2
% 3.43/1.84 => ( eta2
% 3.43/1.84 => ( iffalseProp1
% 3.43/1.84 => ( iffalseProp2
% 3.43/1.84 => ( iftrueProp1
% 3.43/1.84 => ( iftrueProp2
% 3.43/1.84 => ( ifSingleton
% 3.43/1.84 => ( ifp
% 3.43/1.84 => ( theeq
% 3.43/1.84 => ( iftrue
% 3.43/1.84 => ! [A: $i,Xphi: $o,Xx: $i] :
% 3.43/1.84 ( ( in @ Xx @ A )
% 3.43/1.84 => ! [Xy: $i] :
% 3.43/1.84 ( ( in @ Xy @ A )
% 3.43/1.84 => ( ~ Xphi
% 3.43/1.84 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.43/1.84 = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.43/1.84 ------------------------
% 3.43/1.88 % (21473)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_21305 for (2999ds/275Mi)
% 3.43/1.88 % (21472)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_21305 for (2999ds/2Mi)
% 3.43/1.88 % (21471)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/2Mi)
% 3.43/1.88 % (21468)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/183Mi)
% 3.43/1.88 % (21470)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_21305 for (2999ds/27Mi)
% 3.43/1.88 % (21469)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_21305 for (2999ds/4Mi)
% 3.43/1.88 % (21474)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/18Mi)
% 3.43/1.88 % (21471)Instruction limit reached!
% 3.43/1.88 % (21471)------------------------------
% 3.43/1.88 % (21471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.88 % (21471)Termination reason: Unknown
% 3.43/1.88 % (21471)Termination phase: shuffling
% 3.43/1.88 % (21472)Instruction limit reached!
% 3.43/1.88 % (21472)------------------------------
% 3.43/1.88 % (21472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.88 % (21472)Termination reason: Unknown
% 3.43/1.88 % (21472)Termination phase: shuffling
% 3.43/1.88
% 3.43/1.88 % (21472)Memory used [KB]: 1407
% 3.43/1.88 % (21472)Time elapsed: 0.003 s
% 3.43/1.88 % (21472)Instructions burned: 3 (million)
% 3.43/1.88 % (21472)------------------------------
% 3.43/1.88 % (21472)------------------------------
% 3.43/1.88
% 3.43/1.88 % (21471)Memory used [KB]: 1407
% 3.43/1.88 % (21471)Time elapsed: 0.003 s
% 3.43/1.88 % (21471)Instructions burned: 3 (million)
% 3.43/1.88 % (21471)------------------------------
% 3.43/1.88 % (21471)------------------------------
% 3.43/1.88 % (21469)Instruction limit reached!
% 3.43/1.88 % (21469)------------------------------
% 3.43/1.88 % (21469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.88 % (21469)Termination reason: Unknown
% 3.43/1.88 % (21469)Termination phase: shuffling
% 3.43/1.88
% 3.43/1.88 % (21469)Memory used [KB]: 1407
% 3.43/1.88 % (21469)Time elapsed: 0.004 s
% 3.43/1.88 % (21469)Instructions burned: 4 (million)
% 3.43/1.88 % (21469)------------------------------
% 3.43/1.88 % (21469)------------------------------
% 3.43/1.89 % (21474)Instruction limit reached!
% 3.43/1.89 % (21474)------------------------------
% 3.43/1.89 % (21474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.89 % (21474)Termination reason: Unknown
% 3.43/1.89 % (21474)Termination phase: shuffling
% 3.43/1.89
% 3.43/1.89 % (21474)Memory used [KB]: 1791
% 3.43/1.89 % (21474)Time elapsed: 0.011 s
% 3.43/1.89 % (21474)Instructions burned: 19 (million)
% 3.43/1.89 % (21474)------------------------------
% 3.43/1.89 % (21474)------------------------------
% 3.43/1.89 % (21470)Instruction limit reached!
% 3.43/1.89 % (21470)------------------------------
% 3.43/1.89 % (21470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.43/1.89 % (21470)Termination reason: Unknown
% 3.43/1.89 % (21470)Termination phase: shuffling
% 3.43/1.89
% 3.43/1.89 % (21470)Memory used [KB]: 1918
% 3.43/1.89 % (21470)Time elapsed: 0.015 s
% 3.43/1.89 % (21470)Instructions burned: 28 (million)
% 3.43/1.89 % (21470)------------------------------
% 3.43/1.89 % (21470)------------------------------
% 3.43/1.90 % (21476)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_21305 for (2999ds/37Mi)
% 3.43/1.90 % (21475)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/3Mi)
% 3.43/1.90 % (21477)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_21305 for (2999ds/15Mi)
% 3.47/1.90 % (21475)Instruction limit reached!
% 3.47/1.90 % (21475)------------------------------
% 3.47/1.90 % (21475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.90 % (21475)Termination reason: Unknown
% 3.47/1.90 % (21475)Termination phase: shuffling
% 3.47/1.90
% 3.47/1.90 % (21475)Memory used [KB]: 1407
% 3.47/1.90 % (21475)Time elapsed: 0.004 s
% 3.47/1.90 % (21475)Instructions burned: 4 (million)
% 3.47/1.90 % (21475)------------------------------
% 3.47/1.90 % (21475)------------------------------
% 3.47/1.90 % (21478)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_21305 for (2999ds/3Mi)
% 3.47/1.90 % (21477)Instruction limit reached!
% 3.47/1.90 % (21477)------------------------------
% 3.47/1.90 % (21477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.90 % (21477)Termination reason: Unknown
% 3.47/1.90 % (21477)Termination phase: shuffling
% 3.47/1.90
% 3.47/1.90 % (21477)Memory used [KB]: 1663
% 3.47/1.90 % (21477)Time elapsed: 0.010 s
% 3.47/1.90 % (21477)Instructions burned: 16 (million)
% 3.47/1.90 % (21477)------------------------------
% 3.47/1.90 % (21477)------------------------------
% 3.47/1.91 % (21478)Instruction limit reached!
% 3.47/1.91 % (21478)------------------------------
% 3.47/1.91 % (21478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.91 % (21478)Termination reason: Unknown
% 3.47/1.91 % (21478)Termination phase: shuffling
% 3.47/1.91
% 3.47/1.91 % (21478)Memory used [KB]: 1535
% 3.47/1.91 % (21478)Time elapsed: 0.004 s
% 3.47/1.91 % (21478)Instructions burned: 5 (million)
% 3.47/1.91 % (21478)------------------------------
% 3.47/1.91 % (21478)------------------------------
% 3.47/1.91 % (21479)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_21305 for (2999ds/1041Mi)
% 3.47/1.91 % (21476)Instruction limit reached!
% 3.47/1.91 % (21476)------------------------------
% 3.47/1.91 % (21476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.91 % (21476)Termination reason: Unknown
% 3.47/1.91 % (21476)Termination phase: shuffling
% 3.47/1.91
% 3.47/1.91 % (21476)Memory used [KB]: 2174
% 3.47/1.91 % (21476)Time elapsed: 0.019 s
% 3.47/1.91 % (21476)Instructions burned: 38 (million)
% 3.47/1.91 % (21476)------------------------------
% 3.47/1.91 % (21476)------------------------------
% 3.47/1.91 % (21480)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/7Mi)
% 3.47/1.92 % (21480)Instruction limit reached!
% 3.47/1.92 % (21480)------------------------------
% 3.47/1.92 % (21480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.92 % (21480)Termination reason: Unknown
% 3.47/1.92 % (21480)Termination phase: shuffling
% 3.47/1.92
% 3.47/1.92 % (21480)Memory used [KB]: 1535
% 3.47/1.92 % (21480)Time elapsed: 0.005 s
% 3.47/1.92 % (21480)Instructions burned: 7 (million)
% 3.47/1.92 % (21480)------------------------------
% 3.47/1.92 % (21480)------------------------------
% 3.47/1.92 % (21481)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/16Mi)
% 3.47/1.92 % (21482)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_21305 for (2999ds/3Mi)
% 3.47/1.92 % (21482)Instruction limit reached!
% 3.47/1.92 % (21482)------------------------------
% 3.47/1.92 % (21482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.92 % (21482)Termination reason: Unknown
% 3.47/1.92 % (21482)Termination phase: shuffling
% 3.47/1.92
% 3.47/1.92 % (21482)Memory used [KB]: 1407
% 3.47/1.92 % (21482)Time elapsed: 0.003 s
% 3.47/1.92 % (21482)Instructions burned: 3 (million)
% 3.47/1.92 % (21482)------------------------------
% 3.47/1.92 % (21482)------------------------------
% 3.47/1.93 % (21481)Instruction limit reached!
% 3.47/1.93 % (21481)------------------------------
% 3.47/1.93 % (21481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.93 % (21481)Termination reason: Unknown
% 3.47/1.93 % (21481)Termination phase: shuffling
% 3.47/1.93
% 3.47/1.93 % (21481)Memory used [KB]: 1791
% 3.47/1.93 % (21481)Time elapsed: 0.010 s
% 3.47/1.93 % (21481)Instructions burned: 17 (million)
% 3.47/1.93 % (21481)------------------------------
% 3.47/1.93 % (21481)------------------------------
% 3.47/1.93 % (21483)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_21305 for (2999ds/3Mi)
% 3.47/1.93 % (21483)Instruction limit reached!
% 3.47/1.93 % (21483)------------------------------
% 3.47/1.93 % (21483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.93 % (21483)Termination reason: Unknown
% 3.47/1.93 % (21483)Termination phase: shuffling
% 3.47/1.93
% 3.47/1.93 % (21483)Memory used [KB]: 1407
% 3.47/1.93 % (21483)Time elapsed: 0.004 s
% 3.47/1.93 % (21483)Instructions burned: 5 (million)
% 3.47/1.93 % (21483)------------------------------
% 3.47/1.93 % (21483)------------------------------
% 3.47/1.93 % (21484)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/7Mi)
% 3.47/1.94 % (21484)Instruction limit reached!
% 3.47/1.94 % (21484)------------------------------
% 3.47/1.94 % (21484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.94 % (21484)Termination reason: Unknown
% 3.47/1.94 % (21484)Termination phase: shuffling
% 3.47/1.94
% 3.47/1.94 % (21484)Memory used [KB]: 1535
% 3.47/1.94 % (21484)Time elapsed: 0.005 s
% 3.47/1.94 % (21484)Instructions burned: 7 (million)
% 3.47/1.94 % (21484)------------------------------
% 3.47/1.94 % (21484)------------------------------
% 3.47/1.94 % (21485)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/3Mi)
% 3.47/1.94 % (21485)Instruction limit reached!
% 3.47/1.94 % (21485)------------------------------
% 3.47/1.94 % (21485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.94 % (21485)Termination reason: Unknown
% 3.47/1.94 % (21485)Termination phase: shuffling
% 3.47/1.94
% 3.47/1.94 % (21485)Memory used [KB]: 1407
% 3.47/1.94 % (21485)Time elapsed: 0.003 s
% 3.47/1.94 % (21485)Instructions burned: 3 (million)
% 3.47/1.94 % (21485)------------------------------
% 3.47/1.94 % (21485)------------------------------
% 3.47/1.95 % (21486)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/4Mi)
% 3.47/1.95 % (21486)Instruction limit reached!
% 3.47/1.95 % (21486)------------------------------
% 3.47/1.95 % (21486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.95 % (21486)Termination reason: Unknown
% 3.47/1.95 % (21487)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/18Mi)
% 3.47/1.95 % (21486)Termination phase: shuffling
% 3.47/1.95
% 3.47/1.95 % (21486)Memory used [KB]: 1407
% 3.47/1.95 % (21486)Time elapsed: 0.004 s
% 3.47/1.95 % (21486)Instructions burned: 4 (million)
% 3.47/1.95 % (21486)------------------------------
% 3.47/1.95 % (21486)------------------------------
% 3.47/1.95 % (21488)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/710Mi)
% 3.47/1.96 % (21487)Instruction limit reached!
% 3.47/1.96 % (21487)------------------------------
% 3.47/1.96 % (21487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.96 % (21487)Termination reason: Unknown
% 3.47/1.96 % (21487)Termination phase: shuffling
% 3.47/1.96
% 3.47/1.96 % (21487)Memory used [KB]: 1791
% 3.47/1.96 % (21487)Time elapsed: 0.010 s
% 3.47/1.96 % (21487)Instructions burned: 18 (million)
% 3.47/1.96 % (21487)------------------------------
% 3.47/1.96 % (21487)------------------------------
% 3.47/1.96 % (21489)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_21305 for (2999ds/6Mi)
% 3.47/1.96 % (21489)Instruction limit reached!
% 3.47/1.96 % (21489)------------------------------
% 3.47/1.96 % (21489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.96 % (21489)Termination reason: Unknown
% 3.47/1.96 % (21489)Termination phase: shuffling
% 3.47/1.96
% 3.47/1.96 % (21468)Instruction limit reached!
% 3.47/1.96 % (21468)------------------------------
% 3.47/1.96 % (21468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.96 % (21468)Termination reason: Unknown
% 3.47/1.96 % (21468)Termination phase: Saturation
% 3.47/1.96
% 3.47/1.96 % (21468)Memory used [KB]: 7547
% 3.47/1.96 % (21468)Time elapsed: 0.085 s
% 3.47/1.96 % (21468)Instructions burned: 184 (million)
% 3.47/1.96 % (21468)------------------------------
% 3.47/1.96 % (21468)------------------------------
% 3.47/1.96 % (21489)Memory used [KB]: 1535
% 3.47/1.96 % (21489)Time elapsed: 0.005 s
% 3.47/1.96 % (21489)Instructions burned: 8 (million)
% 3.47/1.96 % (21489)------------------------------
% 3.47/1.96 % (21489)------------------------------
% 3.47/1.96 % (21490)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_21305 for (2999ds/902Mi)
% 3.47/1.97 % (21491)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_21305 for (2998ds/21Mi)
% 3.47/1.98 % (21493)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_21305 for (2998ds/6Mi)
% 3.47/1.98 % (21492)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_21305 for (2998ds/5Mi)
% 3.47/1.98 % (21492)Instruction limit reached!
% 3.47/1.98 % (21492)------------------------------
% 3.47/1.98 % (21492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.98 % (21492)Termination reason: Unknown
% 3.47/1.98 % (21492)Termination phase: shuffling
% 3.47/1.98
% 3.47/1.98 % (21492)Memory used [KB]: 1535
% 3.47/1.98 % (21492)Time elapsed: 0.004 s
% 3.47/1.98 % (21492)Instructions burned: 6 (million)
% 3.47/1.98 % (21492)------------------------------
% 3.47/1.98 % (21492)------------------------------
% 3.47/1.98 % (21493)Instruction limit reached!
% 3.47/1.98 % (21493)------------------------------
% 3.47/1.98 % (21493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.98 % (21493)Termination reason: Unknown
% 3.47/1.98 % (21493)Termination phase: shuffling
% 3.47/1.98
% 3.47/1.98 % (21493)Memory used [KB]: 1535
% 3.47/1.98 % (21493)Time elapsed: 0.005 s
% 3.47/1.98 % (21493)Instructions burned: 7 (million)
% 3.47/1.98 % (21493)------------------------------
% 3.47/1.98 % (21493)------------------------------
% 3.47/1.98 % (21491)Instruction limit reached!
% 3.47/1.98 % (21491)------------------------------
% 3.47/1.98 % (21491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/1.98 % (21491)Termination reason: Unknown
% 3.47/1.98 % (21491)Termination phase: shuffling
% 3.47/1.98
% 3.47/1.98 % (21491)Memory used [KB]: 1791
% 3.47/1.98 % (21491)Time elapsed: 0.012 s
% 3.47/1.98 % (21491)Instructions burned: 21 (million)
% 3.47/1.98 % (21491)------------------------------
% 3.47/1.98 % (21491)------------------------------
% 3.47/2.00 % (21495)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_21305 for (2998ds/779Mi)
% 3.47/2.00 % (21494)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_21305 for (2998ds/377Mi)
% 3.47/2.00 % (21496)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_21305 for (2998ds/19Mi)
% 3.47/2.01 % (21496)Instruction limit reached!
% 3.47/2.01 % (21496)------------------------------
% 3.47/2.01 % (21496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/2.01 % (21496)Termination reason: Unknown
% 3.47/2.01 % (21496)Termination phase: shuffling
% 3.47/2.01
% 3.47/2.01 % (21496)Memory used [KB]: 1791
% 3.47/2.01 % (21496)Time elapsed: 0.011 s
% 3.47/2.01 % (21496)Instructions burned: 21 (million)
% 3.47/2.01 % (21496)------------------------------
% 3.47/2.01 % (21496)------------------------------
% 3.47/2.01 % (21473)Instruction limit reached!
% 3.47/2.01 % (21473)------------------------------
% 3.47/2.01 % (21473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.47/2.01 % (21473)Termination reason: Unknown
% 3.47/2.01 % (21473)Termination phase: Saturation
% 3.47/2.01
% 3.47/2.01 % (21473)Memory used [KB]: 9210
% 3.47/2.01 % (21473)Time elapsed: 0.134 s
% 3.47/2.01 % (21473)Instructions burned: 276 (million)
% 3.47/2.01 % (21473)------------------------------
% 3.47/2.01 % (21473)------------------------------
% 3.47/2.03 % (21497)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_21305 for (2998ds/879Mi)
% 4.55/2.03 % (21498)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_21305 for (2998ds/17Mi)
% 4.55/2.03 % (21498)Instruction limit reached!
% 4.55/2.03 % (21498)------------------------------
% 4.55/2.03 % (21498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.55/2.03 % (21498)Termination reason: Unknown
% 4.55/2.03 % (21498)Termination phase: shuffling
% 4.55/2.03
% 4.55/2.03 % (21498)Memory used [KB]: 1791
% 4.55/2.03 % (21498)Time elapsed: 0.009 s
% 4.55/2.03 % (21498)Instructions burned: 17 (million)
% 4.55/2.03 % (21498)------------------------------
% 4.55/2.03 % (21498)------------------------------
% 4.55/2.05 % (21499)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_21305 for (2998ds/3Mi)
% 4.55/2.05 % (21499)Instruction limit reached!
% 4.55/2.05 % (21499)------------------------------
% 4.55/2.05 % (21499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.55/2.05 % (21499)Termination reason: Unknown
% 4.55/2.05 % (21499)Termination phase: shuffling
% 4.55/2.05
% 4.55/2.05 % (21499)Memory used [KB]: 1407
% 4.55/2.05 % (21499)Time elapsed: 0.003 s
% 4.55/2.05 % (21499)Instructions burned: 3 (million)
% 4.55/2.05 % (21499)------------------------------
% 4.55/2.05 % (21499)------------------------------
% 4.55/2.07 % (21500)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_21305 for (2997ds/30Mi)
% 4.55/2.08 % (21500)Instruction limit reached!
% 4.55/2.08 % (21500)------------------------------
% 4.55/2.08 % (21500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.55/2.08 % (21500)Termination reason: Unknown
% 4.55/2.08 % (21500)Termination phase: shuffling
% 4.55/2.08
% 4.55/2.08 % (21500)Memory used [KB]: 2046
% 4.55/2.08 % (21500)Time elapsed: 0.016 s
% 4.55/2.08 % (21500)Instructions burned: 31 (million)
% 4.55/2.08 % (21500)------------------------------
% 4.55/2.08 % (21500)------------------------------
% 4.55/2.10 % (21501)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_21305 for (2997ds/127Mi)
% 4.55/2.11 % (21490)Refutation not found, incomplete strategy
% 4.55/2.11 % (21490)------------------------------
% 4.55/2.11 % (21490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.55/2.11 % (21490)Termination reason: Refutation not found, incomplete strategy
% 4.55/2.11
% 4.55/2.11
% 4.55/2.11 % (21490)Memory used [KB]: 9083
% 4.55/2.11 % (21490)Time elapsed: 0.165 s
% 4.55/2.11 % (21490)Instructions burned: 291 (million)
% 4.55/2.11 % (21490)------------------------------
% 4.55/2.11 % (21490)------------------------------
% 4.55/2.12 % (21502)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_21305 for (2997ds/100Mi)
% 5.12/2.16 % (21501)Instruction limit reached!
% 5.12/2.16 % (21501)------------------------------
% 5.12/2.16 % (21501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.12/2.16 % (21501)Termination reason: Unknown
% 5.12/2.16 % (21501)Termination phase: Property scanning
% 5.12/2.16
% 5.12/2.16 % (21501)Memory used [KB]: 2430
% 5.12/2.16 % (21501)Time elapsed: 0.057 s
% 5.12/2.16 % (21501)Instructions burned: 128 (million)
% 5.12/2.16 % (21501)------------------------------
% 5.12/2.16 % (21501)------------------------------
% 5.12/2.17 % (21502)Instruction limit reached!
% 5.12/2.17 % (21502)------------------------------
% 5.12/2.17 % (21502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.12/2.17 % (21502)Termination reason: Unknown
% 5.12/2.17 % (21502)Termination phase: Property scanning
% 5.12/2.17
% 5.12/2.17 % (21502)Memory used [KB]: 2430
% 5.12/2.17 % (21502)Time elapsed: 0.045 s
% 5.12/2.17 % (21502)Instructions burned: 100 (million)
% 5.12/2.17 % (21502)------------------------------
% 5.12/2.17 % (21502)------------------------------
% 5.12/2.17 % (21503)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_21305 for (2996ds/3Mi)
% 5.12/2.17 % (21503)Instruction limit reached!
% 5.12/2.17 % (21503)------------------------------
% 5.12/2.17 % (21503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.12/2.17 % (21503)Termination reason: Unknown
% 5.12/2.17 % (21503)Termination phase: shuffling
% 5.12/2.17
% 5.12/2.17 % (21503)Memory used [KB]: 1407
% 5.12/2.17 % (21503)Time elapsed: 0.003 s
% 5.12/2.17 % (21503)Instructions burned: 3 (million)
% 5.12/2.17 % (21503)------------------------------
% 5.12/2.17 % (21503)------------------------------
% 5.12/2.18 % (21494)Instruction limit reached!
% 5.12/2.18 % (21494)------------------------------
% 5.12/2.18 % (21494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.12/2.18 % (21494)Termination reason: Unknown
% 5.12/2.18 % (21494)Termination phase: Saturation
% 5.12/2.18
% 5.12/2.18 % (21494)Memory used [KB]: 8699
% 5.12/2.18 % (21494)Time elapsed: 0.182 s
% 5.12/2.18 % (21494)Instructions burned: 378 (million)
% 5.12/2.18 % (21494)------------------------------
% 5.12/2.18 % (21494)------------------------------
% 5.12/2.18 % (21504)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_21305 for (2996ds/20Mi)
% 5.12/2.19 % (21505)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_21305 for (2996ds/86Mi)
% 5.12/2.19 % (21504)Instruction limit reached!
% 5.12/2.19 % (21504)------------------------------
% 5.12/2.19 % (21504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.12/2.19 % (21504)Termination reason: Unknown
% 5.12/2.19 % (21504)Termination phase: shuffling
% 5.12/2.19
% 5.12/2.19 % (21504)Memory used [KB]: 1791
% 5.12/2.19 % (21504)Time elapsed: 0.012 s
% 5.12/2.19 % (21504)Instructions burned: 21 (million)
% 5.12/2.19 % (21504)------------------------------
% 5.12/2.19 % (21504)------------------------------
% 5.12/2.20 % (21506)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_21305 for (2996ds/107Mi)
% 5.55/2.21 % (21507)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_21305 for (2996ds/39Mi)
% 5.55/2.23 % (21505)Instruction limit reached!
% 5.55/2.23 % (21505)------------------------------
% 5.55/2.23 % (21505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.23 % (21505)Termination reason: Unknown
% 5.55/2.23 % (21505)Termination phase: Property scanning
% 5.55/2.23
% 5.55/2.23 % (21505)Memory used [KB]: 2430
% 5.55/2.23 % (21505)Time elapsed: 0.040 s
% 5.55/2.23 % (21505)Instructions burned: 86 (million)
% 5.55/2.23 % (21505)------------------------------
% 5.55/2.23 % (21505)------------------------------
% 5.55/2.23 % (21507)Instruction limit reached!
% 5.55/2.23 % (21507)------------------------------
% 5.55/2.23 % (21507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.23 % (21507)Termination reason: Unknown
% 5.55/2.23 % (21507)Termination phase: shuffling
% 5.55/2.23
% 5.55/2.23 % (21507)Memory used [KB]: 2174
% 5.55/2.23 % (21507)Time elapsed: 0.019 s
% 5.55/2.23 % (21507)Instructions burned: 39 (million)
% 5.55/2.23 % (21507)------------------------------
% 5.55/2.23 % (21507)------------------------------
% 5.55/2.24 % (21506)Instruction limit reached!
% 5.55/2.24 % (21506)------------------------------
% 5.55/2.24 % (21506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.24 % (21506)Termination reason: Unknown
% 5.55/2.24 % (21506)Termination phase: Function definition elimination
% 5.55/2.24
% 5.55/2.24 % (21506)Memory used [KB]: 2430
% 5.55/2.24 % (21506)Time elapsed: 0.050 s
% 5.55/2.24 % (21506)Instructions burned: 107 (million)
% 5.55/2.24 % (21506)------------------------------
% 5.55/2.24 % (21506)------------------------------
% 5.55/2.24 % (21508)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_21305 for (2996ds/448Mi)
% 5.55/2.24 % (21509)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_21305 for (2996ds/46Mi)
% 5.55/2.26 % (21510)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_21305 for (2996ds/98Mi)
% 5.55/2.27 % (21509)Instruction limit reached!
% 5.55/2.27 % (21509)------------------------------
% 5.55/2.27 % (21509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.27 % (21509)Termination reason: Unknown
% 5.55/2.27 % (21509)Termination phase: Property scanning
% 5.55/2.27
% 5.55/2.27 % (21509)Memory used [KB]: 2302
% 5.55/2.27 % (21509)Time elapsed: 0.023 s
% 5.55/2.27 % (21509)Instructions burned: 47 (million)
% 5.55/2.27 % (21509)------------------------------
% 5.55/2.27 % (21509)------------------------------
% 5.55/2.28 % (21511)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_21305 for (2995ds/507Mi)
% 5.55/2.29 % (21495)Instruction limit reached!
% 5.55/2.29 % (21495)------------------------------
% 5.55/2.29 % (21495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.55/2.29 % (21495)Termination reason: Unknown
% 5.55/2.29 % (21495)Termination phase: Saturation
% 5.55/2.29
% 5.55/2.29 % (21495)Memory used [KB]: 7291
% 5.55/2.29 % (21495)Time elapsed: 0.298 s
% 5.55/2.29 % (21495)Instructions burned: 779 (million)
% 5.55/2.29 % (21495)------------------------------
% 5.55/2.29 % (21495)------------------------------
% 6.04/2.30 % (21510)Instruction limit reached!
% 6.04/2.30 % (21510)------------------------------
% 6.04/2.30 % (21510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.04/2.30 % (21510)Termination reason: Unknown
% 6.04/2.30 % (21510)Termination phase: Function definition elimination
% 6.04/2.30
% 6.04/2.30 % (21510)Memory used [KB]: 2430
% 6.04/2.30 % (21510)Time elapsed: 0.046 s
% 6.04/2.30 % (21510)Instructions burned: 99 (million)
% 6.04/2.30 % (21510)------------------------------
% 6.04/2.30 % (21510)------------------------------
% 6.04/2.31 % (21479)Instruction limit reached!
% 6.04/2.31 % (21479)------------------------------
% 6.04/2.31 % (21479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.04/2.31 % (21479)Termination reason: Unknown
% 6.04/2.31 % (21479)Termination phase: Saturation
% 6.04/2.31
% 6.04/2.31 % (21479)Memory used [KB]: 12920
% 6.04/2.31 % (21479)Time elapsed: 0.402 s
% 6.04/2.31 % (21479)Instructions burned: 1042 (million)
% 6.04/2.31 % (21479)------------------------------
% 6.04/2.31 % (21479)------------------------------
% 6.04/2.31 % (21512)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_21305 for (2995ds/149Mi)
% 6.04/2.32 % (21513)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_21305 for (2995ds/39Mi)
% 6.04/2.32 % (21514)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_21305 for (2995ds/6Mi)
% 6.04/2.33 % (21514)Instruction limit reached!
% 6.04/2.33 % (21514)------------------------------
% 6.04/2.33 % (21514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.04/2.33 % (21514)Termination reason: Unknown
% 6.04/2.33 % (21514)Termination phase: shuffling
% 6.04/2.33
% 6.04/2.33 % (21514)Memory used [KB]: 1535
% 6.04/2.33 % (21514)Time elapsed: 0.005 s
% 6.04/2.33 % (21514)Instructions burned: 7 (million)
% 6.04/2.33 % (21514)------------------------------
% 6.04/2.33 % (21514)------------------------------
% 6.04/2.33 % (21488)Instruction limit reached!
% 6.04/2.33 % (21488)------------------------------
% 6.04/2.33 % (21488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.04/2.33 % (21488)Termination reason: Unknown
% 6.04/2.33 % (21488)Termination phase: Saturation
% 6.04/2.33
% 6.04/2.33 % (21488)Memory used [KB]: 10106
% 6.04/2.33 % (21488)Time elapsed: 0.380 s
% 6.04/2.33 % (21488)Instructions burned: 710 (million)
% 6.04/2.33 % (21488)------------------------------
% 6.04/2.33 % (21488)------------------------------
% 6.04/2.34 % (21513)Instruction limit reached!
% 6.04/2.34 % (21513)------------------------------
% 6.04/2.34 % (21513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.04/2.34 % (21513)Termination reason: Unknown
% 6.04/2.34 % (21513)Termination phase: Property scanning
% 6.04/2.34
% 6.04/2.34 % (21513)Memory used [KB]: 2302
% 6.04/2.34 % (21513)Time elapsed: 0.021 s
% 6.04/2.34 % (21513)Instructions burned: 40 (million)
% 6.04/2.34 % (21513)------------------------------
% 6.04/2.34 % (21513)------------------------------
% 6.04/2.34 % (21515)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_21305 for (2995ds/1087Mi)
% 6.04/2.36 % (21517)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_21305 for (2994ds/44Mi)
% 6.04/2.36 % (21516)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_21305 for (2995ds/47Mi)
% 6.04/2.38 % (21517)Instruction limit reached!
% 6.04/2.38 % (21517)------------------------------
% 6.04/2.38 % (21517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.04/2.38 % (21517)Termination reason: Unknown
% 6.04/2.38 % (21517)Termination phase: Property scanning
% 6.04/2.38
% 6.04/2.38 % (21517)Memory used [KB]: 2174
% 6.04/2.38 % (21517)Time elapsed: 0.023 s
% 6.04/2.38 % (21517)Instructions burned: 45 (million)
% 6.04/2.38 % (21517)------------------------------
% 6.04/2.38 % (21517)------------------------------
% 6.04/2.38 % (21512)Instruction limit reached!
% 6.04/2.38 % (21512)------------------------------
% 6.04/2.38 % (21512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.04/2.38 % (21512)Termination reason: Unknown
% 6.04/2.38 % (21512)Termination phase: Saturation
% 6.04/2.38
% 6.04/2.38 % (21512)Memory used [KB]: 7419
% 6.04/2.38 % (21512)Time elapsed: 0.070 s
% 6.04/2.38 % (21512)Instructions burned: 151 (million)
% 6.04/2.38 % (21512)------------------------------
% 6.04/2.38 % (21512)------------------------------
% 6.04/2.38 % (21516)Instruction limit reached!
% 6.04/2.38 % (21516)------------------------------
% 6.04/2.38 % (21516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.04/2.38 % (21516)Termination reason: Unknown
% 6.04/2.38 % (21516)Termination phase: Property scanning
% 6.04/2.38
% 6.04/2.38 % (21516)Memory used [KB]: 2302
% 6.04/2.38 % (21516)Time elapsed: 0.023 s
% 6.04/2.38 % (21516)Instructions burned: 47 (million)
% 6.04/2.38 % (21516)------------------------------
% 6.04/2.38 % (21516)------------------------------
% 6.04/2.39 % (21518)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_21305 for (2994ds/432Mi)
% 6.04/2.39 % (21519)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_21305 for (2994ds/62Mi)
% 6.40/2.40 % (21520)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_21305 for (2994ds/31Mi)
% 6.40/2.41 % (21520)Instruction limit reached!
% 6.40/2.41 % (21520)------------------------------
% 6.40/2.41 % (21520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.41 % (21520)Termination reason: Unknown
% 6.40/2.41 % (21520)Termination phase: shuffling
% 6.40/2.41
% 6.40/2.41 % (21520)Memory used [KB]: 2046
% 6.40/2.41 % (21520)Time elapsed: 0.017 s
% 6.40/2.41 % (21520)Instructions burned: 32 (million)
% 6.40/2.41 % (21520)------------------------------
% 6.40/2.41 % (21520)------------------------------
% 6.40/2.42 % (21519)Instruction limit reached!
% 6.40/2.42 % (21519)------------------------------
% 6.40/2.42 % (21519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.42 % (21519)Termination reason: Unknown
% 6.40/2.42 % (21519)Termination phase: Property scanning
% 6.40/2.42
% 6.40/2.42 % (21519)Memory used [KB]: 2302
% 6.40/2.42 % (21519)Time elapsed: 0.030 s
% 6.40/2.42 % (21519)Instructions burned: 63 (million)
% 6.40/2.42 % (21519)------------------------------
% 6.40/2.42 % (21519)------------------------------
% 6.40/2.42 % (21508)Instruction limit reached!
% 6.40/2.42 % (21508)------------------------------
% 6.40/2.42 % (21508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.42 % (21508)Termination reason: Unknown
% 6.40/2.42 % (21508)Termination phase: Saturation
% 6.40/2.42
% 6.40/2.42 % (21508)Memory used [KB]: 7291
% 6.40/2.42 % (21508)Time elapsed: 0.182 s
% 6.40/2.42 % (21508)Instructions burned: 449 (million)
% 6.40/2.42 % (21508)------------------------------
% 6.40/2.42 % (21508)------------------------------
% 6.40/2.43 % (21521)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_21305 for (2994ds/96Mi)
% 6.40/2.44 % (21522)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_21305 for (2994ds/453Mi)
% 6.40/2.44 % (21523)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_21305 for (2994ds/71Mi)
% 6.40/2.45 % (21497)Refutation not found, non-redundant clauses discarded% (21497)------------------------------
% 6.40/2.45 % (21497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.45 % (21497)Termination reason: Refutation not found, non-redundant clauses discarded
% 6.40/2.45
% 6.40/2.45 % (21497)Memory used [KB]: 9466
% 6.40/2.45 % (21497)Time elapsed: 0.425 s
% 6.40/2.45 % (21497)Instructions burned: 863 (million)
% 6.40/2.45 % (21497)------------------------------
% 6.40/2.45 % (21497)------------------------------
% 6.40/2.47 % (21524)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_21305 for (2993ds/83Mi)
% 6.40/2.47 % (21523)Instruction limit reached!
% 6.40/2.47 % (21523)------------------------------
% 6.40/2.47 % (21523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.47 % (21523)Termination reason: Unknown
% 6.40/2.47 % (21523)Termination phase: Property scanning
% 6.40/2.47
% 6.40/2.47 % (21523)Memory used [KB]: 2430
% 6.40/2.47 % (21523)Time elapsed: 0.034 s
% 6.40/2.47 % (21523)Instructions burned: 71 (million)
% 6.40/2.47 % (21523)------------------------------
% 6.40/2.47 % (21523)------------------------------
% 6.40/2.47 % (21521)Instruction limit reached!
% 6.40/2.47 % (21521)------------------------------
% 6.40/2.47 % (21521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.47 % (21521)Termination reason: Unknown
% 6.40/2.47 % (21521)Termination phase: Property scanning
% 6.40/2.47
% 6.40/2.47 % (21521)Memory used [KB]: 2558
% 6.40/2.47 % (21521)Time elapsed: 0.046 s
% 6.40/2.47 % (21521)Instructions burned: 98 (million)
% 6.40/2.47 % (21521)------------------------------
% 6.40/2.47 % (21521)------------------------------
% 6.40/2.49 % (21525)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_21305 for (2993ds/11Mi)
% 6.40/2.49 % (21526)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_21305 for (2993ds/41Mi)
% 6.40/2.49 % (21525)Instruction limit reached!
% 6.40/2.49 % (21525)------------------------------
% 6.40/2.49 % (21525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.40/2.49 % (21525)Termination reason: Unknown
% 6.40/2.49 % (21525)Termination phase: shuffling
% 6.40/2.49
% 6.40/2.49 % (21525)Memory used [KB]: 1535
% 6.40/2.49 % (21525)Time elapsed: 0.007 s
% 6.40/2.49 % (21525)Instructions burned: 12 (million)
% 6.40/2.49 % (21525)------------------------------
% 6.40/2.49 % (21525)------------------------------
% 7.25/2.50 % (21524)Instruction limit reached!
% 7.25/2.50 % (21524)------------------------------
% 7.25/2.50 % (21524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.25/2.50 % (21524)Termination reason: Unknown
% 7.25/2.50 % (21524)Termination phase: SInE selection
% 7.25/2.50
% 7.25/2.50 % (21524)Memory used [KB]: 2302
% 7.25/2.50 % (21524)Time elapsed: 0.039 s
% 7.25/2.50 % (21524)Instructions burned: 85 (million)
% 7.25/2.50 % (21524)------------------------------
% 7.25/2.50 % (21524)------------------------------
% 7.25/2.51 % (21527)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_21305 for (2993ds/6Mi)
% 7.25/2.51 % (21526)Instruction limit reached!
% 7.25/2.51 % (21526)------------------------------
% 7.25/2.51 % (21526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.25/2.51 % (21526)Termination reason: Unknown
% 7.25/2.51 % (21526)Termination phase: Property scanning
% 7.25/2.51
% 7.25/2.51 % (21526)Memory used [KB]: 2174
% 7.25/2.51 % (21526)Time elapsed: 0.022 s
% 7.25/2.51 % (21526)Instructions burned: 42 (million)
% 7.25/2.51 % (21526)------------------------------
% 7.25/2.51 % (21526)------------------------------
% 7.25/2.51 % (21527)Instruction limit reached!
% 7.25/2.51 % (21527)------------------------------
% 7.25/2.51 % (21527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.25/2.51 % (21527)Termination reason: Unknown
% 7.25/2.51 % (21527)Termination phase: shuffling
% 7.25/2.51
% 7.25/2.51 % (21527)Memory used [KB]: 1535
% 7.25/2.51 % (21527)Time elapsed: 0.005 s
% 7.25/2.51 % (21527)Instructions burned: 7 (million)
% 7.25/2.51 % (21527)------------------------------
% 7.25/2.51 % (21527)------------------------------
% 7.25/2.52 % (21511)Instruction limit reached!
% 7.25/2.52 % (21511)------------------------------
% 7.25/2.52 % (21511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.25/2.52 % (21511)Termination reason: Unknown
% 7.25/2.52 % (21511)Termination phase: Saturation
% 7.25/2.52
% 7.25/2.52 % (21511)Memory used [KB]: 7931
% 7.25/2.52 % (21511)Time elapsed: 0.259 s
% 7.25/2.52 % (21511)Instructions burned: 507 (million)
% 7.25/2.52 % (21511)------------------------------
% 7.25/2.52 % (21511)------------------------------
% 7.25/2.52 % (21528)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_21305 for (2993ds/150Mi)
% 7.25/2.53 % (21529)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_21305 for (2993ds/343Mi)
% 7.25/2.53 % (21530)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_21305 for (2993ds/109Mi)
% 7.25/2.54 % (21531)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_21305 for (2993ds/253Mi)
% 7.25/2.57 % (21518)Instruction limit reached!
% 7.25/2.57 % (21518)------------------------------
% 7.25/2.57 % (21518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.25/2.57 % (21518)Termination reason: Unknown
% 7.25/2.57 % (21518)Termination phase: Saturation
% 7.25/2.57
% 7.25/2.57 % (21518)Memory used [KB]: 2814
% 7.25/2.57 % (21518)Time elapsed: 0.182 s
% 7.25/2.57 % (21518)Instructions burned: 433 (million)
% 7.25/2.57 % (21518)------------------------------
% 7.25/2.57 % (21518)------------------------------
% 7.25/2.58 % (21530)Instruction limit reached!
% 7.25/2.58 % (21530)------------------------------
% 7.25/2.58 % (21530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.25/2.58 % (21530)Termination reason: Unknown
% 7.25/2.58 % (21530)Termination phase: Function definition elimination
% 7.25/2.58
% 7.25/2.58 % (21530)Memory used [KB]: 2430
% 7.25/2.58 % (21530)Time elapsed: 0.049 s
% 7.25/2.58 % (21530)Instructions burned: 109 (million)
% 7.25/2.58 % (21530)------------------------------
% 7.25/2.58 % (21530)------------------------------
% 7.25/2.59 % (21528)Instruction limit reached!
% 7.25/2.59 % (21528)------------------------------
% 7.25/2.59 % (21528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.25/2.59 % (21528)Termination reason: Unknown
% 7.25/2.59 % (21528)Termination phase: Property scanning
% 7.25/2.59
% 7.25/2.59 % (21528)Memory used [KB]: 4093
% 7.25/2.59 % (21528)Time elapsed: 0.074 s
% 7.25/2.59 % (21528)Instructions burned: 151 (million)
% 7.25/2.59 % (21528)------------------------------
% 7.25/2.59 % (21528)------------------------------
% 7.25/2.59 % (21532)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_21305 for (2992ds/140Mi)
% 7.25/2.59 % (21533)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_21305 for (2992ds/614Mi)
% 7.96/2.61 % (21534)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_21305 for (2992ds/202Mi)
% 7.96/2.62 % (21522)Instruction limit reached!
% 7.96/2.62 % (21522)------------------------------
% 7.96/2.62 % (21522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.96/2.62 % (21522)Termination reason: Unknown
% 7.96/2.62 % (21522)Termination phase: Saturation
% 7.96/2.62
% 7.96/2.62 % (21522)Memory used [KB]: 7291
% 7.96/2.62 % (21522)Time elapsed: 0.183 s
% 7.96/2.62 % (21522)Instructions burned: 453 (million)
% 7.96/2.62 % (21522)------------------------------
% 7.96/2.62 % (21522)------------------------------
% 7.96/2.64 % (21535)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_21305 for (2992ds/339Mi)
% 7.96/2.66 % (21532)Instruction limit reached!
% 7.96/2.66 % (21532)------------------------------
% 7.96/2.66 % (21532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.96/2.66 % (21532)Termination reason: Unknown
% 7.96/2.66 % (21532)Termination phase: Saturation
% 7.96/2.66
% 7.96/2.66 % (21532)Memory used [KB]: 7036
% 7.96/2.66 % (21532)Time elapsed: 0.065 s
% 7.96/2.66 % (21532)Instructions burned: 141 (million)
% 7.96/2.66 % (21532)------------------------------
% 7.96/2.66 % (21532)------------------------------
% 7.96/2.66 % (21531)Instruction limit reached!
% 7.96/2.66 % (21531)------------------------------
% 7.96/2.66 % (21531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.96/2.66 % (21531)Termination reason: Unknown
% 7.96/2.66 % (21531)Termination phase: Saturation
% 7.96/2.66
% 7.96/2.66 % (21531)Memory used [KB]: 9338
% 7.96/2.66 % (21531)Time elapsed: 0.127 s
% 7.96/2.66 % (21531)Instructions burned: 253 (million)
% 7.96/2.66 % (21531)------------------------------
% 7.96/2.66 % (21531)------------------------------
% 7.96/2.67 % (21536)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_21305 for (2991ds/283Mi)
% 7.96/2.68 % (21537)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_21305 for (2991ds/134Mi)
% 7.96/2.69 % (21529)Instruction limit reached!
% 7.96/2.69 % (21529)------------------------------
% 7.96/2.69 % (21529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.96/2.69 % (21529)Termination reason: Unknown
% 7.96/2.69 % (21529)Termination phase: Saturation
% 7.96/2.69
% 7.96/2.69 % (21529)Memory used [KB]: 8443
% 7.96/2.69 % (21529)Time elapsed: 0.160 s
% 7.96/2.69 % (21529)Instructions burned: 344 (million)
% 7.96/2.69 % (21529)------------------------------
% 7.96/2.69 % (21529)------------------------------
% 8.56/2.70 % (21538)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_21305 for (2991ds/298Mi)
% 8.56/2.71 % (21534)Instruction limit reached!
% 8.56/2.71 % (21534)------------------------------
% 8.56/2.71 % (21534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.56/2.71 % (21534)Termination reason: Unknown
% 8.56/2.71 % (21534)Termination phase: Saturation
% 8.56/2.71
% 8.56/2.71 % (21534)Memory used [KB]: 4861
% 8.56/2.71 % (21534)Time elapsed: 0.099 s
% 8.56/2.71 % (21534)Instructions burned: 203 (million)
% 8.56/2.71 % (21534)------------------------------
% 8.56/2.71 % (21534)------------------------------
% 8.56/2.72 % (21539)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_21305 for (2991ds/192Mi)
% 8.56/2.74 % (21537)Instruction limit reached!
% 8.56/2.74 % (21537)------------------------------
% 8.56/2.74 % (21537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.56/2.74 % (21537)Termination reason: Unknown
% 8.56/2.74 % (21537)Termination phase: Saturation
% 8.56/2.74
% 8.56/2.74 % (21537)Memory used [KB]: 7036
% 8.56/2.74 % (21537)Time elapsed: 0.063 s
% 8.56/2.74 % (21537)Instructions burned: 135 (million)
% 8.56/2.74 % (21537)------------------------------
% 8.56/2.74 % (21537)------------------------------
% 8.56/2.76 % (21540)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_21305 for (2991ds/258Mi)
% 8.56/2.79 % (21536)Instruction limit reached!
% 8.56/2.79 % (21536)------------------------------
% 8.56/2.79 % (21536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.56/2.79 % (21536)Termination reason: Unknown
% 8.56/2.79 % (21536)Termination phase: Saturation
% 8.56/2.79
% 8.56/2.79 % (21536)Memory used [KB]: 7291
% 8.56/2.79 % (21536)Time elapsed: 0.122 s
% 8.56/2.79 % (21536)Instructions burned: 284 (million)
% 8.56/2.79 % (21536)------------------------------
% 8.56/2.79 % (21536)------------------------------
% 8.56/2.79 % (21535)Instruction limit reached!
% 8.56/2.79 % (21535)------------------------------
% 8.56/2.79 % (21535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.56/2.79 % (21535)Termination reason: Unknown
% 8.56/2.79 % (21535)Termination phase: Saturation
% 8.56/2.79
% 8.56/2.79 % (21535)Memory used [KB]: 7931
% 8.56/2.79 % (21535)Time elapsed: 0.159 s
% 8.56/2.79 % (21535)Instructions burned: 340 (million)
% 8.56/2.79 % (21535)------------------------------
% 8.56/2.79 % (21535)------------------------------
% 9.05/2.81 % (21541)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_21305 for (2990ds/90Mi)
% 9.05/2.81 % (21542)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_21305 for (2990ds/231Mi)
% 9.05/2.82 % (21539)Instruction limit reached!
% 9.05/2.82 % (21539)------------------------------
% 9.05/2.82 % (21539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.05/2.82 % (21539)Termination reason: Unknown
% 9.05/2.82 % (21539)Termination phase: Saturation
% 9.05/2.82
% 9.05/2.82 % (21539)Memory used [KB]: 7547
% 9.05/2.82 % (21539)Time elapsed: 0.102 s
% 9.05/2.82 % (21539)Instructions burned: 192 (million)
% 9.05/2.82 % (21539)------------------------------
% 9.05/2.82 % (21539)------------------------------
% 9.05/2.83 % (21538)Instruction limit reached!
% 9.05/2.83 % (21538)------------------------------
% 9.05/2.83 % (21538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.05/2.83 % (21538)Termination reason: Unknown
% 9.05/2.83 % (21538)Termination phase: Saturation
% 9.05/2.83
% 9.05/2.83 % (21538)Memory used [KB]: 7291
% 9.05/2.83 % (21538)Time elapsed: 0.130 s
% 9.05/2.83 % (21538)Instructions burned: 300 (million)
% 9.05/2.83 % (21538)------------------------------
% 9.05/2.83 % (21538)------------------------------
% 9.05/2.84 % (21543)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_21305 for (2990ds/458Mi)
% 9.05/2.85 % (21544)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_21305 for (2990ds/31Mi)
% 9.05/2.85 % (21541)Instruction limit reached!
% 9.05/2.85 % (21541)------------------------------
% 9.05/2.85 % (21541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.05/2.85 % (21541)Termination reason: Unknown
% 9.05/2.85 % (21541)Termination phase: Function definition elimination
% 9.05/2.85
% 9.05/2.85 % (21541)Memory used [KB]: 2430
% 9.05/2.85 % (21541)Time elapsed: 0.045 s
% 9.05/2.85 % (21541)Instructions burned: 90 (million)
% 9.05/2.85 % (21541)------------------------------
% 9.05/2.85 % (21541)------------------------------
% 9.05/2.86 % (21544)Instruction limit reached!
% 9.05/2.86 % (21544)------------------------------
% 9.05/2.86 % (21544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.05/2.86 % (21544)Termination reason: Unknown
% 9.05/2.86 % (21544)Termination phase: shuffling
% 9.05/2.86
% 9.05/2.86 % (21544)Memory used [KB]: 2046
% 9.05/2.86 % (21544)Time elapsed: 0.017 s
% 9.05/2.86 % (21544)Instructions burned: 31 (million)
% 9.05/2.86 % (21544)------------------------------
% 9.05/2.86 % (21544)------------------------------
% 9.05/2.87 % (21533)First to succeed.
% 9.05/2.87 % (21545)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_21305 for (2990ds/297Mi)
% 9.05/2.87 % (21540)Instruction limit reached!
% 9.05/2.87 % (21540)------------------------------
% 9.05/2.87 % (21540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.05/2.87 % (21540)Termination reason: Unknown
% 9.05/2.87 % (21540)Termination phase: Saturation
% 9.05/2.87
% 9.05/2.87 % (21540)Memory used [KB]: 7803
% 9.05/2.87 % (21540)Time elapsed: 0.119 s
% 9.05/2.87 % (21540)Instructions burned: 258 (million)
% 9.05/2.87 % (21540)------------------------------
% 9.05/2.87 % (21540)------------------------------
% 9.05/2.87 % (21533)Refutation found. Thanks to Tanya!
% 9.05/2.87 % SZS status Theorem for DTF2THF_21305
% 9.05/2.87 % SZS output start Proof for DTF2THF_21305
% 9.05/2.87 thf(func_def_0, type, in: $i > $i > $o).
% 9.05/2.87 thf(func_def_1, type, exu: ($i > $o) > $o).
% 9.05/2.87 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 9.05/2.87 thf(func_def_8, type, powerset: $i > $i).
% 9.05/2.87 thf(func_def_10, type, setunion: $i > $i).
% 9.05/2.87 thf(func_def_19, type, descr: ($i > $o) > $i).
% 9.05/2.87 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 9.05/2.87 thf(func_def_26, type, prop2set: $o > $i).
% 9.05/2.87 thf(func_def_36, type, nonempty: $i > $o).
% 9.05/2.87 thf(func_def_69, type, set2prop: $i > $o).
% 9.05/2.87 thf(func_def_88, type, subset: $i > $i > $o).
% 9.05/2.87 thf(func_def_89, type, disjoint: $i > $i > $o).
% 9.05/2.87 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 9.05/2.87 thf(func_def_114, type, binunion: $i > $i > $i).
% 9.05/2.87 thf(func_def_122, type, binintersect: $i > $i > $i).
% 9.05/2.87 thf(func_def_135, type, regular: $i > $o).
% 9.05/2.87 thf(func_def_136, type, setminus: $i > $i > $i).
% 9.05/2.87 thf(func_def_147, type, symdiff: $i > $i > $i).
% 9.05/2.87 thf(func_def_153, type, iskpair: $i > $o).
% 9.05/2.87 thf(func_def_158, type, kpair: $i > $i > $i).
% 9.05/2.87 thf(func_def_160, type, cartprod: $i > $i > $i).
% 9.05/2.87 thf(func_def_177, type, singleton: $i > $o).
% 9.05/2.87 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 9.05/2.87 thf(func_def_184, type, atmost1p: $i > $o).
% 9.05/2.87 thf(func_def_185, type, atleast2p: $i > $o).
% 9.05/2.87 thf(func_def_186, type, atmost2p: $i > $o).
% 9.05/2.87 thf(func_def_187, type, upairsetp: $i > $o).
% 9.05/2.87 thf(func_def_191, type, kfst: $i > $i).
% 9.05/2.87 thf(func_def_203, type, ksnd: $i > $i).
% 9.05/2.87 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 9.05/2.87 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 9.05/2.87 thf(func_def_222, type, func: $i > $i > $i > $o).
% 9.05/2.87 thf(func_def_223, type, funcSet: $i > $i > $i).
% 9.05/2.87 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 9.05/2.87 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 9.05/2.87 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 9.05/2.87 thf(func_def_286, type, sK5: $i > ($i > $o) > $i).
% 9.05/2.87 thf(func_def_287, type, sK6: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_288, type, sK7: $i > $i > $i).
% 9.05/2.87 thf(func_def_289, type, sK8: $i > $i > $i).
% 9.05/2.87 thf(func_def_290, type, sK9: $i > $i > $i).
% 9.05/2.87 thf(func_def_291, type, sK10: ($i > $i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_292, type, sK11: ($i > $i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_293, type, sK12: $i > ($i > $i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_294, type, sK13: $i > ($i > $i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_295, type, sK14: ($i > $i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_296, type, sK15: $i > ($i > $i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_297, type, sK16: ($i > $o) > $i).
% 9.05/2.87 thf(func_def_298, type, sK17: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_299, type, sK18: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_300, type, sK19: $i > $i > $i > $i).
% 9.05/2.87 thf(func_def_301, type, sK20: $i > $i > $i > $i).
% 9.05/2.87 thf(func_def_302, type, sK21: ($i > $o) > $i).
% 9.05/2.87 thf(func_def_303, type, sK22: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_304, type, sK23: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_305, type, sK24: $i > $i).
% 9.05/2.87 thf(func_def_306, type, sK25: $i > $i > $i).
% 9.05/2.87 thf(func_def_307, type, sK26: ($i > $o) > ($i > $o) > $i > $i > $i).
% 9.05/2.87 thf(func_def_308, type, sK27: $i > $i > ($i > $o) > ($i > $o) > $i).
% 9.05/2.87 thf(func_def_309, type, sK28: $i > $i > $i > $i > $i).
% 9.05/2.87 thf(func_def_310, type, sK29: $i > $i > $i > $i > $i).
% 9.05/2.87 thf(func_def_311, type, sK30: ($i > $i) > $i > $i > $i).
% 9.05/2.87 thf(func_def_312, type, sK31: $i > $i > ($i > $i) > $i).
% 9.05/2.87 thf(func_def_313, type, sK32: $i > $i).
% 9.05/2.87 thf(func_def_314, type, sK33: $i > $i > $i).
% 9.05/2.87 thf(func_def_315, type, sK34: $i > $i > $i).
% 9.05/2.87 thf(func_def_316, type, sK35: $i > $i > $i).
% 9.05/2.87 thf(func_def_317, type, sK36: $i > $i > $i > $i).
% 9.05/2.87 thf(func_def_318, type, sK37: ($i > $o) > ($i > $o) > $i).
% 9.05/2.87 thf(func_def_319, type, sK38: ($i > $o) > ($i > $o) > $i).
% 9.05/2.87 thf(func_def_320, type, sK39: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_321, type, sK40: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_322, type, sK41: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_323, type, sK42: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_324, type, sK43: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_325, type, sK44: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_326, type, sK45: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_327, type, sK46: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_328, type, sK47: $i > $o > $i > $i).
% 9.05/2.87 thf(func_def_329, type, sK48: $i > $i > $i).
% 9.05/2.87 thf(func_def_330, type, sK49: ($i > $i) > $i > $i > $i).
% 9.05/2.87 thf(func_def_331, type, sK50: ($i > $o) > $i).
% 9.05/2.87 thf(func_def_332, type, sK51: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_333, type, sK52: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_334, type, sK53: ($i > $o) > $i).
% 9.05/2.87 thf(func_def_335, type, sK54: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_336, type, sK55: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_337, type, sK56: $i > $i > ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_338, type, sK57: $i > $i > $i > ($i > $o) > $i).
% 9.05/2.87 thf(func_def_339, type, sK58: $i > $i > $i).
% 9.05/2.87 thf(func_def_340, type, sK59: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_341, type, sK60: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_342, type, sK61: $i > $i > $i).
% 9.05/2.87 thf(func_def_343, type, sK62: ($i > $o) > $i).
% 9.05/2.87 thf(func_def_344, type, sK63: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_345, type, sK64: $i > ($i > $o) > $i).
% 9.05/2.87 thf(func_def_346, type, sK65: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_347, type, sK66: ($i > $o) > $i > $i).
% 9.05/2.87 thf(func_def_348, type, sK67: $i > ($i > $i) > $i > $i).
% 9.05/2.87 thf(func_def_349, type, sK68: $i > $i > ($i > $o) > $i).
% 9.05/2.88 thf(func_def_350, type, sK69: $i > ($i > $i) > $i > $i).
% 9.05/2.88 thf(func_def_351, type, sK70: $i > $i).
% 9.05/2.88 thf(func_def_352, type, sK71: $i > ($i > $o) > $i).
% 9.05/2.88 thf(func_def_353, type, sK72: $i > ($i > $o) > $i).
% 9.05/2.88 thf(func_def_354, type, sK73: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_355, type, sK74: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_356, type, sK75: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_357, type, sK76: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_358, type, sK77: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_359, type, sK78: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_360, type, sK79: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_361, type, sK80: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_362, type, sK81: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_363, type, sK82: $i > $i > $i).
% 9.05/2.88 thf(func_def_364, type, sK83: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_365, type, sK84: $i > $i).
% 9.05/2.88 thf(func_def_366, type, sK85: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_367, type, sK86: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_368, type, sK87: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_369, type, sK88: $i > ($i > $o) > $i).
% 9.05/2.88 thf(func_def_370, type, sK89: $i > $i).
% 9.05/2.88 thf(func_def_371, type, sK90: ($i > $o) > ($i > $o) > $i).
% 9.05/2.88 thf(func_def_372, type, sK91: ($i > $o) > ($i > $o) > $i).
% 9.05/2.88 thf(func_def_373, type, sK92: ($i > $o) > $i).
% 9.05/2.88 thf(func_def_374, type, sK93: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_375, type, sK94: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_376, type, sK95: ($i > $o) > $i).
% 9.05/2.88 thf(func_def_377, type, sK96: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_378, type, sK97: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_379, type, sK98: ($i > $o) > $i).
% 9.05/2.88 thf(func_def_380, type, sK99: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_381, type, sK100: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_382, type, sK101: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_383, type, sK102: ($i > $o) > $i).
% 9.05/2.88 thf(func_def_384, type, sK103: ($i > $o) > $i).
% 9.05/2.88 thf(func_def_385, type, sK104: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_386, type, sK105: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_387, type, sK106: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_388, type, sK107: ($i > $o) > $i).
% 9.05/2.88 thf(func_def_389, type, sK108: $i > $i > $i > $i > $i).
% 9.05/2.88 thf(func_def_390, type, sK109: $i > $i > $i).
% 9.05/2.88 thf(func_def_391, type, sK110: ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_392, type, sK111: ($i > $o) > $i).
% 9.05/2.88 thf(func_def_393, type, sK112: ($i > $o) > $i).
% 9.05/2.88 thf(func_def_394, type, sK113: ($i > $o) > $i).
% 9.05/2.88 thf(func_def_395, type, sK114: $i > $i > $i).
% 9.05/2.88 thf(func_def_396, type, sK115: $i > $i).
% 9.05/2.88 thf(func_def_397, type, sK116: $i > $i > ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_398, type, sK117: $i > $i > ($i > $o) > $i > $i).
% 9.05/2.88 thf(func_def_399, type, sK118: $i > $i).
% 9.05/2.88 thf(func_def_400, type, sK119: $i > $i > $i).
% 9.05/2.88 thf(func_def_401, type, ph120: !>[X0: $tType]:(X0)).
% 9.05/2.88 thf(f5029,plain,(
% 9.05/2.88 $false),
% 9.05/2.88 inference(avatar_sat_refutation,[status(thm)],[f3985,f3995,f4055,f4061,f4076,f4999,f5028])).
% 9.05/2.88 thf(f5028,plain,(
% 9.05/2.88 ($false != (in @ sK3 @ sK2)) | ($true != (in @ sK3 @ sK2)) | ($false = $true)),
% 9.05/2.88 introduced(definition,[],[theory_tautology_sat_conflict])).
% 9.05/2.88 thf(f4999,plain,(
% 9.05/2.88 spl4_37 | spl4_1 | ~spl4_19 | spl4_22),
% 9.05/2.88 inference(avatar_split_clause,[status(thm)],[f4998,f4073,f4058,f3982,f4902])).
% 9.05/2.88 thf(f4902,definition,(
% 9.05/2.88 spl4_37 <=> ($false = (in @ sK3 @ sK2))),
% 9.05/2.88 introduced(definition,[new_symbols(naming,[spl4_37])],[avatar_definition])).
% 9.05/2.88 thf(f3982,definition,(
% 9.05/2.88 spl4_1 <=> ((setunion @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (((sK1 = Y0) & sK0) | ((~ sK0) & (sK3 = Y0)))))) = sK3)),
% 9.05/2.88 introduced(definition,[new_symbols(naming,[spl4_1])],[avatar_definition])).
% 9.05/2.88 thf(f4058,definition,(
% 9.05/2.88 spl4_19 <=> ($true = (in @ sK1 @ sK2))),
% 9.05/2.88 introduced(definition,[new_symbols(naming,[spl4_19])],[avatar_definition])).
% 9.05/2.88 thf(f4073,definition,(
% 9.05/2.88 spl4_22 <=> ($true = sK0)),
% 9.05/2.88 introduced(definition,[new_symbols(naming,[spl4_22])],[avatar_definition])).
% 9.05/2.88 thf(f4998,plain,(
% 9.05/2.88 ($false = (in @ sK3 @ sK2)) | (spl4_1 | ~spl4_19 | spl4_22)),
% 9.05/2.88 inference(trivial_inequality_removal,[status(thm)],[f4997])).
% 9.05/2.88 thf(f4997,plain,(
% 9.05/2.88 ($false = (in @ sK3 @ sK2)) | ($false = $true) | (spl4_1 | ~spl4_19 | spl4_22)),
% 9.05/2.88 inference(forward_demodulation,[status(thm)],[f4996,f4060])).
% 9.05/2.88 thf(f4060,plain,(
% 9.05/2.88 ($true = (in @ sK1 @ sK2)) | ~spl4_19),
% 9.05/2.88 inference(avatar_component_clause,[status(thm)],[f4058])).
% 9.05/2.88 thf(f4996,plain,(
% 9.05/2.88 ($false = (in @ sK1 @ sK2)) | ($false = (in @ sK3 @ sK2)) | (spl4_1 | spl4_22)),
% 9.05/2.88 inference(subsumption_resolution,[status(thm)],[f4995,f3288])).
% 9.05/2.88 thf(f3288,plain,(
% 9.05/2.88 ( ! [X1 : $i] : (((setunion @ (setadjoin @ X1 @ emptyset)) = X1)) )),
% 9.05/2.88 inference(equality_proxy_clausification,[status(thm)],[f3287])).
% 9.05/2.88 thf(f3287,plain,(
% 9.05/2.88 ( ! [X1 : $i] : ((((setunion @ (setadjoin @ X1 @ emptyset)) = X1) = $true)) )),
% 9.05/2.88 inference(beta_eta_normalization,[status(thm)],[f3286])).
% 9.05/2.88 thf(f3286,plain,(
% 9.05/2.88 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: ((setunion @ (setadjoin @ Y0 @ emptyset)) = Y0)) @ X1))) )),
% 9.05/2.88 inference(pi_clausification,[status(thm)],[f1238])).
% 9.05/2.88 thf(f1238,plain,(
% 9.05/2.88 ($true = (!! @ $i @ (^[Y0 : $i]: ((setunion @ (setadjoin @ Y0 @ emptyset)) = Y0))))),
% 9.05/2.88 inference(definition_unfolding,[status(thm)],[f1006,f768])).
% 9.05/2.88 thf(f768,plain,(
% 9.05/2.88 (setunionsingleton = $true)),
% 9.05/2.88 inference(cnf_transformation,[status(thm)],[f692])).
% 9.05/2.88 thf(f692,plain,(
% 9.05/2.88 (eqinunit = $true) & (setunionE = $true) & (cartprodpairmemER = $true) & (powersetI1 = $true) & (binunionE = $true) & (powersetsubset = $true) & (kfstpairEq = $true) & (powersetAx = $true) & (emptyinPowerset = $true) & (upairinpowunion = $true) & (dpsetconstrEL1 = $true) & (emptyI = $true) & (nonemptyImpWitness = $true) & (exu__Cong = $true) & (ex1E1 = $true) & (iffalseProp2 = $true) & (notequalI2 = $true) & (eqimpsubset1 = $true) & (subset2powerset = $true) & (setunion__Cong = $true) & (binunionIR = $true) & (binintersectER = $true) & (upairsetIL = $true) & (upairsetIR = $true) & (funcGraphProp1 = $true) & (dsetconstrI = $true) & (foundationAx = $true) & (singletoninpowunion = $true) & (omega__Cong = $true) & (($true = (in @ sK1 @ sK2)) & ((sK3 != (if @ sK2 @ sK0 @ sK1 @ sK3)) & ($true = (in @ sK3 @ sK2)) & ($true != sK0))) & (setukpairIL = $true) & (emptyset__Cong = $true) & (upairset2E = $true) & (app = $true) & (funcinfuncset = $true) & (lam2p = $true) & (brelnall2 = $true) & (setadjoin__Cong = $true) & (subbreln = $true) & (setadjoinIR = $true) & (singletoninpowerset = $true) & (setadjoinIL = $true) & (ubforcartprodlem1 = $true) & (setukpairIR = $true) & (singletonsuniq = $true) & (inCongP = $true) & (symdiffI1 = $true) & (powersetE = $true) & (emptyE1 = $true) & (prop2set2propI = $true) & (setOfPairsIsBReln = $true) & (exuI3 = $true) & (eta2 = $true) & (omega0Ax = $true) & (binunionRsub = $true) & (binunionIL = $true) & (setadjoinSub2 = $true) & (ex1E2 = $true) & (uniqinunit = $true) & (upairsetE = $true) & (ifSingleton = $true) & (prop2setE = $true) & (ubforcartprodlem2 = $true) & (cartprodmempaircEq = $true) & (iftrue = $true) & (in__Cong = $true) & (powersetE1 = $true) & (setunionI = $true) & (binintersectSubset3 = $true) & (quantDeMorgan1 = $true) & (subsetE2 = $true) & (setminusELneg = $true) & (exuI2 = $true) & (binintersectSubset1 = $true) & (ex1I = $true) & (lamProp = $true) & (replAx = $true) & (binintersectI = $true) & (emptysetE = $true) & (beta2 = $true) & (powerset__Cong = $true) & (theeq = $true) & (exuEu = $true) & (ksndsingleton = $true) & (notdexE = $true) & (cartprodpairsurjEq = $true) & (kpairsurjEq = $true) & (setext = $true) & (exuI1 = $true) & (eqimpsubset2 = $true) & (binintersectLsub = $true) & (exuE3e = $true) & (subsetI2 = $true) & (setukpairinjL2 = $true) & (dpsetconstrEL2 = $true) & (exuE1 = $true) & (cartprodfstin = $true) & (descr__Cong = $true) & (emptysetsubset = $true) & (setminusI = $true) & (symdiffIneg1 = $true) & (setunionsingleton2 = $true) & (omegaSAx = $true) & (setukpairinjL = $true) & (setukpairinjR2 = $true) & (inPowerset = $true) & (notinsingleton = $true) & (setminusIRneg = $true) & (noeltsimpempty = $true) & (emptysetimpfalse = $true) & (setukpairinjR1 = $true) & (cartprodfstpairEq = $true) & (exuE3u = $true) & (subsetI1 = $true) & (emptyinunitempty = $true) & (funcext2 = $true) & (funcGraphProp2 = $true) & (notinemptyset = $true) & (dsetconstrER = $true) & (dpsetconstrSub = $true) & (binintersectSubset2 = $true) & (dpsetconstrER = $true) & (kpairiskpair = $true) & (ksndpairEq = $true) & (cartprodmempair1 = $true) & (prop2setI = $true) & (setbeta = $true) & (eqbreln = $true) & (funcGraphProp4 = $true) & (subPowSU = $true) & (cartprodpairin = $true) & (subsetemptysetimpeq = $true) & (iffalseProp1 = $true) & (omegaIndAx = $true) & (dpsetconstrERa = $true) & (symdiffI2 = $true) & (cartprodsndpairEq = $true) & (setadjoinE = $true) & (setminusILneg = $true) & (upairset2IR = $true) & (sepInPowerset = $true) & (beta1 = $true) & (emptyInPowerset = $true) & (funcGraphProp3 = $true) & (setunionE2 = $true) & (descrp = $true) & (setminusSubset2 = $true) & (setminusSubset1 = $true) & (singletonsubset = $true) & (theprop = $true) & (eta1 = $true) & (notequalI1 = $true) & (kfstsingleton = $true) & (ifp = $true) & (funcext = $true) & (ap2apEq2 = $true) & (quantDeMorgan4 = $true) & (ap2apEq1 = $true) & (subsetTrans = $true) & (notdallE = $true) & (cartprodpairmemEL = $true) & (dsetconstr__Cong = $true) & (lam2lamEq = $true) & (nonemptyI = $true) & (setukpairinjL1 = $true) & (notsubsetI = $true) & (ubforcartprodlem3 = $true) & (dsetconstrEL = $true) & (binunionLsub = $true) & (symdiffIneg2 = $true) & (upairsubunion = $true) & (singletonprop = $true) & (setadjoinAx = $true) & (nonemptyI1 = $true) & (powersetI = $true) & (binintersectEL = $true) & (ap2p = $true) & (quantDeMorgan3 = $true) & (ex1I2 = $true) & (subsetRefl = $true) & (exuE2 = $true) & (setadjoinSub = $true) & (apProp = $true) & (lamp = $true) & (subsetE = $true) & (funcImageSingleton = $true) & (setunionsingleton = $true) & (setukpairinjR = $true) & (setminusER = $true) & (binintersectRsub = $true) & (iftrueProp1 = $true) & (setextsub = $true) & (upairequniteq = $true) & (wellorderingAx = $true) & (brelnall1 = $true) & (funcextLem = $true) & (setminusERneg = $true) & (setukpairinjR12 = $true) & (quantDeMorgan2 = $true) & (binunionEcases = $true) & (symdiffE = $true) & (secondinupair = $true) & (vacuousDall = $true) & (setoftrueEq = $true) & (setunionsingleton1 = $true) & (singletonsswitch = $true) & (disjointsetsI1 = $true) & (binintersectSubset5 = $true) & (sepSubset = $true) & (dpsetconstrI = $true) & (cartprodsndin = $true) & (kpairp = $true) & (iftrueProp2 = $true) & (binintersectSubset4 = $true) & (setunionAx = $true) & (infuncsetfunc = $true) & (setminusEL = $true) & (setminusLsub = $true) & (bs114d = $true) & (setextAx = $true) & (cartprodmempair = $true) & (emptysetAx = $true) & (nonemptyE1 = $true) & (setadjoinOr = $true) & (setukpairinjR11 = $true)),
% 9.05/2.88 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2,sK3])],[f689,f691,f690])).
% 9.05/2.88 thf(f690,plain,(
% 9.05/2.88 ? [X0 : $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) & ? [X3 : $i] : (((if @ X2 @ X0 @ X1 @ X3) != X3) & ((in @ X3 @ X2) = $true) & ($true != X0))) => (($true = (in @ sK1 @ sK2)) & ? [X3 : $i] : (((if @ sK2 @ sK0 @ sK1 @ X3) != X3) & ((in @ X3 @ sK2) = $true) & ($true != sK0)))),
% 9.05/2.88 introduced(definition,[],[choice_axiom])).
% 9.05/2.88 thf(f691,plain,(
% 9.05/2.88 ? [X3 : $i] : (((if @ sK2 @ sK0 @ sK1 @ X3) != X3) & ((in @ X3 @ sK2) = $true) & ($true != sK0)) => ((sK3 != (if @ sK2 @ sK0 @ sK1 @ sK3)) & ($true = (in @ sK3 @ sK2)) & ($true != sK0))),
% 9.05/2.88 introduced(definition,[],[choice_axiom])).
% 9.05/2.88 thf(f689,plain,(
% 9.05/2.88 (eqinunit = $true) & (setunionE = $true) & (cartprodpairmemER = $true) & (powersetI1 = $true) & (binunionE = $true) & (powersetsubset = $true) & (kfstpairEq = $true) & (powersetAx = $true) & (emptyinPowerset = $true) & (upairinpowunion = $true) & (dpsetconstrEL1 = $true) & (emptyI = $true) & (nonemptyImpWitness = $true) & (exu__Cong = $true) & (ex1E1 = $true) & (iffalseProp2 = $true) & (notequalI2 = $true) & (eqimpsubset1 = $true) & (subset2powerset = $true) & (setunion__Cong = $true) & (binunionIR = $true) & (binintersectER = $true) & (upairsetIL = $true) & (upairsetIR = $true) & (funcGraphProp1 = $true) & (dsetconstrI = $true) & (foundationAx = $true) & (singletoninpowunion = $true) & (omega__Cong = $true) & ? [X0 : $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) & ? [X3 : $i] : (((if @ X2 @ X0 @ X1 @ X3) != X3) & ((in @ X3 @ X2) = $true) & ($true != X0))) & (setukpairIL = $true) & (emptyset__Cong = $true) & (upairset2E = $true) & (app = $true) & (funcinfuncset = $true) & (lam2p = $true) & (brelnall2 = $true) & (setadjoin__Cong = $true) & (subbreln = $true) & (setadjoinIR = $true) & (singletoninpowerset = $true) & (setadjoinIL = $true) & (ubforcartprodlem1 = $true) & (setukpairIR = $true) & (singletonsuniq = $true) & (inCongP = $true) & (symdiffI1 = $true) & (powersetE = $true) & (emptyE1 = $true) & (prop2set2propI = $true) & (setOfPairsIsBReln = $true) & (exuI3 = $true) & (eta2 = $true) & (omega0Ax = $true) & (binunionRsub = $true) & (binunionIL = $true) & (setadjoinSub2 = $true) & (ex1E2 = $true) & (uniqinunit = $true) & (upairsetE = $true) & (ifSingleton = $true) & (prop2setE = $true) & (ubforcartprodlem2 = $true) & (cartprodmempaircEq = $true) & (iftrue = $true) & (in__Cong = $true) & (powersetE1 = $true) & (setunionI = $true) & (binintersectSubset3 = $true) & (quantDeMorgan1 = $true) & (subsetE2 = $true) & (setminusELneg = $true) & (exuI2 = $true) & (binintersectSubset1 = $true) & (ex1I = $true) & (lamProp = $true) & (replAx = $true) & (binintersectI = $true) & (emptysetE = $true) & (beta2 = $true) & (powerset__Cong = $true) & (theeq = $true) & (exuEu = $true) & (ksndsingleton = $true) & (notdexE = $true) & (cartprodpairsurjEq = $true) & (kpairsurjEq = $true) & (setext = $true) & (exuI1 = $true) & (eqimpsubset2 = $true) & (binintersectLsub = $true) & (exuE3e = $true) & (subsetI2 = $true) & (setukpairinjL2 = $true) & (dpsetconstrEL2 = $true) & (exuE1 = $true) & (cartprodfstin = $true) & (descr__Cong = $true) & (emptysetsubset = $true) & (setminusI = $true) & (symdiffIneg1 = $true) & (setunionsingleton2 = $true) & (omegaSAx = $true) & (setukpairinjL = $true) & (setukpairinjR2 = $true) & (inPowerset = $true) & (notinsingleton = $true) & (setminusIRneg = $true) & (noeltsimpempty = $true) & (emptysetimpfalse = $true) & (setukpairinjR1 = $true) & (cartprodfstpairEq = $true) & (exuE3u = $true) & (subsetI1 = $true) & (emptyinunitempty = $true) & (funcext2 = $true) & (funcGraphProp2 = $true) & (notinemptyset = $true) & (dsetconstrER = $true) & (dpsetconstrSub = $true) & (binintersectSubset2 = $true) & (dpsetconstrER = $true) & (kpairiskpair = $true) & (ksndpairEq = $true) & (cartprodmempair1 = $true) & (prop2setI = $true) & (setbeta = $true) & (eqbreln = $true) & (funcGraphProp4 = $true) & (subPowSU = $true) & (cartprodpairin = $true) & (subsetemptysetimpeq = $true) & (iffalseProp1 = $true) & (omegaIndAx = $true) & (dpsetconstrERa = $true) & (symdiffI2 = $true) & (cartprodsndpairEq = $true) & (setadjoinE = $true) & (setminusILneg = $true) & (upairset2IR = $true) & (sepInPowerset = $true) & (beta1 = $true) & (emptyInPowerset = $true) & (funcGraphProp3 = $true) & (setunionE2 = $true) & (descrp = $true) & (setminusSubset2 = $true) & (setminusSubset1 = $true) & (singletonsubset = $true) & (theprop = $true) & (eta1 = $true) & (notequalI1 = $true) & (kfstsingleton = $true) & (ifp = $true) & (funcext = $true) & (ap2apEq2 = $true) & (quantDeMorgan4 = $true) & (ap2apEq1 = $true) & (subsetTrans = $true) & (notdallE = $true) & (cartprodpairmemEL = $true) & (dsetconstr__Cong = $true) & (lam2lamEq = $true) & (nonemptyI = $true) & (setukpairinjL1 = $true) & (notsubsetI = $true) & (ubforcartprodlem3 = $true) & (dsetconstrEL = $true) & (binunionLsub = $true) & (symdiffIneg2 = $true) & (upairsubunion = $true) & (singletonprop = $true) & (setadjoinAx = $true) & (nonemptyI1 = $true) & (powersetI = $true) & (binintersectEL = $true) & (ap2p = $true) & (quantDeMorgan3 = $true) & (ex1I2 = $true) & (subsetRefl = $true) & (exuE2 = $true) & (setadjoinSub = $true) & (apProp = $true) & (lamp = $true) & (subsetE = $true) & (funcImageSingleton = $true) & (setunionsingleton = $true) & (setukpairinjR = $true) & (setminusER = $true) & (binintersectRsub = $true) & (iftrueProp1 = $true) & (setextsub = $true) & (upairequniteq = $true) & (wellorderingAx = $true) & (brelnall1 = $true) & (funcextLem = $true) & (setminusERneg = $true) & (setukpairinjR12 = $true) & (quantDeMorgan2 = $true) & (binunionEcases = $true) & (symdiffE = $true) & (secondinupair = $true) & (vacuousDall = $true) & (setoftrueEq = $true) & (setunionsingleton1 = $true) & (singletonsswitch = $true) & (disjointsetsI1 = $true) & (binintersectSubset5 = $true) & (sepSubset = $true) & (dpsetconstrI = $true) & (cartprodsndin = $true) & (kpairp = $true) & (iftrueProp2 = $true) & (binintersectSubset4 = $true) & (setunionAx = $true) & (infuncsetfunc = $true) & (setminusEL = $true) & (setminusLsub = $true) & (bs114d = $true) & (setextAx = $true) & (cartprodmempair = $true) & (emptysetAx = $true) & (nonemptyE1 = $true) & (setadjoinOr = $true) & (setukpairinjR11 = $true)),
% 9.05/2.88 inference(flattening,[status(thm)],[f688])).
% 9.05/2.88 thf(f688,plain,(
% 9.05/2.88 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $o,X1 : $i,X2 : $i] : (? [X3 : $i] : ((((if @ X2 @ X0 @ X1 @ X3) != X3) & ($true != X0)) & ((in @ X3 @ X2) = $true)) & ((in @ X1 @ X2) = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 9.05/2.88 inference(ennf_transformation,[status(thm)],[f687])).
% 9.05/2.88 thf(f687,plain,(
% 9.05/2.88 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ! [X0 : $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => (($true != X0) => ((if @ X2 @ X0 @ X1 @ X3) = X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.05/2.88 inference(flattening,[status(thm)],[f471])).
% 9.05/2.88 thf(f471,plain,(
% 9.05/2.88 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ! [X0 : $o,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => ! [X3 : $i] : (((in @ X3 @ X2) = $true) => (~($true = X0) => ((if @ X2 @ X0 @ X1 @ X3) = X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.05/2.88 inference(fool_elimination,[status(thm)],[f470])).
% 9.05/2.88 thf(f470,plain,(
% 9.05/2.88 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => ! [X0 : $o,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => ! [X3 : $i] : ((in @ X3 @ X2) => (~X0 => ((if @ X2 @ X0 @ X1 @ X3) = X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.05/2.88 inference(rectify,[status(thm)],[f231])).
% 9.05/2.88 thf(f231,negated_conjecture,(
% 9.05/2.88 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => ! [X0 : $o,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (~X0 => ((if @ X3 @ X0 @ X1 @ X2) = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.05/2.88 inference(negated_conjecture,[status(cth)],[f230])).
% 9.05/2.88 thf(f230,conjecture,(
% 9.05/2.88 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => ! [X0 : $o,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (~X0 => ((if @ X3 @ X0 @ X1 @ X2) = X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 9.05/2.88 file('/export/starexec/sandbox/tmp/tmp.K93CCo4jzK/DTF2THF_21305.p',iffalse)).
% 9.05/2.88 thf(f1006,plain,(
% 9.05/2.88 (setunionsingleton = (!! @ $i @ (^[Y0 : $i]: ((setunion @ (setadjoin @ Y0 @ emptyset)) = Y0))))),
% 9.05/2.88 inference(cnf_transformation,[status(thm)],[f469])).
% 9.05/2.88 thf(f469,plain,(
% 9.05/2.88 (setunionsingleton = (!! @ $i @ (^[Y0 : $i]: ((setunion @ (setadjoin @ Y0 @ emptyset)) = Y0))))),
% 9.05/2.88 inference(fool_elimination,[status(thm)],[f468])).
% 9.05/2.88 thf(f468,plain,(
% 9.05/2.88 (setunionsingleton = ! [X0 : $i] : ((setunion @ (setadjoin @ X0 @ emptyset)) = X0))),
% 9.05/2.88 inference(rectify,[status(thm)],[f155])).
% 9.05/2.88 thf(f155,axiom,(
% 9.05/2.88 (setunionsingleton = ! [X1 : $i] : ((setunion @ (setadjoin @ X1 @ emptyset)) = X1))),
% 9.05/2.88 file('/export/starexec/sandbox/tmp/tmp.K93CCo4jzK/DTF2THF_21305.p',setunionsingleton)).
% 9.05/2.88 thf(f4995,plain,(
% 9.05/2.88 ($false = (in @ sK1 @ sK2)) | ($false = (in @ sK3 @ sK2)) | ((setunion @ (setadjoin @ sK3 @ emptyset)) != sK3) | (spl4_1 | spl4_22)),
% 9.05/2.88 inference(subsumption_resolution,[status(thm)],[f4951,f4075])).
% 9.05/2.88 thf(f4075,plain,(
% 9.05/2.88 ($true != sK0) | spl4_22),
% 9.05/2.88 inference(avatar_component_clause,[status(thm)],[f4073])).
% 9.05/2.88 thf(f4951,plain,(
% 9.05/2.88 ($false = (in @ sK1 @ sK2)) | ($true = sK0) | ((setunion @ (setadjoin @ sK3 @ emptyset)) != sK3) | ($false = (in @ sK3 @ sK2)) | spl4_1),
% 9.05/2.88 inference(superposition,[status(thm)],[f3984,f2695])).
% 9.05/2.88 thf(f2695,plain,(
% 9.05/2.88 ( ! [X2 : $i,X3 : $o,X1 : $i,X4 : $i] : (((dsetconstr @ X2 @ (^[Y0 : $i]: (((X1 = Y0) & X3) | ((~ X3) & (X4 = Y0))))) = (setadjoin @ X4 @ emptyset)) | ($false = (in @ X4 @ X2)) | ($true = X3) | ((in @ X1 @ X2) = $false)) )),
% 9.05/2.88 inference(not_proxy_clausification,[status(thm)],[f2694])).
% 9.05/2.88 thf(f2694,plain,(
% 9.05/2.88 ( ! [X2 : $i,X3 : $o,X1 : $i,X4 : $i] : (($false = (~ X3)) | ($false = (in @ X4 @ X2)) | ((dsetconstr @ X2 @ (^[Y0 : $i]: (((X1 = Y0) & X3) | ((~ X3) & (X4 = Y0))))) = (setadjoin @ X4 @ emptyset)) | ((in @ X1 @ X2) = $false)) )),
% 9.05/2.88 inference(equality_proxy_clausification,[status(thm)],[f2693])).
% 9.05/2.88 thf(f2693,plain,(
% 9.05/2.88 ( ! [X2 : $i,X3 : $o,X1 : $i,X4 : $i] : (($true = ((setadjoin @ X4 @ emptyset) = (dsetconstr @ X2 @ (^[Y0 : $i]: (((X1 = Y0) & X3) | ((~ X3) & (X4 = Y0))))))) | ($false = (~ X3)) | ((in @ X1 @ X2) = $false) | ($false = (in @ X4 @ X2))) )),
% 9.05/2.88 inference(binary_proxy_clausification,[status(thm)],[f2692])).
% 9.05/2.88 thf(f2692,plain,(
% 9.05/2.88 ( ! [X2 : $i,X3 : $o,X1 : $i,X4 : $i] : (((in @ X1 @ X2) = $false) | ($true = ((~ X3) => ((setadjoin @ X4 @ emptyset) = (dsetconstr @ X2 @ (^[Y0 : $i]: (((X1 = Y0) & X3) | ((~ X3) & (X4 = Y0)))))))) | ($false = (in @ X4 @ X2))) )),
% 9.05/2.88 inference(binary_proxy_clausification,[status(thm)],[f2691])).
% 9.05/2.88 thf(f2691,plain,(
% 9.05/2.88 ( ! [X2 : $i,X3 : $o,X1 : $i,X4 : $i] : ((((in @ X4 @ X2) => ((~ X3) => ((setadjoin @ X4 @ emptyset) = (dsetconstr @ X2 @ (^[Y0 : $i]: (((X1 = Y0) & X3) | ((~ X3) & (X4 = Y0)))))))) = $true) | ((in @ X1 @ X2) = $false)) )),
% 9.05/2.88 inference(beta_eta_normalization,[status(thm)],[f2690])).
% 9.05/2.88 thf(f2690,plain,(
% 9.05/2.88 ( ! [X2 : $i,X3 : $o,X1 : $i,X4 : $i] : (((in @ X1 @ X2) = $false) | (((^[Y0 : $i]: ((in @ Y0 @ X2) => ((~ X3) => ((setadjoin @ Y0 @ emptyset) = (dsetconstr @ X2 @ (^[Y1 : $i]: (((X1 = Y1) & X3) | ((~ X3) & (Y0 = Y1))))))))) @ X4) = $true)) )),
% 9.05/2.88 inference(pi_clausification,[status(thm)],[f2689])).
% 9.05/2.88 thf(f2689,plain,(
% 9.05/2.88 ( ! [X2 : $i,X3 : $o,X1 : $i] : (((in @ X1 @ X2) = $false) | ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X2) => ((~ X3) => ((setadjoin @ Y0 @ emptyset) = (dsetconstr @ X2 @ (^[Y1 : $i]: (((X1 = Y1) & X3) | ((~ X3) & (Y0 = Y1)))))))))) = $true)) )),
% 9.05/2.88 inference(binary_proxy_clausification,[status(thm)],[f2688])).
% 9.05/2.88 thf(f2688,plain,(
% 9.05/2.88 ( ! [X2 : $i,X3 : $o,X1 : $i] : ((((in @ X1 @ X2) => (!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ X2) => ((~ X3) => ((setadjoin @ Y0 @ emptyset) = (dsetconstr @ X2 @ (^[Y1 : $i]: (((X1 = Y1) & X3) | ((~ X3) & (Y0 = Y1))))))))))) = $true)) )),
% 9.05/2.88 inference(beta_eta_normalization,[status(thm)],[f2687])).
% 9.05/2.88 thf(f2687,plain,(
% 9.05/2.88 ( ! [X2 : $i,X3 : $o,X1 : $i] : ((((^[Y0 : $o]: ((in @ X1 @ X2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => ((~ Y0) => ((setadjoin @ Y1 @ emptyset) = (dsetconstr @ X2 @ (^[Y2 : $i]: (((X1 = Y2) & Y0) | ((~ Y0) & (Y1 = Y2)))))))))))) @ X3) = $true)) )),
% 9.05/2.88 inference(pi_clausification,[status(thm)],[f2686])).
% 9.05/2.88 thf(f2686,plain,(
% 9.05/2.88 ( ! [X2 : $i,X1 : $i] : (($true = (!! @ $o @ (^[Y0 : $o]: ((in @ X1 @ X2) => (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ X2) => ((~ Y0) => ((setadjoin @ Y1 @ emptyset) = (dsetconstr @ X2 @ (^[Y2 : $i]: (((X1 = Y2) & Y0) | ((~ Y0) & (Y1 = Y2))))))))))))))) )),
% 9.05/2.88 inference(beta_eta_normalization,[status(thm)],[f2685])).
% 9.05/2.88 thf(f2685,plain,(
% 9.05/2.88 ( ! [X2 : $i,X1 : $i] : ((((^[Y0 : $i]: (!! @ $o @ (^[Y1 : $o]: ((in @ X1 @ Y0) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => ((~ Y1) => ((setadjoin @ Y2 @ emptyset) = (dsetconstr @ Y0 @ (^[Y3 : $i]: (((X1 = Y3) & Y1) | ((~ Y1) & (Y2 = Y3)))))))))))))) @ X2) = $true)) )),
% 9.05/2.88 inference(pi_clausification,[status(thm)],[f2684])).
% 9.05/2.88 thf(f2684,plain,(
% 9.05/2.88 ( ! [X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $o @ (^[Y1 : $o]: ((in @ X1 @ Y0) => (!! @ $i @ (^[Y2 : $i]: ((in @ Y2 @ Y0) => ((~ Y1) => ((setadjoin @ Y2 @ emptyset) = (dsetconstr @ Y0 @ (^[Y3 : $i]: (((X1 = Y3) & Y1) | ((~ Y1) & (Y2 = Y3))))))))))))))))) )),
% 9.05/2.88 inference(beta_eta_normalization,[status(thm)],[f2683])).
% 9.05/2.88 thf(f2683,plain,(
% 9.05/2.88 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $o @ (^[Y2 : $o]: ((in @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => ((~ Y2) => ((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y1 @ (^[Y4 : $i]: (((Y0 = Y4) & Y2) | ((~ Y2) & (Y3 = Y4)))))))))))))))) @ X1))) )),
% 9.05/2.88 inference(pi_clausification,[status(thm)],[f1160])).
% 9.05/2.88 thf(f1160,plain,(
% 9.05/2.88 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $o @ (^[Y2 : $o]: ((in @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => ((~ Y2) => ((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y1 @ (^[Y4 : $i]: (((Y0 = Y4) & Y2) | ((~ Y2) & (Y3 = Y4))))))))))))))))))),
% 9.05/2.88 inference(definition_unfolding,[status(thm)],[f942,f711])).
% 9.05/2.88 thf(f711,plain,(
% 9.05/2.88 (iffalseProp2 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $o @ (^[Y2 : $o]: ((in @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => ((~ Y2) => ((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y1 @ (^[Y4 : $i]: (((Y0 = Y4) & Y2) | ((~ Y2) & (Y3 = Y4))))))))))))))))))),
% 9.05/2.88 inference(cnf_transformation,[status(thm)],[f621])).
% 9.05/2.88 thf(f621,plain,(
% 9.05/2.88 (iffalseProp2 = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (!! @ $o @ (^[Y2 : $o]: ((in @ Y0 @ Y1) => (!! @ $i @ (^[Y3 : $i]: ((in @ Y3 @ Y1) => ((~ Y2) => ((setadjoin @ Y3 @ emptyset) = (dsetconstr @ Y1 @ (^[Y4 : $i]: (((Y0 = Y4) & Y2) | ((~ Y2) & (Y3 = Y4))))))))))))))))))),
% 9.05/2.88 inference(fool_elimination,[status(thm)],[f620])).
% 9.05/2.88 thf(f620,plain,(
% 9.05/2.88 (iffalseProp2 = ! [X0 : $o,X1 : $i,X2 : $i] : ((in @ X2 @ X1) => ! [X3 : $i] : ((in @ X3 @ X1) => (~X0 => ((setadjoin @ X3 @ emptyset) = (dsetconstr @ X1 @ (^[X4 : $i] : (((X3 = X4) & ~X0) | (X0 & (X2 = X4))))))))))),
% 9.05/2.88 inference(rectify,[status(thm)],[f222])).
% 9.05/2.88 thf(f222,axiom,(
% 9.05/2.88 (iffalseProp2 = ! [X0 : $o,X3 : $i,X1 : $i] : ((in @ X1 @ X3) => ! [X2 : $i] : ((in @ X2 @ X3) => (~X0 => ((setadjoin @ X2 @ emptyset) = (dsetconstr @ X3 @ (^[X8 : $i] : (((X2 = X8) & ~X0) | (X0 & (X1 = X8))))))))))),
% 9.05/2.88 file('/export/starexec/sandbox/tmp/tmp.K93CCo4jzK/DTF2THF_21305.p',iffalseProp2)).
% 9.05/2.88 thf(f942,plain,(
% 9.05/2.88 (iffalseProp2 = $true)),
% 9.05/2.88 inference(cnf_transformation,[status(thm)],[f692])).
% 9.05/2.88 thf(f3984,plain,(
% 9.05/2.88 ((setunion @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (((sK1 = Y0) & sK0) | ((~ sK0) & (sK3 = Y0)))))) != sK3) | spl4_1),
% 9.05/2.88 inference(avatar_component_clause,[status(thm)],[f3982])).
% 9.05/2.88 thf(f4076,plain,(
% 9.05/2.88 ~spl4_22),
% 9.05/2.88 inference(avatar_split_clause,[status(thm)],[f925,f4073])).
% 9.05/2.88 thf(f925,plain,(
% 9.05/2.88 ($true != sK0)),
% 9.05/2.88 inference(cnf_transformation,[status(thm)],[f692])).
% 9.05/2.88 thf(f4061,plain,(
% 9.05/2.88 spl4_19),
% 9.05/2.88 inference(avatar_split_clause,[status(thm)],[f928,f4058])).
% 9.05/2.88 thf(f928,plain,(
% 9.05/2.88 ($true = (in @ sK1 @ sK2))),
% 9.05/2.88 inference(cnf_transformation,[status(thm)],[f692])).
% 9.05/2.88 thf(f4055,plain,(
% 9.05/2.88 spl4_18),
% 9.05/2.88 inference(avatar_split_clause,[status(thm)],[f926,f4052])).
% 9.05/2.88 thf(f4052,definition,(
% 9.05/2.88 spl4_18 <=> ($true = (in @ sK3 @ sK2))),
% 9.05/2.88 introduced(definition,[new_symbols(naming,[spl4_18])],[avatar_definition])).
% 9.05/2.88 thf(f926,plain,(
% 9.05/2.88 ($true = (in @ sK3 @ sK2))),
% 9.05/2.88 inference(cnf_transformation,[status(thm)],[f692])).
% 9.05/2.88 thf(f3995,plain,(
% 9.05/2.88 ~spl4_3),
% 9.05/2.88 inference(avatar_split_clause,[status(thm)],[f232,f3992])).
% 9.05/2.88 thf(f3992,definition,(
% 9.05/2.88 spl4_3 <=> ($false = $true)),
% 9.05/2.88 introduced(definition,[new_symbols(naming,[spl4_3])],[avatar_definition])).
% 9.05/2.88 thf(f232,plain,(
% 9.05/2.88 ($false != $true)),
% 9.05/2.88 introduced(definition,[],[fool_axiom])).
% 9.05/2.88 thf(f3985,plain,(
% 9.05/2.88 ~spl4_1),
% 9.05/2.88 inference(avatar_split_clause,[status(thm)],[f3802,f3982])).
% 9.05/2.88 thf(f3802,plain,(
% 9.05/2.88 ((setunion @ (dsetconstr @ sK2 @ (^[Y0 : $i]: (((sK1 = Y0) & sK0) | ((~ sK0) & (sK3 = Y0)))))) != sK3)),
% 9.05/2.88 inference(beta_eta_normalization,[status(thm)],[f1165])).
% 9.05/2.88 thf(f1165,plain,(
% 9.05/2.88 (sK3 != ((^[Y0 : $i]: ((^[Y1 : $o]: ((^[Y2 : $i]: ((^[Y3 : $i]: (setunion @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (((Y2 = Y4) & Y1) | ((~ Y1) & (Y3 = Y4))))))))))))) @ sK2 @ sK0 @ sK1 @ sK3))),
% 9.05/2.88 inference(definition_unfolding,[status(thm)],[f927,f965])).
% 9.05/2.88 thf(f965,plain,(
% 9.05/2.88 (if = (^[Y0 : $i]: ((^[Y1 : $o]: ((^[Y2 : $i]: ((^[Y3 : $i]: (setunion @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (((Y2 = Y4) & Y1) | ((~ Y1) & (Y3 = Y4))))))))))))))),
% 9.05/2.88 inference(cnf_transformation,[status(thm)],[f272])).
% 9.05/2.88 thf(f272,plain,(
% 9.05/2.88 (if = (^[Y0 : $i]: ((^[Y1 : $o]: ((^[Y2 : $i]: ((^[Y3 : $i]: (setunion @ (dsetconstr @ Y0 @ (^[Y4 : $i]: (((Y2 = Y4) & Y1) | ((~ Y1) & (Y3 = Y4))))))))))))))),
% 9.05/2.88 inference(fool_elimination,[status(thm)],[f271])).
% 9.05/2.88 thf(f271,plain,(
% 9.05/2.88 ((^[X0 : $i, X1 : $o, X2 : $i, X3 : $i] : (setunion @ (dsetconstr @ X0 @ (^[X4 : $i] : (((X3 = X4) & ~X1) | (X1 & (X2 = X4))))))) = if)),
% 9.05/2.88 inference(rectify,[status(thm)],[f226])).
% 9.05/2.88 thf(f226,axiom,(
% 9.05/2.88 ((^[X3 : $i, X0 : $o, X1 : $i, X2 : $i] : (setunion @ (dsetconstr @ X3 @ (^[X8 : $i] : (((X2 = X8) & ~X0) | (X0 & (X1 = X8))))))) = if)),
% 9.05/2.88 file('/export/starexec/sandbox/tmp/tmp.K93CCo4jzK/DTF2THF_21305.p',if)).
% 9.05/2.88 thf(f927,plain,(
% 9.05/2.88 (sK3 != (if @ sK2 @ sK0 @ sK1 @ sK3))),
% 9.05/2.88 inference(cnf_transformation,[status(thm)],[f692])).
% 9.05/2.88 % SZS output end Proof for DTF2THF_21305
% 9.05/2.88 % (21533)------------------------------
% 9.05/2.88 % (21533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.05/2.88 % (21533)Termination reason: Refutation
% 9.05/2.88
% 9.05/2.88 % (21533)Memory used [KB]: 9594
% 9.05/2.88 % (21533)Time elapsed: 0.282 s
% 9.05/2.88 % (21533)Instructions burned: 579 (million)
% 9.05/2.88 % (21533)------------------------------
% 9.05/2.88 % (21533)------------------------------
% 9.05/2.88 % (21467)Success in time 1.02 s
%------------------------------------------------------------------------------