%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU778^1 : 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 : n005.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 09:37:06 AM UTC 2026
% Result : Theorem 3.21s 2.00s
% Output : Refutation 3.21s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 6
% Syntax : Number of formulae : 32 ( 23 unt; 0 typ; 0 def)
% Number of atoms : 4615 (1220 equ; 0 cnn)
% Maximal formula atoms : 297 ( 144 avg)
% Number of connectives : 2418 ( 13 ~; 0 |; 906 &; 299 @)
% ( 0 <=>;1186 =>; 0 <=; 0 <~>)
% Maximal formula depth : 305 ( 67 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 30 ( 30 >; 0 *; 0 +; 0 <<)
% Number of symbols : 444 ( 439 usr; 301 con; 0-4 aty)
% ( 10 !!; 4 ??; 0 @@+; 0 @@-)
% Number of variables : 100 ( 65 ^; 22 !; 13 ?; 100 :)
% 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_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_365,type,
sK4: $i > $i > ( $i > $o ) > $i ).
thf(func_def_366,type,
sK5: $i > $i > $i ).
thf(func_def_367,type,
sK6: ( $i > $o ) > $i > $i ).
thf(func_def_368,type,
sK7: $i > $i ).
thf(func_def_369,type,
sK8: $i > $i ).
thf(func_def_370,type,
sK9: $i > $i > $i > $i > $i ).
thf(func_def_371,type,
sK10: $i > $i > $i > $i > $i ).
thf(func_def_372,type,
sK11: $i > $i > $i > $i ).
thf(func_def_373,type,
sK12: $i > $i > $i > $i ).
thf(func_def_374,type,
sK13: $i > ( $i > $i ) > $i > $i ).
thf(func_def_375,type,
sK14: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_376,type,
sK15: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_377,type,
sK16: $i > $i ).
thf(func_def_378,type,
sK17: $i > $i > $i > $i ).
thf(func_def_379,type,
sK18: $i > $i > $i ).
thf(func_def_380,type,
sK19: $i > $i > $i ).
thf(func_def_381,type,
sK20: $i > $i > $i ).
thf(func_def_382,type,
sK21: $i > $i > $i ).
thf(func_def_383,type,
sK22: $i > $i > $i ).
thf(func_def_384,type,
sK23: ( $i > $o ) > $i ).
thf(func_def_385,type,
sK24: ( $i > $o ) > $i > $i ).
thf(func_def_386,type,
sK25: ( $i > $o ) > $i > $i ).
thf(func_def_387,type,
sK26: ( $i > $o ) > $i > $i ).
thf(func_def_388,type,
sK27: $i > $i > $i ).
thf(func_def_389,type,
sK28: $i > ( $i > $o ) > $i ).
thf(func_def_390,type,
sK29: ( $i > $o ) > $i > $i ).
thf(func_def_391,type,
sK30: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_392,type,
sK31: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_393,type,
sK32: $i > $i > $i > $i > $i ).
thf(func_def_394,type,
sK33: $i > ( $i > $o ) > $i > $i ).
thf(func_def_395,type,
sK34: $i > ( $i > $o ) > $i > $i ).
thf(func_def_396,type,
sK35: $i > $i > $i > $i ).
thf(func_def_397,type,
sK36: $i > $i > $i > $i ).
thf(func_def_398,type,
sK37: $i > ( $i > $o ) > $i > $i ).
thf(func_def_399,type,
sK38: $i > ( $i > $o ) > $i > $i ).
thf(func_def_400,type,
sK39: $i > $i ).
thf(func_def_401,type,
sK40: ( $i > $o ) > $i ).
thf(func_def_402,type,
sK41: ( $i > $o ) > $i ).
thf(func_def_403,type,
sK42: $i > $i > $i ).
thf(func_def_404,type,
sK43: ( $i > $i ) > $i > $i > $i ).
thf(func_def_405,type,
sK44: $i > $i > $i ).
thf(func_def_406,type,
sK45: ( $i > $o ) > $i > $i ).
thf(func_def_407,type,
sK46: $i > $i ).
thf(func_def_408,type,
sK47: ( $i > $i ) > $i > $i > $i ).
thf(func_def_409,type,
sK48: $i > $i > $i > $i > $i ).
thf(func_def_410,type,
sK49: $i > $i > $i > $i > $i ).
thf(func_def_411,type,
sK50: $i > $i > $i > $i > $i ).
thf(func_def_412,type,
sK51: $i > $i > $i > $i > $i ).
thf(func_def_413,type,
sK52: $i > $i > $i > $i > $i ).
thf(func_def_414,type,
sK53: $i > $i > $i > $i > $i ).
thf(func_def_415,type,
sK54: $i > $i > $i > $i > $i ).
thf(func_def_416,type,
sK55: $i > ( $i > $i ) > $i > $i ).
thf(func_def_417,type,
sK56: $i > $i > ( $i > $i ) > $i ).
thf(func_def_418,type,
sK57: $i > ( $i > $o ) > $i ).
thf(func_def_419,type,
sK58: $i > $i > $i > $i ).
thf(func_def_420,type,
sK59: $i > $i > $i > $i ).
thf(func_def_421,type,
sK60: $i > $i > $i > $i ).
thf(func_def_422,type,
sK61: $i > $i > $i > $i ).
thf(func_def_423,type,
sK62: $i > $i > $i > $i ).
thf(func_def_424,type,
sK63: $i > $i > $i > $i ).
thf(func_def_425,type,
sK64: $i > $i > $i > $i ).
thf(func_def_426,type,
sK65: $i > $i > $i > $i > $i ).
thf(func_def_427,type,
sK66: $i > $i > $i ).
thf(func_def_428,type,
sK67: ( $i > $o ) > $i ).
thf(func_def_429,type,
sK68: ( $i > $o ) > $i ).
thf(func_def_430,type,
sK69: ( $i > $o ) > $i > $i ).
thf(func_def_431,type,
sK70: $i > $i > $i > $i ).
thf(func_def_432,type,
sK71: $i > $i > $i > $i ).
thf(func_def_433,type,
sK72: $i > $i > $i > $i > $i ).
thf(func_def_434,type,
sK73: $i > $i ).
thf(func_def_435,type,
sK74: $i > $i > $o > $i ).
thf(func_def_436,type,
sK75: ( $i > $o ) > $i > $i ).
thf(func_def_437,type,
sK76: $i > $i > $i ).
thf(func_def_438,type,
sK77: $i > $i > $i ).
thf(func_def_439,type,
sK78: $i > $i > $i ).
thf(func_def_440,type,
sK79: $i > $i > $i > $i ).
thf(func_def_441,type,
sK80: $i > $i > $i > $i ).
thf(func_def_442,type,
sK81: $i > $i > $i > $i ).
thf(func_def_443,type,
sK82: $i > $i > $i > $i ).
thf(func_def_444,type,
sK83: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_445,type,
sK84: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_446,type,
sK85: $i > ( $i > $o ) > $i ).
thf(func_def_447,type,
sK86: $i > ( $i > $o ) > $i ).
thf(func_def_448,type,
sK87: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_449,type,
sK88: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_450,type,
sK89: ( $i > $o ) > $i > $i ).
thf(func_def_451,type,
sK90: $i > $i > $i ).
thf(func_def_452,type,
sK91: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_453,type,
sK92: $i > $i ).
thf(func_def_454,type,
sK93: $i > $i > $i ).
thf(func_def_455,type,
sK94: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_456,type,
sK95: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_457,type,
sK96: ( $i > $i > $o ) > $i > $i ).
thf(func_def_458,type,
sK97: ( $i > $i > $o ) > $i > $i ).
thf(func_def_459,type,
sK98: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_460,type,
sK99: ( $i > $i > $o ) > $i > $i > $i ).
thf(f5021,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f3288,f4843]) ).
thf(f4843,plain,
! [X2: $i,X1: $i > $i > $o] :
( $true
= ( breln @ X2 @ X2 @ ( dpsetconstr @ X2 @ X2 @ X1 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f4842]) ).
thf(f4842,plain,
! [X2: $i,X1: $i > $i > $o] :
( $true
= ( ^ [Y0: $i] : ( breln @ Y0 @ Y0 @ ( dpsetconstr @ Y0 @ Y0 @ X1 ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f4841]) ).
thf(f4841,plain,
! [X1: $i > $i > $o] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] : ( breln @ Y0 @ Y0 @ ( dpsetconstr @ Y0 @ Y0 @ X1 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f4840]) ).
thf(f4840,plain,
! [X1: $i > $i > $o] :
( $true
= ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( breln @ Y1 @ Y1 @ ( dpsetconstr @ Y1 @ Y1 @ Y0 ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f4839]) ).
thf(f4839,plain,
( ( !! @ ( $i > $i > $o )
@ ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( breln @ Y1 @ Y1 @ ( dpsetconstr @ Y1 @ Y1 @ Y0 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1787]) ).
thf(f1787,plain,
( $true
= ( !! @ ( $i > $i > $o )
@ ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( ^ [Y2: $i,Y3: $i] : ( breln @ Y2 @ Y2 @ Y3 )
@ Y1
@ ( dpsetconstr @ Y1 @ Y1
@ ^ [Y2: $i,Y3: $i] : ( Y0 @ Y2 @ Y3 ) ) ) ) ) ),
inference(definition_unfolding,[status(thm)],[f1483,f1365,f1083]) ).
thf(f1083,plain,
( breln1
= ( ^ [Y0: $i,Y1: $i] : ( breln @ Y0 @ Y0 @ Y1 ) ) ),
inference(cnf_transformation,[status(thm)],[f486]) ).
thf(f486,plain,
( breln1
= ( ^ [Y0: $i,Y1: $i] : ( breln @ Y0 @ Y0 @ Y1 ) ) ),
inference(fool_elimination,[status(thm)],[f485]) ).
thf(f485,plain,
( ( ^ [X0: $i,X1: $i] : ( breln @ X0 @ X0 @ X1 ) )
= breln1 ),
inference(rectify,[status(thm)],[f288]) ).
thf(f288,axiom,
( ( ^ [X3: $i,X13: $i] : ( breln @ X3 @ X3 @ X13 ) )
= breln1 ),
file('/export/starexec/sandbox/tmp/tmp.TJ114YmswG/DTF2THF_22699.p',breln1) ).
thf(f1365,plain,
setOfPairsIsBReln1 = $true,
inference(cnf_transformation,[status(thm)],[f901]) ).
thf(f901,plain,
( ( setminusILneg = $true )
& ( dpsetconstrERa = $true )
& ( cartprodpairmemER = $true )
& ( symdiffI2 = $true )
& ( kpairp = $true )
& ( notdexE = $true )
& ( doubleComplementSub1 = $true )
& ( omega0Ax = $true )
& ( apProp = $true )
& ( complementT_lem = $true )
& ( ( breln1 @ sK0 @ sK1 )
= $true )
& ( ( breln1 @ sK0 @ ( breln1compset @ sK0 @ sK1 @ sK2 ) )
!= $true )
& ( $true
= ( breln1 @ sK0 @ sK2 ) )
& ( kfstsingleton = $true )
& ( emptysetAx = $true )
& ( cartprodpairsurjEq = $true )
& ( iftrueProp2 = $true )
& ( cartprodfstpairEq = $true )
& ( binintersectTELcontra = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodpairmemEL = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan3 = $true )
& ( woz1_1 = $true )
& ( funcGraphProp1 = $true )
& ( setukpairinjR2 = $true )
& ( theeq = $true )
& ( prop2set2propI = $true )
& ( setminusLsub = $true )
& ( ex1I = $true )
& ( subsetTI = $true )
& ( setukpairinjR12 = $true )
& ( in__Cong = $true )
& ( setunionAx = $true )
& ( exuI1 = $true )
& ( brelnall2 = $true )
& ( exuE3e = $true )
& ( binintersectSubset2 = $true )
& ( emptyset__Cong = $true )
& ( binintersectSubset4 = $true )
& ( bs114d = $true )
& ( theprop = $true )
& ( setbeta = $true )
& ( notequalI2 = $true )
& ( doubleComplementEq = $true )
& ( dpsetconstrI = $true )
& ( contrasubsetT2 = $true )
& ( upairsetE = $true )
& ( cartprodmempair = $true )
& ( binunionIL = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( subPowSU = $true )
& ( demorgan2a2 = $true )
& ( setOfPairsIsBReln = $true )
& ( quantDeMorgan1 = $true )
& ( binunionEcases = $true )
& ( lam2lamEq = $true )
& ( subbreln = $true )
& ( lamProp = $true )
& ( descrp = $true )
& ( complementImpComplementIntersect = $true )
& ( sepSubset = $true )
& ( setoftrueEq = $true )
& ( iffalseProp2 = $true )
& ( secondinupair = $true )
& ( powersetsubset = $true )
& ( ap2p = $true )
& ( setunionsingleton = $true )
& ( iftrue = $true )
& ( ex1I2 = $true )
& ( subsetI1 = $true )
& ( binintersectT_lem = $true )
& ( setminusSubset1 = $true )
& ( setukpairinjR1 = $true )
& ( eqbreln = $true )
& ( setukpairIR = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( dsetconstrEL = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( woz13rule0 = $true )
& ( funcGraphProp4 = $true )
& ( nonemptyImpWitness = $true )
& ( setunionsingleton1 = $true )
& ( beta1 = $true )
& ( breln1invprop = $true )
& ( noeltsimpempty = $true )
& ( upairsetIR = $true )
& ( setminusER = $true )
& ( quantDeMorgan2 = $true )
& ( foundationAx = $true )
& ( uniqinunit = $true )
& ( doubleComplementSub2 = $true )
& ( emptysetsubset = $true )
& ( contrasubsetT3 = $true )
& ( iffalse = $true )
& ( dpsetconstrEL1 = $true )
& ( setunion__Cong = $true )
& ( demorgan2b2 = $true )
& ( powersetAx = $true )
& ( ifp = $true )
& ( complementTE1 = $true )
& ( demorgan1b = $true )
& ( ap2apEq2 = $true )
& ( subsetRefl = $true )
& ( eta2 = $true )
& ( infuncsetfunc = $true )
& ( eqimpsubset2 = $true )
& ( woz13rule3 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( powersetTE1 = $true )
& ( singletonsubset = $true )
& ( breln1invE = $true )
& ( doubleComplementI1 = $true )
& ( notinemptyset = $true )
& ( upairset2IR = $true )
& ( ubforcartprodlem2 = $true )
& ( setadjoinE = $true )
& ( exuEu = $true )
& ( upairinpowunion = $true )
& ( omega__Cong = $true )
& ( setextT = $true )
& ( binintersectSubset3 = $true )
& ( setminusSubset2 = $true )
& ( binunionIR = $true )
& ( eta1 = $true )
& ( setextsub = $true )
& ( inPowerset = $true )
& ( funcGraphProp2 = $true )
& ( demorgan2a1 = $true )
& ( ex1E1 = $true )
& ( iftrueorfalse = $true )
& ( subset2powerset = $true )
& ( cartprodpairin = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setukpairinjL = $true )
& ( ap2apEq1 = $true )
& ( emptysetE = $true )
& ( nonemptyI1 = $true )
& ( emptyinPowerset = $true )
& ( upairsetIL = $true )
& ( complementTcontraSubset = $true )
& ( upairsubunion = $true )
& ( exuI3 = $true )
& ( choice2fnsingleton = $true )
& ( setukpairinjL2 = $true )
& ( ksndpairEq = $true )
& ( iftrueProp1 = $true )
& ( emptyinunitempty = $true )
& ( notdallE = $true )
& ( ksndsingleton = $true )
& ( vacuousDall = $true )
& ( omegaIndAx = $true )
& ( setadjoinSub = $true )
& ( singletonsswitch = $true )
& ( kfstpairEq = $true )
& ( funcext2 = $true )
& ( inIntersectImpInUnion = $true )
& ( ubforcartprodlem1 = $true )
& ( singletonsuniq = $true )
& ( dpsetconstrER = $true )
& ( setukpairIL = $true )
& ( singletonprop = $true )
& ( prop2setI = $true )
& ( setunionE = $true )
& ( contrasubsetT = $true )
& ( cartprodsndpairEq = $true )
& ( eqbreln1 = $true )
& ( lam2p = $true )
& ( contraSubsetComplement = $true )
& ( powersetT_lem = $true )
& ( demorgan2b = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( dpsetconstrSub = $true )
& ( powerset__Cong = $true )
& ( contrasubsetT1 = $true )
& ( binunionE = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinIR = $true )
& ( binunionLsub = $true )
& ( nonemptyI = $true )
& ( symdiffIneg1 = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinOr = $true )
& ( subbreln1 = $true )
& ( ex1E2 = $true )
& ( cartprodsndin = $true )
& ( lamp = $true )
& ( subsetI2 = $true )
& ( powersetE1 = $true )
& ( setadjoinIL = $true )
& ( disjointsetsI1 = $true )
& ( binintersectRsub = $true )
& ( cartprodfstin = $true )
& ( setminusEL = $true )
& ( subsetTrans = $true )
& ( setunionI = $true )
& ( eqinunit = $true )
& ( app = $true )
& ( powersetI = $true )
& ( subsetE = $true )
& ( complementTnotintersectT = $true )
& ( symdiffI1 = $true )
& ( funcImageSingleton = $true )
& ( funcext = $true )
& ( upairset2E = $true )
& ( funcGraphProp3 = $true )
& ( woz13rule4 = $true )
& ( eqimpsubset1 = $true )
& ( emptyI = $true )
& ( demorgan2 = $true )
& ( funcinfuncset = $true )
& ( setextAx = $true )
& ( powersetE = $true )
& ( woz1_2 = $true )
& ( brelnall1 = $true )
& ( binintersectSubset5 = $true )
& ( powersetTI1 = $true )
& ( singletoninpowerset = $true )
& ( woz1_3 = $true )
& ( breln1all1 = $true )
& ( replAx = $true )
& ( beta2 = $true )
& ( descr__Cong = $true )
& ( setminusT_lem = $true )
& ( inIntersectImpInUnion2 = $true )
& ( upairequniteq = $true )
& ( kpairsurjEq = $true )
& ( binunionTIRcontra = $true )
& ( funcextLem = $true )
& ( kpairiskpair = $true )
& ( subsetE2 = $true )
& ( subsetemptysetimpeq = $true )
& ( symdiffIneg2 = $true )
& ( complementTI1 = $true )
& ( exuE1 = $true )
& ( setunionsingleton2 = $true )
& ( exuE2 = $true )
& ( binintersectEL = $true )
& ( exu__Cong = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( binintersectER = $true )
& ( breln1SetBreln1 = $true )
& ( demorgan1a = $true )
& ( notinsingleton = $true )
& ( emptyInPowerset = $true )
& ( symdiffE = $true )
& ( binintersectI = $true )
& ( woz13rule2 = $true )
& ( setadjoin__Cong = $true )
& ( notsubsetI = $true )
& ( dsetconstr__Cong = $true )
& ( ifSingleton = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( dsetconstrER = $true )
& ( sepInPowerset = $true )
& ( binunionT_lem = $true )
& ( exuE3u = $true )
& ( woz13rule1 = $true )
& ( singletoninpowunion = $true )
& ( setminusELneg = $true )
& ( woz1_5 = $true )
& ( cartprodmempaircEq = $true )
& ( omegaSAx = $true )
& ( setminusERneg = $true )
& ( nonemptyE1 = $true )
& ( binintersectTERcontra = $true )
& ( setunionE2 = $true )
& ( doubleComplementE1 = $true )
& ( woz1_4 = $true )
& ( iffalseProp1 = $true )
& ( demorgan1 = $true )
& ( demorgan2a = $true )
& ( cartprodmempair1 = $true )
& ( setukpairinjR11 = $true )
& ( powersetI1 = $true )
& ( quantDeMorgan4 = $true )
& ( binunionRsub = $true )
& ( binintersectLsub = $true )
& ( setminusI = $true )
& ( setminusIRneg = $true )
& ( setadjoinAx = $true )
& ( wellorderingAx = $true )
& ( breln1invI = $true )
& ( complementUnionInPowersetComplement = $true )
& ( notequalI1 = $true )
& ( exuI2 = $true )
& ( binunionTILcontra = $true )
& ( emptyE1 = $true )
& ( breln1all2 = $true )
& ( dsetconstrI = $true )
& ( setadjoinSub2 = $true )
& ( emptysetimpfalse = $true )
& ( setext = $true )
& ( prop2setE = $true )
& ( inCongP = $true )
& ( setukpairinjR = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f898,f900,f899]) ).
thf(f899,plain,
( ? [X0: $i,X1: $i] :
( ( ( breln1 @ X0 @ X1 )
= $true )
& ? [X2: $i] :
( ( $true
!= ( breln1 @ X0 @ ( breln1compset @ X0 @ X1 @ X2 ) ) )
& ( ( breln1 @ X0 @ X2 )
= $true ) ) )
=> ( ( ( breln1 @ sK0 @ sK1 )
= $true )
& ? [X2: $i] :
( ( $true
!= ( breln1 @ sK0 @ ( breln1compset @ sK0 @ sK1 @ X2 ) ) )
& ( ( breln1 @ sK0 @ X2 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f900,plain,
( ? [X2: $i] :
( ( $true
!= ( breln1 @ sK0 @ ( breln1compset @ sK0 @ sK1 @ X2 ) ) )
& ( ( breln1 @ sK0 @ X2 )
= $true ) )
=> ( ( ( breln1 @ sK0 @ ( breln1compset @ sK0 @ sK1 @ sK2 ) )
!= $true )
& ( $true
= ( breln1 @ sK0 @ sK2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f898,plain,
( ( setminusILneg = $true )
& ( dpsetconstrERa = $true )
& ( cartprodpairmemER = $true )
& ( symdiffI2 = $true )
& ( kpairp = $true )
& ( notdexE = $true )
& ( doubleComplementSub1 = $true )
& ( omega0Ax = $true )
& ( apProp = $true )
& ( complementT_lem = $true )
& ? [X0: $i,X1: $i] :
( ( ( breln1 @ X0 @ X1 )
= $true )
& ? [X2: $i] :
( ( $true
!= ( breln1 @ X0 @ ( breln1compset @ X0 @ X1 @ X2 ) ) )
& ( ( breln1 @ X0 @ X2 )
= $true ) ) )
& ( kfstsingleton = $true )
& ( emptysetAx = $true )
& ( cartprodpairsurjEq = $true )
& ( iftrueProp2 = $true )
& ( cartprodfstpairEq = $true )
& ( binintersectTELcontra = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodpairmemEL = $true )
& ( setukpairinjL1 = $true )
& ( quantDeMorgan3 = $true )
& ( woz1_1 = $true )
& ( funcGraphProp1 = $true )
& ( setukpairinjR2 = $true )
& ( theeq = $true )
& ( prop2set2propI = $true )
& ( setminusLsub = $true )
& ( ex1I = $true )
& ( subsetTI = $true )
& ( setukpairinjR12 = $true )
& ( in__Cong = $true )
& ( setunionAx = $true )
& ( exuI1 = $true )
& ( brelnall2 = $true )
& ( exuE3e = $true )
& ( binintersectSubset2 = $true )
& ( emptyset__Cong = $true )
& ( binintersectSubset4 = $true )
& ( bs114d = $true )
& ( theprop = $true )
& ( setbeta = $true )
& ( notequalI2 = $true )
& ( doubleComplementEq = $true )
& ( dpsetconstrI = $true )
& ( contrasubsetT2 = $true )
& ( upairsetE = $true )
& ( cartprodmempair = $true )
& ( binunionIL = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( subPowSU = $true )
& ( demorgan2a2 = $true )
& ( setOfPairsIsBReln = $true )
& ( quantDeMorgan1 = $true )
& ( binunionEcases = $true )
& ( lam2lamEq = $true )
& ( subbreln = $true )
& ( lamProp = $true )
& ( descrp = $true )
& ( complementImpComplementIntersect = $true )
& ( sepSubset = $true )
& ( setoftrueEq = $true )
& ( iffalseProp2 = $true )
& ( secondinupair = $true )
& ( powersetsubset = $true )
& ( ap2p = $true )
& ( setunionsingleton = $true )
& ( iftrue = $true )
& ( ex1I2 = $true )
& ( subsetI1 = $true )
& ( binintersectT_lem = $true )
& ( setminusSubset1 = $true )
& ( setukpairinjR1 = $true )
& ( eqbreln = $true )
& ( setukpairIR = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( dsetconstrEL = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( woz13rule0 = $true )
& ( funcGraphProp4 = $true )
& ( nonemptyImpWitness = $true )
& ( setunionsingleton1 = $true )
& ( beta1 = $true )
& ( breln1invprop = $true )
& ( noeltsimpempty = $true )
& ( upairsetIR = $true )
& ( setminusER = $true )
& ( quantDeMorgan2 = $true )
& ( foundationAx = $true )
& ( uniqinunit = $true )
& ( doubleComplementSub2 = $true )
& ( emptysetsubset = $true )
& ( contrasubsetT3 = $true )
& ( iffalse = $true )
& ( dpsetconstrEL1 = $true )
& ( setunion__Cong = $true )
& ( demorgan2b2 = $true )
& ( powersetAx = $true )
& ( ifp = $true )
& ( complementTE1 = $true )
& ( demorgan1b = $true )
& ( ap2apEq2 = $true )
& ( subsetRefl = $true )
& ( eta2 = $true )
& ( infuncsetfunc = $true )
& ( eqimpsubset2 = $true )
& ( woz13rule3 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( powersetTE1 = $true )
& ( singletonsubset = $true )
& ( breln1invE = $true )
& ( doubleComplementI1 = $true )
& ( notinemptyset = $true )
& ( upairset2IR = $true )
& ( ubforcartprodlem2 = $true )
& ( setadjoinE = $true )
& ( exuEu = $true )
& ( upairinpowunion = $true )
& ( omega__Cong = $true )
& ( setextT = $true )
& ( binintersectSubset3 = $true )
& ( setminusSubset2 = $true )
& ( binunionIR = $true )
& ( eta1 = $true )
& ( setextsub = $true )
& ( inPowerset = $true )
& ( funcGraphProp2 = $true )
& ( demorgan2a1 = $true )
& ( ex1E1 = $true )
& ( iftrueorfalse = $true )
& ( subset2powerset = $true )
& ( cartprodpairin = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setukpairinjL = $true )
& ( ap2apEq1 = $true )
& ( emptysetE = $true )
& ( nonemptyI1 = $true )
& ( emptyinPowerset = $true )
& ( upairsetIL = $true )
& ( complementTcontraSubset = $true )
& ( upairsubunion = $true )
& ( exuI3 = $true )
& ( choice2fnsingleton = $true )
& ( setukpairinjL2 = $true )
& ( ksndpairEq = $true )
& ( iftrueProp1 = $true )
& ( emptyinunitempty = $true )
& ( notdallE = $true )
& ( ksndsingleton = $true )
& ( vacuousDall = $true )
& ( omegaIndAx = $true )
& ( setadjoinSub = $true )
& ( singletonsswitch = $true )
& ( kfstpairEq = $true )
& ( funcext2 = $true )
& ( inIntersectImpInUnion = $true )
& ( ubforcartprodlem1 = $true )
& ( singletonsuniq = $true )
& ( dpsetconstrER = $true )
& ( setukpairIL = $true )
& ( singletonprop = $true )
& ( prop2setI = $true )
& ( setunionE = $true )
& ( contrasubsetT = $true )
& ( cartprodsndpairEq = $true )
& ( eqbreln1 = $true )
& ( lam2p = $true )
& ( contraSubsetComplement = $true )
& ( powersetT_lem = $true )
& ( demorgan2b = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( dpsetconstrSub = $true )
& ( powerset__Cong = $true )
& ( contrasubsetT1 = $true )
& ( binunionE = $true )
& ( binintersectSubset1 = $true )
& ( setadjoinIR = $true )
& ( binunionLsub = $true )
& ( nonemptyI = $true )
& ( symdiffIneg1 = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinOr = $true )
& ( subbreln1 = $true )
& ( ex1E2 = $true )
& ( cartprodsndin = $true )
& ( lamp = $true )
& ( subsetI2 = $true )
& ( powersetE1 = $true )
& ( setadjoinIL = $true )
& ( disjointsetsI1 = $true )
& ( binintersectRsub = $true )
& ( cartprodfstin = $true )
& ( setminusEL = $true )
& ( subsetTrans = $true )
& ( setunionI = $true )
& ( eqinunit = $true )
& ( app = $true )
& ( powersetI = $true )
& ( subsetE = $true )
& ( complementTnotintersectT = $true )
& ( symdiffI1 = $true )
& ( funcImageSingleton = $true )
& ( funcext = $true )
& ( upairset2E = $true )
& ( funcGraphProp3 = $true )
& ( woz13rule4 = $true )
& ( eqimpsubset1 = $true )
& ( emptyI = $true )
& ( demorgan2 = $true )
& ( funcinfuncset = $true )
& ( setextAx = $true )
& ( powersetE = $true )
& ( woz1_2 = $true )
& ( brelnall1 = $true )
& ( binintersectSubset5 = $true )
& ( powersetTI1 = $true )
& ( singletoninpowerset = $true )
& ( woz1_3 = $true )
& ( breln1all1 = $true )
& ( replAx = $true )
& ( beta2 = $true )
& ( descr__Cong = $true )
& ( setminusT_lem = $true )
& ( inIntersectImpInUnion2 = $true )
& ( upairequniteq = $true )
& ( kpairsurjEq = $true )
& ( binunionTIRcontra = $true )
& ( funcextLem = $true )
& ( kpairiskpair = $true )
& ( subsetE2 = $true )
& ( subsetemptysetimpeq = $true )
& ( symdiffIneg2 = $true )
& ( complementTI1 = $true )
& ( exuE1 = $true )
& ( setunionsingleton2 = $true )
& ( exuE2 = $true )
& ( binintersectEL = $true )
& ( exu__Cong = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( binintersectER = $true )
& ( breln1SetBreln1 = $true )
& ( demorgan1a = $true )
& ( notinsingleton = $true )
& ( emptyInPowerset = $true )
& ( symdiffE = $true )
& ( binintersectI = $true )
& ( woz13rule2 = $true )
& ( setadjoin__Cong = $true )
& ( notsubsetI = $true )
& ( dsetconstr__Cong = $true )
& ( ifSingleton = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( dsetconstrER = $true )
& ( sepInPowerset = $true )
& ( binunionT_lem = $true )
& ( exuE3u = $true )
& ( woz13rule1 = $true )
& ( singletoninpowunion = $true )
& ( setminusELneg = $true )
& ( woz1_5 = $true )
& ( cartprodmempaircEq = $true )
& ( omegaSAx = $true )
& ( setminusERneg = $true )
& ( nonemptyE1 = $true )
& ( binintersectTERcontra = $true )
& ( setunionE2 = $true )
& ( doubleComplementE1 = $true )
& ( woz1_4 = $true )
& ( iffalseProp1 = $true )
& ( demorgan1 = $true )
& ( demorgan2a = $true )
& ( cartprodmempair1 = $true )
& ( setukpairinjR11 = $true )
& ( powersetI1 = $true )
& ( quantDeMorgan4 = $true )
& ( binunionRsub = $true )
& ( binintersectLsub = $true )
& ( setminusI = $true )
& ( setminusIRneg = $true )
& ( setadjoinAx = $true )
& ( wellorderingAx = $true )
& ( breln1invI = $true )
& ( complementUnionInPowersetComplement = $true )
& ( notequalI1 = $true )
& ( exuI2 = $true )
& ( binunionTILcontra = $true )
& ( emptyE1 = $true )
& ( breln1all2 = $true )
& ( dsetconstrI = $true )
& ( setadjoinSub2 = $true )
& ( emptysetimpfalse = $true )
& ( setext = $true )
& ( prop2setE = $true )
& ( inCongP = $true )
& ( setukpairinjR = $true ) ),
inference(flattening,[status(thm)],[f897]) ).
thf(f897,plain,
( ? [X0: $i,X1: $i] :
( ( ( breln1 @ X0 @ X1 )
= $true )
& ? [X2: $i] :
( ( $true
!= ( breln1 @ X0 @ ( breln1compset @ X0 @ X1 @ X2 ) ) )
& ( ( breln1 @ X0 @ X2 )
= $true ) ) )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f480]) ).
thf(f480,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ! [X0: $i,X1: $i] :
( ( ( breln1 @ X0 @ X1 )
= $true )
=> ! [X2: $i] :
( ( ( breln1 @ X0 @ X2 )
= $true )
=> ( $true
= ( breln1 @ X0 @ ( breln1compset @ X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f479]) ).
thf(f479,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ! [X0: $i,X1: $i] :
( ( breln1 @ X0 @ X1 )
=> ! [X2: $i] :
( ( breln1 @ X0 @ X2 )
=> ( breln1 @ X0 @ ( breln1compset @ X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f301]) ).
thf(f301,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ! [X3: $i,X13: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X15: $i] :
( ( breln1 @ X3 @ X15 )
=> ( breln1 @ X3 @ ( breln1compset @ X3 @ X13 @ X15 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f300]) ).
thf(f300,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ! [X3: $i,X13: $i] :
( ( breln1 @ X3 @ X13 )
=> ! [X15: $i] :
( ( breln1 @ X3 @ X15 )
=> ( breln1 @ X3 @ ( breln1compset @ X3 @ X13 @ X15 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.TJ114YmswG/DTF2THF_22699.p',breln1compprop) ).
thf(f1483,plain,
( setOfPairsIsBReln1
= ( !! @ ( $i > $i > $o )
@ ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( breln1 @ Y1
@ ( dpsetconstr @ Y1 @ Y1
@ ^ [Y2: $i,Y3: $i] : ( Y0 @ Y2 @ Y3 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f756]) ).
thf(f756,plain,
( setOfPairsIsBReln1
= ( !! @ ( $i > $i > $o )
@ ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] :
( breln1 @ Y1
@ ( dpsetconstr @ Y1 @ Y1
@ ^ [Y2: $i,Y3: $i] : ( Y0 @ Y2 @ Y3 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f755]) ).
thf(f755,plain,
! [X0: $i,X1: $i > $i > $o] :
( ( breln1 @ X0
@ ( dpsetconstr @ X0 @ X0
@ ^ [X2: $i,X3: $i] : ( X1 @ X2 @ X3 ) ) )
= setOfPairsIsBReln1 ),
inference(rectify,[status(thm)],[f292]) ).
thf(f292,axiom,
! [X3: $i,X0: $i > $i > $o] :
( ( breln1 @ X3
@ ( dpsetconstr @ X3 @ X3
@ ^ [X1: $i,X2: $i] : ( X0 @ X1 @ X2 ) ) )
= setOfPairsIsBReln1 ),
file('/export/starexec/sandbox/tmp/tmp.TJ114YmswG/DTF2THF_22699.p',setOfPairsIsBReln1) ).
thf(f3288,plain,
( ( breln @ sK0 @ sK0
@ ( dpsetconstr @ sK0 @ sK0
@ ^ [Y0: $i,Y1: $i] :
( ?? @ $i
@ ^ [Y2: $i] :
( ( in @ Y2 @ sK0 )
& ( in @ ( kpair @ Y0 @ Y2 ) @ sK1 )
& ( in @ ( kpair @ Y2 @ Y1 ) @ sK2 ) ) ) ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f1513]) ).
thf(f1513,plain,
( $true
!= ( ^ [Y0: $i,Y1: $i] : ( breln @ Y0 @ Y0 @ Y1 )
@ sK0
@ ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( dpsetconstr @ Y0 @ Y0
@ ^ [Y3: $i,Y4: $i] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y0 )
& ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 )
& ( in @ ( kpair @ Y5 @ Y4 ) @ Y2 ) ) ) )
@ sK0
@ sK1
@ sK2 ) ) ),
inference(definition_unfolding,[status(thm)],[f1462,f1083,f1042]) ).
thf(f1042,plain,
( breln1compset
= ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( dpsetconstr @ Y0 @ Y0
@ ^ [Y3: $i,Y4: $i] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y0 )
& ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 )
& ( in @ ( kpair @ Y5 @ Y4 ) @ Y2 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f864]) ).
thf(f864,plain,
( breln1compset
= ( ^ [Y0: $i,Y1: $i,Y2: $i] :
( dpsetconstr @ Y0 @ Y0
@ ^ [Y3: $i,Y4: $i] :
( ?? @ $i
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y0 )
& ( in @ ( kpair @ Y3 @ Y5 ) @ Y1 )
& ( in @ ( kpair @ Y5 @ Y4 ) @ Y2 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f863]) ).
thf(f863,plain,
( ( ^ [X0: $i,X1: $i,X2: $i] :
( dpsetconstr @ X0 @ X0
@ ^ [X3: $i,X4: $i] :
? [X5: $i] :
( ( in @ ( kpair @ X5 @ X4 ) @ X2 )
& ( in @ ( kpair @ X3 @ X5 ) @ X1 )
& ( in @ X5 @ X0 ) ) ) )
= breln1compset ),
inference(rectify,[status(thm)],[f299]) ).
thf(f299,axiom,
( ( ^ [X3: $i,X13: $i,X15: $i] :
( dpsetconstr @ X3 @ X3
@ ^ [X1: $i,X2: $i] :
? [X8: $i] :
( ( in @ ( kpair @ X8 @ X2 ) @ X15 )
& ( in @ ( kpair @ X1 @ X8 ) @ X13 )
& ( in @ X8 @ X3 ) ) ) )
= breln1compset ),
file('/export/starexec/sandbox/tmp/tmp.TJ114YmswG/DTF2THF_22699.p',breln1compset) ).
thf(f1462,plain,
( ( breln1 @ sK0 @ ( breln1compset @ sK0 @ sK1 @ sK2 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f901]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU778^1 : 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
% 0.14/0.32 % Computer : n005.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 12:33:29 EST 2026
% 0.14/0.33 % CPUTime :
% 0.14/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.44 ---- Original DTF file ---
% 0.17/0.44 thf(spec,logic,$$dhol).
% 0.17/0.44 %------------------------------------------------------------------------------
% 0.17/0.44 % File : SEU778^1 : TPTP v9.2.1. Released v3.7.0.
% 0.17/0.44 % Domain : Set Theory
% 0.17/0.44 % Problem : Binary Relations on a Set
% 0.17/0.44 % Version : Especial.
% 0.17/0.44 % English : (! A:i.! R:i.breln1 A R -> (! S:i.breln1 A S ->
% 0.17/0.44 % breln1 A (breln1compset A R S)))
% 0.17/0.44
% 0.17/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.17/0.44 % Source : [Bro08]
% 0.17/0.44 % Names : ZFC280g [Bro08]
% 0.17/0.44
% 0.17/0.44 % Status : Theorem
% 0.17/0.44 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.54 v8.1.0, 0.45 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.17/0.44 % Syntax : Number of formulae : 642 ( 299 unt; 342 typ; 299 def)
% 0.17/0.44 % Number of atoms : 1809 ( 429 equ; 0 cnn)
% 0.17/0.44 % Maximal formula atoms : 297 ( 6 avg)
% 0.17/0.44 % Number of connectives : 3912 ( 73 ~; 12 |; 48 &;2838 @)
% 0.17/0.44 % ( 14 <=>; 927 =>; 0 <=; 0 <~>)
% 0.17/0.44 % Maximal formula depth : 304 ( 2 avg)
% 0.17/0.44 % Number of types : 2 ( 0 usr)
% 0.17/0.44 % Number of type conns : 157 ( 157 >; 0 *; 0 +; 0 <<)
% 0.17/0.44 % Number of symbols : 345 ( 342 usr; 298 con; 0-4 aty)
% 0.17/0.44 % Number of variables : 1040 ( 81 ^; 922 !; 37 ?;1040 :)
% 0.17/0.44 % SPC : TH0_THM_EQU_NAR
% 0.17/0.44
% 0.17/0.44 % Comments : http://mathgate.info/detsetitem.php?id=344
% 0.17/0.44 %------------------------------------------------------------------------------
% 0.17/0.44 thf(in_type,type,
% 0.17/0.44 in: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(exu_type,type,
% 0.17/0.44 exu: ( $i > $o ) > $o ).
% 0.17/0.44
% 0.17/0.44 thf(exu,definition,
% 0.17/0.44 ( exu
% 0.17/0.44 = ( ^ [Xphi: $i > $o] :
% 0.17/0.44 ? [Xx: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 & ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setextAx_type,type,
% 0.17/0.44 setextAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setextAx,definition,
% 0.17/0.44 ( setextAx
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 <=> ( in @ Xx @ B ) )
% 0.17/0.44 => ( A = B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyset_type,type,
% 0.17/0.44 emptyset: $i ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetAx_type,type,
% 0.17/0.44 emptysetAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetAx,definition,
% 0.17/0.44 ( emptysetAx
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoin_type,type,
% 0.17/0.44 setadjoin: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinAx_type,type,
% 0.17/0.44 setadjoinAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinAx,definition,
% 0.17/0.44 ( setadjoinAx
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.44 <=> ( ( Xy = Xx )
% 0.17/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powerset_type,type,
% 0.17/0.44 powerset: $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(powersetAx_type,type,
% 0.17/0.44 powersetAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetAx,definition,
% 0.17/0.44 ( powersetAx
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.44 <=> ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunion_type,type,
% 0.17/0.44 setunion: $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(setunionAx_type,type,
% 0.17/0.44 setunionAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionAx,definition,
% 0.17/0.44 ( setunionAx
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.44 <=> ? [B: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(omega_type,type,
% 0.17/0.44 omega: $i ).
% 0.17/0.44
% 0.17/0.44 thf(omega0Ax_type,type,
% 0.17/0.44 omega0Ax: $o ).
% 0.17/0.44
% 0.17/0.44 thf(omega0Ax,definition,
% 0.17/0.44 ( omega0Ax
% 0.17/0.44 = ( in @ emptyset @ omega ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(omegaSAx_type,type,
% 0.17/0.44 omegaSAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(omegaSAx,definition,
% 0.17/0.44 ( omegaSAx
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ omega )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(omegaIndAx_type,type,
% 0.17/0.44 omegaIndAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(omegaIndAx,definition,
% 0.17/0.44 ( omegaIndAx
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( ( in @ emptyset @ A )
% 0.17/0.44 & ! [Xx: $i] :
% 0.17/0.44 ( ( ( in @ Xx @ omega )
% 0.17/0.44 & ( in @ Xx @ A ) )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ omega )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(replAx_type,type,
% 0.17/0.44 replAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(replAx,definition,
% 0.17/0.44 ( replAx
% 0.17/0.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.17/0.44 => ? [B: $i] :
% 0.17/0.44 ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 <=> ? [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(foundationAx_type,type,
% 0.17/0.44 foundationAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(foundationAx,definition,
% 0.17/0.44 ( foundationAx
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.44 => ? [B: $i] :
% 0.17/0.44 ( ( in @ B @ A )
% 0.17/0.44 & ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(wellorderingAx_type,type,
% 0.17/0.44 wellorderingAx: $o ).
% 0.17/0.44
% 0.17/0.44 thf(wellorderingAx,definition,
% 0.17/0.44 ( wellorderingAx
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ? [B: $i] :
% 0.17/0.44 ( ! [C: $i] :
% 0.17/0.44 ( ( in @ C @ B )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ C )
% 0.17/0.44 => ( in @ Xx @ A ) ) )
% 0.17/0.44 & ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( ( in @ Xx @ A )
% 0.17/0.44 & ( in @ Xy @ A ) )
% 0.17/0.44 => ( ! [C: $i] :
% 0.17/0.44 ( ( in @ C @ B )
% 0.17/0.44 => ( ( in @ Xx @ C )
% 0.17/0.44 <=> ( in @ Xy @ C ) ) )
% 0.17/0.44 => ( Xx = Xy ) ) )
% 0.17/0.44 & ! [C: $i,D: $i] :
% 0.17/0.44 ( ( ( in @ C @ B )
% 0.17/0.44 & ( in @ D @ B ) )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ C )
% 0.17/0.44 => ( in @ Xx @ D ) )
% 0.17/0.44 | ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ D )
% 0.17/0.44 => ( in @ Xx @ C ) ) ) )
% 0.17/0.44 & ! [C: $i] :
% 0.17/0.44 ( ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ C )
% 0.17/0.44 => ( in @ Xx @ A ) )
% 0.17/0.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.17/0.44 => ? [D: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ D @ B )
% 0.17/0.44 & ( in @ Xx @ C )
% 0.17/0.44 & ~ ? [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ D )
% 0.17/0.44 & ( in @ Xy @ C ) )
% 0.17/0.44 & ! [E: $i] :
% 0.17/0.44 ( ( in @ E @ B )
% 0.17/0.44 => ( ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ E )
% 0.17/0.44 => ( in @ Xy @ D ) )
% 0.17/0.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(descr_type,type,
% 0.17/0.44 descr: ( $i > $o ) > $i ).
% 0.17/0.44
% 0.17/0.44 thf(descrp_type,type,
% 0.17/0.44 descrp: $o ).
% 0.17/0.44
% 0.17/0.44 thf(descrp,definition,
% 0.17/0.44 ( descrp
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ( Xphi
% 0.17/0.44 @ ( descr
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstr_type,type,
% 0.17/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrI_type,type,
% 0.17/0.44 dsetconstrI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrI,definition,
% 0.17/0.44 ( dsetconstrI
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrEL_type,type,
% 0.17/0.44 dsetconstrEL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrEL,definition,
% 0.17/0.44 ( dsetconstrEL
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrER_type,type,
% 0.17/0.44 dsetconstrER: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstrER,definition,
% 0.17/0.44 ( dsetconstrER
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE1_type,type,
% 0.17/0.44 exuE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE1,definition,
% 0.17/0.44 ( exuE1
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 & ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(prop2set_type,type,
% 0.17/0.44 prop2set: $o > $i ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setE_type,type,
% 0.17/0.44 prop2setE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setE,definition,
% 0.17/0.44 ( prop2setE
% 0.17/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.17/0.44 => Xphi ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetE_type,type,
% 0.17/0.44 emptysetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetE,definition,
% 0.17/0.44 ( emptysetE
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ emptyset )
% 0.17/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetimpfalse_type,type,
% 0.17/0.44 emptysetimpfalse: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetimpfalse,definition,
% 0.17/0.44 ( emptysetimpfalse
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ emptyset )
% 0.17/0.44 => $false ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notinemptyset_type,type,
% 0.17/0.44 notinemptyset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notinemptyset,definition,
% 0.17/0.44 ( notinemptyset
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3e_type,type,
% 0.17/0.44 exuE3e: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3e,definition,
% 0.17/0.44 ( exuE3e
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setext_type,type,
% 0.17/0.44 setext: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setext,definition,
% 0.17/0.44 ( setext
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) )
% 0.17/0.44 => ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) )
% 0.17/0.44 => ( A = B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyI_type,type,
% 0.17/0.44 emptyI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyI,definition,
% 0.17/0.44 ( emptyI
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ A )
% 0.17/0.44 => ( A = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(noeltsimpempty_type,type,
% 0.17/0.44 noeltsimpempty: $o ).
% 0.17/0.44
% 0.17/0.44 thf(noeltsimpempty,definition,
% 0.17/0.44 ( noeltsimpempty
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ~ ( in @ Xx @ A )
% 0.17/0.44 => ( A = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setbeta_type,type,
% 0.17/0.44 setbeta: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setbeta,definition,
% 0.17/0.44 ( setbeta
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonempty_type,type,
% 0.17/0.44 nonempty: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonempty,definition,
% 0.17/0.44 ( nonempty
% 0.17/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyE1_type,type,
% 0.17/0.44 nonemptyE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyE1,definition,
% 0.17/0.44 ( nonemptyE1
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( nonempty @ A )
% 0.17/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI_type,type,
% 0.17/0.44 nonemptyI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI,definition,
% 0.17/0.44 ( nonemptyI
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( nonempty
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI1_type,type,
% 0.17/0.44 nonemptyI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyI1,definition,
% 0.17/0.44 ( nonemptyI1
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIL_type,type,
% 0.17/0.44 setadjoinIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIL,definition,
% 0.17/0.44 ( setadjoinIL
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinunitempty_type,type,
% 0.17/0.44 emptyinunitempty: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinunitempty,definition,
% 0.17/0.44 ( emptyinunitempty
% 0.17/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIR_type,type,
% 0.17/0.44 setadjoinIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinIR,definition,
% 0.17/0.44 ( setadjoinIR
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinE_type,type,
% 0.17/0.44 setadjoinE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinE,definition,
% 0.17/0.44 ( setadjoinE
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.44 => ! [Xphi: $o] :
% 0.17/0.44 ( ( ( Xy = Xx )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => ( ( ( in @ Xy @ A )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinOr_type,type,
% 0.17/0.44 setadjoinOr: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinOr,definition,
% 0.17/0.44 ( setadjoinOr
% 0.17/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.44 => ( ( Xy = Xx )
% 0.17/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setoftrueEq_type,type,
% 0.17/0.44 setoftrueEq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setoftrueEq,definition,
% 0.17/0.44 ( setoftrueEq
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : $true )
% 0.17/0.44 = A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI_type,type,
% 0.17/0.44 powersetI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI,definition,
% 0.17/0.44 ( powersetI
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) )
% 0.17/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinPowerset_type,type,
% 0.17/0.44 emptyinPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyinPowerset,definition,
% 0.17/0.44 ( emptyinPowerset
% 0.17/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyInPowerset_type,type,
% 0.17/0.44 emptyInPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyInPowerset,definition,
% 0.17/0.44 ( emptyInPowerset
% 0.17/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE_type,type,
% 0.17/0.44 powersetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE,definition,
% 0.17/0.44 ( powersetE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionI_type,type,
% 0.17/0.44 setunionI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionI,definition,
% 0.17/0.44 ( setunionI
% 0.17/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( ( in @ B @ A )
% 0.17/0.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE_type,type,
% 0.17/0.44 setunionE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE,definition,
% 0.17/0.44 ( setunionE
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.44 => ! [Xphi: $o] :
% 0.17/0.44 ( ! [B: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( ( in @ B @ A )
% 0.17/0.44 => Xphi ) )
% 0.17/0.44 => Xphi ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subPowSU_type,type,
% 0.17/0.44 subPowSU: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subPowSU,definition,
% 0.17/0.44 ( subPowSU
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE2_type,type,
% 0.17/0.44 exuE2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE2,definition,
% 0.17/0.44 ( exuE2
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyImpWitness_type,type,
% 0.17/0.44 nonemptyImpWitness: $o ).
% 0.17/0.44
% 0.17/0.44 thf(nonemptyImpWitness,definition,
% 0.17/0.44 ( nonemptyImpWitness
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( nonempty @ A )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & $true ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(uniqinunit_type,type,
% 0.17/0.44 uniqinunit: $o ).
% 0.17/0.44
% 0.17/0.44 thf(uniqinunit,definition,
% 0.17/0.44 ( uniqinunit
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notinsingleton_type,type,
% 0.17/0.44 notinsingleton: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notinsingleton,definition,
% 0.17/0.44 ( notinsingleton
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx != Xy )
% 0.17/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqinunit_type,type,
% 0.17/0.44 eqinunit: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqinunit,definition,
% 0.17/0.44 ( eqinunit
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsswitch_type,type,
% 0.17/0.44 singletonsswitch: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsswitch,definition,
% 0.17/0.44 ( singletonsswitch
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetE_type,type,
% 0.17/0.44 upairsetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetE,definition,
% 0.17/0.44 ( upairsetE
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.44 => ( ( Xz = Xx )
% 0.17/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIL_type,type,
% 0.17/0.44 upairsetIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIL,definition,
% 0.17/0.44 ( upairsetIL
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIR_type,type,
% 0.17/0.44 upairsetIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsetIR,definition,
% 0.17/0.44 ( upairsetIR
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyE1_type,type,
% 0.17/0.44 emptyE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyE1,definition,
% 0.17/0.44 ( emptyE1
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ( ( ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 = emptyset )
% 0.17/0.44 => $false ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(vacuousDall_type,type,
% 0.17/0.44 vacuousDall: $o ).
% 0.17/0.44
% 0.17/0.44 thf(vacuousDall,definition,
% 0.17/0.44 ( vacuousDall
% 0.17/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ emptyset )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan1_type,type,
% 0.17/0.44 quantDeMorgan1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan1,definition,
% 0.17/0.44 ( quantDeMorgan1
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan2_type,type,
% 0.17/0.44 quantDeMorgan2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan2,definition,
% 0.17/0.44 ( quantDeMorgan2
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ~ ( Xphi @ Xx ) )
% 0.17/0.44 => ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan3_type,type,
% 0.17/0.44 quantDeMorgan3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan3,definition,
% 0.17/0.44 ( quantDeMorgan3
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan4_type,type,
% 0.17/0.44 quantDeMorgan4: $o ).
% 0.17/0.44
% 0.17/0.44 thf(quantDeMorgan4,definition,
% 0.17/0.44 ( quantDeMorgan4
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ~ ( Xphi @ Xx ) )
% 0.17/0.44 => ~ ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setI_type,type,
% 0.17/0.44 prop2setI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2setI,definition,
% 0.17/0.44 ( prop2setI
% 0.17/0.44 = ( ! [Xphi: $o] :
% 0.17/0.44 ( Xphi
% 0.17/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(set2prop_type,type,
% 0.17/0.44 set2prop: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2set2propI_type,type,
% 0.17/0.44 prop2set2propI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(prop2set2propI,definition,
% 0.17/0.44 ( prop2set2propI
% 0.17/0.44 = ( ! [Xphi: $o] :
% 0.17/0.44 ( Xphi
% 0.17/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notdexE_type,type,
% 0.17/0.44 notdexE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notdexE,definition,
% 0.17/0.44 ( notdexE
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notdallE_type,type,
% 0.17/0.44 notdallE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notdallE,definition,
% 0.17/0.44 ( notdallE
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ~ ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuI1_type,type,
% 0.17/0.44 exuI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuI1,definition,
% 0.17/0.44 ( exuI1
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 & ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuI3_type,type,
% 0.17/0.44 exuI3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuI3,definition,
% 0.17/0.44 ( exuI3
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.17/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuI2_type,type,
% 0.17/0.44 exuI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuI2,definition,
% 0.17/0.44 ( exuI2
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ? [Xx: $i] :
% 0.17/0.44 ! [Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xy )
% 0.17/0.44 <=> ( Xy = Xx ) )
% 0.17/0.44 => ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(inCongP_type,type,
% 0.17/0.44 inCongP: $o ).
% 0.17/0.44
% 0.17/0.44 thf(inCongP,definition,
% 0.17/0.44 ( inCongP
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(in__Cong_type,type,
% 0.17/0.44 in__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(in__Cong,definition,
% 0.17/0.44 ( in__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3u_type,type,
% 0.17/0.44 exuE3u: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuE3u,definition,
% 0.17/0.44 ( exuE3u
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exu__Cong_type,type,
% 0.17/0.44 exu__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exu__Cong,definition,
% 0.17/0.44 ( exu__Cong
% 0.17/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.44 => ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 <=> ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptyset__Cong_type,type,
% 0.17/0.44 emptyset__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptyset__Cong,definition,
% 0.17/0.44 ( emptyset__Cong
% 0.17/0.44 = ( emptyset = emptyset ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoin__Cong_type,type,
% 0.17/0.44 setadjoin__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoin__Cong,definition,
% 0.17/0.44 ( setadjoin__Cong
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ! [Xz: $i,Xu: $i] :
% 0.17/0.44 ( ( Xz = Xu )
% 0.17/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.17/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powerset__Cong_type,type,
% 0.17/0.44 powerset__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powerset__Cong,definition,
% 0.17/0.44 ( powerset__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( ( powerset @ A )
% 0.17/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunion__Cong_type,type,
% 0.17/0.44 setunion__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunion__Cong,definition,
% 0.17/0.44 ( setunion__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( ( setunion @ A )
% 0.17/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(omega__Cong_type,type,
% 0.17/0.44 omega__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(omega__Cong,definition,
% 0.17/0.44 ( omega__Cong
% 0.17/0.44 = ( omega = omega ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(exuEu_type,type,
% 0.17/0.44 exuEu: $o ).
% 0.17/0.44
% 0.17/0.44 thf(exuEu,definition,
% 0.17/0.44 ( exuEu
% 0.17/0.44 = ( ! [Xphi: $i > $o] :
% 0.17/0.44 ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(descr__Cong_type,type,
% 0.17/0.44 descr__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(descr__Cong,definition,
% 0.17/0.44 ( descr__Cong
% 0.17/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.44 ( ( Xx = Xy )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.44 => ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ( ( exu
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.17/0.44 => ( ( descr
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 = ( descr
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstr__Cong_type,type,
% 0.17/0.44 dsetconstr__Cong: $o ).
% 0.17/0.44
% 0.17/0.44 thf(dsetconstr__Cong,definition,
% 0.17/0.44 ( dsetconstr__Cong
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( ( Xx = Xy )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.17/0.44 => ( ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 = ( dsetconstr @ B
% 0.17/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subset_type,type,
% 0.17/0.44 subset: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(disjoint_type,type,
% 0.17/0.44 disjoint: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(setsmeet_type,type,
% 0.17/0.44 setsmeet: $i > $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI1_type,type,
% 0.17/0.44 subsetI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI1,definition,
% 0.17/0.44 ( subsetI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset2_type,type,
% 0.17/0.44 eqimpsubset2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset2,definition,
% 0.17/0.44 ( eqimpsubset2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset1_type,type,
% 0.17/0.44 eqimpsubset1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(eqimpsubset1,definition,
% 0.17/0.44 ( eqimpsubset1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( A = B )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI2_type,type,
% 0.17/0.44 subsetI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetI2,definition,
% 0.17/0.44 ( subsetI2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetsubset_type,type,
% 0.17/0.44 emptysetsubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(emptysetsubset,definition,
% 0.17/0.44 ( emptysetsubset
% 0.17/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE_type,type,
% 0.17/0.44 subsetE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE,definition,
% 0.17/0.44 ( subsetE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE2_type,type,
% 0.17/0.44 subsetE2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetE2,definition,
% 0.17/0.44 ( subsetE2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notsubsetI_type,type,
% 0.17/0.44 notsubsetI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notsubsetI,definition,
% 0.17/0.44 ( notsubsetI
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI1_type,type,
% 0.17/0.44 notequalI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI1,definition,
% 0.17/0.44 ( notequalI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ~ ( subset @ A @ B )
% 0.17/0.44 => ( A != B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI2_type,type,
% 0.17/0.44 notequalI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(notequalI2,definition,
% 0.17/0.44 ( notequalI2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ( A != B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetRefl_type,type,
% 0.17/0.44 subsetRefl: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetRefl,definition,
% 0.17/0.44 ( subsetRefl
% 0.17/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetTrans_type,type,
% 0.17/0.44 subsetTrans: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetTrans,definition,
% 0.17/0.44 ( subsetTrans
% 0.17/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( subset @ B @ C )
% 0.17/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub_type,type,
% 0.17/0.44 setadjoinSub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub,definition,
% 0.17/0.44 ( setadjoinSub
% 0.17/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub2_type,type,
% 0.17/0.44 setadjoinSub2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setadjoinSub2,definition,
% 0.17/0.44 ( setadjoinSub2
% 0.17/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subset2powerset_type,type,
% 0.17/0.44 subset2powerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subset2powerset,definition,
% 0.17/0.44 ( subset2powerset
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setextsub_type,type,
% 0.17/0.44 setextsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setextsub,definition,
% 0.17/0.44 ( setextsub
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( subset @ B @ A )
% 0.17/0.44 => ( A = B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(subsetemptysetimpeq_type,type,
% 0.17/0.44 subsetemptysetimpeq: $o ).
% 0.17/0.44
% 0.17/0.44 thf(subsetemptysetimpeq,definition,
% 0.17/0.44 ( subsetemptysetimpeq
% 0.17/0.44 = ( ! [A: $i] :
% 0.17/0.44 ( ( subset @ A @ emptyset )
% 0.17/0.44 => ( A = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI1_type,type,
% 0.17/0.44 powersetI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetI1,definition,
% 0.17/0.44 ( powersetI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ B @ A )
% 0.17/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE1_type,type,
% 0.17/0.44 powersetE1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetE1,definition,
% 0.17/0.44 ( powersetE1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(inPowerset_type,type,
% 0.17/0.44 inPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(inPowerset,definition,
% 0.17/0.44 ( inPowerset
% 0.17/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(powersetsubset_type,type,
% 0.17/0.44 powersetsubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(powersetsubset,definition,
% 0.17/0.44 ( powersetsubset
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sepInPowerset_type,type,
% 0.17/0.44 sepInPowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(sepInPowerset,definition,
% 0.17/0.44 ( sepInPowerset
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( in
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(sepSubset_type,type,
% 0.17/0.44 sepSubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(sepSubset,definition,
% 0.17/0.44 ( sepSubset
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( subset
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunion_type,type,
% 0.17/0.44 binunion: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIL_type,type,
% 0.17/0.44 binunionIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIL,definition,
% 0.17/0.44 ( binunionIL
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2IR_type,type,
% 0.17/0.44 upairset2IR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2IR,definition,
% 0.17/0.44 ( upairset2IR
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIR_type,type,
% 0.17/0.44 binunionIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionIR,definition,
% 0.17/0.44 ( binunionIR
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionEcases_type,type,
% 0.17/0.44 binunionEcases: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionEcases,definition,
% 0.17/0.44 ( binunionEcases
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.17/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.44 => ( ( ( in @ Xx @ A )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => ( ( ( in @ Xx @ B )
% 0.17/0.44 => Xphi )
% 0.17/0.44 => Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionE_type,type,
% 0.17/0.44 binunionE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionE,definition,
% 0.17/0.44 ( binunionE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionLsub_type,type,
% 0.17/0.44 binunionLsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionLsub,definition,
% 0.17/0.44 ( binunionLsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binunionRsub_type,type,
% 0.17/0.44 binunionRsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binunionRsub,definition,
% 0.17/0.44 ( binunionRsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersect_type,type,
% 0.17/0.44 binintersect: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectI_type,type,
% 0.17/0.44 binintersectI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectI,definition,
% 0.17/0.44 ( binintersectI
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset5_type,type,
% 0.17/0.44 binintersectSubset5: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset5,definition,
% 0.17/0.44 ( binintersectSubset5
% 0.17/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.44 ( ( subset @ C @ A )
% 0.17/0.44 => ( ( subset @ C @ B )
% 0.17/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectEL_type,type,
% 0.17/0.44 binintersectEL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectEL,definition,
% 0.17/0.44 ( binintersectEL
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectLsub_type,type,
% 0.17/0.44 binintersectLsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectLsub,definition,
% 0.17/0.44 ( binintersectLsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset2_type,type,
% 0.17/0.44 binintersectSubset2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset2,definition,
% 0.17/0.44 ( binintersectSubset2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( binintersect @ A @ B )
% 0.17/0.44 = A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset3_type,type,
% 0.17/0.44 binintersectSubset3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset3,definition,
% 0.17/0.44 ( binintersectSubset3
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( ( binintersect @ A @ B )
% 0.17/0.44 = B )
% 0.17/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectER_type,type,
% 0.17/0.44 binintersectER: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectER,definition,
% 0.17/0.44 ( binintersectER
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.17/0.44 => ( in @ Xx @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(disjointsetsI1_type,type,
% 0.17/0.44 disjointsetsI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(disjointsetsI1,definition,
% 0.17/0.44 ( disjointsetsI1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ~ ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( in @ Xx @ B ) )
% 0.17/0.44 => ( ( binintersect @ A @ B )
% 0.17/0.44 = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectRsub_type,type,
% 0.17/0.44 binintersectRsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectRsub,definition,
% 0.17/0.44 ( binintersectRsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset4_type,type,
% 0.17/0.44 binintersectSubset4: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset4,definition,
% 0.17/0.44 ( binintersectSubset4
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ B @ A )
% 0.17/0.44 => ( ( binintersect @ A @ B )
% 0.17/0.44 = B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset1_type,type,
% 0.17/0.44 binintersectSubset1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(binintersectSubset1,definition,
% 0.17/0.44 ( binintersectSubset1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( ( binintersect @ A @ B )
% 0.17/0.44 = A )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(bs114d_type,type,
% 0.17/0.44 bs114d: $o ).
% 0.17/0.44
% 0.17/0.44 thf(bs114d,definition,
% 0.17/0.44 ( bs114d
% 0.17/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.17/0.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.17/0.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(regular_type,type,
% 0.17/0.44 regular: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminus_type,type,
% 0.17/0.44 setminus: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(setminusI_type,type,
% 0.17/0.44 setminusI: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusI,definition,
% 0.17/0.44 ( setminusI
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusEL_type,type,
% 0.17/0.44 setminusEL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusEL,definition,
% 0.17/0.44 ( setminusEL
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusER_type,type,
% 0.17/0.44 setminusER: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusER,definition,
% 0.17/0.44 ( setminusER
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.44 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusSubset2_type,type,
% 0.17/0.44 setminusSubset2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusSubset2,definition,
% 0.17/0.44 ( setminusSubset2
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( subset @ A @ B )
% 0.17/0.44 => ( ( setminus @ A @ B )
% 0.17/0.44 = emptyset ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusERneg_type,type,
% 0.17/0.44 setminusERneg: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusERneg,definition,
% 0.17/0.44 ( setminusERneg
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.44 => ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusELneg_type,type,
% 0.17/0.44 setminusELneg: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusELneg,definition,
% 0.17/0.44 ( setminusELneg
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusILneg_type,type,
% 0.17/0.44 setminusILneg: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusILneg,definition,
% 0.17/0.44 ( setminusILneg
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ A )
% 0.17/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusIRneg_type,type,
% 0.17/0.44 setminusIRneg: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusIRneg,definition,
% 0.17/0.44 ( setminusIRneg
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusLsub_type,type,
% 0.17/0.44 setminusLsub: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusLsub,definition,
% 0.17/0.44 ( setminusLsub
% 0.17/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setminusSubset1_type,type,
% 0.17/0.44 setminusSubset1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setminusSubset1,definition,
% 0.17/0.44 ( setminusSubset1
% 0.17/0.44 = ( ! [A: $i,B: $i] :
% 0.17/0.44 ( ( ( setminus @ A @ B )
% 0.17/0.44 = emptyset )
% 0.17/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiff_type,type,
% 0.17/0.44 symdiff: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffE_type,type,
% 0.17/0.44 symdiffE: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffE,definition,
% 0.17/0.44 ( symdiffE
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.17/0.44 => ! [Xphi: $o] :
% 0.17/0.44 ( ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => Xphi ) )
% 0.17/0.44 => ( ( ~ ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => Xphi ) )
% 0.17/0.44 => Xphi ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffI1_type,type,
% 0.17/0.44 symdiffI1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffI1,definition,
% 0.17/0.44 ( symdiffI1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffI2_type,type,
% 0.17/0.44 symdiffI2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffI2,definition,
% 0.17/0.44 ( symdiffI2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffIneg1_type,type,
% 0.17/0.44 symdiffIneg1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffIneg1,definition,
% 0.17/0.44 ( symdiffIneg1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffIneg2_type,type,
% 0.17/0.44 symdiffIneg2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(symdiffIneg2,definition,
% 0.17/0.44 ( symdiffIneg2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ~ ( in @ Xx @ A )
% 0.17/0.44 => ( ~ ( in @ Xx @ B )
% 0.17/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(iskpair_type,type,
% 0.17/0.44 iskpair: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(secondinupair_type,type,
% 0.17/0.44 secondinupair: $o ).
% 0.17/0.44
% 0.17/0.44 thf(secondinupair,definition,
% 0.17/0.44 ( secondinupair
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairIL_type,type,
% 0.17/0.44 setukpairIL: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairIL,definition,
% 0.17/0.44 ( setukpairIL
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairIR_type,type,
% 0.17/0.44 setukpairIR: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setukpairIR,definition,
% 0.17/0.44 ( setukpairIR
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(kpairiskpair_type,type,
% 0.17/0.44 kpairiskpair: $o ).
% 0.17/0.44
% 0.17/0.44 thf(kpairiskpair,definition,
% 0.17/0.44 ( kpairiskpair
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(kpair_type,type,
% 0.17/0.44 kpair: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(kpairp_type,type,
% 0.17/0.44 kpairp: $o ).
% 0.17/0.44
% 0.17/0.44 thf(kpairp,definition,
% 0.17/0.44 ( kpairp
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprod_type,type,
% 0.17/0.44 cartprod: $i > $i > $i ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsubset_type,type,
% 0.17/0.44 singletonsubset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletonsubset,definition,
% 0.17/0.44 ( singletonsubset
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singletoninpowerset_type,type,
% 0.17/0.44 singletoninpowerset: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletoninpowerset,definition,
% 0.17/0.44 ( singletoninpowerset
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singletoninpowunion_type,type,
% 0.17/0.44 singletoninpowunion: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletoninpowunion,definition,
% 0.17/0.44 ( singletoninpowunion
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2E_type,type,
% 0.17/0.44 upairset2E: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairset2E,definition,
% 0.17/0.44 ( upairset2E
% 0.17/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.44 => ( ( Xz = Xx )
% 0.17/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairsubunion_type,type,
% 0.17/0.44 upairsubunion: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairsubunion,definition,
% 0.17/0.44 ( upairsubunion
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(upairinpowunion_type,type,
% 0.17/0.44 upairinpowunion: $o ).
% 0.17/0.44
% 0.17/0.44 thf(upairinpowunion,definition,
% 0.17/0.44 ( upairinpowunion
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem1_type,type,
% 0.17/0.44 ubforcartprodlem1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem1,definition,
% 0.17/0.44 ( ubforcartprodlem1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem2_type,type,
% 0.17/0.44 ubforcartprodlem2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem2,definition,
% 0.17/0.44 ( ubforcartprodlem2
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem3_type,type,
% 0.17/0.44 ubforcartprodlem3: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ubforcartprodlem3,definition,
% 0.17/0.44 ( ubforcartprodlem3
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairin_type,type,
% 0.17/0.44 cartprodpairin: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodpairin,definition,
% 0.17/0.44 ( cartprodpairin
% 0.17/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempair1_type,type,
% 0.17/0.44 cartprodmempair1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempair1,definition,
% 0.17/0.44 ( cartprodmempair1
% 0.17/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ? [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ B )
% 0.17/0.44 & ( Xu
% 0.17/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempair_type,type,
% 0.17/0.44 cartprodmempair: $o ).
% 0.17/0.44
% 0.17/0.44 thf(cartprodmempair,definition,
% 0.17/0.44 ( cartprodmempair
% 0.17/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.44 => ( iskpair @ Xu ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE2_type,type,
% 0.17/0.44 setunionE2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionE2,definition,
% 0.17/0.44 ( setunionE2
% 0.17/0.44 = ( ! [A: $i,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.44 => ? [X: $i] :
% 0.17/0.44 ( ( in @ X @ A )
% 0.17/0.44 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton1_type,type,
% 0.17/0.44 setunionsingleton1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton1,definition,
% 0.17/0.44 ( setunionsingleton1
% 0.17/0.44 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton2_type,type,
% 0.17/0.44 setunionsingleton2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton2,definition,
% 0.17/0.44 ( setunionsingleton2
% 0.17/0.44 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton_type,type,
% 0.17/0.44 setunionsingleton: $o ).
% 0.17/0.44
% 0.17/0.44 thf(setunionsingleton,definition,
% 0.17/0.44 ( setunionsingleton
% 0.17/0.44 = ( ! [Xx: $i] :
% 0.17/0.44 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.17/0.44 = Xx ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(singleton_type,type,
% 0.17/0.44 singleton: $i > $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletonprop_type,type,
% 0.17/0.44 singletonprop: $o ).
% 0.17/0.44
% 0.17/0.44 thf(singletonprop,definition,
% 0.17/0.44 ( singletonprop
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xx = Xy ) ) ) ) )
% 0.17/0.44 => ( ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) )
% 0.17/0.44 => ( singleton
% 0.17/0.44 @ ( dsetconstr @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ex1_type,type,
% 0.17/0.44 ex1: $i > ( $i > $o ) > $o ).
% 0.17/0.44
% 0.17/0.44 thf(ex1E1_type,type,
% 0.17/0.44 ex1E1: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ex1E1,definition,
% 0.17/0.44 ( ex1E1
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ( ex1 @ A
% 0.17/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.44 => ? [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ex1I_type,type,
% 0.17/0.44 ex1I: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ex1I,definition,
% 0.17/0.44 ( ex1I
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ( ( Xphi @ Xx )
% 0.17/0.44 => ( ! [Xy: $i] :
% 0.17/0.44 ( ( in @ Xy @ A )
% 0.17/0.44 => ( ( Xphi @ Xy )
% 0.17/0.44 => ( Xy = Xx ) ) )
% 0.17/0.44 => ( ex1 @ A
% 0.17/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.44
% 0.17/0.44 thf(ex1I2_type,type,
% 0.17/0.44 ex1I2: $o ).
% 0.17/0.44
% 0.17/0.44 thf(ex1I2,definition,
% 0.17/0.44 ( ex1I2
% 0.17/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.44 ( ! [Xx: $i] :
% 0.17/0.44 ( ( in @ Xx @ A )
% 0.17/0.44 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ( Xphi @ Xx )
% 0.17/0.45 => ( ( Xphi @ Xy )
% 0.17/0.45 => ( Xx = Xy ) ) ) ) )
% 0.17/0.45 => ( ? [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 & ( Xphi @ Xx ) )
% 0.17/0.45 => ( ex1 @ A
% 0.17/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(singletonsuniq_type,type,
% 0.17/0.45 singletonsuniq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(singletonsuniq,definition,
% 0.17/0.45 ( singletonsuniq
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( ( setadjoin @ Xx @ emptyset )
% 0.17/0.45 = ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(atmost1p_type,type,
% 0.17/0.45 atmost1p: $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(atleast2p_type,type,
% 0.17/0.45 atleast2p: $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(atmost2p_type,type,
% 0.17/0.45 atmost2p: $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(upairsetp_type,type,
% 0.17/0.45 upairsetp: $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjL1_type,type,
% 0.17/0.45 setukpairinjL1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjL1,definition,
% 0.17/0.45 ( setukpairinjL1
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.45 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.45 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(kfstsingleton_type,type,
% 0.17/0.45 kfstsingleton: $o ).
% 0.17/0.45
% 0.17/0.45 thf(kfstsingleton,definition,
% 0.17/0.45 ( kfstsingleton
% 0.17/0.45 = ( ! [Xu: $i] :
% 0.17/0.45 ( ( iskpair @ Xu )
% 0.17/0.45 => ( singleton
% 0.17/0.45 @ ( dsetconstr @ ( setunion @ Xu )
% 0.17/0.45 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(theprop_type,type,
% 0.17/0.45 theprop: $o ).
% 0.17/0.45
% 0.17/0.45 thf(theprop,definition,
% 0.17/0.45 ( theprop
% 0.17/0.45 = ( ! [X: $i] :
% 0.17/0.45 ( ( singleton @ X )
% 0.17/0.45 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(kfst_type,type,
% 0.17/0.45 kfst: $i > $i ).
% 0.17/0.45
% 0.17/0.45 thf(kfstpairEq_type,type,
% 0.17/0.45 kfstpairEq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(kfstpairEq,definition,
% 0.17/0.45 ( kfstpairEq
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.17/0.45 = Xx ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodfstin_type,type,
% 0.17/0.45 cartprodfstin: $o ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodfstin,definition,
% 0.17/0.45 ( cartprodfstin
% 0.17/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.45 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjL2_type,type,
% 0.17/0.45 setukpairinjL2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjL2,definition,
% 0.17/0.45 ( setukpairinjL2
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.45 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjL_type,type,
% 0.17/0.45 setukpairinjL: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjL,definition,
% 0.17/0.45 ( setukpairinjL
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.45 ( ( ( kpair @ Xx @ Xy )
% 0.17/0.45 = ( kpair @ Xz @ Xu ) )
% 0.17/0.45 => ( Xx = Xz ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR11_type,type,
% 0.17/0.45 setukpairinjR11: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR11,definition,
% 0.17/0.45 ( setukpairinjR11
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( Xx = Xy )
% 0.17/0.45 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR12_type,type,
% 0.17/0.45 setukpairinjR12: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR12,definition,
% 0.17/0.45 ( setukpairinjR12
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( Xx = Xy )
% 0.17/0.45 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.45 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR1_type,type,
% 0.17/0.45 setukpairinjR1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR1,definition,
% 0.17/0.45 ( setukpairinjR1
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.45 => ( ( Xz = Xu )
% 0.17/0.45 => ( Xy = Xu ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(upairequniteq_type,type,
% 0.17/0.45 upairequniteq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(upairequniteq,definition,
% 0.17/0.45 ( upairequniteq
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.45 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.45 = ( setadjoin @ Xz @ emptyset ) )
% 0.17/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR2_type,type,
% 0.17/0.45 setukpairinjR2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR2,definition,
% 0.17/0.45 ( setukpairinjR2
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.45 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.17/0.45 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.17/0.45 => ( Xy = Xu ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR_type,type,
% 0.17/0.45 setukpairinjR: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setukpairinjR,definition,
% 0.17/0.45 ( setukpairinjR
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.17/0.45 ( ( ( kpair @ Xx @ Xy )
% 0.17/0.45 = ( kpair @ Xz @ Xu ) )
% 0.17/0.45 => ( Xy = Xu ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(ksndsingleton_type,type,
% 0.17/0.45 ksndsingleton: $o ).
% 0.17/0.45
% 0.17/0.45 thf(ksndsingleton,definition,
% 0.17/0.45 ( ksndsingleton
% 0.17/0.45 = ( ! [Xu: $i] :
% 0.17/0.45 ( ( iskpair @ Xu )
% 0.17/0.45 => ( singleton
% 0.17/0.45 @ ( dsetconstr @ ( setunion @ Xu )
% 0.17/0.45 @ ^ [Xx: $i] :
% 0.17/0.45 ( Xu
% 0.17/0.45 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(ksnd_type,type,
% 0.17/0.45 ksnd: $i > $i ).
% 0.17/0.45
% 0.17/0.45 thf(ksndpairEq_type,type,
% 0.17/0.45 ksndpairEq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(ksndpairEq,definition,
% 0.17/0.45 ( ksndpairEq
% 0.17/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.17/0.45 = Xy ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(kpairsurjEq_type,type,
% 0.17/0.45 kpairsurjEq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(kpairsurjEq,definition,
% 0.17/0.45 ( kpairsurjEq
% 0.17/0.45 = ( ! [Xu: $i] :
% 0.17/0.45 ( ( iskpair @ Xu )
% 0.17/0.45 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.17/0.45 = Xu ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodsndin_type,type,
% 0.17/0.45 cartprodsndin: $o ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodsndin,definition,
% 0.17/0.45 ( cartprodsndin
% 0.17/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.45 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodpairmemEL_type,type,
% 0.17/0.45 cartprodpairmemEL: $o ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodpairmemEL,definition,
% 0.17/0.45 ( cartprodpairmemEL
% 0.17/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.17/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodpairmemER_type,type,
% 0.17/0.45 cartprodpairmemER: $o ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodpairmemER,definition,
% 0.17/0.45 ( cartprodpairmemER
% 0.17/0.45 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.17/0.45 => ( in @ Xy @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodmempaircEq_type,type,
% 0.17/0.45 cartprodmempaircEq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodmempaircEq,definition,
% 0.17/0.45 ( cartprodmempaircEq
% 0.17/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( kpair @ Xx @ Xy )
% 0.17/0.45 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodfstpairEq_type,type,
% 0.17/0.45 cartprodfstpairEq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodfstpairEq,definition,
% 0.17/0.45 ( cartprodfstpairEq
% 0.17/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.17/0.45 = Xx ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodsndpairEq_type,type,
% 0.17/0.45 cartprodsndpairEq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodsndpairEq,definition,
% 0.17/0.45 ( cartprodsndpairEq
% 0.17/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.17/0.45 = Xy ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodpairsurjEq_type,type,
% 0.17/0.45 cartprodpairsurjEq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(cartprodpairsurjEq,definition,
% 0.17/0.45 ( cartprodpairsurjEq
% 0.17/0.45 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.17/0.45 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.17/0.45 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.17/0.45 = Xu ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln_type,type,
% 0.17/0.45 breln: $i > $i > $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(breln,definition,
% 0.17/0.45 ( breln
% 0.17/0.45 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstr_type,type,
% 0.17/0.45 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrI_type,type,
% 0.17/0.45 dpsetconstrI: $o ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrI,definition,
% 0.17/0.45 ( dpsetconstrI
% 0.17/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( Xphi @ Xx @ Xy )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.45 @ ( dpsetconstr @ A @ B
% 0.17/0.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrSub_type,type,
% 0.17/0.45 dpsetconstrSub: $o ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrSub,definition,
% 0.17/0.45 ( dpsetconstrSub
% 0.17/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.17/0.45 ( subset
% 0.17/0.45 @ ( dpsetconstr @ A @ B
% 0.17/0.45 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.17/0.45 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setOfPairsIsBReln_type,type,
% 0.17/0.45 setOfPairsIsBReln: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setOfPairsIsBReln,definition,
% 0.17/0.45 ( setOfPairsIsBReln
% 0.17/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.17/0.45 ( breln @ A @ B
% 0.17/0.45 @ ( dpsetconstr @ A @ B
% 0.17/0.45 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrERa_type,type,
% 0.17/0.45 dpsetconstrERa: $o ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrERa,definition,
% 0.17/0.45 ( dpsetconstrERa
% 0.17/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.45 @ ( dpsetconstr @ A @ B
% 0.17/0.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.45 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrEL1_type,type,
% 0.17/0.45 dpsetconstrEL1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrEL1,definition,
% 0.17/0.45 ( dpsetconstrEL1
% 0.17/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.45 @ ( dpsetconstr @ A @ B
% 0.17/0.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrEL2_type,type,
% 0.17/0.45 dpsetconstrEL2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrEL2,definition,
% 0.17/0.45 ( dpsetconstrEL2
% 0.17/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.45 @ ( dpsetconstr @ A @ B
% 0.17/0.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.45 => ( in @ Xy @ B ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrER_type,type,
% 0.17/0.45 dpsetconstrER: $o ).
% 0.17/0.45
% 0.17/0.45 thf(dpsetconstrER,definition,
% 0.17/0.45 ( dpsetconstrER
% 0.17/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( in @ ( kpair @ Xx @ Xy )
% 0.17/0.45 @ ( dpsetconstr @ A @ B
% 0.17/0.45 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.17/0.45 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(func_type,type,
% 0.17/0.45 func: $i > $i > $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcSet_type,type,
% 0.17/0.45 funcSet: $i > $i > $i ).
% 0.17/0.45
% 0.17/0.45 thf(funcImageSingleton_type,type,
% 0.17/0.45 funcImageSingleton: $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcImageSingleton,definition,
% 0.17/0.45 ( funcImageSingleton
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( singleton
% 0.17/0.45 @ ( dsetconstr @ B
% 0.17/0.45 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(apProp_type,type,
% 0.17/0.45 apProp: $o ).
% 0.17/0.45
% 0.17/0.45 thf(apProp,definition,
% 0.17/0.45 ( apProp
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in
% 0.17/0.45 @ ( setunion
% 0.17/0.45 @ ( dsetconstr @ B
% 0.17/0.45 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.17/0.45 @ B ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(ap_type,type,
% 0.17/0.45 ap: $i > $i > $i > $i > $i ).
% 0.17/0.45
% 0.17/0.45 thf(app_type,type,
% 0.17/0.45 app: $o ).
% 0.17/0.45
% 0.17/0.45 thf(app,definition,
% 0.17/0.45 ( app
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(infuncsetfunc_type,type,
% 0.17/0.45 infuncsetfunc: $o ).
% 0.17/0.45
% 0.17/0.45 thf(infuncsetfunc,definition,
% 0.17/0.45 ( infuncsetfunc
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.45 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(ap2p_type,type,
% 0.17/0.45 ap2p: $o ).
% 0.17/0.45
% 0.17/0.45 thf(ap2p,definition,
% 0.17/0.45 ( ap2p
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(funcinfuncset_type,type,
% 0.17/0.45 funcinfuncset: $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcinfuncset,definition,
% 0.17/0.45 ( funcinfuncset
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(lamProp_type,type,
% 0.17/0.45 lamProp: $o ).
% 0.17/0.45
% 0.17/0.45 thf(lamProp,definition,
% 0.17/0.45 ( lamProp
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.45 => ( func @ A @ B
% 0.17/0.45 @ ( dpsetconstr @ A @ B
% 0.17/0.45 @ ^ [Xx: $i,Xy: $i] :
% 0.17/0.45 ( ( Xf @ Xx )
% 0.17/0.45 = Xy ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(lam_type,type,
% 0.17/0.45 lam: $i > $i > ( $i > $i ) > $i ).
% 0.17/0.45
% 0.17/0.45 thf(lamp_type,type,
% 0.17/0.45 lamp: $o ).
% 0.17/0.45
% 0.17/0.45 thf(lamp,definition,
% 0.17/0.45 ( lamp
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.45 => ( func @ A @ B
% 0.17/0.45 @ ( lam @ A @ B
% 0.17/0.45 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(lam2p_type,type,
% 0.17/0.45 lam2p: $o ).
% 0.17/0.45
% 0.17/0.45 thf(lam2p,definition,
% 0.17/0.45 ( lam2p
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.45 => ( in
% 0.17/0.45 @ ( lam @ A @ B
% 0.17/0.45 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.17/0.45 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(brelnall1_type,type,
% 0.17/0.45 brelnall1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(brelnall1,definition,
% 0.17/0.45 ( brelnall1
% 0.17/0.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.45 ( ( breln @ A @ B @ R )
% 0.17/0.45 => ! [Xphi: $i > $o] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.45 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ R )
% 0.17/0.45 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(brelnall2_type,type,
% 0.17/0.45 brelnall2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(brelnall2,definition,
% 0.17/0.45 ( brelnall2
% 0.17/0.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.45 ( ( breln @ A @ B @ R )
% 0.17/0.45 => ! [Xphi: $i > $o] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.45 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ R )
% 0.17/0.45 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(ex1E2_type,type,
% 0.17/0.45 ex1E2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(ex1E2,definition,
% 0.17/0.45 ( ex1E2
% 0.17/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.45 ( ( ex1 @ A
% 0.17/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ( Xphi @ Xx )
% 0.17/0.45 => ( ( Xphi @ Xy )
% 0.17/0.45 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(funcGraphProp1_type,type,
% 0.17/0.45 funcGraphProp1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcGraphProp1,definition,
% 0.17/0.45 ( funcGraphProp1
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(funcGraphProp3_type,type,
% 0.17/0.45 funcGraphProp3: $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcGraphProp3,definition,
% 0.17/0.45 ( funcGraphProp3
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(funcGraphProp2_type,type,
% 0.17/0.45 funcGraphProp2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcGraphProp2,definition,
% 0.17/0.45 ( funcGraphProp2
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.17/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.45 = Xy ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(funcextLem_type,type,
% 0.17/0.45 funcextLem: $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcextLem,definition,
% 0.17/0.45 ( funcextLem
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ! [Xg: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xg )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.45 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(funcGraphProp4_type,type,
% 0.17/0.45 funcGraphProp4: $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcGraphProp4,definition,
% 0.17/0.45 ( funcGraphProp4
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.17/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.45 = Xy ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(subbreln_type,type,
% 0.17/0.45 subbreln: $o ).
% 0.17/0.45
% 0.17/0.45 thf(subbreln,definition,
% 0.17/0.45 ( subbreln
% 0.17/0.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.45 ( ( breln @ A @ B @ R )
% 0.17/0.45 => ! [S: $i] :
% 0.17/0.45 ( ( breln @ A @ B @ S )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.45 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(eqbreln_type,type,
% 0.17/0.45 eqbreln: $o ).
% 0.17/0.45
% 0.17/0.45 thf(eqbreln,definition,
% 0.17/0.45 ( eqbreln
% 0.17/0.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.17/0.45 ( ( breln @ A @ B @ R )
% 0.17/0.45 => ! [S: $i] :
% 0.17/0.45 ( ( breln @ A @ B @ S )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.17/0.45 => ( R = S ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(funcext_type,type,
% 0.17/0.45 funcext: $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcext,definition,
% 0.17/0.45 ( funcext
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ! [Xg: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xg )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.45 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.45 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(funcext2_type,type,
% 0.17/0.45 funcext2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(funcext2,definition,
% 0.17/0.45 ( funcext2
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.45 => ! [Xg: $i] :
% 0.17/0.45 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.45 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.17/0.45 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(ap2apEq1_type,type,
% 0.17/0.45 ap2apEq1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(ap2apEq1,definition,
% 0.17/0.45 ( ap2apEq1
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.45 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(ap2apEq2_type,type,
% 0.17/0.45 ap2apEq2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(ap2apEq2,definition,
% 0.17/0.45 ( ap2apEq2
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.17/0.45 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(beta1_type,type,
% 0.17/0.45 beta1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(beta1,definition,
% 0.17/0.45 ( beta1
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( ap @ A @ B
% 0.17/0.45 @ ( lam @ A @ B
% 0.17/0.45 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.17/0.45 @ Xx )
% 0.17/0.45 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(eta1_type,type,
% 0.17/0.45 eta1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(eta1,definition,
% 0.17/0.45 ( eta1
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( func @ A @ B @ Xf )
% 0.17/0.45 => ( ( lam @ A @ B
% 0.17/0.45 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.17/0.45 = Xf ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(lam2lamEq_type,type,
% 0.17/0.45 lam2lamEq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(lam2lamEq,definition,
% 0.17/0.45 ( lam2lamEq
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.45 => ( ( lam @ A @ B
% 0.17/0.45 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.17/0.45 = ( lam @ A @ B
% 0.17/0.45 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(beta2_type,type,
% 0.17/0.45 beta2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(beta2,definition,
% 0.17/0.45 ( beta2
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( ap @ A @ B
% 0.17/0.45 @ ( lam @ A @ B
% 0.17/0.45 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.17/0.45 @ Xx )
% 0.17/0.45 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(eta2_type,type,
% 0.17/0.45 eta2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(eta2,definition,
% 0.17/0.45 ( eta2
% 0.17/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.17/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.17/0.45 => ( ( lam @ A @ B
% 0.17/0.45 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.17/0.45 = Xf ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(iffalseProp1_type,type,
% 0.17/0.45 iffalseProp1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(iffalseProp1,definition,
% 0.17/0.45 ( iffalseProp1
% 0.17/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ~ Xphi
% 0.17/0.45 => ( in @ Xy
% 0.17/0.45 @ ( dsetconstr @ A
% 0.17/0.45 @ ^ [Xz: $i] :
% 0.17/0.45 ( ( Xphi
% 0.17/0.45 & ( Xz = Xx ) )
% 0.17/0.45 | ( ~ Xphi
% 0.17/0.45 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(iffalseProp2_type,type,
% 0.17/0.45 iffalseProp2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(iffalseProp2,definition,
% 0.17/0.45 ( iffalseProp2
% 0.17/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ~ Xphi
% 0.17/0.45 => ( ( dsetconstr @ A
% 0.17/0.45 @ ^ [Xz: $i] :
% 0.17/0.45 ( ( Xphi
% 0.17/0.45 & ( Xz = Xx ) )
% 0.17/0.45 | ( ~ Xphi
% 0.17/0.45 & ( Xz = Xy ) ) ) )
% 0.17/0.45 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(iftrueProp1_type,type,
% 0.17/0.45 iftrueProp1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(iftrueProp1,definition,
% 0.17/0.45 ( iftrueProp1
% 0.17/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( Xphi
% 0.17/0.45 => ( in @ Xx
% 0.17/0.45 @ ( dsetconstr @ A
% 0.17/0.45 @ ^ [Xz: $i] :
% 0.17/0.45 ( ( Xphi
% 0.17/0.45 & ( Xz = Xx ) )
% 0.17/0.45 | ( ~ Xphi
% 0.17/0.45 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(iftrueProp2_type,type,
% 0.17/0.45 iftrueProp2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(iftrueProp2,definition,
% 0.17/0.45 ( iftrueProp2
% 0.17/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( Xphi
% 0.17/0.45 => ( ( dsetconstr @ A
% 0.17/0.45 @ ^ [Xz: $i] :
% 0.17/0.45 ( ( Xphi
% 0.17/0.45 & ( Xz = Xx ) )
% 0.17/0.45 | ( ~ Xphi
% 0.17/0.45 & ( Xz = Xy ) ) ) )
% 0.17/0.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(ifSingleton_type,type,
% 0.17/0.45 ifSingleton: $o ).
% 0.17/0.45
% 0.17/0.45 thf(ifSingleton,definition,
% 0.17/0.45 ( ifSingleton
% 0.17/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( singleton
% 0.17/0.45 @ ( dsetconstr @ A
% 0.17/0.45 @ ^ [Xz: $i] :
% 0.17/0.45 ( ( Xphi
% 0.17/0.45 & ( Xz = Xx ) )
% 0.17/0.45 | ( ~ Xphi
% 0.17/0.45 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(if_type,type,
% 0.17/0.45 if: $i > $o > $i > $i > $i ).
% 0.17/0.45
% 0.17/0.45 thf(ifp_type,type,
% 0.17/0.45 ifp: $o ).
% 0.17/0.45
% 0.17/0.45 thf(ifp,definition,
% 0.17/0.45 ( ifp
% 0.17/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(theeq_type,type,
% 0.17/0.45 theeq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(theeq,definition,
% 0.17/0.45 ( theeq
% 0.17/0.45 = ( ! [X: $i] :
% 0.17/0.45 ( ( singleton @ X )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ X )
% 0.17/0.45 => ( ( setunion @ X )
% 0.17/0.45 = Xx ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(iftrue_type,type,
% 0.17/0.45 iftrue: $o ).
% 0.17/0.45
% 0.17/0.45 thf(iftrue,definition,
% 0.17/0.45 ( iftrue
% 0.17/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( Xphi
% 0.17/0.45 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.17/0.45 = Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(iffalse_type,type,
% 0.17/0.45 iffalse: $o ).
% 0.17/0.45
% 0.17/0.45 thf(iffalse,definition,
% 0.17/0.45 ( iffalse
% 0.17/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ~ Xphi
% 0.17/0.45 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.17/0.45 = Xy ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(iftrueorfalse_type,type,
% 0.17/0.45 iftrueorfalse: $o ).
% 0.17/0.45
% 0.17/0.45 thf(iftrueorfalse,definition,
% 0.17/0.45 ( iftrueorfalse
% 0.17/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(binintersectT_lem_type,type,
% 0.17/0.45 binintersectT_lem: $o ).
% 0.17/0.45
% 0.17/0.45 thf(binintersectT_lem,definition,
% 0.17/0.45 ( binintersectT_lem
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(binunionT_lem_type,type,
% 0.17/0.45 binunionT_lem: $o ).
% 0.17/0.45
% 0.17/0.45 thf(binunionT_lem,definition,
% 0.17/0.45 ( binunionT_lem
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(powersetT_lem_type,type,
% 0.17/0.45 powersetT_lem: $o ).
% 0.17/0.45
% 0.17/0.45 thf(powersetT_lem,definition,
% 0.17/0.45 ( powersetT_lem
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setminusT_lem_type,type,
% 0.17/0.45 setminusT_lem: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setminusT_lem,definition,
% 0.17/0.45 ( setminusT_lem
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(complementT_lem_type,type,
% 0.17/0.45 complementT_lem: $o ).
% 0.17/0.45
% 0.17/0.45 thf(complementT_lem,definition,
% 0.17/0.45 ( complementT_lem
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setextT_type,type,
% 0.17/0.45 setextT: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setextT,definition,
% 0.17/0.45 ( setextT
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ X )
% 0.17/0.45 => ( in @ Xx @ Y ) ) )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ Y )
% 0.17/0.45 => ( in @ Xx @ X ) ) )
% 0.17/0.45 => ( X = Y ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(subsetTI_type,type,
% 0.17/0.45 subsetTI: $o ).
% 0.17/0.45
% 0.17/0.45 thf(subsetTI,definition,
% 0.17/0.45 ( subsetTI
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ X )
% 0.17/0.45 => ( in @ Xx @ Y ) ) )
% 0.17/0.45 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(powersetTI1_type,type,
% 0.17/0.45 powersetTI1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(powersetTI1,definition,
% 0.17/0.45 ( powersetTI1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ X )
% 0.17/0.45 => ( in @ Xx @ Y ) ) )
% 0.17/0.45 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(powersetTE1_type,type,
% 0.17/0.45 powersetTE1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(powersetTE1,definition,
% 0.17/0.45 ( powersetTE1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ X @ ( powerset @ Y ) )
% 0.17/0.45 => ( ( in @ Xx @ X )
% 0.17/0.45 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(complementTI1_type,type,
% 0.17/0.45 complementTI1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(complementTI1,definition,
% 0.17/0.45 ( complementTI1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ X )
% 0.17/0.45 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(complementTE1_type,type,
% 0.17/0.45 complementTE1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(complementTE1,definition,
% 0.17/0.45 ( complementTE1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.17/0.45 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(binintersectTELcontra_type,type,
% 0.17/0.45 binintersectTELcontra: $o ).
% 0.17/0.45
% 0.17/0.45 thf(binintersectTELcontra,definition,
% 0.17/0.45 ( binintersectTELcontra
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ~ ( in @ Xx @ X )
% 0.17/0.45 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(binintersectTERcontra_type,type,
% 0.17/0.45 binintersectTERcontra: $o ).
% 0.17/0.45
% 0.17/0.45 thf(binintersectTERcontra,definition,
% 0.17/0.45 ( binintersectTERcontra
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ~ ( in @ Xx @ Y )
% 0.17/0.45 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(contrasubsetT_type,type,
% 0.17/0.45 contrasubsetT: $o ).
% 0.17/0.45
% 0.17/0.45 thf(contrasubsetT,definition,
% 0.17/0.45 ( contrasubsetT
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.17/0.45 => ( ( in @ Xx @ Y )
% 0.17/0.45 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(contrasubsetT1_type,type,
% 0.17/0.45 contrasubsetT1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(contrasubsetT1,definition,
% 0.17/0.45 ( contrasubsetT1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( subset @ X @ Y )
% 0.17/0.45 => ( ~ ( in @ Xx @ Y )
% 0.17/0.45 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(contrasubsetT2_type,type,
% 0.17/0.45 contrasubsetT2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(contrasubsetT2,definition,
% 0.17/0.45 ( contrasubsetT2
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ( subset @ X @ Y )
% 0.17/0.45 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(contrasubsetT3_type,type,
% 0.17/0.45 contrasubsetT3: $o ).
% 0.17/0.45
% 0.17/0.45 thf(contrasubsetT3,definition,
% 0.17/0.45 ( contrasubsetT3
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.17/0.45 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementI1_type,type,
% 0.17/0.45 doubleComplementI1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementI1,definition,
% 0.17/0.45 ( doubleComplementI1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ X )
% 0.17/0.45 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementE1_type,type,
% 0.17/0.45 doubleComplementE1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementE1,definition,
% 0.17/0.45 ( doubleComplementE1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.17/0.45 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementSub1_type,type,
% 0.17/0.45 doubleComplementSub1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementSub1,definition,
% 0.17/0.45 ( doubleComplementSub1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementSub2_type,type,
% 0.17/0.45 doubleComplementSub2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementSub2,definition,
% 0.17/0.45 ( doubleComplementSub2
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementEq_type,type,
% 0.17/0.45 doubleComplementEq: $o ).
% 0.17/0.45
% 0.17/0.45 thf(doubleComplementEq,definition,
% 0.17/0.45 ( doubleComplementEq
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ( X
% 0.17/0.45 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(complementTnotintersectT_type,type,
% 0.17/0.45 complementTnotintersectT: $o ).
% 0.17/0.45
% 0.17/0.45 thf(complementTnotintersectT,definition,
% 0.17/0.45 ( complementTnotintersectT
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.17/0.45 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(complementImpComplementIntersect_type,type,
% 0.17/0.45 complementImpComplementIntersect: $o ).
% 0.17/0.45
% 0.17/0.45 thf(complementImpComplementIntersect,definition,
% 0.17/0.45 ( complementImpComplementIntersect
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.17/0.45 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(complementSubsetComplementIntersect_type,type,
% 0.17/0.45 complementSubsetComplementIntersect: $o ).
% 0.17/0.45
% 0.17/0.45 thf(complementSubsetComplementIntersect,definition,
% 0.17/0.45 ( complementSubsetComplementIntersect
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(complementInPowersetComplementIntersect_type,type,
% 0.17/0.45 complementInPowersetComplementIntersect: $o ).
% 0.17/0.45
% 0.17/0.45 thf(complementInPowersetComplementIntersect,definition,
% 0.17/0.45 ( complementInPowersetComplementIntersect
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(contraSubsetComplement_type,type,
% 0.17/0.45 contraSubsetComplement: $o ).
% 0.17/0.45
% 0.17/0.45 thf(contraSubsetComplement,definition,
% 0.17/0.45 ( contraSubsetComplement
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ Y )
% 0.17/0.45 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(complementTcontraSubset_type,type,
% 0.17/0.45 complementTcontraSubset: $o ).
% 0.17/0.45
% 0.17/0.45 thf(complementTcontraSubset,definition,
% 0.17/0.45 ( complementTcontraSubset
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.17/0.45 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(binunionTILcontra_type,type,
% 0.17/0.45 binunionTILcontra: $o ).
% 0.17/0.45
% 0.17/0.45 thf(binunionTILcontra,definition,
% 0.17/0.45 ( binunionTILcontra
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.17/0.45 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(binunionTIRcontra_type,type,
% 0.17/0.45 binunionTIRcontra: $o ).
% 0.17/0.45
% 0.17/0.45 thf(binunionTIRcontra,definition,
% 0.17/0.45 ( binunionTIRcontra
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.17/0.45 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(inIntersectImpInUnion_type,type,
% 0.17/0.45 inIntersectImpInUnion: $o ).
% 0.17/0.45
% 0.17/0.45 thf(inIntersectImpInUnion,definition,
% 0.17/0.45 ( inIntersectImpInUnion
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.17/0.45 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(inIntersectImpInUnion2_type,type,
% 0.17/0.45 inIntersectImpInUnion2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(inIntersectImpInUnion2,definition,
% 0.17/0.45 ( inIntersectImpInUnion2
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.17/0.45 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(inIntersectImpInIntersectUnions_type,type,
% 0.17/0.45 inIntersectImpInIntersectUnions: $o ).
% 0.17/0.45
% 0.17/0.45 thf(inIntersectImpInIntersectUnions,definition,
% 0.17/0.45 ( inIntersectImpInIntersectUnions
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.17/0.45 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(intersectInPowersetIntersectUnions_type,type,
% 0.17/0.45 intersectInPowersetIntersectUnions: $o ).
% 0.17/0.45
% 0.17/0.45 thf(intersectInPowersetIntersectUnions,definition,
% 0.17/0.45 ( intersectInPowersetIntersectUnions
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(inComplementUnionImpNotIn1_type,type,
% 0.17/0.45 inComplementUnionImpNotIn1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(inComplementUnionImpNotIn1,definition,
% 0.17/0.45 ( inComplementUnionImpNotIn1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.17/0.45 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(inComplementUnionImpInComplement1_type,type,
% 0.17/0.45 inComplementUnionImpInComplement1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(inComplementUnionImpInComplement1,definition,
% 0.17/0.45 ( inComplementUnionImpInComplement1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.17/0.45 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(binunionTE_type,type,
% 0.17/0.45 binunionTE: $o ).
% 0.17/0.45
% 0.17/0.45 thf(binunionTE,definition,
% 0.17/0.45 ( binunionTE
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xphi: $o,Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.17/0.45 => ( ( ( in @ Xx @ X )
% 0.17/0.45 => Xphi )
% 0.17/0.45 => ( ( ( in @ Xx @ Y )
% 0.17/0.45 => Xphi )
% 0.17/0.45 => Xphi ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(binunionTEcontra_type,type,
% 0.17/0.45 binunionTEcontra: $o ).
% 0.17/0.45
% 0.17/0.45 thf(binunionTEcontra,definition,
% 0.17/0.45 ( binunionTEcontra
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ~ ( in @ Xx @ X )
% 0.17/0.45 => ( ~ ( in @ Xx @ Y )
% 0.17/0.45 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2a1_type,type,
% 0.17/0.45 demorgan2a1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2a1,definition,
% 0.17/0.45 ( demorgan2a1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.17/0.45 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(complementUnionInPowersetComplement_type,type,
% 0.17/0.45 complementUnionInPowersetComplement: $o ).
% 0.17/0.45
% 0.17/0.45 thf(complementUnionInPowersetComplement,definition,
% 0.17/0.45 ( complementUnionInPowersetComplement
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2a2_type,type,
% 0.17/0.45 demorgan2a2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2a2,definition,
% 0.17/0.45 ( demorgan2a2
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.17/0.45 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan1a_type,type,
% 0.17/0.45 demorgan1a: $o ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan1a,definition,
% 0.17/0.45 ( demorgan1a
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.17/0.45 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan1b_type,type,
% 0.17/0.45 demorgan1b: $o ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan1b,definition,
% 0.17/0.45 ( demorgan1b
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.17/0.45 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan1_type,type,
% 0.17/0.45 demorgan1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan1,definition,
% 0.17/0.45 ( demorgan1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.17/0.45 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2a_type,type,
% 0.17/0.45 demorgan2a: $o ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2a,definition,
% 0.17/0.45 ( demorgan2a
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.17/0.45 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2b2_type,type,
% 0.17/0.45 demorgan2b2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2b2,definition,
% 0.17/0.45 ( demorgan2b2
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.17/0.45 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.17/0.45 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2b_type,type,
% 0.17/0.45 demorgan2b: $o ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2b,definition,
% 0.17/0.45 ( demorgan2b
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.17/0.45 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2_type,type,
% 0.17/0.45 demorgan2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(demorgan2,definition,
% 0.17/0.45 ( demorgan2
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.17/0.45 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule0_type,type,
% 0.17/0.45 woz13rule0: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule0,definition,
% 0.17/0.45 ( woz13rule0
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.17/0.45 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule1_type,type,
% 0.17/0.45 woz13rule1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule1,definition,
% 0.17/0.45 ( woz13rule1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ( ( subset @ X @ Z )
% 0.17/0.45 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule2_type,type,
% 0.17/0.45 woz13rule2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule2,definition,
% 0.17/0.45 ( woz13rule2
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ( ( subset @ Y @ Z )
% 0.17/0.45 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule3_type,type,
% 0.17/0.45 woz13rule3: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule3,definition,
% 0.17/0.45 ( woz13rule3
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ( ( subset @ X @ Y )
% 0.17/0.45 => ( ( subset @ X @ Z )
% 0.17/0.45 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule4_type,type,
% 0.17/0.45 woz13rule4: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz13rule4,definition,
% 0.17/0.45 ( woz13rule4
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ! [W: $i] :
% 0.17/0.45 ( ( in @ W @ ( powerset @ A ) )
% 0.17/0.45 => ( ( subset @ X @ Z )
% 0.17/0.45 => ( ( subset @ Y @ W )
% 0.17/0.45 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_1_type,type,
% 0.17/0.45 woz1_1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_1,definition,
% 0.17/0.45 ( woz1_1
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_2_type,type,
% 0.17/0.45 woz1_2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_2,definition,
% 0.17/0.45 ( woz1_2
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ! [W: $i] :
% 0.17/0.45 ( ( in @ W @ ( powerset @ A ) )
% 0.17/0.45 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.17/0.45 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_3_type,type,
% 0.17/0.45 woz1_3: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_3,definition,
% 0.17/0.45 ( woz1_3
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ! [Z: $i] :
% 0.17/0.45 ( ( in @ Z @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_4_type,type,
% 0.17/0.45 woz1_4: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_4,definition,
% 0.17/0.45 ( woz1_4
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.17/0.45 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_5_type,type,
% 0.17/0.45 woz1_5: $o ).
% 0.17/0.45
% 0.17/0.45 thf(woz1_5,definition,
% 0.17/0.45 ( woz1_5
% 0.17/0.45 = ( ! [A: $i,X: $i] :
% 0.17/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.17/0.45 => ! [Y: $i] :
% 0.17/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.17/0.45 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln1_type,type,
% 0.17/0.45 breln1: $i > $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(breln1,definition,
% 0.17/0.45 ( breln1
% 0.17/0.45 = ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln1all2_type,type,
% 0.17/0.45 breln1all2: $o ).
% 0.17/0.45
% 0.17/0.45 thf(breln1all2,definition,
% 0.17/0.45 ( breln1all2
% 0.17/0.45 = ( ! [A: $i,R: $i] :
% 0.17/0.45 ( ( breln1 @ A @ R )
% 0.17/0.45 => ! [Xphi: $i > $o] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.45 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ R )
% 0.17/0.45 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln1Set_type,type,
% 0.17/0.45 breln1Set: $i > $i ).
% 0.17/0.45
% 0.17/0.45 thf(breln1SetBreln1_type,type,
% 0.17/0.45 breln1SetBreln1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(breln1SetBreln1,definition,
% 0.17/0.45 ( breln1SetBreln1
% 0.17/0.45 = ( ! [A: $i,R: $i] :
% 0.17/0.45 ( ( in @ R @ ( breln1Set @ A ) )
% 0.17/0.45 => ( breln1 @ A @ R ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(transitive_type,type,
% 0.17/0.45 transitive: $i > $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(antisymmetric_type,type,
% 0.17/0.45 antisymmetric: $i > $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(reflexive_type,type,
% 0.17/0.45 reflexive: $i > $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(refltransitive_type,type,
% 0.17/0.45 refltransitive: $i > $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(refllinearorder_type,type,
% 0.17/0.45 refllinearorder: $i > $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(reflwellordering_type,type,
% 0.17/0.45 reflwellordering: $i > $i > $o ).
% 0.17/0.45
% 0.17/0.45 thf(choice2fnsingleton_type,type,
% 0.17/0.45 choice2fnsingleton: $o ).
% 0.17/0.45
% 0.17/0.45 thf(choice2fnsingleton,definition,
% 0.17/0.45 ( choice2fnsingleton
% 0.17/0.45 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ? [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ B )
% 0.17/0.45 & ( Xphi @ Xx @ Xy ) ) )
% 0.17/0.45 => ! [R: $i] :
% 0.17/0.45 ( ( in @ R @ ( breln1Set @ B ) )
% 0.17/0.45 => ( ( reflwellordering @ B @ R )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ( singleton
% 0.17/0.45 @ ( dsetconstr @ B
% 0.17/0.45 @ ^ [Xy: $i] :
% 0.17/0.45 ( ( Xphi @ Xx @ Xy )
% 0.17/0.45 & ! [Xz: $i] :
% 0.17/0.45 ( ( in @ Xz @ B )
% 0.17/0.45 => ( ( Xphi @ Xx @ Xz )
% 0.17/0.45 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(setOfPairsIsBReln1_type,type,
% 0.17/0.45 setOfPairsIsBReln1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(setOfPairsIsBReln1,definition,
% 0.17/0.45 ( setOfPairsIsBReln1
% 0.17/0.45 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.17/0.45 ( breln1 @ A
% 0.17/0.45 @ ( dpsetconstr @ A @ A
% 0.17/0.45 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln1all1_type,type,
% 0.17/0.45 breln1all1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(breln1all1,definition,
% 0.17/0.45 ( breln1all1
% 0.17/0.45 = ( ! [A: $i,R: $i] :
% 0.17/0.45 ( ( breln1 @ A @ R )
% 0.17/0.45 => ! [Xphi: $i > $o] :
% 0.17/0.45 ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.45 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ R )
% 0.17/0.45 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(subbreln1_type,type,
% 0.17/0.45 subbreln1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(subbreln1,definition,
% 0.17/0.45 ( subbreln1
% 0.17/0.45 = ( ! [A: $i,R: $i] :
% 0.17/0.45 ( ( breln1 @ A @ R )
% 0.17/0.45 => ! [S: $i] :
% 0.17/0.45 ( ( breln1 @ A @ S )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.45 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(eqbreln1_type,type,
% 0.17/0.45 eqbreln1: $o ).
% 0.17/0.45
% 0.17/0.45 thf(eqbreln1,definition,
% 0.17/0.45 ( eqbreln1
% 0.17/0.45 = ( ! [A: $i,R: $i] :
% 0.17/0.45 ( ( breln1 @ A @ R )
% 0.17/0.45 => ! [S: $i] :
% 0.17/0.45 ( ( breln1 @ A @ S )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.17/0.45 => ( ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.17/0.45 => ( R = S ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln1invset_type,type,
% 0.17/0.45 breln1invset: $i > $i > $i ).
% 0.17/0.45
% 0.17/0.45 thf(breln1invprop_type,type,
% 0.17/0.45 breln1invprop: $o ).
% 0.17/0.45
% 0.17/0.45 thf(breln1invprop,definition,
% 0.17/0.45 ( breln1invprop
% 0.17/0.45 = ( ! [A: $i,R: $i] :
% 0.17/0.45 ( ( breln1 @ A @ R )
% 0.17/0.45 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln1invI_type,type,
% 0.17/0.45 breln1invI: $o ).
% 0.17/0.45
% 0.17/0.45 thf(breln1invI,definition,
% 0.17/0.45 ( breln1invI
% 0.17/0.45 = ( ! [A: $i,R: $i] :
% 0.17/0.45 ( ( breln1 @ A @ R )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.17/0.45 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln1invE_type,type,
% 0.17/0.45 breln1invE: $o ).
% 0.17/0.45
% 0.17/0.45 thf(breln1invE,definition,
% 0.17/0.45 ( breln1invE
% 0.17/0.45 = ( ! [A: $i,R: $i] :
% 0.17/0.45 ( ( breln1 @ A @ R )
% 0.17/0.45 => ! [Xx: $i] :
% 0.17/0.45 ( ( in @ Xx @ A )
% 0.17/0.45 => ! [Xy: $i] :
% 0.17/0.45 ( ( in @ Xy @ A )
% 0.17/0.45 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.17/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln1compset_type,type,
% 0.17/0.45 breln1compset: $i > $i > $i > $i ).
% 0.17/0.45
% 0.17/0.45 thf(breln1compset,definition,
% 0.17/0.45 ( breln1compset
% 0.17/0.45 = ( ^ [A: $i,R: $i,S: $i] :
% 0.17/0.45 ( dpsetconstr @ A @ A
% 0.17/0.45 @ ^ [Xx: $i,Xy: $i] :
% 0.17/0.45 ? [Xz: $i] :
% 0.17/0.45 ( ( in @ Xz @ A )
% 0.17/0.45 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.17/0.45 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 thf(breln1compprop,conjecture,
% 0.17/0.45 ( setextAx
% 0.17/0.45 => ( emptysetAx
% 0.17/0.45 => ( setadjoinAx
% 0.17/0.45 => ( powersetAx
% 0.17/0.45 => ( setunionAx
% 0.17/0.45 => ( omega0Ax
% 0.17/0.45 => ( omegaSAx
% 0.17/0.45 => ( omegaIndAx
% 0.17/0.45 => ( replAx
% 0.17/0.45 => ( foundationAx
% 0.17/0.45 => ( wellorderingAx
% 0.17/0.45 => ( descrp
% 0.17/0.45 => ( dsetconstrI
% 0.17/0.45 => ( dsetconstrEL
% 0.17/0.45 => ( dsetconstrER
% 0.17/0.45 => ( exuE1
% 0.17/0.45 => ( prop2setE
% 0.17/0.45 => ( emptysetE
% 0.17/0.45 => ( emptysetimpfalse
% 0.17/0.45 => ( notinemptyset
% 0.17/0.45 => ( exuE3e
% 0.17/0.45 => ( setext
% 0.17/0.45 => ( emptyI
% 0.17/0.45 => ( noeltsimpempty
% 0.17/0.45 => ( setbeta
% 0.17/0.45 => ( nonemptyE1
% 0.17/0.45 => ( nonemptyI
% 0.17/0.45 => ( nonemptyI1
% 0.17/0.45 => ( setadjoinIL
% 0.17/0.45 => ( emptyinunitempty
% 0.17/0.45 => ( setadjoinIR
% 0.17/0.45 => ( setadjoinE
% 0.17/0.45 => ( setadjoinOr
% 0.17/0.45 => ( setoftrueEq
% 0.17/0.45 => ( powersetI
% 0.17/0.45 => ( emptyinPowerset
% 0.17/0.45 => ( emptyInPowerset
% 0.17/0.45 => ( powersetE
% 0.17/0.45 => ( setunionI
% 0.17/0.45 => ( setunionE
% 0.17/0.45 => ( subPowSU
% 0.17/0.45 => ( exuE2
% 0.17/0.45 => ( nonemptyImpWitness
% 0.17/0.45 => ( uniqinunit
% 0.17/0.45 => ( notinsingleton
% 0.17/0.45 => ( eqinunit
% 0.17/0.45 => ( singletonsswitch
% 0.17/0.45 => ( upairsetE
% 0.17/0.45 => ( upairsetIL
% 0.17/0.45 => ( upairsetIR
% 0.17/0.45 => ( emptyE1
% 0.17/0.45 => ( vacuousDall
% 0.17/0.45 => ( quantDeMorgan1
% 0.17/0.45 => ( quantDeMorgan2
% 0.17/0.45 => ( quantDeMorgan3
% 0.17/0.45 => ( quantDeMorgan4
% 0.17/0.45 => ( prop2setI
% 0.17/0.45 => ( prop2set2propI
% 0.17/0.45 => ( notdexE
% 0.17/0.45 => ( notdallE
% 0.17/0.45 => ( exuI1
% 0.17/0.45 => ( exuI3
% 0.17/0.45 => ( exuI2
% 0.17/0.45 => ( inCongP
% 0.17/0.45 => ( in__Cong
% 0.17/0.45 => ( exuE3u
% 0.17/0.45 => ( exu__Cong
% 0.17/0.45 => ( emptyset__Cong
% 0.17/0.45 => ( setadjoin__Cong
% 0.17/0.45 => ( powerset__Cong
% 0.17/0.45 => ( setunion__Cong
% 0.17/0.45 => ( omega__Cong
% 0.17/0.45 => ( exuEu
% 0.17/0.45 => ( descr__Cong
% 0.17/0.45 => ( dsetconstr__Cong
% 0.17/0.45 => ( subsetI1
% 0.17/0.45 => ( eqimpsubset2
% 0.17/0.45 => ( eqimpsubset1
% 0.17/0.45 => ( subsetI2
% 0.17/0.45 => ( emptysetsubset
% 0.17/0.45 => ( subsetE
% 0.17/0.45 => ( subsetE2
% 0.17/0.45 => ( notsubsetI
% 0.17/0.45 => ( notequalI1
% 0.17/0.45 => ( notequalI2
% 0.17/0.45 => ( subsetRefl
% 0.17/0.45 => ( subsetTrans
% 0.17/0.45 => ( setadjoinSub
% 0.17/0.45 => ( setadjoinSub2
% 0.17/0.45 => ( subset2powerset
% 0.17/0.45 => ( setextsub
% 0.17/0.45 => ( subsetemptysetimpeq
% 0.17/0.45 => ( powersetI1
% 0.17/0.45 => ( powersetE1
% 0.17/0.45 => ( inPowerset
% 0.17/0.45 => ( powersetsubset
% 0.17/0.45 => ( sepInPowerset
% 0.17/0.45 => ( sepSubset
% 0.17/0.45 => ( binunionIL
% 0.17/0.45 => ( upairset2IR
% 0.17/0.45 => ( binunionIR
% 0.17/0.45 => ( binunionEcases
% 0.17/0.45 => ( binunionE
% 0.17/0.45 => ( binunionLsub
% 0.17/0.45 => ( binunionRsub
% 0.17/0.45 => ( binintersectI
% 0.17/0.45 => ( binintersectSubset5
% 0.17/0.45 => ( binintersectEL
% 0.17/0.45 => ( binintersectLsub
% 0.17/0.45 => ( binintersectSubset2
% 0.17/0.45 => ( binintersectSubset3
% 0.17/0.45 => ( binintersectER
% 0.17/0.45 => ( disjointsetsI1
% 0.17/0.45 => ( binintersectRsub
% 0.17/0.45 => ( binintersectSubset4
% 0.17/0.45 => ( binintersectSubset1
% 0.17/0.45 => ( bs114d
% 0.17/0.45 => ( setminusI
% 0.17/0.45 => ( setminusEL
% 0.17/0.45 => ( setminusER
% 0.17/0.45 => ( setminusSubset2
% 0.17/0.45 => ( setminusERneg
% 0.17/0.45 => ( setminusELneg
% 0.17/0.45 => ( setminusILneg
% 0.17/0.45 => ( setminusIRneg
% 0.17/0.45 => ( setminusLsub
% 0.17/0.45 => ( setminusSubset1
% 0.17/0.45 => ( symdiffE
% 0.17/0.45 => ( symdiffI1
% 0.17/0.45 => ( symdiffI2
% 0.17/0.45 => ( symdiffIneg1
% 0.17/0.45 => ( symdiffIneg2
% 0.17/0.45 => ( secondinupair
% 0.17/0.45 => ( setukpairIL
% 0.17/0.45 => ( setukpairIR
% 0.17/0.45 => ( kpairiskpair
% 0.17/0.45 => ( kpairp
% 0.17/0.45 => ( singletonsubset
% 0.17/0.45 => ( singletoninpowerset
% 0.17/0.45 => ( singletoninpowunion
% 0.17/0.45 => ( upairset2E
% 0.17/0.45 => ( upairsubunion
% 0.17/0.45 => ( upairinpowunion
% 0.17/0.45 => ( ubforcartprodlem1
% 0.17/0.45 => ( ubforcartprodlem2
% 0.17/0.45 => ( ubforcartprodlem3
% 0.17/0.45 => ( cartprodpairin
% 0.17/0.45 => ( cartprodmempair1
% 0.17/0.45 => ( cartprodmempair
% 0.17/0.45 => ( setunionE2
% 0.17/0.45 => ( setunionsingleton1
% 0.17/0.45 => ( setunionsingleton2
% 0.17/0.45 => ( setunionsingleton
% 0.17/0.45 => ( singletonprop
% 0.17/0.45 => ( ex1E1
% 0.17/0.45 => ( ex1I
% 0.17/0.45 => ( ex1I2
% 0.17/0.45 => ( singletonsuniq
% 0.17/0.45 => ( setukpairinjL1
% 0.17/0.45 => ( kfstsingleton
% 0.17/0.45 => ( theprop
% 0.17/0.45 => ( kfstpairEq
% 0.17/0.45 => ( cartprodfstin
% 0.17/0.45 => ( setukpairinjL2
% 0.17/0.45 => ( setukpairinjL
% 0.17/0.45 => ( setukpairinjR11
% 0.17/0.45 => ( setukpairinjR12
% 0.17/0.45 => ( setukpairinjR1
% 0.17/0.45 => ( upairequniteq
% 0.17/0.45 => ( setukpairinjR2
% 0.17/0.45 => ( setukpairinjR
% 0.17/0.45 => ( ksndsingleton
% 0.17/0.45 => ( ksndpairEq
% 0.17/0.45 => ( kpairsurjEq
% 0.17/0.45 => ( cartprodsndin
% 0.17/0.45 => ( cartprodpairmemEL
% 0.17/0.45 => ( cartprodpairmemER
% 0.17/0.45 => ( cartprodmempaircEq
% 0.17/0.45 => ( cartprodfstpairEq
% 0.17/0.45 => ( cartprodsndpairEq
% 0.17/0.45 => ( cartprodpairsurjEq
% 0.17/0.45 => ( dpsetconstrI
% 0.17/0.45 => ( dpsetconstrSub
% 0.17/0.45 => ( setOfPairsIsBReln
% 0.17/0.45 => ( dpsetconstrERa
% 0.17/0.45 => ( dpsetconstrEL1
% 0.17/0.45 => ( dpsetconstrEL2
% 0.17/0.45 => ( dpsetconstrER
% 0.17/0.45 => ( funcImageSingleton
% 0.17/0.45 => ( apProp
% 0.17/0.45 => ( app
% 0.17/0.45 => ( infuncsetfunc
% 0.17/0.45 => ( ap2p
% 0.17/0.45 => ( funcinfuncset
% 0.17/0.45 => ( lamProp
% 0.17/0.45 => ( lamp
% 0.17/0.45 => ( lam2p
% 0.17/0.45 => ( brelnall1
% 0.17/0.45 => ( brelnall2
% 0.17/0.45 => ( ex1E2
% 0.17/0.45 => ( funcGraphProp1
% 0.17/0.45 => ( funcGraphProp3
% 0.17/0.45 => ( funcGraphProp2
% 0.17/0.45 => ( funcextLem
% 0.17/0.45 => ( funcGraphProp4
% 0.17/0.45 => ( subbreln
% 0.17/0.45 => ( eqbreln
% 0.17/0.45 => ( funcext
% 0.17/0.45 => ( funcext2
% 0.17/0.45 => ( ap2apEq1
% 0.17/0.45 => ( ap2apEq2
% 0.17/0.45 => ( beta1
% 0.17/0.45 => ( eta1
% 0.17/0.45 => ( lam2lamEq
% 0.17/0.45 => ( beta2
% 0.17/0.45 => ( eta2
% 0.17/0.45 => ( iffalseProp1
% 0.17/0.45 => ( iffalseProp2
% 0.17/0.45 => ( iftrueProp1
% 0.17/0.45 => ( iftrueProp2
% 0.17/0.45 => ( ifSingleton
% 0.17/0.45 => ( ifp
% 0.17/0.45 => ( theeq
% 0.17/0.45 => ( iftrue
% 0.17/0.45 => ( iffalse
% 0.17/0.45 => ( iftrueorfalse
% 0.17/0.45 => ( binintersectT_lem
% 0.17/0.45 => ( binunionT_lem
% 0.17/0.45 => ( powersetT_lem
% 0.17/0.45 => ( setminusT_lem
% 0.17/0.45 => ( complementT_lem
% 0.17/0.45 => ( setextT
% 0.17/0.45 => ( subsetTI
% 0.17/0.45 => ( powersetTI1
% 0.17/0.45 => ( powersetTE1
% 0.17/0.45 => ( complementTI1
% 0.17/0.45 => ( complementTE1
% 0.17/0.45 => ( binintersectTELcontra
% 0.17/0.45 => ( binintersectTERcontra
% 0.17/0.45 => ( contrasubsetT
% 0.17/0.45 => ( contrasubsetT1
% 0.17/0.45 => ( contrasubsetT2
% 0.17/0.45 => ( contrasubsetT3
% 0.17/0.45 => ( doubleComplementI1
% 0.17/0.45 => ( doubleComplementE1
% 0.17/0.45 => ( doubleComplementSub1
% 0.17/0.45 => ( doubleComplementSub2
% 0.17/0.45 => ( doubleComplementEq
% 0.17/0.45 => ( complementTnotintersectT
% 0.17/0.45 => ( complementImpComplementIntersect
% 0.17/0.45 => ( complementSubsetComplementIntersect
% 0.17/0.45 => ( complementInPowersetComplementIntersect
% 0.17/0.45 => ( contraSubsetComplement
% 0.17/0.45 => ( complementTcontraSubset
% 0.17/0.45 => ( binunionTILcontra
% 0.17/0.45 => ( binunionTIRcontra
% 0.17/0.45 => ( inIntersectImpInUnion
% 0.17/0.45 => ( inIntersectImpInUnion2
% 0.17/0.45 => ( inIntersectImpInIntersectUnions
% 0.17/0.45 => ( intersectInPowersetIntersectUnions
% 0.17/0.45 => ( inComplementUnionImpNotIn1
% 0.17/0.45 => ( inComplementUnionImpInComplement1
% 0.17/0.45 => ( binunionTE
% 0.17/0.45 => ( binunionTEcontra
% 0.17/0.45 => ( demorgan2a1
% 0.17/0.45 => ( complementUnionInPowersetComplement
% 0.17/0.45 => ( demorgan2a2
% 0.17/0.45 => ( demorgan1a
% 0.17/0.45 => ( demorgan1b
% 0.17/0.45 => ( demorgan1
% 0.17/0.45 => ( demorgan2a
% 0.17/0.45 => ( demorgan2b2
% 0.17/0.45 => ( demorgan2b
% 0.17/0.45 => ( demorgan2
% 0.17/0.45 => ( woz13rule0
% 0.17/0.45 => ( woz13rule1
% 0.17/0.45 => ( woz13rule2
% 0.17/0.45 => ( woz13rule3
% 0.17/0.45 => ( woz13rule4
% 0.17/0.45 => ( woz1_1
% 0.17/0.45 => ( woz1_2
% 0.17/0.45 => ( woz1_3
% 0.17/0.45 => ( woz1_4
% 0.17/0.45 => ( woz1_5
% 0.17/0.45 => ( breln1all2
% 0.17/0.45 => ( breln1SetBreln1
% 0.17/0.45 => ( choice2fnsingleton
% 0.17/0.45 => ( setOfPairsIsBReln1
% 0.17/0.45 => ( breln1all1
% 0.17/0.45 => ( subbreln1
% 0.17/0.45 => ( eqbreln1
% 0.17/0.45 => ( breln1invprop
% 0.17/0.45 => ( breln1invI
% 0.17/0.45 => ( breln1invE
% 0.17/0.45 => ! [A: $i,R: $i] :
% 0.17/0.45 ( ( breln1 @ A @ R )
% 0.17/0.45 => ! [S: $i] :
% 0.17/0.45 ( ( breln1 @ A @ S )
% 0.17/0.45 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.45
% 0.17/0.45 %------------------------------------------------------------------------------
% 0.17/0.45 ------------------------
% 3.03/1.56 ---- Embedded in THF ---
% 3.03/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.03/1.56 %%% Generated on Tue Feb 24 12:33:30 EST 2026
% 3.03/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 3.03/1.56 %%% Logic specification used:
% 3.03/1.56 %%% thf(spec, logic, $$dhol).
% 3.03/1.56
% 3.03/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.TJ114YmswG/DTF2DTF22699.p
% See solution above
% 3.03/1.56 ------------------------
% 3.17/1.60 ---- Cleaned THF ---
% 3.17/1.60 thf(in_type,type,
% 3.17/1.60 in: $i > $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(exu_type,type,
% 3.17/1.60 exu: ( $i > $o ) > $o ).
% 3.17/1.60
% 3.17/1.60 thf(exu,definition,
% 3.17/1.60 ( exu
% 3.17/1.60 = ( ^ [Xphi: $i > $o] :
% 3.17/1.60 ? [Xx: $i] :
% 3.17/1.60 ( ( Xphi @ Xx )
% 3.17/1.60 & ! [Xy: $i] :
% 3.17/1.60 ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setextAx_type,type,
% 3.17/1.60 setextAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setextAx,definition,
% 3.17/1.60 ( setextAx
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 <=> ( in @ Xx @ B ) )
% 3.17/1.60 => ( A = B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptyset_type,type,
% 3.17/1.60 emptyset: $i ).
% 3.17/1.60
% 3.17/1.60 thf(emptysetAx_type,type,
% 3.17/1.60 emptysetAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptysetAx,definition,
% 3.17/1.60 ( emptysetAx
% 3.17/1.60 = ( ! [Xx: $i] :
% 3.17/1.60 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoin_type,type,
% 3.17/1.60 setadjoin: $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinAx_type,type,
% 3.17/1.60 setadjoinAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinAx,definition,
% 3.17/1.60 ( setadjoinAx
% 3.17/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.17/1.60 <=> ( ( Xy = Xx )
% 3.17/1.60 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(powerset_type,type,
% 3.17/1.60 powerset: $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(powersetAx_type,type,
% 3.17/1.60 powersetAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(powersetAx,definition,
% 3.17/1.60 ( powersetAx
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( in @ B @ ( powerset @ A ) )
% 3.17/1.60 <=> ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setunion_type,type,
% 3.17/1.60 setunion: $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(setunionAx_type,type,
% 3.17/1.60 setunionAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setunionAx,definition,
% 3.17/1.60 ( setunionAx
% 3.17/1.60 = ( ! [A: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( setunion @ A ) )
% 3.17/1.60 <=> ? [B: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 & ( in @ B @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(omega_type,type,
% 3.17/1.60 omega: $i ).
% 3.17/1.60
% 3.17/1.60 thf(omega0Ax_type,type,
% 3.17/1.60 omega0Ax: $o ).
% 3.17/1.60
% 3.17/1.60 thf(omega0Ax,definition,
% 3.17/1.60 ( omega0Ax
% 3.17/1.60 = ( in @ emptyset @ omega ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(omegaSAx_type,type,
% 3.17/1.60 omegaSAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(omegaSAx,definition,
% 3.17/1.60 ( omegaSAx
% 3.17/1.60 = ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ omega )
% 3.17/1.60 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(omegaIndAx_type,type,
% 3.17/1.60 omegaIndAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(omegaIndAx,definition,
% 3.17/1.60 ( omegaIndAx
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ( ( ( in @ emptyset @ A )
% 3.17/1.60 & ! [Xx: $i] :
% 3.17/1.60 ( ( ( in @ Xx @ omega )
% 3.17/1.60 & ( in @ Xx @ A ) )
% 3.17/1.60 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ omega )
% 3.17/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(replAx_type,type,
% 3.17/1.60 replAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(replAx,definition,
% 3.17/1.60 ( replAx
% 3.17/1.60 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( exu
% 3.17/1.60 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.17/1.60 => ? [B: $i] :
% 3.17/1.60 ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 <=> ? [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(foundationAx_type,type,
% 3.17/1.60 foundationAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(foundationAx,definition,
% 3.17/1.60 ( foundationAx
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.17/1.60 => ? [B: $i] :
% 3.17/1.60 ( ( in @ B @ A )
% 3.17/1.60 & ~ ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(wellorderingAx_type,type,
% 3.17/1.60 wellorderingAx: $o ).
% 3.17/1.60
% 3.17/1.60 thf(wellorderingAx,definition,
% 3.17/1.60 ( wellorderingAx
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ? [B: $i] :
% 3.17/1.60 ( ! [C: $i] :
% 3.17/1.60 ( ( in @ C @ B )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ C )
% 3.17/1.60 => ( in @ Xx @ A ) ) )
% 3.17/1.60 & ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( ( in @ Xx @ A )
% 3.17/1.60 & ( in @ Xy @ A ) )
% 3.17/1.60 => ( ! [C: $i] :
% 3.17/1.60 ( ( in @ C @ B )
% 3.17/1.60 => ( ( in @ Xx @ C )
% 3.17/1.60 <=> ( in @ Xy @ C ) ) )
% 3.17/1.60 => ( Xx = Xy ) ) )
% 3.17/1.60 & ! [C: $i,D: $i] :
% 3.17/1.60 ( ( ( in @ C @ B )
% 3.17/1.60 & ( in @ D @ B ) )
% 3.17/1.60 => ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ C )
% 3.17/1.60 => ( in @ Xx @ D ) )
% 3.17/1.60 | ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ D )
% 3.17/1.60 => ( in @ Xx @ C ) ) ) )
% 3.17/1.60 & ! [C: $i] :
% 3.17/1.60 ( ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ C )
% 3.17/1.60 => ( in @ Xx @ A ) )
% 3.17/1.60 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.17/1.60 => ? [D: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ D @ B )
% 3.17/1.60 & ( in @ Xx @ C )
% 3.17/1.60 & ~ ? [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ D )
% 3.17/1.60 & ( in @ Xy @ C ) )
% 3.17/1.60 & ! [E: $i] :
% 3.17/1.60 ( ( in @ E @ B )
% 3.17/1.60 => ( ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ E )
% 3.17/1.60 => ( in @ Xy @ D ) )
% 3.17/1.60 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(descr_type,type,
% 3.17/1.60 descr: ( $i > $o ) > $i ).
% 3.17/1.60
% 3.17/1.60 thf(descrp_type,type,
% 3.17/1.60 descrp: $o ).
% 3.17/1.60
% 3.17/1.60 thf(descrp,definition,
% 3.17/1.60 ( descrp
% 3.17/1.60 = ( ! [Xphi: $i > $o] :
% 3.17/1.60 ( ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 => ( Xphi
% 3.17/1.60 @ ( descr
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dsetconstr_type,type,
% 3.17/1.60 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.17/1.60
% 3.17/1.60 thf(dsetconstrI_type,type,
% 3.17/1.60 dsetconstrI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dsetconstrI,definition,
% 3.17/1.60 ( dsetconstrI
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( Xphi @ Xx )
% 3.17/1.60 => ( in @ Xx
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dsetconstrEL_type,type,
% 3.17/1.60 dsetconstrEL: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dsetconstrEL,definition,
% 3.17/1.60 ( dsetconstrEL
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.17/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dsetconstrER_type,type,
% 3.17/1.60 dsetconstrER: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dsetconstrER,definition,
% 3.17/1.60 ( dsetconstrER
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.17/1.60 => ( Xphi @ Xx ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(exuE1_type,type,
% 3.17/1.60 exuE1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(exuE1,definition,
% 3.17/1.60 ( exuE1
% 3.17/1.60 = ( ! [Xphi: $i > $o] :
% 3.17/1.60 ( ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 => ? [Xx: $i] :
% 3.17/1.60 ( ( Xphi @ Xx )
% 3.17/1.60 & ! [Xy: $i] :
% 3.17/1.60 ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(prop2set_type,type,
% 3.17/1.60 prop2set: $o > $i ).
% 3.17/1.60
% 3.17/1.60 thf(prop2setE_type,type,
% 3.17/1.60 prop2setE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(prop2setE,definition,
% 3.17/1.60 ( prop2setE
% 3.17/1.60 = ( ! [Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.17/1.60 => Xphi ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptysetE_type,type,
% 3.17/1.60 emptysetE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptysetE,definition,
% 3.17/1.60 ( emptysetE
% 3.17/1.60 = ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ emptyset )
% 3.17/1.60 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptysetimpfalse_type,type,
% 3.17/1.60 emptysetimpfalse: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptysetimpfalse,definition,
% 3.17/1.60 ( emptysetimpfalse
% 3.17/1.60 = ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ emptyset )
% 3.17/1.60 => $false ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(notinemptyset_type,type,
% 3.17/1.60 notinemptyset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(notinemptyset,definition,
% 3.17/1.60 ( notinemptyset
% 3.17/1.60 = ( ! [Xx: $i] :
% 3.17/1.60 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(exuE3e_type,type,
% 3.17/1.60 exuE3e: $o ).
% 3.17/1.60
% 3.17/1.60 thf(exuE3e,definition,
% 3.17/1.60 ( exuE3e
% 3.17/1.60 = ( ! [Xphi: $i > $o] :
% 3.17/1.60 ( ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setext_type,type,
% 3.17/1.60 setext: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setext,definition,
% 3.17/1.60 ( setext
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ Xx @ B ) )
% 3.17/1.60 => ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 => ( in @ Xx @ A ) )
% 3.17/1.60 => ( A = B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptyI_type,type,
% 3.17/1.60 emptyI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptyI,definition,
% 3.17/1.60 ( emptyI
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ~ ( in @ Xx @ A )
% 3.17/1.60 => ( A = emptyset ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(noeltsimpempty_type,type,
% 3.17/1.60 noeltsimpempty: $o ).
% 3.17/1.60
% 3.17/1.60 thf(noeltsimpempty,definition,
% 3.17/1.60 ( noeltsimpempty
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ~ ( in @ Xx @ A )
% 3.17/1.60 => ( A = emptyset ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setbeta_type,type,
% 3.17/1.60 setbeta: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setbeta,definition,
% 3.17/1.60 ( setbeta
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( in @ Xx
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.17/1.60 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(nonempty_type,type,
% 3.17/1.60 nonempty: $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(nonempty,definition,
% 3.17/1.60 ( nonempty
% 3.17/1.60 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(nonemptyE1_type,type,
% 3.17/1.60 nonemptyE1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(nonemptyE1,definition,
% 3.17/1.60 ( nonemptyE1
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ( ( nonempty @ A )
% 3.17/1.60 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(nonemptyI_type,type,
% 3.17/1.60 nonemptyI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(nonemptyI,definition,
% 3.17/1.60 ( nonemptyI
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( Xphi @ Xx )
% 3.17/1.60 => ( nonempty
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(nonemptyI1_type,type,
% 3.17/1.60 nonemptyI1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(nonemptyI1,definition,
% 3.17/1.60 ( nonemptyI1
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.17/1.60 => ( nonempty @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinIL_type,type,
% 3.17/1.60 setadjoinIL: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinIL,definition,
% 3.17/1.60 ( setadjoinIL
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptyinunitempty_type,type,
% 3.17/1.60 emptyinunitempty: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptyinunitempty,definition,
% 3.17/1.60 ( emptyinunitempty
% 3.17/1.60 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinIR_type,type,
% 3.17/1.60 setadjoinIR: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinIR,definition,
% 3.17/1.60 ( setadjoinIR
% 3.17/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinE_type,type,
% 3.17/1.60 setadjoinE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinE,definition,
% 3.17/1.60 ( setadjoinE
% 3.17/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.17/1.60 => ! [Xphi: $o] :
% 3.17/1.60 ( ( ( Xy = Xx )
% 3.17/1.60 => Xphi )
% 3.17/1.60 => ( ( ( in @ Xy @ A )
% 3.17/1.60 => Xphi )
% 3.17/1.60 => Xphi ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinOr_type,type,
% 3.17/1.60 setadjoinOr: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinOr,definition,
% 3.17/1.60 ( setadjoinOr
% 3.17/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.17/1.60 => ( ( Xy = Xx )
% 3.17/1.60 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setoftrueEq_type,type,
% 3.17/1.60 setoftrueEq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setoftrueEq,definition,
% 3.17/1.60 ( setoftrueEq
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ( ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xx: $i] : $true )
% 3.17/1.60 = A ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(powersetI_type,type,
% 3.17/1.60 powersetI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(powersetI,definition,
% 3.17/1.60 ( powersetI
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 => ( in @ Xx @ A ) )
% 3.17/1.60 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptyinPowerset_type,type,
% 3.17/1.60 emptyinPowerset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptyinPowerset,definition,
% 3.17/1.60 ( emptyinPowerset
% 3.17/1.60 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptyInPowerset_type,type,
% 3.17/1.60 emptyInPowerset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptyInPowerset,definition,
% 3.17/1.60 ( emptyInPowerset
% 3.17/1.60 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(powersetE_type,type,
% 3.17/1.60 powersetE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(powersetE,definition,
% 3.17/1.60 ( powersetE
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ B @ ( powerset @ A ) )
% 3.17/1.60 => ( ( in @ Xx @ B )
% 3.17/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setunionI_type,type,
% 3.17/1.60 setunionI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setunionI,definition,
% 3.17/1.60 ( setunionI
% 3.17/1.60 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 => ( ( in @ B @ A )
% 3.17/1.60 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setunionE_type,type,
% 3.17/1.60 setunionE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setunionE,definition,
% 3.17/1.60 ( setunionE
% 3.17/1.60 = ( ! [A: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( setunion @ A ) )
% 3.17/1.60 => ! [Xphi: $o] :
% 3.17/1.60 ( ! [B: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 => ( ( in @ B @ A )
% 3.17/1.60 => Xphi ) )
% 3.17/1.60 => Xphi ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subPowSU_type,type,
% 3.17/1.60 subPowSU: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subPowSU,definition,
% 3.17/1.60 ( subPowSU
% 3.17/1.60 = ( ! [A: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(exuE2_type,type,
% 3.17/1.60 exuE2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(exuE2,definition,
% 3.17/1.60 ( exuE2
% 3.17/1.60 = ( ! [Xphi: $i > $o] :
% 3.17/1.60 ( ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 => ? [Xx: $i] :
% 3.17/1.60 ! [Xy: $i] :
% 3.17/1.60 ( ( Xphi @ Xy )
% 3.17/1.60 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(nonemptyImpWitness_type,type,
% 3.17/1.60 nonemptyImpWitness: $o ).
% 3.17/1.60
% 3.17/1.60 thf(nonemptyImpWitness,definition,
% 3.17/1.60 ( nonemptyImpWitness
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ( ( nonempty @ A )
% 3.17/1.60 => ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & $true ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(uniqinunit_type,type,
% 3.17/1.60 uniqinunit: $o ).
% 3.17/1.60
% 3.17/1.60 thf(uniqinunit,definition,
% 3.17/1.60 ( uniqinunit
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(notinsingleton_type,type,
% 3.17/1.60 notinsingleton: $o ).
% 3.17/1.60
% 3.17/1.60 thf(notinsingleton,definition,
% 3.17/1.60 ( notinsingleton
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xx != Xy )
% 3.17/1.60 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(eqinunit_type,type,
% 3.17/1.60 eqinunit: $o ).
% 3.17/1.60
% 3.17/1.60 thf(eqinunit,definition,
% 3.17/1.60 ( eqinunit
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xx = Xy )
% 3.17/1.60 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(singletonsswitch_type,type,
% 3.17/1.60 singletonsswitch: $o ).
% 3.17/1.60
% 3.17/1.60 thf(singletonsswitch,definition,
% 3.17/1.60 ( singletonsswitch
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.60 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(upairsetE_type,type,
% 3.17/1.60 upairsetE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(upairsetE,definition,
% 3.17/1.60 ( upairsetE
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.17/1.60 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.17/1.60 => ( ( Xz = Xx )
% 3.17/1.60 | ( Xz = Xy ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(upairsetIL_type,type,
% 3.17/1.60 upairsetIL: $o ).
% 3.17/1.60
% 3.17/1.60 thf(upairsetIL,definition,
% 3.17/1.60 ( upairsetIL
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(upairsetIR_type,type,
% 3.17/1.60 upairsetIR: $o ).
% 3.17/1.60
% 3.17/1.60 thf(upairsetIR,definition,
% 3.17/1.60 ( upairsetIR
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptyE1_type,type,
% 3.17/1.60 emptyE1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptyE1,definition,
% 3.17/1.60 ( emptyE1
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ( Xphi @ Xx ) )
% 3.17/1.60 => ( ( ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 = emptyset )
% 3.17/1.60 => $false ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(vacuousDall_type,type,
% 3.17/1.60 vacuousDall: $o ).
% 3.17/1.60
% 3.17/1.60 thf(vacuousDall,definition,
% 3.17/1.60 ( vacuousDall
% 3.17/1.60 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ emptyset )
% 3.17/1.60 => ( Xphi @ Xx ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(quantDeMorgan1_type,type,
% 3.17/1.60 quantDeMorgan1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(quantDeMorgan1,definition,
% 3.17/1.60 ( quantDeMorgan1
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ~ ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( Xphi @ Xx ) )
% 3.17/1.60 => ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(quantDeMorgan2_type,type,
% 3.17/1.60 quantDeMorgan2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(quantDeMorgan2,definition,
% 3.17/1.60 ( quantDeMorgan2
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ~ ( Xphi @ Xx ) )
% 3.17/1.60 => ~ ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(quantDeMorgan3_type,type,
% 3.17/1.60 quantDeMorgan3: $o ).
% 3.17/1.60
% 3.17/1.60 thf(quantDeMorgan3,definition,
% 3.17/1.60 ( quantDeMorgan3
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ~ ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ( Xphi @ Xx ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(quantDeMorgan4_type,type,
% 3.17/1.60 quantDeMorgan4: $o ).
% 3.17/1.60
% 3.17/1.60 thf(quantDeMorgan4,definition,
% 3.17/1.60 ( quantDeMorgan4
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ~ ( Xphi @ Xx ) )
% 3.17/1.60 => ~ ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(prop2setI_type,type,
% 3.17/1.60 prop2setI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(prop2setI,definition,
% 3.17/1.60 ( prop2setI
% 3.17/1.60 = ( ! [Xphi: $o] :
% 3.17/1.60 ( Xphi
% 3.17/1.60 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(set2prop_type,type,
% 3.17/1.60 set2prop: $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(prop2set2propI_type,type,
% 3.17/1.60 prop2set2propI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(prop2set2propI,definition,
% 3.17/1.60 ( prop2set2propI
% 3.17/1.60 = ( ! [Xphi: $o] :
% 3.17/1.60 ( Xphi
% 3.17/1.60 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(notdexE_type,type,
% 3.17/1.60 notdexE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(notdexE,definition,
% 3.17/1.60 ( notdexE
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ~ ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ( Xphi @ Xx ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(notdallE_type,type,
% 3.17/1.60 notdallE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(notdallE,definition,
% 3.17/1.60 ( notdallE
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ~ ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( Xphi @ Xx ) )
% 3.17/1.60 => ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(exuI1_type,type,
% 3.17/1.60 exuI1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(exuI1,definition,
% 3.17/1.60 ( exuI1
% 3.17/1.60 = ( ! [Xphi: $i > $o] :
% 3.17/1.60 ( ? [Xx: $i] :
% 3.17/1.60 ( ( Xphi @ Xx )
% 3.17/1.60 & ! [Xy: $i] :
% 3.17/1.60 ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xx = Xy ) ) )
% 3.17/1.60 => ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(exuI3_type,type,
% 3.17/1.60 exuI3: $o ).
% 3.17/1.60
% 3.17/1.60 thf(exuI3,definition,
% 3.17/1.60 ( exuI3
% 3.17/1.60 = ( ! [Xphi: $i > $o] :
% 3.17/1.60 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.17/1.60 => ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xphi @ Xx )
% 3.17/1.60 => ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xx = Xy ) ) )
% 3.17/1.60 => ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(exuI2_type,type,
% 3.17/1.60 exuI2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(exuI2,definition,
% 3.17/1.60 ( exuI2
% 3.17/1.60 = ( ! [Xphi: $i > $o] :
% 3.17/1.60 ( ? [Xx: $i] :
% 3.17/1.60 ! [Xy: $i] :
% 3.17/1.60 ( ( Xphi @ Xy )
% 3.17/1.60 <=> ( Xy = Xx ) )
% 3.17/1.60 => ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(inCongP_type,type,
% 3.17/1.60 inCongP: $o ).
% 3.17/1.60
% 3.17/1.60 thf(inCongP,definition,
% 3.17/1.60 ( inCongP
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( A = B )
% 3.17/1.60 => ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xx = Xy )
% 3.17/1.60 => ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(in__Cong_type,type,
% 3.17/1.60 in__Cong: $o ).
% 3.17/1.60
% 3.17/1.60 thf(in__Cong,definition,
% 3.17/1.60 ( in__Cong
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( A = B )
% 3.17/1.60 => ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xx = Xy )
% 3.17/1.60 => ( ( in @ Xx @ A )
% 3.17/1.60 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(exuE3u_type,type,
% 3.17/1.60 exuE3u: $o ).
% 3.17/1.60
% 3.17/1.60 thf(exuE3u,definition,
% 3.17/1.60 ( exuE3u
% 3.17/1.60 = ( ! [Xphi: $i > $o] :
% 3.17/1.60 ( ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 => ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xphi @ Xx )
% 3.17/1.60 => ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(exu__Cong_type,type,
% 3.17/1.60 exu__Cong: $o ).
% 3.17/1.60
% 3.17/1.60 thf(exu__Cong,definition,
% 3.17/1.60 ( exu__Cong
% 3.17/1.60 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.17/1.60 ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xx = Xy )
% 3.17/1.60 => ( ( Xphi @ Xx )
% 3.17/1.60 <=> ( Xpsi @ Xy ) ) )
% 3.17/1.60 => ( ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 <=> ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptyset__Cong_type,type,
% 3.17/1.60 emptyset__Cong: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptyset__Cong,definition,
% 3.17/1.60 ( emptyset__Cong
% 3.17/1.60 = ( emptyset = emptyset ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoin__Cong_type,type,
% 3.17/1.60 setadjoin__Cong: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoin__Cong,definition,
% 3.17/1.60 ( setadjoin__Cong
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xx = Xy )
% 3.17/1.60 => ! [Xz: $i,Xu: $i] :
% 3.17/1.60 ( ( Xz = Xu )
% 3.17/1.60 => ( ( setadjoin @ Xx @ Xz )
% 3.17/1.60 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(powerset__Cong_type,type,
% 3.17/1.60 powerset__Cong: $o ).
% 3.17/1.60
% 3.17/1.60 thf(powerset__Cong,definition,
% 3.17/1.60 ( powerset__Cong
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( A = B )
% 3.17/1.60 => ( ( powerset @ A )
% 3.17/1.60 = ( powerset @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setunion__Cong_type,type,
% 3.17/1.60 setunion__Cong: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setunion__Cong,definition,
% 3.17/1.60 ( setunion__Cong
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( A = B )
% 3.17/1.60 => ( ( setunion @ A )
% 3.17/1.60 = ( setunion @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(omega__Cong_type,type,
% 3.17/1.60 omega__Cong: $o ).
% 3.17/1.60
% 3.17/1.60 thf(omega__Cong,definition,
% 3.17/1.60 ( omega__Cong
% 3.17/1.60 = ( omega = omega ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(exuEu_type,type,
% 3.17/1.60 exuEu: $o ).
% 3.17/1.60
% 3.17/1.60 thf(exuEu,definition,
% 3.17/1.60 ( exuEu
% 3.17/1.60 = ( ! [Xphi: $i > $o] :
% 3.17/1.60 ( ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 => ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xphi @ Xx )
% 3.17/1.60 => ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(descr__Cong_type,type,
% 3.17/1.60 descr__Cong: $o ).
% 3.17/1.60
% 3.17/1.60 thf(descr__Cong,definition,
% 3.17/1.60 ( descr__Cong
% 3.17/1.60 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.17/1.60 ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xx = Xy )
% 3.17/1.60 => ( ( Xphi @ Xx )
% 3.17/1.60 <=> ( Xpsi @ Xy ) ) )
% 3.17/1.60 => ( ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 => ( ( exu
% 3.17/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.17/1.60 => ( ( descr
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 = ( descr
% 3.17/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dsetconstr__Cong_type,type,
% 3.17/1.60 dsetconstr__Cong: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dsetconstr__Cong,definition,
% 3.17/1.60 ( dsetconstr__Cong
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( A = B )
% 3.17/1.60 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( Xx = Xy )
% 3.17/1.60 => ( ( Xphi @ Xx )
% 3.17/1.60 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.17/1.60 => ( ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 = ( dsetconstr @ B
% 3.17/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subset_type,type,
% 3.17/1.60 subset: $i > $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(disjoint_type,type,
% 3.17/1.60 disjoint: $i > $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(setsmeet_type,type,
% 3.17/1.60 setsmeet: $i > $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(subsetI1_type,type,
% 3.17/1.60 subsetI1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subsetI1,definition,
% 3.17/1.60 ( subsetI1
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ Xx @ B ) )
% 3.17/1.60 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(eqimpsubset2_type,type,
% 3.17/1.60 eqimpsubset2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(eqimpsubset2,definition,
% 3.17/1.60 ( eqimpsubset2
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( A = B )
% 3.17/1.60 => ( subset @ B @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(eqimpsubset1_type,type,
% 3.17/1.60 eqimpsubset1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(eqimpsubset1,definition,
% 3.17/1.60 ( eqimpsubset1
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( A = B )
% 3.17/1.60 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subsetI2_type,type,
% 3.17/1.60 subsetI2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subsetI2,definition,
% 3.17/1.60 ( subsetI2
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ Xx @ B ) )
% 3.17/1.60 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(emptysetsubset_type,type,
% 3.17/1.60 emptysetsubset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(emptysetsubset,definition,
% 3.17/1.60 ( emptysetsubset
% 3.17/1.60 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subsetE_type,type,
% 3.17/1.60 subsetE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subsetE,definition,
% 3.17/1.60 ( subsetE
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( subset @ A @ B )
% 3.17/1.60 => ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subsetE2_type,type,
% 3.17/1.60 subsetE2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subsetE2,definition,
% 3.17/1.60 ( subsetE2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( subset @ A @ B )
% 3.17/1.60 => ( ~ ( in @ Xx @ B )
% 3.17/1.60 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(notsubsetI_type,type,
% 3.17/1.60 notsubsetI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(notsubsetI,definition,
% 3.17/1.60 ( notsubsetI
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ~ ( in @ Xx @ B )
% 3.17/1.60 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(notequalI1_type,type,
% 3.17/1.60 notequalI1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(notequalI1,definition,
% 3.17/1.60 ( notequalI1
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ~ ( subset @ A @ B )
% 3.17/1.60 => ( A != B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(notequalI2_type,type,
% 3.17/1.60 notequalI2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(notequalI2,definition,
% 3.17/1.60 ( notequalI2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ~ ( in @ Xx @ B )
% 3.17/1.60 => ( A != B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subsetRefl_type,type,
% 3.17/1.60 subsetRefl: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subsetRefl,definition,
% 3.17/1.60 ( subsetRefl
% 3.17/1.60 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subsetTrans_type,type,
% 3.17/1.60 subsetTrans: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subsetTrans,definition,
% 3.17/1.60 ( subsetTrans
% 3.17/1.60 = ( ! [A: $i,B: $i,C: $i] :
% 3.17/1.60 ( ( subset @ A @ B )
% 3.17/1.60 => ( ( subset @ B @ C )
% 3.17/1.60 => ( subset @ A @ C ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinSub_type,type,
% 3.17/1.60 setadjoinSub: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinSub,definition,
% 3.17/1.60 ( setadjoinSub
% 3.17/1.60 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinSub2_type,type,
% 3.17/1.60 setadjoinSub2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setadjoinSub2,definition,
% 3.17/1.60 ( setadjoinSub2
% 3.17/1.60 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.17/1.60 ( ( subset @ A @ B )
% 3.17/1.60 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subset2powerset_type,type,
% 3.17/1.60 subset2powerset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subset2powerset,definition,
% 3.17/1.60 ( subset2powerset
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( subset @ A @ B )
% 3.17/1.60 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setextsub_type,type,
% 3.17/1.60 setextsub: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setextsub,definition,
% 3.17/1.60 ( setextsub
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( subset @ A @ B )
% 3.17/1.60 => ( ( subset @ B @ A )
% 3.17/1.60 => ( A = B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subsetemptysetimpeq_type,type,
% 3.17/1.60 subsetemptysetimpeq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subsetemptysetimpeq,definition,
% 3.17/1.60 ( subsetemptysetimpeq
% 3.17/1.60 = ( ! [A: $i] :
% 3.17/1.60 ( ( subset @ A @ emptyset )
% 3.17/1.60 => ( A = emptyset ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(powersetI1_type,type,
% 3.17/1.60 powersetI1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(powersetI1,definition,
% 3.17/1.60 ( powersetI1
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( subset @ B @ A )
% 3.17/1.60 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(powersetE1_type,type,
% 3.17/1.60 powersetE1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(powersetE1,definition,
% 3.17/1.60 ( powersetE1
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( in @ B @ ( powerset @ A ) )
% 3.17/1.60 => ( subset @ B @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(inPowerset_type,type,
% 3.17/1.60 inPowerset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(inPowerset,definition,
% 3.17/1.60 ( inPowerset
% 3.17/1.60 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(powersetsubset_type,type,
% 3.17/1.60 powersetsubset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(powersetsubset,definition,
% 3.17/1.60 ( powersetsubset
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( subset @ A @ B )
% 3.17/1.60 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(sepInPowerset_type,type,
% 3.17/1.60 sepInPowerset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(sepInPowerset,definition,
% 3.17/1.60 ( sepInPowerset
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( in
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 @ ( powerset @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(sepSubset_type,type,
% 3.17/1.60 sepSubset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(sepSubset,definition,
% 3.17/1.60 ( sepSubset
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( subset
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 @ A ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binunion_type,type,
% 3.17/1.60 binunion: $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(binunionIL_type,type,
% 3.17/1.60 binunionIL: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binunionIL,definition,
% 3.17/1.60 ( binunionIL
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(upairset2IR_type,type,
% 3.17/1.60 upairset2IR: $o ).
% 3.17/1.60
% 3.17/1.60 thf(upairset2IR,definition,
% 3.17/1.60 ( upairset2IR
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binunionIR_type,type,
% 3.17/1.60 binunionIR: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binunionIR,definition,
% 3.17/1.60 ( binunionIR
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binunionEcases_type,type,
% 3.17/1.60 binunionEcases: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binunionEcases,definition,
% 3.17/1.60 ( binunionEcases
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.17/1.60 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.17/1.60 => ( ( ( in @ Xx @ A )
% 3.17/1.60 => Xphi )
% 3.17/1.60 => ( ( ( in @ Xx @ B )
% 3.17/1.60 => Xphi )
% 3.17/1.60 => Xphi ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binunionE_type,type,
% 3.17/1.60 binunionE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binunionE,definition,
% 3.17/1.60 ( binunionE
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.17/1.60 => ( ( in @ Xx @ A )
% 3.17/1.60 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binunionLsub_type,type,
% 3.17/1.60 binunionLsub: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binunionLsub,definition,
% 3.17/1.60 ( binunionLsub
% 3.17/1.60 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binunionRsub_type,type,
% 3.17/1.60 binunionRsub: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binunionRsub,definition,
% 3.17/1.60 ( binunionRsub
% 3.17/1.60 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersect_type,type,
% 3.17/1.60 binintersect: $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectI_type,type,
% 3.17/1.60 binintersectI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectI,definition,
% 3.17/1.60 ( binintersectI
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( in @ Xx @ B )
% 3.17/1.60 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset5_type,type,
% 3.17/1.60 binintersectSubset5: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset5,definition,
% 3.17/1.60 ( binintersectSubset5
% 3.17/1.60 = ( ! [A: $i,B: $i,C: $i] :
% 3.17/1.60 ( ( subset @ C @ A )
% 3.17/1.60 => ( ( subset @ C @ B )
% 3.17/1.60 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectEL_type,type,
% 3.17/1.60 binintersectEL: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectEL,definition,
% 3.17/1.60 ( binintersectEL
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.17/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectLsub_type,type,
% 3.17/1.60 binintersectLsub: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectLsub,definition,
% 3.17/1.60 ( binintersectLsub
% 3.17/1.60 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset2_type,type,
% 3.17/1.60 binintersectSubset2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset2,definition,
% 3.17/1.60 ( binintersectSubset2
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( subset @ A @ B )
% 3.17/1.60 => ( ( binintersect @ A @ B )
% 3.17/1.60 = A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset3_type,type,
% 3.17/1.60 binintersectSubset3: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset3,definition,
% 3.17/1.60 ( binintersectSubset3
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( ( binintersect @ A @ B )
% 3.17/1.60 = B )
% 3.17/1.60 => ( subset @ B @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectER_type,type,
% 3.17/1.60 binintersectER: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectER,definition,
% 3.17/1.60 ( binintersectER
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.17/1.60 => ( in @ Xx @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(disjointsetsI1_type,type,
% 3.17/1.60 disjointsetsI1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(disjointsetsI1,definition,
% 3.17/1.60 ( disjointsetsI1
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ~ ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ( in @ Xx @ B ) )
% 3.17/1.60 => ( ( binintersect @ A @ B )
% 3.17/1.60 = emptyset ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectRsub_type,type,
% 3.17/1.60 binintersectRsub: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectRsub,definition,
% 3.17/1.60 ( binintersectRsub
% 3.17/1.60 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset4_type,type,
% 3.17/1.60 binintersectSubset4: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset4,definition,
% 3.17/1.60 ( binintersectSubset4
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( subset @ B @ A )
% 3.17/1.60 => ( ( binintersect @ A @ B )
% 3.17/1.60 = B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset1_type,type,
% 3.17/1.60 binintersectSubset1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectSubset1,definition,
% 3.17/1.60 ( binintersectSubset1
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( ( binintersect @ A @ B )
% 3.17/1.60 = A )
% 3.17/1.60 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(bs114d_type,type,
% 3.17/1.60 bs114d: $o ).
% 3.17/1.60
% 3.17/1.60 thf(bs114d,definition,
% 3.17/1.60 ( bs114d
% 3.17/1.60 = ( ! [A: $i,B: $i,C: $i] :
% 3.17/1.60 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.17/1.60 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(regular_type,type,
% 3.17/1.60 regular: $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminus_type,type,
% 3.17/1.60 setminus: $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(setminusI_type,type,
% 3.17/1.60 setminusI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusI,definition,
% 3.17/1.60 ( setminusI
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ~ ( in @ Xx @ B )
% 3.17/1.60 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusEL_type,type,
% 3.17/1.60 setminusEL: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusEL,definition,
% 3.17/1.60 ( setminusEL
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.17/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusER_type,type,
% 3.17/1.60 setminusER: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusER,definition,
% 3.17/1.60 ( setminusER
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.17/1.60 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusSubset2_type,type,
% 3.17/1.60 setminusSubset2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusSubset2,definition,
% 3.17/1.60 ( setminusSubset2
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( subset @ A @ B )
% 3.17/1.60 => ( ( setminus @ A @ B )
% 3.17/1.60 = emptyset ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusERneg_type,type,
% 3.17/1.60 setminusERneg: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusERneg,definition,
% 3.17/1.60 ( setminusERneg
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.17/1.60 => ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusELneg_type,type,
% 3.17/1.60 setminusELneg: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusELneg,definition,
% 3.17/1.60 ( setminusELneg
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.17/1.60 => ( ~ ( in @ Xx @ B )
% 3.17/1.60 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusILneg_type,type,
% 3.17/1.60 setminusILneg: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusILneg,definition,
% 3.17/1.60 ( setminusILneg
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ~ ( in @ Xx @ A )
% 3.17/1.60 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusIRneg_type,type,
% 3.17/1.60 setminusIRneg: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusIRneg,definition,
% 3.17/1.60 ( setminusIRneg
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ B )
% 3.17/1.60 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusLsub_type,type,
% 3.17/1.60 setminusLsub: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusLsub,definition,
% 3.17/1.60 ( setminusLsub
% 3.17/1.60 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusSubset1_type,type,
% 3.17/1.60 setminusSubset1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusSubset1,definition,
% 3.17/1.60 ( setminusSubset1
% 3.17/1.60 = ( ! [A: $i,B: $i] :
% 3.17/1.60 ( ( ( setminus @ A @ B )
% 3.17/1.60 = emptyset )
% 3.17/1.60 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(symdiff_type,type,
% 3.17/1.60 symdiff: $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffE_type,type,
% 3.17/1.60 symdiffE: $o ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffE,definition,
% 3.17/1.60 ( symdiffE
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.17/1.60 => ! [Xphi: $o] :
% 3.17/1.60 ( ( ( in @ Xx @ A )
% 3.17/1.60 => ( ~ ( in @ Xx @ B )
% 3.17/1.60 => Xphi ) )
% 3.17/1.60 => ( ( ~ ( in @ Xx @ A )
% 3.17/1.60 => ( ( in @ Xx @ B )
% 3.17/1.60 => Xphi ) )
% 3.17/1.60 => Xphi ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffI1_type,type,
% 3.17/1.60 symdiffI1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffI1,definition,
% 3.17/1.60 ( symdiffI1
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ~ ( in @ Xx @ B )
% 3.17/1.60 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffI2_type,type,
% 3.17/1.60 symdiffI2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffI2,definition,
% 3.17/1.60 ( symdiffI2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ~ ( in @ Xx @ A )
% 3.17/1.60 => ( ( in @ Xx @ B )
% 3.17/1.60 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffIneg1_type,type,
% 3.17/1.60 symdiffIneg1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffIneg1,definition,
% 3.17/1.60 ( symdiffIneg1
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( in @ Xx @ B )
% 3.17/1.60 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffIneg2_type,type,
% 3.17/1.60 symdiffIneg2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(symdiffIneg2,definition,
% 3.17/1.60 ( symdiffIneg2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ~ ( in @ Xx @ A )
% 3.17/1.60 => ( ~ ( in @ Xx @ B )
% 3.17/1.60 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(iskpair_type,type,
% 3.17/1.60 iskpair: $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(secondinupair_type,type,
% 3.17/1.60 secondinupair: $o ).
% 3.17/1.60
% 3.17/1.60 thf(secondinupair,definition,
% 3.17/1.60 ( secondinupair
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairIL_type,type,
% 3.17/1.60 setukpairIL: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairIL,definition,
% 3.17/1.60 ( setukpairIL
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairIR_type,type,
% 3.17/1.60 setukpairIR: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairIR,definition,
% 3.17/1.60 ( setukpairIR
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(kpairiskpair_type,type,
% 3.17/1.60 kpairiskpair: $o ).
% 3.17/1.60
% 3.17/1.60 thf(kpairiskpair,definition,
% 3.17/1.60 ( kpairiskpair
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(kpair_type,type,
% 3.17/1.60 kpair: $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(kpairp_type,type,
% 3.17/1.60 kpairp: $o ).
% 3.17/1.60
% 3.17/1.60 thf(kpairp,definition,
% 3.17/1.60 ( kpairp
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprod_type,type,
% 3.17/1.60 cartprod: $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(singletonsubset_type,type,
% 3.17/1.60 singletonsubset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(singletonsubset,definition,
% 3.17/1.60 ( singletonsubset
% 3.17/1.60 = ( ! [A: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(singletoninpowerset_type,type,
% 3.17/1.60 singletoninpowerset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(singletoninpowerset,definition,
% 3.17/1.60 ( singletoninpowerset
% 3.17/1.60 = ( ! [A: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(singletoninpowunion_type,type,
% 3.17/1.60 singletoninpowunion: $o ).
% 3.17/1.60
% 3.17/1.60 thf(singletoninpowunion,definition,
% 3.17/1.60 ( singletoninpowunion
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(upairset2E_type,type,
% 3.17/1.60 upairset2E: $o ).
% 3.17/1.60
% 3.17/1.60 thf(upairset2E,definition,
% 3.17/1.60 ( upairset2E
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.17/1.60 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.17/1.60 => ( ( Xz = Xx )
% 3.17/1.60 | ( Xz = Xy ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(upairsubunion_type,type,
% 3.17/1.60 upairsubunion: $o ).
% 3.17/1.60
% 3.17/1.60 thf(upairsubunion,definition,
% 3.17/1.60 ( upairsubunion
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(upairinpowunion_type,type,
% 3.17/1.60 upairinpowunion: $o ).
% 3.17/1.60
% 3.17/1.60 thf(upairinpowunion,definition,
% 3.17/1.60 ( upairinpowunion
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ubforcartprodlem1_type,type,
% 3.17/1.60 ubforcartprodlem1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ubforcartprodlem1,definition,
% 3.17/1.60 ( ubforcartprodlem1
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ubforcartprodlem2_type,type,
% 3.17/1.60 ubforcartprodlem2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ubforcartprodlem2,definition,
% 3.17/1.60 ( ubforcartprodlem2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ubforcartprodlem3_type,type,
% 3.17/1.60 ubforcartprodlem3: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ubforcartprodlem3,definition,
% 3.17/1.60 ( ubforcartprodlem3
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodpairin_type,type,
% 3.17/1.60 cartprodpairin: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodpairin,definition,
% 3.17/1.60 ( cartprodpairin
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodmempair1_type,type,
% 3.17/1.60 cartprodmempair1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodmempair1,definition,
% 3.17/1.60 ( cartprodmempair1
% 3.17/1.60 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.60 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.60 => ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ? [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 & ( Xu
% 3.17/1.60 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodmempair_type,type,
% 3.17/1.60 cartprodmempair: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodmempair,definition,
% 3.17/1.60 ( cartprodmempair
% 3.17/1.60 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.60 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.60 => ( iskpair @ Xu ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setunionE2_type,type,
% 3.17/1.60 setunionE2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setunionE2,definition,
% 3.17/1.60 ( setunionE2
% 3.17/1.60 = ( ! [A: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ ( setunion @ A ) )
% 3.17/1.60 => ? [X: $i] :
% 3.17/1.60 ( ( in @ X @ A )
% 3.17/1.60 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setunionsingleton1_type,type,
% 3.17/1.60 setunionsingleton1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setunionsingleton1,definition,
% 3.17/1.60 ( setunionsingleton1
% 3.17/1.60 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setunionsingleton2_type,type,
% 3.17/1.60 setunionsingleton2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setunionsingleton2,definition,
% 3.17/1.60 ( setunionsingleton2
% 3.17/1.60 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setunionsingleton_type,type,
% 3.17/1.60 setunionsingleton: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setunionsingleton,definition,
% 3.17/1.60 ( setunionsingleton
% 3.17/1.60 = ( ! [Xx: $i] :
% 3.17/1.60 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.17/1.60 = Xx ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(singleton_type,type,
% 3.17/1.60 singleton: $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(singletonprop_type,type,
% 3.17/1.60 singletonprop: $o ).
% 3.17/1.60
% 3.17/1.60 thf(singletonprop,definition,
% 3.17/1.60 ( singletonprop
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( ( Xphi @ Xx )
% 3.17/1.60 => ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) )
% 3.17/1.60 => ( ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ( Xphi @ Xx ) )
% 3.17/1.60 => ( singleton
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ex1_type,type,
% 3.17/1.60 ex1: $i > ( $i > $o ) > $o ).
% 3.17/1.60
% 3.17/1.60 thf(ex1E1_type,type,
% 3.17/1.60 ex1E1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ex1E1,definition,
% 3.17/1.60 ( ex1E1
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ( ex1 @ A
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 => ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ex1I_type,type,
% 3.17/1.60 ex1I: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ex1I,definition,
% 3.17/1.60 ( ex1I
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( Xphi @ Xx )
% 3.17/1.60 => ( ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xy = Xx ) ) )
% 3.17/1.60 => ( ex1 @ A
% 3.17/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ex1I2_type,type,
% 3.17/1.60 ex1I2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ex1I2,definition,
% 3.17/1.60 ( ex1I2
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( ( Xphi @ Xx )
% 3.17/1.60 => ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) )
% 3.17/1.60 => ( ? [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 & ( Xphi @ Xx ) )
% 3.17/1.60 => ( ex1 @ A
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(singletonsuniq_type,type,
% 3.17/1.60 singletonsuniq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(singletonsuniq,definition,
% 3.17/1.60 ( singletonsuniq
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( ( setadjoin @ Xx @ emptyset )
% 3.17/1.60 = ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(atmost1p_type,type,
% 3.17/1.60 atmost1p: $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(atleast2p_type,type,
% 3.17/1.60 atleast2p: $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(atmost2p_type,type,
% 3.17/1.60 atmost2p: $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(upairsetp_type,type,
% 3.17/1.60 upairsetp: $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjL1_type,type,
% 3.17/1.60 setukpairinjL1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjL1,definition,
% 3.17/1.60 ( setukpairinjL1
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.17/1.60 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.17/1.60 => ( Xx = Xz ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(kfstsingleton_type,type,
% 3.17/1.60 kfstsingleton: $o ).
% 3.17/1.60
% 3.17/1.60 thf(kfstsingleton,definition,
% 3.17/1.60 ( kfstsingleton
% 3.17/1.60 = ( ! [Xu: $i] :
% 3.17/1.60 ( ( iskpair @ Xu )
% 3.17/1.60 => ( singleton
% 3.17/1.60 @ ( dsetconstr @ ( setunion @ Xu )
% 3.17/1.60 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(theprop_type,type,
% 3.17/1.60 theprop: $o ).
% 3.17/1.60
% 3.17/1.60 thf(theprop,definition,
% 3.17/1.60 ( theprop
% 3.17/1.60 = ( ! [X: $i] :
% 3.17/1.60 ( ( singleton @ X )
% 3.17/1.60 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(kfst_type,type,
% 3.17/1.60 kfst: $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(kfstpairEq_type,type,
% 3.17/1.60 kfstpairEq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(kfstpairEq,definition,
% 3.17/1.60 ( kfstpairEq
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.17/1.60 = Xx ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodfstin_type,type,
% 3.17/1.60 cartprodfstin: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodfstin,definition,
% 3.17/1.60 ( cartprodfstin
% 3.17/1.60 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.60 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.60 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjL2_type,type,
% 3.17/1.60 setukpairinjL2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjL2,definition,
% 3.17/1.60 ( setukpairinjL2
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.60 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.17/1.60 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.17/1.60 => ( Xx = Xz ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjL_type,type,
% 3.17/1.60 setukpairinjL: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjL,definition,
% 3.17/1.60 ( setukpairinjL
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.60 ( ( ( kpair @ Xx @ Xy )
% 3.17/1.60 = ( kpair @ Xz @ Xu ) )
% 3.17/1.60 => ( Xx = Xz ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR11_type,type,
% 3.17/1.60 setukpairinjR11: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR11,definition,
% 3.17/1.60 ( setukpairinjR11
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xx = Xy )
% 3.17/1.60 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.60 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR12_type,type,
% 3.17/1.60 setukpairinjR12: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR12,definition,
% 3.17/1.60 ( setukpairinjR12
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xx = Xy )
% 3.17/1.60 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.17/1.60 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR1_type,type,
% 3.17/1.60 setukpairinjR1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR1,definition,
% 3.17/1.60 ( setukpairinjR1
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.60 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.17/1.60 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.17/1.60 => ( ( Xz = Xu )
% 3.17/1.60 => ( Xy = Xu ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(upairequniteq_type,type,
% 3.17/1.60 upairequniteq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(upairequniteq,definition,
% 3.17/1.60 ( upairequniteq
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.17/1.60 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.60 = ( setadjoin @ Xz @ emptyset ) )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR2_type,type,
% 3.17/1.60 setukpairinjR2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR2,definition,
% 3.17/1.60 ( setukpairinjR2
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.60 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.17/1.60 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.17/1.60 => ( Xy = Xu ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR_type,type,
% 3.17/1.60 setukpairinjR: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setukpairinjR,definition,
% 3.17/1.60 ( setukpairinjR
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.60 ( ( ( kpair @ Xx @ Xy )
% 3.17/1.60 = ( kpair @ Xz @ Xu ) )
% 3.17/1.60 => ( Xy = Xu ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ksndsingleton_type,type,
% 3.17/1.60 ksndsingleton: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ksndsingleton,definition,
% 3.17/1.60 ( ksndsingleton
% 3.17/1.60 = ( ! [Xu: $i] :
% 3.17/1.60 ( ( iskpair @ Xu )
% 3.17/1.60 => ( singleton
% 3.17/1.60 @ ( dsetconstr @ ( setunion @ Xu )
% 3.17/1.60 @ ^ [Xx: $i] :
% 3.17/1.60 ( Xu
% 3.17/1.60 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ksnd_type,type,
% 3.17/1.60 ksnd: $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(ksndpairEq_type,type,
% 3.17/1.60 ksndpairEq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ksndpairEq,definition,
% 3.17/1.60 ( ksndpairEq
% 3.17/1.60 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.17/1.60 = Xy ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(kpairsurjEq_type,type,
% 3.17/1.60 kpairsurjEq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(kpairsurjEq,definition,
% 3.17/1.60 ( kpairsurjEq
% 3.17/1.60 = ( ! [Xu: $i] :
% 3.17/1.60 ( ( iskpair @ Xu )
% 3.17/1.60 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.17/1.60 = Xu ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodsndin_type,type,
% 3.17/1.60 cartprodsndin: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodsndin,definition,
% 3.17/1.60 ( cartprodsndin
% 3.17/1.60 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.60 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.60 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodpairmemEL_type,type,
% 3.17/1.60 cartprodpairmemEL: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodpairmemEL,definition,
% 3.17/1.60 ( cartprodpairmemEL
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.17/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodpairmemER_type,type,
% 3.17/1.60 cartprodpairmemER: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodpairmemER,definition,
% 3.17/1.60 ( cartprodpairmemER
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.17/1.60 => ( in @ Xy @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodmempaircEq_type,type,
% 3.17/1.60 cartprodmempaircEq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodmempaircEq,definition,
% 3.17/1.60 ( cartprodmempaircEq
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( kpair @ Xx @ Xy )
% 3.17/1.60 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodfstpairEq_type,type,
% 3.17/1.60 cartprodfstpairEq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodfstpairEq,definition,
% 3.17/1.60 ( cartprodfstpairEq
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.17/1.60 = Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodsndpairEq_type,type,
% 3.17/1.60 cartprodsndpairEq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodsndpairEq,definition,
% 3.17/1.60 ( cartprodsndpairEq
% 3.17/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.17/1.60 = Xy ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodpairsurjEq_type,type,
% 3.17/1.60 cartprodpairsurjEq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(cartprodpairsurjEq,definition,
% 3.17/1.60 ( cartprodpairsurjEq
% 3.17/1.60 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.60 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.60 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.17/1.60 = Xu ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(breln_type,type,
% 3.17/1.60 breln: $i > $i > $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(breln,definition,
% 3.17/1.60 ( breln
% 3.17/1.60 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstr_type,type,
% 3.17/1.60 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrI_type,type,
% 3.17/1.60 dpsetconstrI: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrI,definition,
% 3.17/1.60 ( dpsetconstrI
% 3.17/1.60 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( Xphi @ Xx @ Xy )
% 3.17/1.60 => ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.60 @ ( dpsetconstr @ A @ B
% 3.17/1.60 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrSub_type,type,
% 3.17/1.60 dpsetconstrSub: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrSub,definition,
% 3.17/1.60 ( dpsetconstrSub
% 3.17/1.60 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.17/1.60 ( subset
% 3.17/1.60 @ ( dpsetconstr @ A @ B
% 3.17/1.60 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.17/1.60 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setOfPairsIsBReln_type,type,
% 3.17/1.60 setOfPairsIsBReln: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setOfPairsIsBReln,definition,
% 3.17/1.60 ( setOfPairsIsBReln
% 3.17/1.60 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.17/1.60 ( breln @ A @ B
% 3.17/1.60 @ ( dpsetconstr @ A @ B
% 3.17/1.60 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrERa_type,type,
% 3.17/1.60 dpsetconstrERa: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrERa,definition,
% 3.17/1.60 ( dpsetconstrERa
% 3.17/1.60 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.60 @ ( dpsetconstr @ A @ B
% 3.17/1.60 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.60 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrEL1_type,type,
% 3.17/1.60 dpsetconstrEL1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrEL1,definition,
% 3.17/1.60 ( dpsetconstrEL1
% 3.17/1.60 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.60 @ ( dpsetconstr @ A @ B
% 3.17/1.60 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrEL2_type,type,
% 3.17/1.60 dpsetconstrEL2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrEL2,definition,
% 3.17/1.60 ( dpsetconstrEL2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.60 @ ( dpsetconstr @ A @ B
% 3.17/1.60 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.60 => ( in @ Xy @ B ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrER_type,type,
% 3.17/1.60 dpsetconstrER: $o ).
% 3.17/1.60
% 3.17/1.60 thf(dpsetconstrER,definition,
% 3.17/1.60 ( dpsetconstrER
% 3.17/1.60 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.60 @ ( dpsetconstr @ A @ B
% 3.17/1.60 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.60 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(func_type,type,
% 3.17/1.60 func: $i > $i > $i > $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcSet_type,type,
% 3.17/1.60 funcSet: $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(funcImageSingleton_type,type,
% 3.17/1.60 funcImageSingleton: $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcImageSingleton,definition,
% 3.17/1.60 ( funcImageSingleton
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( singleton
% 3.17/1.60 @ ( dsetconstr @ B
% 3.17/1.60 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(apProp_type,type,
% 3.17/1.60 apProp: $o ).
% 3.17/1.60
% 3.17/1.60 thf(apProp,definition,
% 3.17/1.60 ( apProp
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in
% 3.17/1.60 @ ( setunion
% 3.17/1.60 @ ( dsetconstr @ B
% 3.17/1.60 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.17/1.60 @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ap_type,type,
% 3.17/1.60 ap: $i > $i > $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(app_type,type,
% 3.17/1.60 app: $o ).
% 3.17/1.60
% 3.17/1.60 thf(app,definition,
% 3.17/1.60 ( app
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(infuncsetfunc_type,type,
% 3.17/1.60 infuncsetfunc: $o ).
% 3.17/1.60
% 3.17/1.60 thf(infuncsetfunc,definition,
% 3.17/1.60 ( infuncsetfunc
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.60 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ap2p_type,type,
% 3.17/1.60 ap2p: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ap2p,definition,
% 3.17/1.60 ( ap2p
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(funcinfuncset_type,type,
% 3.17/1.60 funcinfuncset: $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcinfuncset,definition,
% 3.17/1.60 ( funcinfuncset
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(lamProp_type,type,
% 3.17/1.60 lamProp: $o ).
% 3.17/1.60
% 3.17/1.60 thf(lamProp,definition,
% 3.17/1.60 ( lamProp
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.60 => ( func @ A @ B
% 3.17/1.60 @ ( dpsetconstr @ A @ B
% 3.17/1.60 @ ^ [Xx: $i,Xy: $i] :
% 3.17/1.60 ( ( Xf @ Xx )
% 3.17/1.60 = Xy ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(lam_type,type,
% 3.17/1.60 lam: $i > $i > ( $i > $i ) > $i ).
% 3.17/1.60
% 3.17/1.60 thf(lamp_type,type,
% 3.17/1.60 lamp: $o ).
% 3.17/1.60
% 3.17/1.60 thf(lamp,definition,
% 3.17/1.60 ( lamp
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.60 => ( func @ A @ B
% 3.17/1.60 @ ( lam @ A @ B
% 3.17/1.60 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(lam2p_type,type,
% 3.17/1.60 lam2p: $o ).
% 3.17/1.60
% 3.17/1.60 thf(lam2p,definition,
% 3.17/1.60 ( lam2p
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.60 => ( in
% 3.17/1.60 @ ( lam @ A @ B
% 3.17/1.60 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.17/1.60 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(brelnall1_type,type,
% 3.17/1.60 brelnall1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(brelnall1,definition,
% 3.17/1.60 ( brelnall1
% 3.17/1.60 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.60 ( ( breln @ A @ B @ R )
% 3.17/1.60 => ! [Xphi: $i > $o] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.60 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ R )
% 3.17/1.60 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(brelnall2_type,type,
% 3.17/1.60 brelnall2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(brelnall2,definition,
% 3.17/1.60 ( brelnall2
% 3.17/1.60 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.60 ( ( breln @ A @ B @ R )
% 3.17/1.60 => ! [Xphi: $i > $o] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.60 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ R )
% 3.17/1.60 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ex1E2_type,type,
% 3.17/1.60 ex1E2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ex1E2,definition,
% 3.17/1.60 ( ex1E2
% 3.17/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.60 ( ( ex1 @ A
% 3.17/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( ( Xphi @ Xx )
% 3.17/1.60 => ( ( Xphi @ Xy )
% 3.17/1.60 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(funcGraphProp1_type,type,
% 3.17/1.60 funcGraphProp1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcGraphProp1,definition,
% 3.17/1.60 ( funcGraphProp1
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(funcGraphProp3_type,type,
% 3.17/1.60 funcGraphProp3: $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcGraphProp3,definition,
% 3.17/1.60 ( funcGraphProp3
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(funcGraphProp2_type,type,
% 3.17/1.60 funcGraphProp2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcGraphProp2,definition,
% 3.17/1.60 ( funcGraphProp2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.17/1.60 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.60 = Xy ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(funcextLem_type,type,
% 3.17/1.60 funcextLem: $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcextLem,definition,
% 3.17/1.60 ( funcextLem
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ! [Xg: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xg )
% 3.17/1.60 => ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.60 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.17/1.60 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(funcGraphProp4_type,type,
% 3.17/1.60 funcGraphProp4: $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcGraphProp4,definition,
% 3.17/1.60 ( funcGraphProp4
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.17/1.60 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.60 = Xy ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(subbreln_type,type,
% 3.17/1.60 subbreln: $o ).
% 3.17/1.60
% 3.17/1.60 thf(subbreln,definition,
% 3.17/1.60 ( subbreln
% 3.17/1.60 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.60 ( ( breln @ A @ B @ R )
% 3.17/1.60 => ! [S: $i] :
% 3.17/1.60 ( ( breln @ A @ B @ S )
% 3.17/1.60 => ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.60 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.60 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(eqbreln_type,type,
% 3.17/1.60 eqbreln: $o ).
% 3.17/1.60
% 3.17/1.60 thf(eqbreln,definition,
% 3.17/1.60 ( eqbreln
% 3.17/1.60 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.60 ( ( breln @ A @ B @ R )
% 3.17/1.60 => ! [S: $i] :
% 3.17/1.60 ( ( breln @ A @ B @ S )
% 3.17/1.60 => ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.60 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.60 => ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ B )
% 3.17/1.60 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.17/1.60 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.17/1.60 => ( R = S ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(funcext_type,type,
% 3.17/1.60 funcext: $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcext,definition,
% 3.17/1.60 ( funcext
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ! [Xg: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xg )
% 3.17/1.60 => ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.60 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.17/1.60 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(funcext2_type,type,
% 3.17/1.60 funcext2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(funcext2,definition,
% 3.17/1.60 ( funcext2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.60 => ! [Xg: $i] :
% 3.17/1.60 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.17/1.60 => ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.60 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.17/1.60 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ap2apEq1_type,type,
% 3.17/1.60 ap2apEq1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ap2apEq1,definition,
% 3.17/1.60 ( ap2apEq1
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.60 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ap2apEq2_type,type,
% 3.17/1.60 ap2apEq2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ap2apEq2,definition,
% 3.17/1.60 ( ap2apEq2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.60 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(beta1_type,type,
% 3.17/1.60 beta1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(beta1,definition,
% 3.17/1.60 ( beta1
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( ap @ A @ B
% 3.17/1.60 @ ( lam @ A @ B
% 3.17/1.60 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.17/1.60 @ Xx )
% 3.17/1.60 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(eta1_type,type,
% 3.17/1.60 eta1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(eta1,definition,
% 3.17/1.60 ( eta1
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( func @ A @ B @ Xf )
% 3.17/1.60 => ( ( lam @ A @ B
% 3.17/1.60 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.17/1.60 = Xf ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(lam2lamEq_type,type,
% 3.17/1.60 lam2lamEq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(lam2lamEq,definition,
% 3.17/1.60 ( lam2lamEq
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.60 => ( ( lam @ A @ B
% 3.17/1.60 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.17/1.60 = ( lam @ A @ B
% 3.17/1.60 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(beta2_type,type,
% 3.17/1.60 beta2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(beta2,definition,
% 3.17/1.60 ( beta2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.60 ( ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ( ( ap @ A @ B
% 3.17/1.60 @ ( lam @ A @ B
% 3.17/1.60 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.17/1.60 @ Xx )
% 3.17/1.60 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(eta2_type,type,
% 3.17/1.60 eta2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(eta2,definition,
% 3.17/1.60 ( eta2
% 3.17/1.60 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.60 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.60 => ( ( lam @ A @ B
% 3.17/1.60 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.17/1.60 = Xf ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(iffalseProp1_type,type,
% 3.17/1.60 iffalseProp1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(iffalseProp1,definition,
% 3.17/1.60 ( iffalseProp1
% 3.17/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( ~ Xphi
% 3.17/1.60 => ( in @ Xy
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xz: $i] :
% 3.17/1.60 ( ( Xphi
% 3.17/1.60 & ( Xz = Xx ) )
% 3.17/1.60 | ( ~ Xphi
% 3.17/1.60 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(iffalseProp2_type,type,
% 3.17/1.60 iffalseProp2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(iffalseProp2,definition,
% 3.17/1.60 ( iffalseProp2
% 3.17/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( ~ Xphi
% 3.17/1.60 => ( ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xz: $i] :
% 3.17/1.60 ( ( Xphi
% 3.17/1.60 & ( Xz = Xx ) )
% 3.17/1.60 | ( ~ Xphi
% 3.17/1.60 & ( Xz = Xy ) ) ) )
% 3.17/1.60 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(iftrueProp1_type,type,
% 3.17/1.60 iftrueProp1: $o ).
% 3.17/1.60
% 3.17/1.60 thf(iftrueProp1,definition,
% 3.17/1.60 ( iftrueProp1
% 3.17/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( Xphi
% 3.17/1.60 => ( in @ Xx
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xz: $i] :
% 3.17/1.60 ( ( Xphi
% 3.17/1.60 & ( Xz = Xx ) )
% 3.17/1.60 | ( ~ Xphi
% 3.17/1.60 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(iftrueProp2_type,type,
% 3.17/1.60 iftrueProp2: $o ).
% 3.17/1.60
% 3.17/1.60 thf(iftrueProp2,definition,
% 3.17/1.60 ( iftrueProp2
% 3.17/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( Xphi
% 3.17/1.60 => ( ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xz: $i] :
% 3.17/1.60 ( ( Xphi
% 3.17/1.60 & ( Xz = Xx ) )
% 3.17/1.60 | ( ~ Xphi
% 3.17/1.60 & ( Xz = Xy ) ) ) )
% 3.17/1.60 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(ifSingleton_type,type,
% 3.17/1.60 ifSingleton: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ifSingleton,definition,
% 3.17/1.60 ( ifSingleton
% 3.17/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( singleton
% 3.17/1.60 @ ( dsetconstr @ A
% 3.17/1.60 @ ^ [Xz: $i] :
% 3.17/1.60 ( ( Xphi
% 3.17/1.60 & ( Xz = Xx ) )
% 3.17/1.60 | ( ~ Xphi
% 3.17/1.60 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(if_type,type,
% 3.17/1.60 if: $i > $o > $i > $i > $i ).
% 3.17/1.60
% 3.17/1.60 thf(ifp_type,type,
% 3.17/1.60 ifp: $o ).
% 3.17/1.60
% 3.17/1.60 thf(ifp,definition,
% 3.17/1.60 ( ifp
% 3.17/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(theeq_type,type,
% 3.17/1.60 theeq: $o ).
% 3.17/1.60
% 3.17/1.60 thf(theeq,definition,
% 3.17/1.60 ( theeq
% 3.17/1.60 = ( ! [X: $i] :
% 3.17/1.60 ( ( singleton @ X )
% 3.17/1.60 => ! [Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ X )
% 3.17/1.60 => ( ( setunion @ X )
% 3.17/1.60 = Xx ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(iftrue_type,type,
% 3.17/1.60 iftrue: $o ).
% 3.17/1.60
% 3.17/1.60 thf(iftrue,definition,
% 3.17/1.60 ( iftrue
% 3.17/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( Xphi
% 3.17/1.60 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.17/1.60 = Xx ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(iffalse_type,type,
% 3.17/1.60 iffalse: $o ).
% 3.17/1.60
% 3.17/1.60 thf(iffalse,definition,
% 3.17/1.60 ( iffalse
% 3.17/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( ~ Xphi
% 3.17/1.60 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.17/1.60 = Xy ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(iftrueorfalse_type,type,
% 3.17/1.60 iftrueorfalse: $o ).
% 3.17/1.60
% 3.17/1.60 thf(iftrueorfalse,definition,
% 3.17/1.60 ( iftrueorfalse
% 3.17/1.60 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.60 ( ( in @ Xx @ A )
% 3.17/1.60 => ! [Xy: $i] :
% 3.17/1.60 ( ( in @ Xy @ A )
% 3.17/1.60 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectT_lem_type,type,
% 3.17/1.60 binintersectT_lem: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binintersectT_lem,definition,
% 3.17/1.60 ( binintersectT_lem
% 3.17/1.60 = ( ! [A: $i,X: $i] :
% 3.17/1.60 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.60 => ! [Y: $i] :
% 3.17/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.60 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(binunionT_lem_type,type,
% 3.17/1.60 binunionT_lem: $o ).
% 3.17/1.60
% 3.17/1.60 thf(binunionT_lem,definition,
% 3.17/1.60 ( binunionT_lem
% 3.17/1.60 = ( ! [A: $i,X: $i] :
% 3.17/1.60 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.60 => ! [Y: $i] :
% 3.17/1.60 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.60 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(powersetT_lem_type,type,
% 3.17/1.60 powersetT_lem: $o ).
% 3.17/1.60
% 3.17/1.60 thf(powersetT_lem,definition,
% 3.17/1.60 ( powersetT_lem
% 3.17/1.60 = ( ! [A: $i,X: $i] :
% 3.17/1.60 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.60 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.60
% 3.17/1.60 thf(setminusT_lem_type,type,
% 3.17/1.60 setminusT_lem: $o ).
% 3.17/1.60
% 3.17/1.60 thf(setminusT_lem,definition,
% 3.17/1.61 ( setminusT_lem
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(complementT_lem_type,type,
% 3.17/1.61 complementT_lem: $o ).
% 3.17/1.61
% 3.17/1.61 thf(complementT_lem,definition,
% 3.17/1.61 ( complementT_lem
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(setextT_type,type,
% 3.17/1.61 setextT: $o ).
% 3.17/1.61
% 3.17/1.61 thf(setextT,definition,
% 3.17/1.61 ( setextT
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ X )
% 3.17/1.61 => ( in @ Xx @ Y ) ) )
% 3.17/1.61 => ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ Y )
% 3.17/1.61 => ( in @ Xx @ X ) ) )
% 3.17/1.61 => ( X = Y ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(subsetTI_type,type,
% 3.17/1.61 subsetTI: $o ).
% 3.17/1.61
% 3.17/1.61 thf(subsetTI,definition,
% 3.17/1.61 ( subsetTI
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ X )
% 3.17/1.61 => ( in @ Xx @ Y ) ) )
% 3.17/1.61 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(powersetTI1_type,type,
% 3.17/1.61 powersetTI1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(powersetTI1,definition,
% 3.17/1.61 ( powersetTI1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ X )
% 3.17/1.61 => ( in @ Xx @ Y ) ) )
% 3.17/1.61 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(powersetTE1_type,type,
% 3.17/1.61 powersetTE1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(powersetTE1,definition,
% 3.17/1.61 ( powersetTE1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ X @ ( powerset @ Y ) )
% 3.17/1.61 => ( ( in @ Xx @ X )
% 3.17/1.61 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(complementTI1_type,type,
% 3.17/1.61 complementTI1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(complementTI1,definition,
% 3.17/1.61 ( complementTI1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ X )
% 3.17/1.61 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(complementTE1_type,type,
% 3.17/1.61 complementTE1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(complementTE1,definition,
% 3.17/1.61 ( complementTE1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.61 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(binintersectTELcontra_type,type,
% 3.17/1.61 binintersectTELcontra: $o ).
% 3.17/1.61
% 3.17/1.61 thf(binintersectTELcontra,definition,
% 3.17/1.61 ( binintersectTELcontra
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ~ ( in @ Xx @ X )
% 3.17/1.61 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(binintersectTERcontra_type,type,
% 3.17/1.61 binintersectTERcontra: $o ).
% 3.17/1.61
% 3.17/1.61 thf(binintersectTERcontra,definition,
% 3.17/1.61 ( binintersectTERcontra
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ~ ( in @ Xx @ Y )
% 3.17/1.61 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(contrasubsetT_type,type,
% 3.17/1.61 contrasubsetT: $o ).
% 3.17/1.61
% 3.17/1.61 thf(contrasubsetT,definition,
% 3.17/1.61 ( contrasubsetT
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.61 => ( ( in @ Xx @ Y )
% 3.17/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(contrasubsetT1_type,type,
% 3.17/1.61 contrasubsetT1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(contrasubsetT1,definition,
% 3.17/1.61 ( contrasubsetT1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( subset @ X @ Y )
% 3.17/1.61 => ( ~ ( in @ Xx @ Y )
% 3.17/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(contrasubsetT2_type,type,
% 3.17/1.61 contrasubsetT2: $o ).
% 3.17/1.61
% 3.17/1.61 thf(contrasubsetT2,definition,
% 3.17/1.61 ( contrasubsetT2
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ( subset @ X @ Y )
% 3.17/1.61 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(contrasubsetT3_type,type,
% 3.17/1.61 contrasubsetT3: $o ).
% 3.17/1.61
% 3.17/1.61 thf(contrasubsetT3,definition,
% 3.17/1.61 ( contrasubsetT3
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.17/1.61 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementI1_type,type,
% 3.17/1.61 doubleComplementI1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementI1,definition,
% 3.17/1.61 ( doubleComplementI1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ X )
% 3.17/1.61 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementE1_type,type,
% 3.17/1.61 doubleComplementE1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementE1,definition,
% 3.17/1.61 ( doubleComplementE1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.17/1.61 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementSub1_type,type,
% 3.17/1.61 doubleComplementSub1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementSub1,definition,
% 3.17/1.61 ( doubleComplementSub1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementSub2_type,type,
% 3.17/1.61 doubleComplementSub2: $o ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementSub2,definition,
% 3.17/1.61 ( doubleComplementSub2
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementEq_type,type,
% 3.17/1.61 doubleComplementEq: $o ).
% 3.17/1.61
% 3.17/1.61 thf(doubleComplementEq,definition,
% 3.17/1.61 ( doubleComplementEq
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ( X
% 3.17/1.61 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(complementTnotintersectT_type,type,
% 3.17/1.61 complementTnotintersectT: $o ).
% 3.17/1.61
% 3.17/1.61 thf(complementTnotintersectT,definition,
% 3.17/1.61 ( complementTnotintersectT
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.61 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(complementImpComplementIntersect_type,type,
% 3.17/1.61 complementImpComplementIntersect: $o ).
% 3.17/1.61
% 3.17/1.61 thf(complementImpComplementIntersect,definition,
% 3.17/1.61 ( complementImpComplementIntersect
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.61 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(complementSubsetComplementIntersect_type,type,
% 3.17/1.61 complementSubsetComplementIntersect: $o ).
% 3.17/1.61
% 3.17/1.61 thf(complementSubsetComplementIntersect,definition,
% 3.17/1.61 ( complementSubsetComplementIntersect
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(complementInPowersetComplementIntersect_type,type,
% 3.17/1.61 complementInPowersetComplementIntersect: $o ).
% 3.17/1.61
% 3.17/1.61 thf(complementInPowersetComplementIntersect,definition,
% 3.17/1.61 ( complementInPowersetComplementIntersect
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(contraSubsetComplement_type,type,
% 3.17/1.61 contraSubsetComplement: $o ).
% 3.17/1.61
% 3.17/1.61 thf(contraSubsetComplement,definition,
% 3.17/1.61 ( contraSubsetComplement
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ Y )
% 3.17/1.61 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(complementTcontraSubset_type,type,
% 3.17/1.61 complementTcontraSubset: $o ).
% 3.17/1.61
% 3.17/1.61 thf(complementTcontraSubset,definition,
% 3.17/1.61 ( complementTcontraSubset
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.61 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(binunionTILcontra_type,type,
% 3.17/1.61 binunionTILcontra: $o ).
% 3.17/1.61
% 3.17/1.61 thf(binunionTILcontra,definition,
% 3.17/1.61 ( binunionTILcontra
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.17/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(binunionTIRcontra_type,type,
% 3.17/1.61 binunionTIRcontra: $o ).
% 3.17/1.61
% 3.17/1.61 thf(binunionTIRcontra,definition,
% 3.17/1.61 ( binunionTIRcontra
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.17/1.61 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(inIntersectImpInUnion_type,type,
% 3.17/1.61 inIntersectImpInUnion: $o ).
% 3.17/1.61
% 3.17/1.61 thf(inIntersectImpInUnion,definition,
% 3.17/1.61 ( inIntersectImpInUnion
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.61 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(inIntersectImpInUnion2_type,type,
% 3.17/1.61 inIntersectImpInUnion2: $o ).
% 3.17/1.61
% 3.17/1.61 thf(inIntersectImpInUnion2,definition,
% 3.17/1.61 ( inIntersectImpInUnion2
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.61 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(inIntersectImpInIntersectUnions_type,type,
% 3.17/1.61 inIntersectImpInIntersectUnions: $o ).
% 3.17/1.61
% 3.17/1.61 thf(inIntersectImpInIntersectUnions,definition,
% 3.17/1.61 ( inIntersectImpInIntersectUnions
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.61 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(intersectInPowersetIntersectUnions_type,type,
% 3.17/1.61 intersectInPowersetIntersectUnions: $o ).
% 3.17/1.61
% 3.17/1.61 thf(intersectInPowersetIntersectUnions,definition,
% 3.17/1.61 ( intersectInPowersetIntersectUnions
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(inComplementUnionImpNotIn1_type,type,
% 3.17/1.61 inComplementUnionImpNotIn1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(inComplementUnionImpNotIn1,definition,
% 3.17/1.61 ( inComplementUnionImpNotIn1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(inComplementUnionImpInComplement1_type,type,
% 3.17/1.61 inComplementUnionImpInComplement1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(inComplementUnionImpInComplement1,definition,
% 3.17/1.61 ( inComplementUnionImpInComplement1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.61 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(binunionTE_type,type,
% 3.17/1.61 binunionTE: $o ).
% 3.17/1.61
% 3.17/1.61 thf(binunionTE,definition,
% 3.17/1.61 ( binunionTE
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xphi: $o,Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.17/1.61 => ( ( ( in @ Xx @ X )
% 3.17/1.61 => Xphi )
% 3.17/1.61 => ( ( ( in @ Xx @ Y )
% 3.17/1.61 => Xphi )
% 3.17/1.61 => Xphi ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(binunionTEcontra_type,type,
% 3.17/1.61 binunionTEcontra: $o ).
% 3.17/1.61
% 3.17/1.61 thf(binunionTEcontra,definition,
% 3.17/1.61 ( binunionTEcontra
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ~ ( in @ Xx @ X )
% 3.17/1.61 => ( ~ ( in @ Xx @ Y )
% 3.17/1.61 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2a1_type,type,
% 3.17/1.61 demorgan2a1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2a1,definition,
% 3.17/1.61 ( demorgan2a1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.61 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(complementUnionInPowersetComplement_type,type,
% 3.17/1.61 complementUnionInPowersetComplement: $o ).
% 3.17/1.61
% 3.17/1.61 thf(complementUnionInPowersetComplement,definition,
% 3.17/1.61 ( complementUnionInPowersetComplement
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2a2_type,type,
% 3.17/1.61 demorgan2a2: $o ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2a2,definition,
% 3.17/1.61 ( demorgan2a2
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.61 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan1a_type,type,
% 3.17/1.61 demorgan1a: $o ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan1a,definition,
% 3.17/1.61 ( demorgan1a
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.17/1.61 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan1b_type,type,
% 3.17/1.61 demorgan1b: $o ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan1b,definition,
% 3.17/1.61 ( demorgan1b
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.17/1.61 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan1_type,type,
% 3.17/1.61 demorgan1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan1,definition,
% 3.17/1.61 ( demorgan1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.17/1.61 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2a_type,type,
% 3.17/1.61 demorgan2a: $o ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2a,definition,
% 3.17/1.61 ( demorgan2a
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.17/1.61 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2b2_type,type,
% 3.17/1.61 demorgan2b2: $o ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2b2,definition,
% 3.17/1.61 ( demorgan2b2
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.61 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.17/1.61 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2b_type,type,
% 3.17/1.61 demorgan2b: $o ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2b,definition,
% 3.17/1.61 ( demorgan2b
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.17/1.61 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2_type,type,
% 3.17/1.61 demorgan2: $o ).
% 3.17/1.61
% 3.17/1.61 thf(demorgan2,definition,
% 3.17/1.61 ( demorgan2
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.17/1.61 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule0_type,type,
% 3.17/1.61 woz13rule0: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule0,definition,
% 3.17/1.61 ( woz13rule0
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.61 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule1_type,type,
% 3.17/1.61 woz13rule1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule1,definition,
% 3.17/1.61 ( woz13rule1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ( ( subset @ X @ Z )
% 3.17/1.61 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule2_type,type,
% 3.17/1.61 woz13rule2: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule2,definition,
% 3.17/1.61 ( woz13rule2
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ( ( subset @ Y @ Z )
% 3.17/1.61 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule3_type,type,
% 3.17/1.61 woz13rule3: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule3,definition,
% 3.17/1.61 ( woz13rule3
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ( ( subset @ X @ Y )
% 3.17/1.61 => ( ( subset @ X @ Z )
% 3.17/1.61 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule4_type,type,
% 3.17/1.61 woz13rule4: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz13rule4,definition,
% 3.17/1.61 ( woz13rule4
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ! [W: $i] :
% 3.17/1.61 ( ( in @ W @ ( powerset @ A ) )
% 3.17/1.61 => ( ( subset @ X @ Z )
% 3.17/1.61 => ( ( subset @ Y @ W )
% 3.17/1.61 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_1_type,type,
% 3.17/1.61 woz1_1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_1,definition,
% 3.17/1.61 ( woz1_1
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_2_type,type,
% 3.17/1.61 woz1_2: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_2,definition,
% 3.17/1.61 ( woz1_2
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ! [W: $i] :
% 3.17/1.61 ( ( in @ W @ ( powerset @ A ) )
% 3.17/1.61 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.17/1.61 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_3_type,type,
% 3.17/1.61 woz1_3: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_3,definition,
% 3.17/1.61 ( woz1_3
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ! [Z: $i] :
% 3.17/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.61 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_4_type,type,
% 3.17/1.61 woz1_4: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_4,definition,
% 3.17/1.61 ( woz1_4
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.61 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_5_type,type,
% 3.17/1.61 woz1_5: $o ).
% 3.17/1.61
% 3.17/1.61 thf(woz1_5,definition,
% 3.17/1.61 ( woz1_5
% 3.17/1.61 = ( ! [A: $i,X: $i] :
% 3.17/1.61 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.61 => ! [Y: $i] :
% 3.17/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.61 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(breln1_type,type,
% 3.17/1.61 breln1: $i > $i > $o ).
% 3.17/1.61
% 3.17/1.61 thf(breln1,definition,
% 3.17/1.61 ( breln1
% 3.17/1.61 = ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(breln1all2_type,type,
% 3.17/1.61 breln1all2: $o ).
% 3.17/1.61
% 3.17/1.61 thf(breln1all2,definition,
% 3.17/1.61 ( breln1all2
% 3.17/1.61 = ( ! [A: $i,R: $i] :
% 3.17/1.61 ( ( breln1 @ A @ R )
% 3.17/1.61 => ! [Xphi: $i > $o] :
% 3.17/1.61 ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ! [Xy: $i] :
% 3.17/1.61 ( ( in @ Xy @ A )
% 3.17/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.61 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ R )
% 3.17/1.61 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(breln1Set_type,type,
% 3.17/1.61 breln1Set: $i > $i ).
% 3.17/1.61
% 3.17/1.61 thf(breln1SetBreln1_type,type,
% 3.17/1.61 breln1SetBreln1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(breln1SetBreln1,definition,
% 3.17/1.61 ( breln1SetBreln1
% 3.17/1.61 = ( ! [A: $i,R: $i] :
% 3.17/1.61 ( ( in @ R @ ( breln1Set @ A ) )
% 3.17/1.61 => ( breln1 @ A @ R ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(transitive_type,type,
% 3.17/1.61 transitive: $i > $i > $o ).
% 3.17/1.61
% 3.17/1.61 thf(antisymmetric_type,type,
% 3.17/1.61 antisymmetric: $i > $i > $o ).
% 3.17/1.61
% 3.17/1.61 thf(reflexive_type,type,
% 3.17/1.61 reflexive: $i > $i > $o ).
% 3.17/1.61
% 3.17/1.61 thf(refltransitive_type,type,
% 3.17/1.61 refltransitive: $i > $i > $o ).
% 3.17/1.61
% 3.17/1.61 thf(refllinearorder_type,type,
% 3.17/1.61 refllinearorder: $i > $i > $o ).
% 3.17/1.61
% 3.17/1.61 thf(reflwellordering_type,type,
% 3.17/1.61 reflwellordering: $i > $i > $o ).
% 3.17/1.61
% 3.17/1.61 thf(choice2fnsingleton_type,type,
% 3.17/1.61 choice2fnsingleton: $o ).
% 3.17/1.61
% 3.17/1.61 thf(choice2fnsingleton,definition,
% 3.17/1.61 ( choice2fnsingleton
% 3.17/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.17/1.61 ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ? [Xy: $i] :
% 3.17/1.61 ( ( in @ Xy @ B )
% 3.17/1.61 & ( Xphi @ Xx @ Xy ) ) )
% 3.17/1.61 => ! [R: $i] :
% 3.17/1.61 ( ( in @ R @ ( breln1Set @ B ) )
% 3.17/1.61 => ( ( reflwellordering @ B @ R )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ( singleton
% 3.17/1.61 @ ( dsetconstr @ B
% 3.17/1.61 @ ^ [Xy: $i] :
% 3.17/1.61 ( ( Xphi @ Xx @ Xy )
% 3.17/1.61 & ! [Xz: $i] :
% 3.17/1.61 ( ( in @ Xz @ B )
% 3.17/1.61 => ( ( Xphi @ Xx @ Xz )
% 3.17/1.61 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(setOfPairsIsBReln1_type,type,
% 3.17/1.61 setOfPairsIsBReln1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(setOfPairsIsBReln1,definition,
% 3.17/1.61 ( setOfPairsIsBReln1
% 3.17/1.61 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.17/1.61 ( breln1 @ A
% 3.17/1.61 @ ( dpsetconstr @ A @ A
% 3.17/1.61 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(breln1all1_type,type,
% 3.17/1.61 breln1all1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(breln1all1,definition,
% 3.17/1.61 ( breln1all1
% 3.17/1.61 = ( ! [A: $i,R: $i] :
% 3.17/1.61 ( ( breln1 @ A @ R )
% 3.17/1.61 => ! [Xphi: $i > $o] :
% 3.17/1.61 ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ! [Xy: $i] :
% 3.17/1.61 ( ( in @ Xy @ A )
% 3.17/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.61 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ R )
% 3.17/1.61 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(subbreln1_type,type,
% 3.17/1.61 subbreln1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(subbreln1,definition,
% 3.17/1.61 ( subbreln1
% 3.17/1.61 = ( ! [A: $i,R: $i] :
% 3.17/1.61 ( ( breln1 @ A @ R )
% 3.17/1.61 => ! [S: $i] :
% 3.17/1.61 ( ( breln1 @ A @ S )
% 3.17/1.61 => ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ! [Xy: $i] :
% 3.17/1.61 ( ( in @ Xy @ A )
% 3.17/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.61 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(eqbreln1_type,type,
% 3.17/1.61 eqbreln1: $o ).
% 3.17/1.61
% 3.17/1.61 thf(eqbreln1,definition,
% 3.17/1.61 ( eqbreln1
% 3.17/1.61 = ( ! [A: $i,R: $i] :
% 3.17/1.61 ( ( breln1 @ A @ R )
% 3.17/1.61 => ! [S: $i] :
% 3.17/1.61 ( ( breln1 @ A @ S )
% 3.17/1.61 => ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ! [Xy: $i] :
% 3.17/1.61 ( ( in @ Xy @ A )
% 3.17/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.61 => ( ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ! [Xy: $i] :
% 3.17/1.61 ( ( in @ Xy @ A )
% 3.17/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.17/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.17/1.61 => ( R = S ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(breln1invset_type,type,
% 3.17/1.61 breln1invset: $i > $i > $i ).
% 3.17/1.61
% 3.17/1.61 thf(breln1invprop_type,type,
% 3.17/1.61 breln1invprop: $o ).
% 3.17/1.61
% 3.17/1.61 thf(breln1invprop,definition,
% 3.17/1.61 ( breln1invprop
% 3.17/1.61 = ( ! [A: $i,R: $i] :
% 3.17/1.61 ( ( breln1 @ A @ R )
% 3.17/1.61 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(breln1invI_type,type,
% 3.17/1.61 breln1invI: $o ).
% 3.17/1.61
% 3.17/1.61 thf(breln1invI,definition,
% 3.17/1.61 ( breln1invI
% 3.17/1.61 = ( ! [A: $i,R: $i] :
% 3.17/1.61 ( ( breln1 @ A @ R )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ! [Xy: $i] :
% 3.17/1.61 ( ( in @ Xy @ A )
% 3.17/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.61 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(breln1invE_type,type,
% 3.17/1.61 breln1invE: $o ).
% 3.17/1.61
% 3.17/1.61 thf(breln1invE,definition,
% 3.17/1.61 ( breln1invE
% 3.17/1.61 = ( ! [A: $i,R: $i] :
% 3.17/1.61 ( ( breln1 @ A @ R )
% 3.17/1.61 => ! [Xx: $i] :
% 3.17/1.61 ( ( in @ Xx @ A )
% 3.17/1.61 => ! [Xy: $i] :
% 3.17/1.61 ( ( in @ Xy @ A )
% 3.17/1.61 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.17/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(breln1compset_type,type,
% 3.17/1.61 breln1compset: $i > $i > $i > $i ).
% 3.17/1.61
% 3.17/1.61 thf(breln1compset,definition,
% 3.17/1.61 ( breln1compset
% 3.17/1.61 = ( ^ [A: $i,R: $i,S: $i] :
% 3.17/1.61 ( dpsetconstr @ A @ A
% 3.17/1.61 @ ^ [Xx: $i,Xy: $i] :
% 3.17/1.61 ? [Xz: $i] :
% 3.17/1.61 ( ( in @ Xz @ A )
% 3.17/1.61 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.17/1.61 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ).
% 3.17/1.61
% 3.17/1.61 thf(breln1compprop,conjecture,
% 3.17/1.61 ( setextAx
% 3.17/1.61 => ( emptysetAx
% 3.17/1.61 => ( setadjoinAx
% 3.17/1.61 => ( powersetAx
% 3.17/1.61 => ( setunionAx
% 3.17/1.61 => ( omega0Ax
% 3.17/1.61 => ( omegaSAx
% 3.17/1.61 => ( omegaIndAx
% 3.17/1.61 => ( replAx
% 3.17/1.61 => ( foundationAx
% 3.17/1.61 => ( wellorderingAx
% 3.17/1.61 => ( descrp
% 3.17/1.61 => ( dsetconstrI
% 3.17/1.61 => ( dsetconstrEL
% 3.17/1.61 => ( dsetconstrER
% 3.17/1.61 => ( exuE1
% 3.17/1.61 => ( prop2setE
% 3.17/1.61 => ( emptysetE
% 3.17/1.61 => ( emptysetimpfalse
% 3.17/1.61 => ( notinemptyset
% 3.17/1.61 => ( exuE3e
% 3.17/1.61 => ( setext
% 3.17/1.61 => ( emptyI
% 3.17/1.61 => ( noeltsimpempty
% 3.17/1.61 => ( setbeta
% 3.17/1.61 => ( nonemptyE1
% 3.17/1.61 => ( nonemptyI
% 3.17/1.61 => ( nonemptyI1
% 3.17/1.61 => ( setadjoinIL
% 3.17/1.61 => ( emptyinunitempty
% 3.17/1.61 => ( setadjoinIR
% 3.17/1.61 => ( setadjoinE
% 3.17/1.61 => ( setadjoinOr
% 3.17/1.61 => ( setoftrueEq
% 3.17/1.61 => ( powersetI
% 3.17/1.61 => ( emptyinPowerset
% 3.17/1.61 => ( emptyInPowerset
% 3.17/1.61 => ( powersetE
% 3.17/1.61 => ( setunionI
% 3.17/1.61 => ( setunionE
% 3.17/1.61 => ( subPowSU
% 3.17/1.61 => ( exuE2
% 3.17/1.61 => ( nonemptyImpWitness
% 3.17/1.61 => ( uniqinunit
% 3.17/1.61 => ( notinsingleton
% 3.17/1.61 => ( eqinunit
% 3.17/1.61 => ( singletonsswitch
% 3.17/1.61 => ( upairsetE
% 3.17/1.61 => ( upairsetIL
% 3.17/1.61 => ( upairsetIR
% 3.17/1.61 => ( emptyE1
% 3.17/1.61 => ( vacuousDall
% 3.17/1.61 => ( quantDeMorgan1
% 3.17/1.61 => ( quantDeMorgan2
% 3.17/1.61 => ( quantDeMorgan3
% 3.17/1.61 => ( quantDeMorgan4
% 3.17/1.61 => ( prop2setI
% 3.17/1.61 => ( prop2set2propI
% 3.17/1.61 => ( notdexE
% 3.17/1.61 => ( notdallE
% 3.17/1.61 => ( exuI1
% 3.17/1.61 => ( exuI3
% 3.17/1.61 => ( exuI2
% 3.17/1.61 => ( inCongP
% 3.17/1.61 => ( in__Cong
% 3.17/1.61 => ( exuE3u
% 3.17/1.61 => ( exu__Cong
% 3.17/1.61 => ( emptyset__Cong
% 3.17/1.61 => ( setadjoin__Cong
% 3.17/1.61 => ( powerset__Cong
% 3.17/1.61 => ( setunion__Cong
% 3.17/1.61 => ( omega__Cong
% 3.17/1.61 => ( exuEu
% 3.17/1.61 => ( descr__Cong
% 3.17/1.61 => ( dsetconstr__Cong
% 3.17/1.61 => ( subsetI1
% 3.17/1.61 => ( eqimpsubset2
% 3.17/1.61 => ( eqimpsubset1
% 3.17/1.61 => ( subsetI2
% 3.17/1.61 => ( emptysetsubset
% 3.17/1.61 => ( subsetE
% 3.17/1.61 => ( subsetE2
% 3.17/1.61 => ( notsubsetI
% 3.17/1.61 => ( notequalI1
% 3.17/1.61 => ( notequalI2
% 3.17/1.61 => ( subsetRefl
% 3.17/1.61 => ( subsetTrans
% 3.17/1.61 => ( setadjoinSub
% 3.17/1.61 => ( setadjoinSub2
% 3.17/1.61 => ( subset2powerset
% 3.17/1.61 => ( setextsub
% 3.17/1.61 => ( subsetemptysetimpeq
% 3.17/1.61 => ( powersetI1
% 3.17/1.61 => ( powersetE1
% 3.17/1.61 => ( inPowerset
% 3.17/1.61 => ( powersetsubset
% 3.17/1.61 => ( sepInPowerset
% 3.17/1.61 => ( sepSubset
% 3.17/1.61 => ( binunionIL
% 3.17/1.61 => ( upairset2IR
% 3.17/1.61 => ( binunionIR
% 3.17/1.61 => ( binunionEcases
% 3.17/1.61 => ( binunionE
% 3.17/1.61 => ( binunionLsub
% 3.17/1.61 => ( binunionRsub
% 3.17/1.61 => ( binintersectI
% 3.17/1.61 => ( binintersectSubset5
% 3.17/1.61 => ( binintersectEL
% 3.17/1.61 => ( binintersectLsub
% 3.17/1.61 => ( binintersectSubset2
% 3.17/1.61 => ( binintersectSubset3
% 3.17/1.61 => ( binintersectER
% 3.17/1.61 => ( disjointsetsI1
% 3.17/1.61 => ( binintersectRsub
% 3.17/1.61 => ( binintersectSubset4
% 3.17/1.61 => ( binintersectSubset1
% 3.17/1.61 => ( bs114d
% 3.17/1.61 => ( setminusI
% 3.17/1.61 => ( setminusEL
% 3.17/1.61 => ( setminusER
% 3.17/1.61 => ( setminusSubset2
% 3.17/1.61 => ( setminusERneg
% 3.17/1.61 => ( setminusELneg
% 3.17/1.61 => ( setminusILneg
% 3.17/1.61 => ( setminusIRneg
% 3.17/1.61 => ( setminusLsub
% 3.17/1.61 => ( setminusSubset1
% 3.17/1.61 => ( symdiffE
% 3.17/1.61 => ( symdiffI1
% 3.17/1.61 => ( symdiffI2
% 3.17/1.61 => ( symdiffIneg1
% 3.17/1.61 => ( symdiffIneg2
% 3.17/1.61 => ( secondinupair
% 3.17/1.61 => ( setukpairIL
% 3.17/1.61 => ( setukpairIR
% 3.17/1.61 => ( kpairiskpair
% 3.17/1.61 => ( kpairp
% 3.17/1.61 => ( singletonsubset
% 3.17/1.61 => ( singletoninpowerset
% 3.17/1.61 => ( singletoninpowunion
% 3.17/1.61 => ( upairset2E
% 3.17/1.61 => ( upairsubunion
% 3.17/1.61 => ( upairinpowunion
% 3.17/1.61 => ( ubforcartprodlem1
% 3.17/1.61 => ( ubforcartprodlem2
% 3.17/1.61 => ( ubforcartprodlem3
% 3.17/1.61 => ( cartprodpairin
% 3.17/1.61 => ( cartprodmempair1
% 3.17/1.61 => ( cartprodmempair
% 3.17/1.61 => ( setunionE2
% 3.17/1.61 => ( setunionsingleton1
% 3.17/1.61 => ( setunionsingleton2
% 3.17/1.61 => ( setunionsingleton
% 3.17/1.61 => ( singletonprop
% 3.17/1.61 => ( ex1E1
% 3.17/1.61 => ( ex1I
% 3.17/1.61 => ( ex1I2
% 3.17/1.61 => ( singletonsuniq
% 3.17/1.61 => ( setukpairinjL1
% 3.17/1.61 => ( kfstsingleton
% 3.17/1.61 => ( theprop
% 3.17/1.61 => ( kfstpairEq
% 3.17/1.61 => ( cartprodfstin
% 3.17/1.61 => ( setukpairinjL2
% 3.17/1.61 => ( setukpairinjL
% 3.17/1.61 => ( setukpairinjR11
% 3.17/1.61 => ( setukpairinjR12
% 3.17/1.61 => ( setukpairinjR1
% 3.17/1.61 => ( upairequniteq
% 3.17/1.61 => ( setukpairinjR2
% 3.17/1.61 => ( setukpairinjR
% 3.17/1.61 => ( ksndsingleton
% 3.17/1.61 => ( ksndpairEq
% 3.17/1.61 => ( kpairsurjEq
% 3.17/1.61 => ( cartprodsndin
% 3.17/1.61 => ( cartprodpairmemEL
% 3.17/1.61 => ( cartprodpairmemER
% 3.17/1.61 => ( cartprodmempaircEq
% 3.17/1.61 => ( cartprodfstpairEq
% 3.17/1.61 => ( cartprodsndpairEq
% 3.17/1.61 => ( cartprodpairsurjEq
% 3.17/1.61 => ( dpsetconstrI
% 3.17/1.61 => ( dpsetconstrSub
% 3.17/1.61 => ( setOfPairsIsBReln
% 3.17/1.61 => ( dpsetconstrERa
% 3.17/1.61 => ( dpsetconstrEL1
% 3.17/1.61 => ( dpsetconstrEL2
% 3.17/1.61 => ( dpsetconstrER
% 3.17/1.61 => ( funcImageSingleton
% 3.17/1.61 => ( apProp
% 3.17/1.61 => ( app
% 3.17/1.61 => ( infuncsetfunc
% 3.17/1.61 => ( ap2p
% 3.17/1.61 => ( funcinfuncset
% 3.17/1.61 => ( lamProp
% 3.17/1.61 => ( lamp
% 3.17/1.61 => ( lam2p
% 3.17/1.61 => ( brelnall1
% 3.17/1.61 => ( brelnall2
% 3.17/1.61 => ( ex1E2
% 3.17/1.61 => ( funcGraphProp1
% 3.17/1.61 => ( funcGraphProp3
% 3.17/1.61 => ( funcGraphProp2
% 3.17/1.61 => ( funcextLem
% 3.17/1.61 => ( funcGraphProp4
% 3.17/1.61 => ( subbreln
% 3.17/1.61 => ( eqbreln
% 3.17/1.61 => ( funcext
% 3.17/1.61 => ( funcext2
% 3.17/1.61 => ( ap2apEq1
% 3.17/1.61 => ( ap2apEq2
% 3.17/1.61 => ( beta1
% 3.17/1.61 => ( eta1
% 3.17/1.61 => ( lam2lamEq
% 3.17/1.61 => ( beta2
% 3.17/1.61 => ( eta2
% 3.17/1.61 => ( iffalseProp1
% 3.17/1.61 => ( iffalseProp2
% 3.17/1.61 => ( iftrueProp1
% 3.17/1.61 => ( iftrueProp2
% 3.17/1.61 => ( ifSingleton
% 3.17/1.61 => ( ifp
% 3.17/1.61 => ( theeq
% 3.17/1.61 => ( iftrue
% 3.17/1.61 => ( iffalse
% 3.17/1.61 => ( iftrueorfalse
% 3.17/1.61 => ( binintersectT_lem
% 3.17/1.61 => ( binunionT_lem
% 3.17/1.61 => ( powersetT_lem
% 3.17/1.61 => ( setminusT_lem
% 3.17/1.61 => ( complementT_lem
% 3.17/1.61 => ( setextT
% 3.17/1.61 => ( subsetTI
% 3.17/1.61 => ( powersetTI1
% 3.17/1.61 => ( powersetTE1
% 3.17/1.61 => ( complementTI1
% 3.17/1.61 => ( complementTE1
% 3.17/1.61 => ( binintersectTELcontra
% 3.17/1.61 => ( binintersectTERcontra
% 3.17/1.61 => ( contrasubsetT
% 3.17/1.61 => ( contrasubsetT1
% 3.17/1.61 => ( contrasubsetT2
% 3.17/1.61 => ( contrasubsetT3
% 3.17/1.61 => ( doubleComplementI1
% 3.17/1.61 => ( doubleComplementE1
% 3.17/1.61 => ( doubleComplementSub1
% 3.17/1.61 => ( doubleComplementSub2
% 3.17/1.61 => ( doubleComplementEq
% 3.17/1.61 => ( complementTnotintersectT
% 3.17/1.61 => ( complementImpComplementIntersect
% 3.17/1.61 => ( complementSubsetComplementIntersect
% 3.17/1.61 => ( complementInPowersetComplementIntersect
% 3.17/1.61 => ( contraSubsetComplement
% 3.17/1.61 => ( complementTcontraSubset
% 3.17/1.61 => ( binunionTILcontra
% 3.17/1.61 => ( binunionTIRcontra
% 3.17/1.61 => ( inIntersectImpInUnion
% 3.17/1.61 => ( inIntersectImpInUnion2
% 3.17/1.61 => ( inIntersectImpInIntersectUnions
% 3.17/1.61 => ( intersectInPowersetIntersectUnions
% 3.17/1.61 => ( inComplementUnionImpNotIn1
% 3.17/1.61 => ( inComplementUnionImpInComplement1
% 3.17/1.61 => ( binunionTE
% 3.17/1.61 => ( binunionTEcontra
% 3.17/1.61 => ( demorgan2a1
% 3.17/1.61 => ( complementUnionInPowersetComplement
% 3.17/1.61 => ( demorgan2a2
% 3.17/1.61 => ( demorgan1a
% 3.17/1.61 => ( demorgan1b
% 3.17/1.61 => ( demorgan1
% 3.17/1.61 => ( demorgan2a
% 3.17/1.61 => ( demorgan2b2
% 3.17/1.61 => ( demorgan2b
% 3.17/1.61 => ( demorgan2
% 3.17/1.61 => ( woz13rule0
% 3.17/1.61 => ( woz13rule1
% 3.17/1.61 => ( woz13rule2
% 3.17/1.61 => ( woz13rule3
% 3.17/1.61 => ( woz13rule4
% 3.17/1.61 => ( woz1_1
% 3.17/1.61 => ( woz1_2
% 3.17/1.61 => ( woz1_3
% 3.17/1.61 => ( woz1_4
% 3.17/1.61 => ( woz1_5
% 3.17/1.61 => ( breln1all2
% 3.17/1.61 => ( breln1SetBreln1
% 3.17/1.61 => ( choice2fnsingleton
% 3.17/1.61 => ( setOfPairsIsBReln1
% 3.17/1.61 => ( breln1all1
% 3.17/1.61 => ( subbreln1
% 3.17/1.61 => ( eqbreln1
% 3.17/1.61 => ( breln1invprop
% 3.17/1.61 => ( breln1invI
% 3.17/1.61 => ( breln1invE
% 3.17/1.61 => ! [A: $i,R: $i] :
% 3.17/1.61 ( ( breln1 @ A @ R )
% 3.17/1.61 => ! [S: $i] :
% 3.17/1.61 ( ( breln1 @ A @ S )
% 3.17/1.61 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.61 ------------------------
% 3.17/1.64 % (22902)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_22699 for (2999ds/275Mi)
% 3.17/1.64 % (22899)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_22699 for (2999ds/27Mi)
% 3.17/1.64 % (22898)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_22699 for (2999ds/4Mi)
% 3.17/1.64 % (22901)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_22699 for (2999ds/2Mi)
% 3.17/1.64 % (22900)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/2Mi)
% 3.17/1.64 % (22903)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/18Mi)
% 3.17/1.64 % (22897)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/183Mi)
% 3.17/1.64 % (22898)Instruction limit reached!
% 3.17/1.64 % (22898)------------------------------
% 3.17/1.64 % (22898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.64 % (22898)Termination reason: Unknown
% 3.17/1.64 % (22898)Termination phase: shuffling
% 3.17/1.64
% 3.17/1.64 % (22898)Memory used [KB]: 1535
% 3.17/1.64 % (22898)Time elapsed: 0.003 s
% 3.17/1.64 % (22898)Instructions burned: 4 (million)
% 3.17/1.64 % (22898)------------------------------
% 3.17/1.64 % (22898)------------------------------
% 3.17/1.64 % (22900)Instruction limit reached!
% 3.17/1.64 % (22900)------------------------------
% 3.17/1.64 % (22900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.64 % (22900)Termination reason: Unknown
% 3.17/1.64 % (22900)Termination phase: shuffling
% 3.17/1.64
% 3.17/1.64 % (22900)Memory used [KB]: 1535
% 3.17/1.64 % (22900)Time elapsed: 0.002 s
% 3.17/1.64 % (22901)Instruction limit reached!
% 3.17/1.64 % (22901)------------------------------
% 3.17/1.64 % (22901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.64 % (22901)Termination reason: Unknown
% 3.17/1.64 % (22901)Termination phase: shuffling
% 3.17/1.64
% 3.17/1.64 % (22901)Memory used [KB]: 1535
% 3.17/1.64 % (22901)Time elapsed: 0.003 s
% 3.17/1.64 % (22901)Instructions burned: 5 (million)
% 3.17/1.64 % (22901)------------------------------
% 3.17/1.64 % (22901)------------------------------
% 3.17/1.64 % (22900)Instructions burned: 2 (million)
% 3.17/1.64 % (22900)------------------------------
% 3.17/1.64 % (22900)------------------------------
% 3.17/1.64 % (22899)Instruction limit reached!
% 3.17/1.64 % (22899)------------------------------
% 3.17/1.64 % (22899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.64 % (22899)Termination reason: Unknown
% 3.17/1.64 % (22899)Termination phase: shuffling
% 3.17/1.64
% 3.17/1.64 % (22899)Memory used [KB]: 1918
% 3.17/1.64 % (22899)Time elapsed: 0.009 s
% 3.17/1.64 % (22899)Instructions burned: 27 (million)
% 3.17/1.64 % (22899)------------------------------
% 3.17/1.64 % (22899)------------------------------
% 3.17/1.64 % (22903)Instruction limit reached!
% 3.17/1.64 % (22903)------------------------------
% 3.17/1.64 % (22903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.64 % (22903)Termination reason: Unknown
% 3.17/1.64 % (22903)Termination phase: shuffling
% 3.17/1.64
% 3.17/1.64 % (22903)Memory used [KB]: 1918
% 3.17/1.64 % (22903)Time elapsed: 0.008 s
% 3.17/1.64 % (22903)Instructions burned: 20 (million)
% 3.17/1.64 % (22903)------------------------------
% 3.17/1.64 % (22903)------------------------------
% 3.17/1.65 % (22906)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_22699 for (2999ds/15Mi)
% 3.17/1.65 % (22905)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_22699 for (2999ds/37Mi)
% 3.17/1.65 % (22904)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/3Mi)
% 3.17/1.65 % (22904)Instruction limit reached!
% 3.17/1.65 % (22904)------------------------------
% 3.17/1.65 % (22904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.65 % (22904)Termination reason: Unknown
% 3.17/1.65 % (22904)Termination phase: shuffling
% 3.17/1.65
% 3.17/1.65 % (22904)Memory used [KB]: 1535
% 3.17/1.65 % (22904)Time elapsed: 0.003 s
% 3.17/1.65 % (22904)Instructions burned: 3 (million)
% 3.17/1.65 % (22904)------------------------------
% 3.17/1.65 % (22904)------------------------------
% 3.17/1.65 % (22906)Instruction limit reached!
% 3.17/1.65 % (22906)------------------------------
% 3.17/1.65 % (22906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.65 % (22906)Termination reason: Unknown
% 3.17/1.65 % (22906)Termination phase: shuffling
% 3.17/1.65
% 3.17/1.65 % (22906)Memory used [KB]: 1791
% 3.17/1.65 % (22906)Time elapsed: 0.006 s
% 3.17/1.65 % (22906)Instructions burned: 15 (million)
% 3.17/1.65 % (22906)------------------------------
% 3.17/1.65 % (22906)------------------------------
% 3.17/1.65 % (22908)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_22699 for (2999ds/3Mi)
% 3.17/1.65 % (22909)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_22699 for (2999ds/1041Mi)
% 3.17/1.65 % (22908)Instruction limit reached!
% 3.17/1.65 % (22908)------------------------------
% 3.17/1.65 % (22908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.65 % (22908)Termination reason: Unknown
% 3.17/1.65 % (22908)Termination phase: shuffling
% 3.17/1.65
% 3.17/1.65 % (22908)Memory used [KB]: 1535
% 3.17/1.65 % (22908)Time elapsed: 0.003 s
% 3.17/1.65 % (22908)Instructions burned: 6 (million)
% 3.17/1.65 % (22908)------------------------------
% 3.17/1.65 % (22908)------------------------------
% 3.17/1.66 % (22905)Instruction limit reached!
% 3.17/1.66 % (22905)------------------------------
% 3.17/1.66 % (22905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.66 % (22905)Termination reason: Unknown
% 3.17/1.66 % (22905)Termination phase: shuffling
% 3.17/1.66
% 3.17/1.66 % (22905)Memory used [KB]: 2302
% 3.17/1.66 % (22905)Time elapsed: 0.012 s
% 3.17/1.66 % (22905)Instructions burned: 38 (million)
% 3.17/1.66 % (22905)------------------------------
% 3.17/1.66 % (22905)------------------------------
% 3.17/1.66 % (22910)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/7Mi)
% 3.17/1.66 % (22911)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/16Mi)
% 3.17/1.66 % (22910)Instruction limit reached!
% 3.17/1.66 % (22910)------------------------------
% 3.17/1.66 % (22910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.66 % (22910)Termination reason: Unknown
% 3.17/1.66 % (22910)Termination phase: shuffling
% 3.17/1.66
% 3.17/1.66 % (22910)Memory used [KB]: 1663
% 3.17/1.66 % (22910)Time elapsed: 0.004 s
% 3.17/1.66 % (22910)Instructions burned: 10 (million)
% 3.17/1.66 % (22910)------------------------------
% 3.17/1.66 % (22910)------------------------------
% 3.17/1.66 % (22913)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_22699 for (2999ds/3Mi)
% 3.17/1.66 % (22913)Instruction limit reached!
% 3.17/1.66 % (22913)------------------------------
% 3.17/1.66 % (22913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.66 % (22913)Termination reason: Unknown
% 3.17/1.66 % (22913)Termination phase: shuffling
% 3.17/1.66
% 3.17/1.66 % (22913)Memory used [KB]: 1535
% 3.17/1.66 % (22913)Time elapsed: 0.002 s
% 3.17/1.66 % (22913)Instructions burned: 3 (million)
% 3.17/1.66 % (22913)------------------------------
% 3.17/1.66 % (22913)------------------------------
% 3.17/1.67 % (22911)Instruction limit reached!
% 3.17/1.67 % (22911)------------------------------
% 3.17/1.67 % (22911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.67 % (22911)Termination reason: Unknown
% 3.17/1.67 % (22911)Termination phase: shuffling
% 3.17/1.67
% 3.17/1.67 % (22911)Memory used [KB]: 1791
% 3.17/1.67 % (22911)Time elapsed: 0.006 s
% 3.17/1.67 % (22911)Instructions burned: 17 (million)
% 3.17/1.67 % (22911)------------------------------
% 3.17/1.67 % (22911)------------------------------
% 3.17/1.67 % (22914)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_22699 for (2999ds/3Mi)
% 3.17/1.67 % (22914)Instruction limit reached!
% 3.17/1.67 % (22914)------------------------------
% 3.17/1.67 % (22914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.67 % (22914)Termination reason: Unknown
% 3.17/1.67 % (22914)Termination phase: shuffling
% 3.17/1.67
% 3.17/1.67 % (22914)Memory used [KB]: 1535
% 3.17/1.67 % (22914)Time elapsed: 0.002 s
% 3.17/1.67 % (22914)Instructions burned: 4 (million)
% 3.17/1.67 % (22914)------------------------------
% 3.17/1.67 % (22914)------------------------------
% 3.17/1.67 % (22915)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/7Mi)
% 3.17/1.67 % (22915)Instruction limit reached!
% 3.17/1.67 % (22915)------------------------------
% 3.17/1.67 % (22915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.67 % (22915)Termination reason: Unknown
% 3.17/1.67 % (22915)Termination phase: shuffling
% 3.17/1.67
% 3.17/1.67 % (22915)Memory used [KB]: 1535
% 3.17/1.67 % (22915)Time elapsed: 0.003 s
% 3.17/1.67 % (22915)Instructions burned: 7 (million)
% 3.17/1.67 % (22915)------------------------------
% 3.17/1.67 % (22915)------------------------------
% 3.17/1.68 % (22916)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/3Mi)
% 3.17/1.68 % (22916)Instruction limit reached!
% 3.17/1.68 % (22916)------------------------------
% 3.17/1.68 % (22916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.68 % (22916)Termination reason: Unknown
% 3.17/1.68 % (22916)Termination phase: shuffling
% 3.17/1.68
% 3.17/1.68 % (22916)Memory used [KB]: 1535
% 3.17/1.68 % (22916)Time elapsed: 0.003 s
% 3.17/1.68 % (22916)Instructions burned: 3 (million)
% 3.17/1.68 % (22916)------------------------------
% 3.17/1.68 % (22916)------------------------------
% 3.17/1.68 % (22917)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/4Mi)
% 3.17/1.68 % (22917)Instruction limit reached!
% 3.17/1.68 % (22917)------------------------------
% 3.17/1.68 % (22917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.68 % (22917)Termination reason: Unknown
% 3.17/1.68 % (22917)Termination phase: shuffling
% 3.17/1.68
% 3.17/1.68 % (22917)Memory used [KB]: 1535
% 3.17/1.68 % (22917)Time elapsed: 0.004 s
% 3.17/1.68 % (22917)Instructions burned: 5 (million)
% 3.17/1.68 % (22917)------------------------------
% 3.17/1.68 % (22917)------------------------------
% 3.17/1.68 % (22926)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/710Mi)
% 3.17/1.69 % (22921)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/18Mi)
% 3.17/1.69 % (22927)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/6Mi)
% 3.17/1.69 % (22897)Instruction limit reached!
% 3.17/1.69 % (22897)------------------------------
% 3.17/1.69 % (22897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.69 % (22897)Termination reason: Unknown
% 3.17/1.69 % (22897)Termination phase: Saturation
% 3.17/1.69
% 3.17/1.69 % (22897)Memory used [KB]: 7419
% 3.17/1.69 % (22897)Time elapsed: 0.058 s
% 3.17/1.69 % (22897)Instructions burned: 183 (million)
% 3.17/1.69 % (22897)------------------------------
% 3.17/1.69 % (22897)------------------------------
% 3.17/1.69 % (22927)Instruction limit reached!
% 3.17/1.69 % (22927)------------------------------
% 3.17/1.69 % (22927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.17/1.69 % (22927)Termination reason: Unknown
% 3.17/1.69 % (22927)Termination phase: shuffling
% 3.17/1.69
% 3.17/1.69 % (22927)Memory used [KB]: 1535
% 3.17/1.69 % (22927)Time elapsed: 0.003 s
% 3.17/1.69 % (22927)Instructions burned: 6 (million)
% 3.17/1.69 % (22927)------------------------------
% 3.17/1.69 % (22927)------------------------------
% 3.21/1.70 % (22921)Instruction limit reached!
% 3.21/1.70 % (22921)------------------------------
% 3.21/1.70 % (22921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.70 % (22921)Termination reason: Unknown
% 3.21/1.70 % (22921)Termination phase: shuffling
% 3.21/1.70
% 3.21/1.70 % (22921)Memory used [KB]: 1791
% 3.21/1.70 % (22921)Time elapsed: 0.010 s
% 3.21/1.70 % (22921)Instructions burned: 18 (million)
% 3.21/1.70 % (22921)------------------------------
% 3.21/1.70 % (22921)------------------------------
% 3.21/1.70 % (22928)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_22699 for (2999ds/902Mi)
% 3.21/1.70 % (22933)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/5Mi)
% 3.21/1.71 % (22933)Instruction limit reached!
% 3.21/1.71 % (22933)------------------------------
% 3.21/1.71 % (22933)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.71 % (22933)Termination reason: Unknown
% 3.21/1.71 % (22933)Termination phase: shuffling
% 3.21/1.71
% 3.21/1.71 % (22933)Memory used [KB]: 1535
% 3.21/1.71 % (22933)Time elapsed: 0.003 s
% 3.21/1.71 % (22933)Instructions burned: 6 (million)
% 3.21/1.71 % (22933)------------------------------
% 3.21/1.71 % (22933)------------------------------
% 3.21/1.71 % (22932)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_22699 for (2999ds/21Mi)
% 3.21/1.71 % (22934)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_22699 for (2999ds/6Mi)
% 3.21/1.71 % (22937)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22699 for (2999ds/377Mi)
% 3.21/1.72 % (22934)Instruction limit reached!
% 3.21/1.72 % (22934)------------------------------
% 3.21/1.72 % (22934)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.72 % (22934)Termination reason: Unknown
% 3.21/1.72 % (22934)Termination phase: shuffling
% 3.21/1.72
% 3.21/1.72 % (22934)Memory used [KB]: 1663
% 3.21/1.72 % (22934)Time elapsed: 0.005 s
% 3.21/1.72 % (22934)Instructions burned: 8 (million)
% 3.21/1.72 % (22934)------------------------------
% 3.21/1.72 % (22934)------------------------------
% 3.21/1.72 % (22932)Instruction limit reached!
% 3.21/1.72 % (22932)------------------------------
% 3.21/1.72 % (22932)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.72 % (22932)Termination reason: Unknown
% 3.21/1.72 % (22932)Termination phase: shuffling
% 3.21/1.72
% 3.21/1.72 % (22932)Memory used [KB]: 1918
% 3.21/1.72 % (22932)Time elapsed: 0.011 s
% 3.21/1.72 % (22932)Instructions burned: 22 (million)
% 3.21/1.72 % (22932)------------------------------
% 3.21/1.72 % (22932)------------------------------
% 3.21/1.74 % (22942)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_22699 for (2998ds/19Mi)
% 3.21/1.74 % (22941)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_22699 for (2998ds/779Mi)
% 3.21/1.74 % (22902)Instruction limit reached!
% 3.21/1.74 % (22902)------------------------------
% 3.21/1.74 % (22902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.74 % (22902)Termination reason: Unknown
% 3.21/1.74 % (22902)Termination phase: Saturation
% 3.21/1.74
% 3.21/1.74 % (22902)Memory used [KB]: 10234
% 3.21/1.74 % (22902)Time elapsed: 0.108 s
% 3.21/1.74 % (22902)Instructions burned: 275 (million)
% 3.21/1.74 % (22902)------------------------------
% 3.21/1.74 % (22902)------------------------------
% 3.21/1.75 % (22942)Instruction limit reached!
% 3.21/1.75 % (22942)------------------------------
% 3.21/1.75 % (22942)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.75 % (22942)Termination reason: Unknown
% 3.21/1.75 % (22942)Termination phase: shuffling
% 3.21/1.75
% 3.21/1.75 % (22942)Memory used [KB]: 1791
% 3.21/1.75 % (22942)Time elapsed: 0.011 s
% 3.21/1.75 % (22942)Instructions burned: 19 (million)
% 3.21/1.75 % (22942)------------------------------
% 3.21/1.75 % (22942)------------------------------
% 3.21/1.76 % (22948)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22699 for (2998ds/879Mi)
% 3.21/1.77 % (22950)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_22699 for (2998ds/17Mi)
% 3.21/1.77 % (22950)Instruction limit reached!
% 3.21/1.77 % (22950)------------------------------
% 3.21/1.77 % (22950)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.77 % (22950)Termination reason: Unknown
% 3.21/1.77 % (22950)Termination phase: shuffling
% 3.21/1.77
% 3.21/1.77 % (22950)Memory used [KB]: 1791
% 3.21/1.77 % (22950)Time elapsed: 0.009 s
% 3.21/1.77 % (22950)Instructions burned: 17 (million)
% 3.21/1.77 % (22950)------------------------------
% 3.21/1.77 % (22950)------------------------------
% 3.21/1.79 % (22956)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22699 for (2998ds/3Mi)
% 3.21/1.79 % (22956)Instruction limit reached!
% 3.21/1.79 % (22956)------------------------------
% 3.21/1.79 % (22956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.79 % (22956)Termination reason: Unknown
% 3.21/1.79 % (22956)Termination phase: shuffling
% 3.21/1.79
% 3.21/1.79 % (22956)Memory used [KB]: 1535
% 3.21/1.79 % (22956)Time elapsed: 0.003 s
% 3.21/1.79 % (22956)Instructions burned: 3 (million)
% 3.21/1.79 % (22956)------------------------------
% 3.21/1.79 % (22956)------------------------------
% 3.21/1.81 % (22961)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_22699 for (2998ds/30Mi)
% 3.21/1.82 % (22961)Instruction limit reached!
% 3.21/1.82 % (22961)------------------------------
% 3.21/1.82 % (22961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.82 % (22961)Termination reason: Unknown
% 3.21/1.82 % (22961)Termination phase: shuffling
% 3.21/1.82
% 3.21/1.82 % (22961)Memory used [KB]: 2046
% 3.21/1.82 % (22961)Time elapsed: 0.016 s
% 3.21/1.82 % (22961)Instructions burned: 30 (million)
% 3.21/1.82 % (22961)------------------------------
% 3.21/1.82 % (22961)------------------------------
% 3.21/1.84 % (22963)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_22699 for (2997ds/127Mi)
% 3.21/1.88 % (22937)Instruction limit reached!
% 3.21/1.88 % (22937)------------------------------
% 3.21/1.88 % (22937)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.88 % (22937)Termination reason: Unknown
% 3.21/1.88 % (22937)Termination phase: Saturation
% 3.21/1.88
% 3.21/1.88 % (22937)Memory used [KB]: 9338
% 3.21/1.88 % (22937)Time elapsed: 0.166 s
% 3.21/1.88 % (22937)Instructions burned: 377 (million)
% 3.21/1.88 % (22937)------------------------------
% 3.21/1.88 % (22937)------------------------------
% 3.21/1.90 % (22966)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_22699 for (2997ds/100Mi)
% 3.21/1.90 % (22963)Instruction limit reached!
% 3.21/1.90 % (22963)------------------------------
% 3.21/1.90 % (22963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.90 % (22963)Termination reason: Unknown
% 3.21/1.90 % (22963)Termination phase: Property scanning
% 3.21/1.90
% 3.21/1.90 % (22963)Memory used [KB]: 2814
% 3.21/1.90 % (22963)Time elapsed: 0.059 s
% 3.21/1.90 % (22963)Instructions burned: 128 (million)
% 3.21/1.90 % (22963)------------------------------
% 3.21/1.90 % (22963)------------------------------
% 3.21/1.90 % (22928)Refutation not found, incomplete strategy
% 3.21/1.90 % (22928)------------------------------
% 3.21/1.90 % (22928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.90 % (22928)Termination reason: Refutation not found, incomplete strategy
% 3.21/1.90
% 3.21/1.90
% 3.21/1.90 % (22928)Memory used [KB]: 10874
% 3.21/1.90 % (22928)Time elapsed: 0.228 s
% 3.21/1.90 % (22928)Instructions burned: 427 (million)
% 3.21/1.90 % (22928)------------------------------
% 3.21/1.90 % (22928)------------------------------
% 3.21/1.92 % (22968)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_22699 for (2997ds/3Mi)
% 3.21/1.92 % (22968)Instruction limit reached!
% 3.21/1.92 % (22968)------------------------------
% 3.21/1.92 % (22968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.92 % (22968)Termination reason: Unknown
% 3.21/1.92 % (22968)Termination phase: shuffling
% 3.21/1.92
% 3.21/1.92 % (22968)Memory used [KB]: 1535
% 3.21/1.92 % (22968)Time elapsed: 0.004 s
% 3.21/1.92 % (22968)Instructions burned: 4 (million)
% 3.21/1.92 % (22968)------------------------------
% 3.21/1.92 % (22968)------------------------------
% 3.21/1.92 % (22969)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_22699 for (2997ds/20Mi)
% 3.21/1.93 % (22969)Instruction limit reached!
% 3.21/1.93 % (22969)------------------------------
% 3.21/1.93 % (22969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.93 % (22969)Termination reason: Unknown
% 3.21/1.93 % (22969)Termination phase: shuffling
% 3.21/1.93
% 3.21/1.93 % (22969)Memory used [KB]: 1918
% 3.21/1.93 % (22969)Time elapsed: 0.012 s
% 3.21/1.93 % (22969)Instructions burned: 21 (million)
% 3.21/1.93 % (22969)------------------------------
% 3.21/1.93 % (22969)------------------------------
% 3.21/1.93 % (22970)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_22699 for (2996ds/86Mi)
% 3.21/1.94 % (22966)Instruction limit reached!
% 3.21/1.94 % (22966)------------------------------
% 3.21/1.94 % (22966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.94 % (22966)Termination reason: Unknown
% 3.21/1.94 % (22966)Termination phase: Property scanning
% 3.21/1.94
% 3.21/1.94 % (22966)Memory used [KB]: 2686
% 3.21/1.94 % (22966)Time elapsed: 0.046 s
% 3.21/1.94 % (22966)Instructions burned: 101 (million)
% 3.21/1.94 % (22966)------------------------------
% 3.21/1.94 % (22966)------------------------------
% 3.21/1.95 % (22971)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_22699 for (2996ds/107Mi)
% 3.21/1.96 % (22972)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_22699 for (2996ds/39Mi)
% 3.21/1.97 % (22970)Instruction limit reached!
% 3.21/1.97 % (22970)------------------------------
% 3.21/1.97 % (22970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.97 % (22970)Termination reason: Unknown
% 3.21/1.97 % (22970)Termination phase: Property scanning
% 3.21/1.97
% 3.21/1.97 % (22970)Memory used [KB]: 2558
% 3.21/1.97 % (22970)Time elapsed: 0.034 s
% 3.21/1.97 % (22970)Instructions burned: 87 (million)
% 3.21/1.97 % (22970)------------------------------
% 3.21/1.97 % (22970)------------------------------
% 3.21/1.98 % (22972)Instruction limit reached!
% 3.21/1.98 % (22972)------------------------------
% 3.21/1.98 % (22972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.98 % (22972)Termination reason: Unknown
% 3.21/1.98 % (22972)Termination phase: shuffling
% 3.21/1.98
% 3.21/1.98 % (22972)Memory used [KB]: 2302
% 3.21/1.98 % (22972)Time elapsed: 0.019 s
% 3.21/1.98 % (22972)Instructions burned: 39 (million)
% 3.21/1.98 % (22972)------------------------------
% 3.21/1.98 % (22972)------------------------------
% 3.21/1.98 % (22975)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_22699 for (2996ds/448Mi)
% 3.21/1.99 % (22941)Instruction limit reached!
% 3.21/1.99 % (22941)------------------------------
% 3.21/1.99 % (22941)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.99 % (22941)Termination reason: Unknown
% 3.21/1.99 % (22941)Termination phase: Saturation
% 3.21/1.99
% 3.21/1.99 % (22941)Memory used [KB]: 7675
% 3.21/1.99 % (22941)Time elapsed: 0.245 s
% 3.21/1.99 % (22941)Instructions burned: 780 (million)
% 3.21/1.99 % (22941)------------------------------
% 3.21/1.99 % (22941)------------------------------
% 3.21/1.99 % (22948)First to succeed.
% 3.21/1.99 % (22909)Instruction limit reached!
% 3.21/1.99 % (22909)------------------------------
% 3.21/1.99 % (22909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/1.99 % (22909)Termination reason: Unknown
% 3.21/1.99 % (22909)Termination phase: Saturation
% 3.21/1.99
% 3.21/1.99 % (22909)Memory used [KB]: 16630
% 3.21/1.99 % (22909)Time elapsed: 0.321 s
% 3.21/1.99 % (22909)Instructions burned: 1041 (million)
% 3.21/1.99 % (22909)------------------------------
% 3.21/1.99 % (22909)------------------------------
% 3.21/2.00 % (22976)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_22699 for (2996ds/46Mi)
% 3.21/2.00 % (22971)Instruction limit reached!
% 3.21/2.00 % (22971)------------------------------
% 3.21/2.00 % (22971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/2.00 % (22971)Termination reason: Unknown
% 3.21/2.00 % (22971)Termination phase: SInE selection
% 3.21/2.00
% 3.21/2.00 % (22971)Memory used [KB]: 2558
% 3.21/2.00 % (22971)Time elapsed: 0.046 s
% 3.21/2.00 % (22971)Instructions burned: 107 (million)
% 3.21/2.00 % (22971)------------------------------
% 3.21/2.00 % (22971)------------------------------
% 3.21/2.00 % (22948)Refutation found. Thanks to Tanya!
% 3.21/2.00 % SZS status Theorem for DTF2THF_22699
% 3.21/2.00 % SZS output start Proof for DTF2THF_22699
% 3.21/2.00 thf(func_def_0, type, in: $i > $i > $o).
% 3.21/2.00 thf(func_def_1, type, exu: ($i > $o) > $o).
% 3.21/2.00 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 3.21/2.00 thf(func_def_8, type, powerset: $i > $i).
% 3.21/2.00 thf(func_def_10, type, setunion: $i > $i).
% 3.21/2.00 thf(func_def_19, type, descr: ($i > $o) > $i).
% 3.21/2.00 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 3.21/2.00 thf(func_def_26, type, prop2set: $o > $i).
% 3.21/2.00 thf(func_def_36, type, nonempty: $i > $o).
% 3.21/2.00 thf(func_def_69, type, set2prop: $i > $o).
% 3.21/2.00 thf(func_def_88, type, subset: $i > $i > $o).
% 3.21/2.00 thf(func_def_89, type, disjoint: $i > $i > $o).
% 3.21/2.00 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 3.21/2.00 thf(func_def_114, type, binunion: $i > $i > $i).
% 3.21/2.00 thf(func_def_122, type, binintersect: $i > $i > $i).
% 3.21/2.00 thf(func_def_135, type, regular: $i > $o).
% 3.21/2.00 thf(func_def_136, type, setminus: $i > $i > $i).
% 3.21/2.00 thf(func_def_147, type, symdiff: $i > $i > $i).
% 3.21/2.00 thf(func_def_153, type, iskpair: $i > $o).
% 3.21/2.00 thf(func_def_158, type, kpair: $i > $i > $i).
% 3.21/2.00 thf(func_def_160, type, cartprod: $i > $i > $i).
% 3.21/2.00 thf(func_def_177, type, singleton: $i > $o).
% 3.21/2.00 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 3.21/2.00 thf(func_def_184, type, atmost1p: $i > $o).
% 3.21/2.00 thf(func_def_185, type, atleast2p: $i > $o).
% 3.21/2.00 thf(func_def_186, type, atmost2p: $i > $o).
% 3.21/2.00 thf(func_def_187, type, upairsetp: $i > $o).
% 3.21/2.00 thf(func_def_191, type, kfst: $i > $i).
% 3.21/2.00 thf(func_def_203, type, ksnd: $i > $i).
% 3.21/2.00 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 3.21/2.00 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 3.21/2.00 thf(func_def_222, type, func: $i > $i > $i > $o).
% 3.21/2.00 thf(func_def_223, type, funcSet: $i > $i > $i).
% 3.21/2.00 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 3.21/2.00 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 3.21/2.00 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 3.21/2.00 thf(func_def_323, type, breln1: $i > $i > $o).
% 3.21/2.00 thf(func_def_325, type, breln1Set: $i > $i).
% 3.21/2.00 thf(func_def_327, type, transitive: $i > $i > $o).
% 3.21/2.00 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 3.21/2.00 thf(func_def_329, type, reflexive: $i > $i > $o).
% 3.21/2.01 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 3.21/2.01 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 3.21/2.01 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 3.21/2.01 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 3.21/2.01 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_365, type, sK4: $i > $i > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_366, type, sK5: $i > $i > $i).
% 3.21/2.01 thf(func_def_367, type, sK6: ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_368, type, sK7: $i > $i).
% 3.21/2.01 thf(func_def_369, type, sK8: $i > $i).
% 3.21/2.01 thf(func_def_370, type, sK9: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_371, type, sK10: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_372, type, sK11: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_373, type, sK12: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_374, type, sK13: $i > ($i > $i) > $i > $i).
% 3.21/2.01 thf(func_def_375, type, sK14: ($i > $o) > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_376, type, sK15: ($i > $o) > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_377, type, sK16: $i > $i).
% 3.21/2.01 thf(func_def_378, type, sK17: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_379, type, sK18: $i > $i > $i).
% 3.21/2.01 thf(func_def_380, type, sK19: $i > $i > $i).
% 3.21/2.01 thf(func_def_381, type, sK20: $i > $i > $i).
% 3.21/2.01 thf(func_def_382, type, sK21: $i > $i > $i).
% 3.21/2.01 thf(func_def_383, type, sK22: $i > $i > $i).
% 3.21/2.01 thf(func_def_384, type, sK23: ($i > $o) > $i).
% 3.21/2.01 thf(func_def_385, type, sK24: ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_386, type, sK25: ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_387, type, sK26: ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_388, type, sK27: $i > $i > $i).
% 3.21/2.01 thf(func_def_389, type, sK28: $i > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_390, type, sK29: ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_391, type, sK30: ($i > $o) > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_392, type, sK31: ($i > $o) > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_393, type, sK32: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_394, type, sK33: $i > ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_395, type, sK34: $i > ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_396, type, sK35: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_397, type, sK36: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_398, type, sK37: $i > ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_399, type, sK38: $i > ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_400, type, sK39: $i > $i).
% 3.21/2.01 thf(func_def_401, type, sK40: ($i > $o) > $i).
% 3.21/2.01 thf(func_def_402, type, sK41: ($i > $o) > $i).
% 3.21/2.01 thf(func_def_403, type, sK42: $i > $i > $i).
% 3.21/2.01 thf(func_def_404, type, sK43: ($i > $i) > $i > $i > $i).
% 3.21/2.01 thf(func_def_405, type, sK44: $i > $i > $i).
% 3.21/2.01 thf(func_def_406, type, sK45: ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_407, type, sK46: $i > $i).
% 3.21/2.01 thf(func_def_408, type, sK47: ($i > $i) > $i > $i > $i).
% 3.21/2.01 thf(func_def_409, type, sK48: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_410, type, sK49: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_411, type, sK50: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_412, type, sK51: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_413, type, sK52: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_414, type, sK53: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_415, type, sK54: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_416, type, sK55: $i > ($i > $i) > $i > $i).
% 3.21/2.01 thf(func_def_417, type, sK56: $i > $i > ($i > $i) > $i).
% 3.21/2.01 thf(func_def_418, type, sK57: $i > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_419, type, sK58: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_420, type, sK59: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_421, type, sK60: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_422, type, sK61: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_423, type, sK62: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_424, type, sK63: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_425, type, sK64: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_426, type, sK65: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_427, type, sK66: $i > $i > $i).
% 3.21/2.01 thf(func_def_428, type, sK67: ($i > $o) > $i).
% 3.21/2.01 thf(func_def_429, type, sK68: ($i > $o) > $i).
% 3.21/2.01 thf(func_def_430, type, sK69: ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_431, type, sK70: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_432, type, sK71: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_433, type, sK72: $i > $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_434, type, sK73: $i > $i).
% 3.21/2.01 thf(func_def_435, type, sK74: $i > $i > $o > $i).
% 3.21/2.01 thf(func_def_436, type, sK75: ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_437, type, sK76: $i > $i > $i).
% 3.21/2.01 thf(func_def_438, type, sK77: $i > $i > $i).
% 3.21/2.01 thf(func_def_439, type, sK78: $i > $i > $i).
% 3.21/2.01 thf(func_def_440, type, sK79: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_441, type, sK80: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_442, type, sK81: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_443, type, sK82: $i > $i > $i > $i).
% 3.21/2.01 thf(func_def_444, type, sK83: $i > ($i > $o) > ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_445, type, sK84: $i > $i > ($i > $o) > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_446, type, sK85: $i > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_447, type, sK86: $i > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_448, type, sK87: $i > $i > ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_449, type, sK88: $i > $i > ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_450, type, sK89: ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_451, type, sK90: $i > $i > $i).
% 3.21/2.01 thf(func_def_452, type, sK91: ($i > $i > $o) > $i > $i > $i).
% 3.21/2.01 thf(func_def_453, type, sK92: $i > $i).
% 3.21/2.01 thf(func_def_454, type, sK93: $i > $i > $i).
% 3.21/2.01 thf(func_def_455, type, sK94: $i > $i > $i > ($i > $o) > $i).
% 3.21/2.01 thf(func_def_456, type, sK95: $i > $i > ($i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_457, type, sK96: ($i > $i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_458, type, sK97: ($i > $i > $o) > $i > $i).
% 3.21/2.01 thf(func_def_459, type, sK98: ($i > $i > $o) > $i > $i > $i).
% 3.21/2.01 thf(func_def_460, type, sK99: ($i > $i > $o) > $i > $i > $i).
% 3.21/2.01 thf(f5021,plain,(
% 3.21/2.01 $false),
% 3.21/2.01 inference(subsumption_resolution,[status(thm)],[f3288,f4843])).
% 3.21/2.01 thf(f4843,plain,(
% 3.21/2.01 ( ! [X2 : $i,X1 : $i > $i > $o] : (($true = (breln @ X2 @ X2 @ (dpsetconstr @ X2 @ X2 @ X1)))) )),
% 3.21/2.01 inference(beta_eta_normalization,[status(thm)],[f4842])).
% 3.21/2.01 thf(f4842,plain,(
% 3.21/2.01 ( ! [X2 : $i,X1 : $i > $i > $o] : (($true = ((^[Y0 : $i]: (breln @ Y0 @ Y0 @ (dpsetconstr @ Y0 @ Y0 @ X1))) @ X2))) )),
% 3.21/2.01 inference(pi_clausification,[status(thm)],[f4841])).
% 3.21/2.01 thf(f4841,plain,(
% 3.21/2.01 ( ! [X1 : $i > $i > $o] : (($true = (!! @ $i @ (^[Y0 : $i]: (breln @ Y0 @ Y0 @ (dpsetconstr @ Y0 @ Y0 @ X1)))))) )),
% 3.21/2.01 inference(beta_eta_normalization,[status(thm)],[f4840])).
% 3.21/2.01 thf(f4840,plain,(
% 3.21/2.01 ( ! [X1 : $i > $i > $o] : (($true = ((^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (breln @ Y1 @ Y1 @ (dpsetconstr @ Y1 @ Y1 @ Y0))))) @ X1))) )),
% 3.21/2.01 inference(pi_clausification,[status(thm)],[f4839])).
% 3.21/2.01 thf(f4839,plain,(
% 3.21/2.01 ((!! @ ($i > $i > $o) @ (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (breln @ Y1 @ Y1 @ (dpsetconstr @ Y1 @ Y1 @ Y0)))))) = $true)),
% 3.21/2.01 inference(beta_eta_normalization,[status(thm)],[f1787])).
% 3.21/2.01 thf(f1787,plain,(
% 3.21/2.01 ($true = (!! @ ($i > $i > $o) @ (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: ((^[Y2 : $i]: ((^[Y3 : $i]: (breln @ Y2 @ Y2 @ Y3)))) @ Y1 @ (dpsetconstr @ Y1 @ Y1 @ (^[Y2 : $i]: ((^[Y3 : $i]: (Y0 @ Y2 @ Y3)))))))))))),
% 3.21/2.01 inference(definition_unfolding,[status(thm)],[f1483,f1365,f1083])).
% 3.21/2.01 thf(f1083,plain,(
% 3.21/2.01 (breln1 = (^[Y0 : $i]: ((^[Y1 : $i]: (breln @ Y0 @ Y0 @ Y1)))))),
% 3.21/2.01 inference(cnf_transformation,[status(thm)],[f486])).
% 3.21/2.01 thf(f486,plain,(
% 3.21/2.01 (breln1 = (^[Y0 : $i]: ((^[Y1 : $i]: (breln @ Y0 @ Y0 @ Y1)))))),
% 3.21/2.01 inference(fool_elimination,[status(thm)],[f485])).
% 3.21/2.01 thf(f485,plain,(
% 3.21/2.01 ((^[X0 : $i, X1 : $i] : (breln @ X0 @ X0 @ X1)) = breln1)),
% 3.21/2.01 inference(rectify,[status(thm)],[f288])).
% 3.21/2.01 thf(f288,axiom,(
% 3.21/2.01 ((^[X3 : $i, X13 : $i] : (breln @ X3 @ X3 @ X13)) = breln1)),
% 3.21/2.01 file('/export/starexec/sandbox/tmp/tmp.TJ114YmswG/DTF2THF_22699.p',breln1)).
% 3.21/2.01 thf(f1365,plain,(
% 3.21/2.01 (setOfPairsIsBReln1 = $true)),
% 3.21/2.01 inference(cnf_transformation,[status(thm)],[f901])).
% 3.21/2.01 thf(f901,plain,(
% 3.21/2.01 (setminusILneg = $true) & (dpsetconstrERa = $true) & (cartprodpairmemER = $true) & (symdiffI2 = $true) & (kpairp = $true) & (notdexE = $true) & (doubleComplementSub1 = $true) & (omega0Ax = $true) & (apProp = $true) & (complementT_lem = $true) & (((breln1 @ sK0 @ sK1) = $true) & (((breln1 @ sK0 @ (breln1compset @ sK0 @ sK1 @ sK2)) != $true) & ($true = (breln1 @ sK0 @ sK2)))) & (kfstsingleton = $true) & (emptysetAx = $true) & (cartprodpairsurjEq = $true) & (iftrueProp2 = $true) & (cartprodfstpairEq = $true) & (binintersectTELcontra = $true) & (dpsetconstrEL2 = $true) & (cartprodpairmemEL = $true) & (setukpairinjL1 = $true) & (quantDeMorgan3 = $true) & (woz1_1 = $true) & (funcGraphProp1 = $true) & (setukpairinjR2 = $true) & (theeq = $true) & (prop2set2propI = $true) & (setminusLsub = $true) & (ex1I = $true) & (subsetTI = $true) & (setukpairinjR12 = $true) & (in__Cong = $true) & (setunionAx = $true) & (exuI1 = $true) & (brelnall2 = $true) & (exuE3e = $true) & (binintersectSubset2 = $true) & (emptyset__Cong = $true) & (binintersectSubset4 = $true) & (bs114d = $true) & (theprop = $true) & (setbeta = $true) & (notequalI2 = $true) & (doubleComplementEq = $true) & (dpsetconstrI = $true) & (contrasubsetT2 = $true) & (upairsetE = $true) & (cartprodmempair = $true) & (binunionIL = $true) & (inComplementUnionImpInComplement1 = $true) & (subPowSU = $true) & (demorgan2a2 = $true) & (setOfPairsIsBReln = $true) & (quantDeMorgan1 = $true) & (binunionEcases = $true) & (lam2lamEq = $true) & (subbreln = $true) & (lamProp = $true) & (descrp = $true) & (complementImpComplementIntersect = $true) & (sepSubset = $true) & (setoftrueEq = $true) & (iffalseProp2 = $true) & (secondinupair = $true) & (powersetsubset = $true) & (ap2p = $true) & (setunionsingleton = $true) & (iftrue = $true) & (ex1I2 = $true) & (subsetI1 = $true) & (binintersectT_lem = $true) & (setminusSubset1 = $true) & (setukpairinjR1 = $true) & (eqbreln = $true) & (setukpairIR = $true) & (complementInPowersetComplementIntersect = $true) & (dsetconstrEL = $true) & (inIntersectImpInIntersectUnions = $true) & (woz13rule0 = $true) & (funcGraphProp4 = $true) & (nonemptyImpWitness = $true) & (setunionsingleton1 = $true) & (beta1 = $true) & (breln1invprop = $true) & (noeltsimpempty = $true) & (upairsetIR = $true) & (setminusER = $true) & (quantDeMorgan2 = $true) & (foundationAx = $true) & (uniqinunit = $true) & (doubleComplementSub2 = $true) & (emptysetsubset = $true) & (contrasubsetT3 = $true) & (iffalse = $true) & (dpsetconstrEL1 = $true) & (setunion__Cong = $true) & (demorgan2b2 = $true) & (powersetAx = $true) & (ifp = $true) & (complementTE1 = $true) & (demorgan1b = $true) & (ap2apEq2 = $true) & (subsetRefl = $true) & (eta2 = $true) & (infuncsetfunc = $true) & (eqimpsubset2 = $true) & (woz13rule3 = $true) & (setOfPairsIsBReln1 = $true) & (powersetTE1 = $true) & (singletonsubset = $true) & (breln1invE = $true) & (doubleComplementI1 = $true) & (notinemptyset = $true) & (upairset2IR = $true) & (ubforcartprodlem2 = $true) & (setadjoinE = $true) & (exuEu = $true) & (upairinpowunion = $true) & (omega__Cong = $true) & (setextT = $true) & (binintersectSubset3 = $true) & (setminusSubset2 = $true) & (binunionIR = $true) & (eta1 = $true) & (setextsub = $true) & (inPowerset = $true) & (funcGraphProp2 = $true) & (demorgan2a1 = $true) & (ex1E1 = $true) & (iftrueorfalse = $true) & (subset2powerset = $true) & (cartprodpairin = $true) & (complementSubsetComplementIntersect = $true) & (setukpairinjL = $true) & (ap2apEq1 = $true) & (emptysetE = $true) & (nonemptyI1 = $true) & (emptyinPowerset = $true) & (upairsetIL = $true) & (complementTcontraSubset = $true) & (upairsubunion = $true) & (exuI3 = $true) & (choice2fnsingleton = $true) & (setukpairinjL2 = $true) & (ksndpairEq = $true) & (iftrueProp1 = $true) & (emptyinunitempty = $true) & (notdallE = $true) & (ksndsingleton = $true) & (vacuousDall = $true) & (omegaIndAx = $true) & (setadjoinSub = $true) & (singletonsswitch = $true) & (kfstpairEq = $true) & (funcext2 = $true) & (inIntersectImpInUnion = $true) & (ubforcartprodlem1 = $true) & (singletonsuniq = $true) & (dpsetconstrER = $true) & (setukpairIL = $true) & (singletonprop = $true) & (prop2setI = $true) & (setunionE = $true) & (contrasubsetT = $true) & (cartprodsndpairEq = $true) & (eqbreln1 = $true) & (lam2p = $true) & (contraSubsetComplement = $true) & (powersetT_lem = $true) & (demorgan2b = $true) & (inComplementUnionImpNotIn1 = $true) & (dpsetconstrSub = $true) & (powerset__Cong = $true) & (contrasubsetT1 = $true) & (binunionE = $true) & (binintersectSubset1 = $true) & (setadjoinIR = $true) & (binunionLsub = $true) & (nonemptyI = $true) & (symdiffIneg1 = $true) & (ubforcartprodlem3 = $true) & (setadjoinOr = $true) & (subbreln1 = $true) & (ex1E2 = $true) & (cartprodsndin = $true) & (lamp = $true) & (subsetI2 = $true) & (powersetE1 = $true) & (setadjoinIL = $true) & (disjointsetsI1 = $true) & (binintersectRsub = $true) & (cartprodfstin = $true) & (setminusEL = $true) & (subsetTrans = $true) & (setunionI = $true) & (eqinunit = $true) & (app = $true) & (powersetI = $true) & (subsetE = $true) & (complementTnotintersectT = $true) & (symdiffI1 = $true) & (funcImageSingleton = $true) & (funcext = $true) & (upairset2E = $true) & (funcGraphProp3 = $true) & (woz13rule4 = $true) & (eqimpsubset1 = $true) & (emptyI = $true) & (demorgan2 = $true) & (funcinfuncset = $true) & (setextAx = $true) & (powersetE = $true) & (woz1_2 = $true) & (brelnall1 = $true) & (binintersectSubset5 = $true) & (powersetTI1 = $true) & (singletoninpowerset = $true) & (woz1_3 = $true) & (breln1all1 = $true) & (replAx = $true) & (beta2 = $true) & (descr__Cong = $true) & (setminusT_lem = $true) & (inIntersectImpInUnion2 = $true) & (upairequniteq = $true) & (kpairsurjEq = $true) & (binunionTIRcontra = $true) & (funcextLem = $true) & (kpairiskpair = $true) & (subsetE2 = $true) & (subsetemptysetimpeq = $true) & (symdiffIneg2 = $true) & (complementTI1 = $true) & (exuE1 = $true) & (setunionsingleton2 = $true) & (exuE2 = $true) & (binintersectEL = $true) & (exu__Cong = $true) & (intersectInPowersetIntersectUnions = $true) & (binintersectER = $true) & (breln1SetBreln1 = $true) & (demorgan1a = $true) & (notinsingleton = $true) & (emptyInPowerset = $true) & (symdiffE = $true) & (binintersectI = $true) & (woz13rule2 = $true) & (setadjoin__Cong = $true) & (notsubsetI = $true) & (dsetconstr__Cong = $true) & (ifSingleton = $true) & (binunionTEcontra = $true) & (binunionTE = $true) & (dsetconstrER = $true) & (sepInPowerset = $true) & (binunionT_lem = $true) & (exuE3u = $true) & (woz13rule1 = $true) & (singletoninpowunion = $true) & (setminusELneg = $true) & (woz1_5 = $true) & (cartprodmempaircEq = $true) & (omegaSAx = $true) & (setminusERneg = $true) & (nonemptyE1 = $true) & (binintersectTERcontra = $true) & (setunionE2 = $true) & (doubleComplementE1 = $true) & (woz1_4 = $true) & (iffalseProp1 = $true) & (demorgan1 = $true) & (demorgan2a = $true) & (cartprodmempair1 = $true) & (setukpairinjR11 = $true) & (powersetI1 = $true) & (quantDeMorgan4 = $true) & (binunionRsub = $true) & (binintersectLsub = $true) & (setminusI = $true) & (setminusIRneg = $true) & (setadjoinAx = $true) & (wellorderingAx = $true) & (breln1invI = $true) & (complementUnionInPowersetComplement = $true) & (notequalI1 = $true) & (exuI2 = $true) & (binunionTILcontra = $true) & (emptyE1 = $true) & (breln1all2 = $true) & (dsetconstrI = $true) & (setadjoinSub2 = $true) & (emptysetimpfalse = $true) & (setext = $true) & (prop2setE = $true) & (inCongP = $true) & (setukpairinjR = $true)),
% 3.21/2.01 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f898,f900,f899])).
% 3.21/2.01 thf(f899,plain,(
% 3.21/2.01 ? [X0 : $i,X1 : $i] : (((breln1 @ X0 @ X1) = $true) & ? [X2 : $i] : (($true != (breln1 @ X0 @ (breln1compset @ X0 @ X1 @ X2))) & ((breln1 @ X0 @ X2) = $true))) => (((breln1 @ sK0 @ sK1) = $true) & ? [X2 : $i] : (($true != (breln1 @ sK0 @ (breln1compset @ sK0 @ sK1 @ X2))) & ((breln1 @ sK0 @ X2) = $true)))),
% 3.21/2.01 introduced(definition,[],[choice_axiom])).
% 3.21/2.01 thf(f900,plain,(
% 3.21/2.01 ? [X2 : $i] : (($true != (breln1 @ sK0 @ (breln1compset @ sK0 @ sK1 @ X2))) & ((breln1 @ sK0 @ X2) = $true)) => (((breln1 @ sK0 @ (breln1compset @ sK0 @ sK1 @ sK2)) != $true) & ($true = (breln1 @ sK0 @ sK2)))),
% 3.21/2.01 introduced(definition,[],[choice_axiom])).
% 3.21/2.01 thf(f898,plain,(
% 3.21/2.01 (setminusILneg = $true) & (dpsetconstrERa = $true) & (cartprodpairmemER = $true) & (symdiffI2 = $true) & (kpairp = $true) & (notdexE = $true) & (doubleComplementSub1 = $true) & (omega0Ax = $true) & (apProp = $true) & (complementT_lem = $true) & ? [X0 : $i,X1 : $i] : (((breln1 @ X0 @ X1) = $true) & ? [X2 : $i] : (($true != (breln1 @ X0 @ (breln1compset @ X0 @ X1 @ X2))) & ((breln1 @ X0 @ X2) = $true))) & (kfstsingleton = $true) & (emptysetAx = $true) & (cartprodpairsurjEq = $true) & (iftrueProp2 = $true) & (cartprodfstpairEq = $true) & (binintersectTELcontra = $true) & (dpsetconstrEL2 = $true) & (cartprodpairmemEL = $true) & (setukpairinjL1 = $true) & (quantDeMorgan3 = $true) & (woz1_1 = $true) & (funcGraphProp1 = $true) & (setukpairinjR2 = $true) & (theeq = $true) & (prop2set2propI = $true) & (setminusLsub = $true) & (ex1I = $true) & (subsetTI = $true) & (setukpairinjR12 = $true) & (in__Cong = $true) & (setunionAx = $true) & (exuI1 = $true) & (brelnall2 = $true) & (exuE3e = $true) & (binintersectSubset2 = $true) & (emptyset__Cong = $true) & (binintersectSubset4 = $true) & (bs114d = $true) & (theprop = $true) & (setbeta = $true) & (notequalI2 = $true) & (doubleComplementEq = $true) & (dpsetconstrI = $true) & (contrasubsetT2 = $true) & (upairsetE = $true) & (cartprodmempair = $true) & (binunionIL = $true) & (inComplementUnionImpInComplement1 = $true) & (subPowSU = $true) & (demorgan2a2 = $true) & (setOfPairsIsBReln = $true) & (quantDeMorgan1 = $true) & (binunionEcases = $true) & (lam2lamEq = $true) & (subbreln = $true) & (lamProp = $true) & (descrp = $true) & (complementImpComplementIntersect = $true) & (sepSubset = $true) & (setoftrueEq = $true) & (iffalseProp2 = $true) & (secondinupair = $true) & (powersetsubset = $true) & (ap2p = $true) & (setunionsingleton = $true) & (iftrue = $true) & (ex1I2 = $true) & (subsetI1 = $true) & (binintersectT_lem = $true) & (setminusSubset1 = $true) & (setukpairinjR1 = $true) & (eqbreln = $true) & (setukpairIR = $true) & (complementInPowersetComplementIntersect = $true) & (dsetconstrEL = $true) & (inIntersectImpInIntersectUnions = $true) & (woz13rule0 = $true) & (funcGraphProp4 = $true) & (nonemptyImpWitness = $true) & (setunionsingleton1 = $true) & (beta1 = $true) & (breln1invprop = $true) & (noeltsimpempty = $true) & (upairsetIR = $true) & (setminusER = $true) & (quantDeMorgan2 = $true) & (foundationAx = $true) & (uniqinunit = $true) & (doubleComplementSub2 = $true) & (emptysetsubset = $true) & (contrasubsetT3 = $true) & (iffalse = $true) & (dpsetconstrEL1 = $true) & (setunion__Cong = $true) & (demorgan2b2 = $true) & (powersetAx = $true) & (ifp = $true) & (complementTE1 = $true) & (demorgan1b = $true) & (ap2apEq2 = $true) & (subsetRefl = $true) & (eta2 = $true) & (infuncsetfunc = $true) & (eqimpsubset2 = $true) & (woz13rule3 = $true) & (setOfPairsIsBReln1 = $true) & (powersetTE1 = $true) & (singletonsubset = $true) & (breln1invE = $true) & (doubleComplementI1 = $true) & (notinemptyset = $true) & (upairset2IR = $true) & (ubforcartprodlem2 = $true) & (setadjoinE = $true) & (exuEu = $true) & (upairinpowunion = $true) & (omega__Cong = $true) & (setextT = $true) & (binintersectSubset3 = $true) & (setminusSubset2 = $true) & (binunionIR = $true) & (eta1 = $true) & (setextsub = $true) & (inPowerset = $true) & (funcGraphProp2 = $true) & (demorgan2a1 = $true) & (ex1E1 = $true) & (iftrueorfalse = $true) & (subset2powerset = $true) & (cartprodpairin = $true) & (complementSubsetComplementIntersect = $true) & (setukpairinjL = $true) & (ap2apEq1 = $true) & (emptysetE = $true) & (nonemptyI1 = $true) & (emptyinPowerset = $true) & (upairsetIL = $true) & (complementTcontraSubset = $true) & (upairsubunion = $true) & (exuI3 = $true) & (choice2fnsingleton = $true) & (setukpairinjL2 = $true) & (ksndpairEq = $true) & (iftrueProp1 = $true) & (emptyinunitempty = $true) & (notdallE = $true) & (ksndsingleton = $true) & (vacuousDall = $true) & (omegaIndAx = $true) & (setadjoinSub = $true) & (singletonsswitch = $true) & (kfstpairEq = $true) & (funcext2 = $true) & (inIntersectImpInUnion = $true) & (ubforcartprodlem1 = $true) & (singletonsuniq = $true) & (dpsetconstrER = $true) & (setukpairIL = $true) & (singletonprop = $true) & (prop2setI = $true) & (setunionE = $true) & (contrasubsetT = $true) & (cartprodsndpairEq = $true) & (eqbreln1 = $true) & (lam2p = $true) & (contraSubsetComplement = $true) & (powersetT_lem = $true) & (demorgan2b = $true) & (inComplementUnionImpNotIn1 = $true) & (dpsetconstrSub = $true) & (powerset__Cong = $true) & (contrasubsetT1 = $true) & (binunionE = $true) & (binintersectSubset1 = $true) & (setadjoinIR = $true) & (binunionLsub = $true) & (nonemptyI = $true) & (symdiffIneg1 = $true) & (ubforcartprodlem3 = $true) & (setadjoinOr = $true) & (subbreln1 = $true) & (ex1E2 = $true) & (cartprodsndin = $true) & (lamp = $true) & (subsetI2 = $true) & (powersetE1 = $true) & (setadjoinIL = $true) & (disjointsetsI1 = $true) & (binintersectRsub = $true) & (cartprodfstin = $true) & (setminusEL = $true) & (subsetTrans = $true) & (setunionI = $true) & (eqinunit = $true) & (app = $true) & (powersetI = $true) & (subsetE = $true) & (complementTnotintersectT = $true) & (symdiffI1 = $true) & (funcImageSingleton = $true) & (funcext = $true) & (upairset2E = $true) & (funcGraphProp3 = $true) & (woz13rule4 = $true) & (eqimpsubset1 = $true) & (emptyI = $true) & (demorgan2 = $true) & (funcinfuncset = $true) & (setextAx = $true) & (powersetE = $true) & (woz1_2 = $true) & (brelnall1 = $true) & (binintersectSubset5 = $true) & (powersetTI1 = $true) & (singletoninpowerset = $true) & (woz1_3 = $true) & (breln1all1 = $true) & (replAx = $true) & (beta2 = $true) & (descr__Cong = $true) & (setminusT_lem = $true) & (inIntersectImpInUnion2 = $true) & (upairequniteq = $true) & (kpairsurjEq = $true) & (binunionTIRcontra = $true) & (funcextLem = $true) & (kpairiskpair = $true) & (subsetE2 = $true) & (subsetemptysetimpeq = $true) & (symdiffIneg2 = $true) & (complementTI1 = $true) & (exuE1 = $true) & (setunionsingleton2 = $true) & (exuE2 = $true) & (binintersectEL = $true) & (exu__Cong = $true) & (intersectInPowersetIntersectUnions = $true) & (binintersectER = $true) & (breln1SetBreln1 = $true) & (demorgan1a = $true) & (notinsingleton = $true) & (emptyInPowerset = $true) & (symdiffE = $true) & (binintersectI = $true) & (woz13rule2 = $true) & (setadjoin__Cong = $true) & (notsubsetI = $true) & (dsetconstr__Cong = $true) & (ifSingleton = $true) & (binunionTEcontra = $true) & (binunionTE = $true) & (dsetconstrER = $true) & (sepInPowerset = $true) & (binunionT_lem = $true) & (exuE3u = $true) & (woz13rule1 = $true) & (singletoninpowunion = $true) & (setminusELneg = $true) & (woz1_5 = $true) & (cartprodmempaircEq = $true) & (omegaSAx = $true) & (setminusERneg = $true) & (nonemptyE1 = $true) & (binintersectTERcontra = $true) & (setunionE2 = $true) & (doubleComplementE1 = $true) & (woz1_4 = $true) & (iffalseProp1 = $true) & (demorgan1 = $true) & (demorgan2a = $true) & (cartprodmempair1 = $true) & (setukpairinjR11 = $true) & (powersetI1 = $true) & (quantDeMorgan4 = $true) & (binunionRsub = $true) & (binintersectLsub = $true) & (setminusI = $true) & (setminusIRneg = $true) & (setadjoinAx = $true) & (wellorderingAx = $true) & (breln1invI = $true) & (complementUnionInPowersetComplement = $true) & (notequalI1 = $true) & (exuI2 = $true) & (binunionTILcontra = $true) & (emptyE1 = $true) & (breln1all2 = $true) & (dsetconstrI = $true) & (setadjoinSub2 = $true) & (emptysetimpfalse = $true) & (setext = $true) & (prop2setE = $true) & (inCongP = $true) & (setukpairinjR = $true)),
% 3.21/2.01 inference(flattening,[status(thm)],[f897])).
% 3.21/2.01 thf(f897,plain,(
% 3.21/2.01 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (((breln1 @ X0 @ X1) = $true) & ? [X2 : $i] : (($true != (breln1 @ X0 @ (breln1compset @ X0 @ X1 @ X2))) & ((breln1 @ X0 @ X2) = $true))) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 3.21/2.01 inference(ennf_transformation,[status(thm)],[f480])).
% 3.21/2.01 thf(f480,plain,(
% 3.21/2.01 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ! [X0 : $i,X1 : $i] : (((breln1 @ X0 @ X1) = $true) => ! [X2 : $i] : (((breln1 @ X0 @ X2) = $true) => ($true = (breln1 @ X0 @ (breln1compset @ X0 @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.21/2.01 inference(fool_elimination,[status(thm)],[f479])).
% 3.21/2.01 thf(f479,plain,(
% 3.21/2.01 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => ! [X0 : $i,X1 : $i] : ((breln1 @ X0 @ X1) => ! [X2 : $i] : ((breln1 @ X0 @ X2) => (breln1 @ X0 @ (breln1compset @ X0 @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.21/2.01 inference(rectify,[status(thm)],[f301])).
% 3.21/2.01 thf(f301,negated_conjecture,(
% 3.21/2.01 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => ! [X3 : $i,X13 : $i] : ((breln1 @ X3 @ X13) => ! [X15 : $i] : ((breln1 @ X3 @ X15) => (breln1 @ X3 @ (breln1compset @ X3 @ X13 @ X15))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.21/2.01 inference(negated_conjecture,[status(cth)],[f300])).
% 3.21/2.01 thf(f300,conjecture,(
% 3.21/2.01 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => ! [X3 : $i,X13 : $i] : ((breln1 @ X3 @ X13) => ! [X15 : $i] : ((breln1 @ X3 @ X15) => (breln1 @ X3 @ (breln1compset @ X3 @ X13 @ X15)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 3.21/2.01 file('/export/starexec/sandbox/tmp/tmp.TJ114YmswG/DTF2THF_22699.p',breln1compprop)).
% 3.21/2.01 thf(f1483,plain,(
% 3.21/2.01 (setOfPairsIsBReln1 = (!! @ ($i > $i > $o) @ (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (breln1 @ Y1 @ (dpsetconstr @ Y1 @ Y1 @ (^[Y2 : $i]: ((^[Y3 : $i]: (Y0 @ Y2 @ Y3)))))))))))),
% 3.21/2.01 inference(cnf_transformation,[status(thm)],[f756])).
% 3.21/2.01 thf(f756,plain,(
% 3.21/2.01 (setOfPairsIsBReln1 = (!! @ ($i > $i > $o) @ (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (breln1 @ Y1 @ (dpsetconstr @ Y1 @ Y1 @ (^[Y2 : $i]: ((^[Y3 : $i]: (Y0 @ Y2 @ Y3)))))))))))),
% 3.21/2.01 inference(fool_elimination,[status(thm)],[f755])).
% 3.21/2.01 thf(f755,plain,(
% 3.21/2.01 (! [X0 : $i,X1 : $i > $i > $o] : (breln1 @ X0 @ (dpsetconstr @ X0 @ X0 @ (^[X2 : $i, X3 : $i] : (X1 @ X2 @ X3)))) = setOfPairsIsBReln1)),
% 3.21/2.01 inference(rectify,[status(thm)],[f292])).
% 3.21/2.01 thf(f292,axiom,(
% 3.21/2.01 (! [X3 : $i,X0 : $i > $i > $o] : (breln1 @ X3 @ (dpsetconstr @ X3 @ X3 @ (^[X1 : $i, X2 : $i] : (X0 @ X1 @ X2)))) = setOfPairsIsBReln1)),
% 3.21/2.01 file('/export/starexec/sandbox/tmp/tmp.TJ114YmswG/DTF2THF_22699.p',setOfPairsIsBReln1)).
% 3.21/2.01 thf(f3288,plain,(
% 3.21/2.01 ((breln @ sK0 @ sK0 @ (dpsetconstr @ sK0 @ sK0 @ (^[Y0 : $i]: ((^[Y1 : $i]: (?? @ $i @ (^[Y2 : $i]: (((in @ Y2 @ sK0) & (in @ (kpair @ Y0 @ Y2) @ sK1)) & (in @ (kpair @ Y2 @ Y1) @ sK2))))))))) != $true)),
% 3.21/2.01 inference(beta_eta_normalization,[status(thm)],[f1513])).
% 3.21/2.01 thf(f1513,plain,(
% 3.21/2.01 ($true != ((^[Y0 : $i]: ((^[Y1 : $i]: (breln @ Y0 @ Y0 @ Y1)))) @ sK0 @ ((^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (dpsetconstr @ Y0 @ Y0 @ (^[Y3 : $i]: ((^[Y4 : $i]: (?? @ $i @ (^[Y5 : $i]: (((in @ Y5 @ Y0) & (in @ (kpair @ Y3 @ Y5) @ Y1)) & (in @ (kpair @ Y5 @ Y4) @ Y2))))))))))))) @ sK0 @ sK1 @ sK2)))),
% 3.21/2.01 inference(definition_unfolding,[status(thm)],[f1462,f1083,f1042])).
% 3.21/2.01 thf(f1042,plain,(
% 3.21/2.01 (breln1compset = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (dpsetconstr @ Y0 @ Y0 @ (^[Y3 : $i]: ((^[Y4 : $i]: (?? @ $i @ (^[Y5 : $i]: (((in @ Y5 @ Y0) & (in @ (kpair @ Y3 @ Y5) @ Y1)) & (in @ (kpair @ Y5 @ Y4) @ Y2))))))))))))))),
% 3.21/2.01 inference(cnf_transformation,[status(thm)],[f864])).
% 3.21/2.01 thf(f864,plain,(
% 3.21/2.01 (breln1compset = (^[Y0 : $i]: ((^[Y1 : $i]: ((^[Y2 : $i]: (dpsetconstr @ Y0 @ Y0 @ (^[Y3 : $i]: ((^[Y4 : $i]: (?? @ $i @ (^[Y5 : $i]: (((in @ Y5 @ Y0) & (in @ (kpair @ Y3 @ Y5) @ Y1)) & (in @ (kpair @ Y5 @ Y4) @ Y2))))))))))))))),
% 3.21/2.01 inference(fool_elimination,[status(thm)],[f863])).
% 3.21/2.01 thf(f863,plain,(
% 3.21/2.01 ((^[X0 : $i, X1 : $i, X2 : $i] : (dpsetconstr @ X0 @ X0 @ (^[X3 : $i, X4 : $i] : (? [X5 : $i] : ((in @ (kpair @ X5 @ X4) @ X2) & (in @ (kpair @ X3 @ X5) @ X1) & (in @ X5 @ X0)))))) = breln1compset)),
% 3.21/2.01 inference(rectify,[status(thm)],[f299])).
% 3.21/2.01 thf(f299,axiom,(
% 3.21/2.01 ((^[X3 : $i, X13 : $i, X15 : $i] : (dpsetconstr @ X3 @ X3 @ (^[X1 : $i, X2 : $i] : (? [X8 : $i] : ((in @ (kpair @ X8 @ X2) @ X15) & (in @ (kpair @ X1 @ X8) @ X13) & (in @ X8 @ X3)))))) = breln1compset)),
% 3.21/2.01 file('/export/starexec/sandbox/tmp/tmp.TJ114YmswG/DTF2THF_22699.p',breln1compset)).
% 3.21/2.01 thf(f1462,plain,(
% 3.21/2.01 ((breln1 @ sK0 @ (breln1compset @ sK0 @ sK1 @ sK2)) != $true)),
% 3.21/2.01 inference(cnf_transformation,[status(thm)],[f901])).
% 3.21/2.01 % SZS output end Proof for DTF2THF_22699
% 3.21/2.01 % (22948)------------------------------
% 3.21/2.01 % (22948)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.21/2.01 % (22948)Termination reason: Refutation
% 3.21/2.01
% 3.21/2.01 % (22948)Memory used [KB]: 10106
% 3.21/2.01 % (22948)Time elapsed: 0.240 s
% 3.21/2.01 % (22948)Instructions burned: 515 (million)
% 3.21/2.01 % (22948)------------------------------
% 3.21/2.01 % (22948)------------------------------
% 3.21/2.01 % (22892)Success in time 0.39 s
%------------------------------------------------------------------------------