%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU771^1 : 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 : n003.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 09:37:03 AM UTC 2026
% Result : Theorem 2.96s 1.78s
% Output : Refutation 2.96s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 4
% Syntax : Number of formulae : 29 ( 21 unt; 0 typ; 0 def)
% Number of atoms : 4426 (1174 equ; 0 cnn)
% Maximal formula atoms : 288 ( 152 avg)
% Number of connectives : 2258 ( 11 ~; 0 |; 861 &; 219 @)
% ( 0 <=>;1149 =>; 0 <=; 0 <~>)
% Maximal formula depth : 297 ( 72 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 52 ( 52 >; 0 *; 0 +; 0 <<)
% Number of symbols : 418 ( 414 usr; 291 con; 0-4 aty)
% ( 18 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 95 ( 63 ^; 26 !; 6 ?; 95 :)
% 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_352,type,
sK0: $i > $i > $o ).
thf(func_def_355,type,
sK3: $i > $i > $i ).
thf(func_def_356,type,
sK4: $i > $i > $i > $i > $i ).
thf(func_def_357,type,
sK5: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_358,type,
sK6: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_359,type,
sK7: $i > $i > $i > $i ).
thf(func_def_360,type,
sK8: $i > $i > $i > $i ).
thf(func_def_361,type,
sK9: $i > ( $i > $o ) > $i ).
thf(func_def_362,type,
sK10: $i > ( $i > $o ) > $i ).
thf(func_def_363,type,
sK11: $i > ( $i > $o ) > $i ).
thf(func_def_364,type,
sK12: $i > ( $i > $o ) > $i > $i ).
thf(func_def_365,type,
sK13: $i > ( $i > $o ) > $i > $i ).
thf(func_def_366,type,
sK14: ( $i > $i > $o ) > $i > $i > $i ).
thf(func_def_367,type,
sK15: ( $i > $o ) > $i > $i ).
thf(func_def_368,type,
sK16: $i > $i > ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_369,type,
sK17: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_370,type,
sK18: $i > $i > $i > $i ).
thf(func_def_371,type,
sK19: $i > $i > $i > $i ).
thf(func_def_372,type,
sK20: $i > ( $i > $o ) > $i ).
thf(func_def_373,type,
sK21: $i > $i ).
thf(func_def_374,type,
sK22: ( $i > $o ) > $i ).
thf(func_def_375,type,
sK23: $i > $i ).
thf(func_def_376,type,
sK24: $i > $i > $i > $i > $i ).
thf(func_def_377,type,
sK25: $i > $i > $i > $i > $i ).
thf(func_def_378,type,
sK26: $i > $i > $i > $i > $i ).
thf(func_def_379,type,
sK27: $i > $i > $i > $i > $i ).
thf(func_def_380,type,
sK28: $i > $i > $i > $i > $i ).
thf(func_def_381,type,
sK29: $i > $i > $i > $i > $i ).
thf(func_def_382,type,
sK30: $i > $i > $i > $i > $i ).
thf(func_def_383,type,
sK31: ( $i > $o ) > $i ).
thf(func_def_384,type,
sK32: ( $i > $o ) > $i ).
thf(func_def_385,type,
sK33: ( $i > $i ) > $i > $i > $i ).
thf(func_def_386,type,
sK34: $i > $i > $i ).
thf(func_def_387,type,
sK35: $i > $i ).
thf(func_def_388,type,
sK36: $i > $i > $i > ( $i > $o ) > $i ).
thf(func_def_389,type,
sK37: $i > $i > ( $i > $o ) > $i > $i ).
thf(func_def_390,type,
sK38: $i > $i > $i ).
thf(func_def_391,type,
sK39: $i > $i > $i ).
thf(func_def_392,type,
sK40: ( $i > $o ) > $i > $i ).
thf(func_def_393,type,
sK41: $i > $i > ( $i > $i ) > $i ).
thf(func_def_394,type,
sK42: $i > $i > $i > $i ).
thf(func_def_395,type,
sK43: ( $i > $o ) > $i ).
thf(func_def_396,type,
sK44: $i > $i > $i ).
thf(func_def_397,type,
sK45: $i > $i > ( $i > $i ) > $i ).
thf(func_def_398,type,
sK46: ( $i > $o ) > $i > $i > $i ).
thf(func_def_399,type,
sK47: ( $i > $o ) > $i > $i ).
thf(func_def_400,type,
sK48: $i > $i ).
thf(func_def_401,type,
sK49: $i > $i > $i > $i ).
thf(func_def_402,type,
sK50: $i > $i > $i ).
thf(func_def_403,type,
sK51: $i > $i > $i ).
thf(func_def_404,type,
sK52: $i > $i > $i ).
thf(func_def_405,type,
sK53: $i > $i > $i > $i > $i ).
thf(func_def_406,type,
sK54: ( $i > $o ) > $i > $i ).
thf(func_def_407,type,
sK55: ( $i > $o ) > $i > $i ).
thf(func_def_408,type,
sK56: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_409,type,
sK57: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_410,type,
sK58: $i > $i ).
thf(func_def_411,type,
sK59: $i > $i > $i ).
thf(func_def_412,type,
sK60: $i > $i > $i ).
thf(func_def_413,type,
sK61: $i > ( $i > $i ) > $i > $i ).
thf(func_def_414,type,
sK62: $i > $i > $i ).
thf(func_def_415,type,
sK63: $i > $i > $i > $i ).
thf(func_def_416,type,
sK64: $i > $i > $i ).
thf(func_def_417,type,
sK65: $i > $i > $o > $i ).
thf(func_def_418,type,
sK66: ( $i > $o ) > $i > $i ).
thf(func_def_419,type,
sK67: ( $i > $o ) > $i ).
thf(func_def_420,type,
sK68: $i > $i > $i > $i > $i ).
thf(func_def_421,type,
sK69: $i > $i ).
thf(func_def_422,type,
sK70: $i > $i ).
thf(func_def_423,type,
sK71: $i > $i > ( $i > $i ) > $i ).
thf(func_def_424,type,
sK72: $i > $i > $i > $i > $i ).
thf(func_def_425,type,
sK73: $i > $i > $i > $i > $i ).
thf(func_def_426,type,
sK74: $i > ( $i > $o ) > $i ).
thf(func_def_427,type,
sK75: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_428,type,
sK76: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_429,type,
sK77: $i > $i > $i ).
thf(func_def_430,type,
sK78: $i > ( $i > $i > $o ) > $i ).
thf(func_def_431,type,
sK79: $i > ( $i > $i > $o ) > $i ).
thf(func_def_432,type,
sK80: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_433,type,
sK81: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_434,type,
sK82: $i > ( $i > $o ) > $i ).
thf(func_def_435,type,
sK83: ( $i > $o ) > $i > $i ).
thf(f4777,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f3803,f4656]) ).
thf(f4656,plain,
! [X2: $i > $i > $o,X3: $i,X1: $i] :
( ( breln @ X1 @ X3 @ ( dpsetconstr @ X1 @ X3 @ X2 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f4655]) ).
thf(f4655,plain,
! [X2: $i > $i > $o,X3: $i,X1: $i] :
( ( ^ [Y0: $i] : ( breln @ X1 @ Y0 @ ( dpsetconstr @ X1 @ Y0 @ X2 ) )
@ X3 )
= $true ),
inference(pi_clausification,[status(thm)],[f4654]) ).
thf(f4654,plain,
! [X2: $i > $i > $o,X1: $i] :
( $true
= ( !! @ $i
@ ^ [Y0: $i] : ( breln @ X1 @ Y0 @ ( dpsetconstr @ X1 @ Y0 @ X2 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f4653]) ).
thf(f4653,plain,
! [X2: $i > $i > $o,X1: $i] :
( ( ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( breln @ X1 @ Y1 @ ( dpsetconstr @ X1 @ Y1 @ Y0 ) ) )
@ X2 )
= $true ),
inference(pi_clausification,[status(thm)],[f4652]) ).
thf(f4652,plain,
! [X1: $i] :
( ( !! @ ( $i > $i > $o )
@ ^ [Y0: $i > $i > $o] :
( !! @ $i
@ ^ [Y1: $i] : ( breln @ X1 @ Y1 @ ( dpsetconstr @ X1 @ Y1 @ Y0 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f4651]) ).
thf(f4651,plain,
! [X1: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ ( $i > $i > $o )
@ ^ [Y1: $i > $i > $o] :
( !! @ $i
@ ^ [Y2: $i] : ( breln @ Y0 @ Y2 @ ( dpsetconstr @ Y0 @ Y2 @ Y1 ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f4650]) ).
thf(f4650,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $i > $o )
@ ^ [Y1: $i > $i > $o] :
( !! @ $i
@ ^ [Y2: $i] : ( breln @ Y0 @ Y2 @ ( dpsetconstr @ Y0 @ Y2 @ Y1 ) ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1742]) ).
thf(f1742,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $i > $o )
@ ^ [Y1: $i > $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( breln @ Y0 @ Y2
@ ( dpsetconstr @ Y0 @ Y2
@ ^ [Y3: $i,Y4: $i] : ( Y1 @ Y3 @ Y4 ) ) ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1452,f1178]) ).
thf(f1178,plain,
setOfPairsIsBReln = $true,
inference(cnf_transformation,[status(thm)],[f876]) ).
thf(f876,plain,
( ( setunionI = $true )
& ( replAx = $true )
& ( complementTE1 = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( doubleComplementI1 = $true )
& ( kfstpairEq = $true )
& ( binintersectSubset1 = $true )
& ( funcext2 = $true )
& ( cartprodsndin = $true )
& ( powersetE1 = $true )
& ( binunionTE = $true )
& ( kpairp = $true )
& ( dpsetconstrERa = $true )
& ( uniqinunit = $true )
& ( upairequniteq = $true )
& ( dsetconstrI = $true )
& ( binintersectSubset2 = $true )
& ( cartprodmempair = $true )
& ( cartprodpairmemEL = $true )
& ( ap2apEq2 = $true )
& ( binintersectSubset4 = $true )
& ( descrp = $true )
& ( upairinpowunion = $true )
& ( kpairsurjEq = $true )
& ( subPowSU = $true )
& ( ubforcartprodlem2 = $true )
& ( singletonsubset = $true )
& ( powersetI = $true )
& ( inCongP = $true )
& ( dpsetconstrEL2 = $true )
& ( ifp = $true )
& ( powersetI1 = $true )
& ( choice2fnsingleton = $true )
& ( demorgan2a = $true )
& ( upairsetIR = $true )
& ( setadjoin__Cong = $true )
& ( app = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( singletonsswitch = $true )
& ( woz13rule2 = $true )
& ( ap2p = $true )
& ( exuE3e = $true )
& ( setminusIRneg = $true )
& ( inPowerset = $true )
& ( setminusERneg = $true )
& ( subsetTrans = $true )
& ( subsetI1 = $true )
& ( upairset2E = $true )
& ( setunionE = $true )
& ( setunion__Cong = $true )
& ( woz1_1 = $true )
& ( notdexE = $true )
& ( setukpairinjR2 = $true )
& ( subbreln = $true )
& ( setminusSubset1 = $true )
& ( beta1 = $true )
& ( setminusSubset2 = $true )
& ( powersetsubset = $true )
& ( ap2apEq1 = $true )
& ( binunionTEcontra = $true )
& ( prop2set2propI = $true )
& ( omegaSAx = $true )
& ( eta2 = $true )
& ( ( breln1 @ sK1
@ ( dpsetconstr @ sK1 @ sK1
@ ^ [Y0: $i,Y1: $i] : ( sK0 @ Y0 @ Y1 ) ) )
!= $true )
& ( dsetconstrER = $true )
& ( powersetTE1 = $true )
& ( inIntersectImpInUnion = $true )
& ( notinsingleton = $true )
& ( exuE3u = $true )
& ( secondinupair = $true )
& ( quantDeMorgan3 = $true )
& ( doubleComplementE1 = $true )
& ( upairsetE = $true )
& ( binunionLsub = $true )
& ( dpsetconstrSub = $true )
& ( binintersectEL = $true )
& ( binintersectRsub = $true )
& ( setminusLsub = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( binunionRsub = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( cartprodfstin = $true )
& ( eqinunit = $true )
& ( demorgan2 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( upairset2IR = $true )
& ( contrasubsetT3 = $true )
& ( dpsetconstrEL1 = $true )
& ( contrasubsetT1 = $true )
& ( sepSubset = $true )
& ( theeq = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( binunionTILcontra = $true )
& ( singletoninpowerset = $true )
& ( woz1_2 = $true )
& ( woz1_4 = $true )
& ( demorgan2b2 = $true )
& ( setextT = $true )
& ( emptysetimpfalse = $true )
& ( funcinfuncset = $true )
& ( iffalseProp1 = $true )
& ( woz1_5 = $true )
& ( prop2setE = $true )
& ( setadjoinE = $true )
& ( ex1E1 = $true )
& ( setadjoinAx = $true )
& ( dsetconstr__Cong = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectLsub = $true )
& ( cartprodpairmemER = $true )
& ( powersetTI1 = $true )
& ( woz13rule4 = $true )
& ( eqbreln = $true )
& ( cartprodpairin = $true )
& ( emptyInPowerset = $true )
& ( singletonsuniq = $true )
& ( brelnall2 = $true )
& ( setminusILneg = $true )
& ( demorgan1b = $true )
& ( disjointsetsI1 = $true )
& ( setOfPairsIsBReln = $true )
& ( powersetAx = $true )
& ( setminusELneg = $true )
& ( demorgan1 = $true )
& ( demorgan1a = $true )
& ( iftrue = $true )
& ( inIntersectImpInUnion2 = $true )
& ( binintersectER = $true )
& ( nonemptyE1 = $true )
& ( upairsubunion = $true )
& ( setunionE2 = $true )
& ( setadjoinSub = $true )
& ( setunionsingleton2 = $true )
& ( binunionIL = $true )
& ( nonemptyImpWitness = $true )
& ( ksndpairEq = $true )
& ( powersetT_lem = $true )
& ( cartprodsndpairEq = $true )
& ( exuI1 = $true )
& ( subsetTI = $true )
& ( setadjoinIR = $true )
& ( woz13rule1 = $true )
& ( funcext = $true )
& ( ubforcartprodlem3 = $true )
& ( cartprodfstpairEq = $true )
& ( setminusEL = $true )
& ( subsetE = $true )
& ( dpsetconstrI = $true )
& ( setukpairIR = $true )
& ( funcGraphProp4 = $true )
& ( emptyI = $true )
& ( dpsetconstrER = $true )
& ( eqimpsubset2 = $true )
& ( funcGraphProp1 = $true )
& ( doubleComplementEq = $true )
& ( prop2setI = $true )
& ( powerset__Cong = $true )
& ( contrasubsetT2 = $true )
& ( setukpairinjL = $true )
& ( emptyinPowerset = $true )
& ( notdallE = $true )
& ( singletonprop = $true )
& ( setukpairinjL1 = $true )
& ( lam2lamEq = $true )
& ( emptyinunitempty = $true )
& ( subset2powerset = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinSub2 = $true )
& ( complementT_lem = $true )
& ( woz13rule3 = $true )
& ( doubleComplementSub1 = $true )
& ( setadjoinIL = $true )
& ( setukpairIL = $true )
& ( emptyset__Cong = $true )
& ( binunionTIRcontra = $true )
& ( subsetI2 = $true )
& ( emptysetAx = $true )
& ( kpairiskpair = $true )
& ( setextAx = $true )
& ( setukpairinjL2 = $true )
& ( binintersectI = $true )
& ( binunionIR = $true )
& ( exuEu = $true )
& ( complementUnionInPowersetComplement = $true )
& ( complementImpComplementIntersect = $true )
& ( binintersectT_lem = $true )
& ( ex1E2 = $true )
& ( upairsetIL = $true )
& ( demorgan2a2 = $true )
& ( iftrueProp2 = $true )
& ( wellorderingAx = $true )
& ( quantDeMorgan2 = $true )
& ( powersetE = $true )
& ( cartprodmempair1 = $true )
& ( complementTnotintersectT = $true )
& ( notequalI2 = $true )
& ( exuI3 = $true )
& ( emptysetE = $true )
& ( setminusT_lem = $true )
& ( symdiffIneg1 = $true )
& ( demorgan2b = $true )
& ( complementTI1 = $true )
& ( nonemptyI = $true )
& ( apProp = $true )
& ( notinemptyset = $true )
& ( woz1_3 = $true )
& ( binintersectTERcontra = $true )
& ( woz13rule0 = $true )
& ( eqimpsubset1 = $true )
& ( emptyE1 = $true )
& ( notequalI1 = $true )
& ( subsetE2 = $true )
& ( setukpairinjR12 = $true )
& ( omega__Cong = $true )
& ( lamProp = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( setukpairinjR1 = $true )
& ( exu__Cong = $true )
& ( exuI2 = $true )
& ( ubforcartprodlem1 = $true )
& ( omega0Ax = $true )
& ( iffalseProp2 = $true )
& ( quantDeMorgan1 = $true )
& ( iftrueorfalse = $true )
& ( ex1I = $true )
& ( setunionsingleton = $true )
& ( setukpairinjR11 = $true )
& ( breln1all2 = $true )
& ( eta1 = $true )
& ( funcImageSingleton = $true )
& ( noeltsimpempty = $true )
& ( setunionsingleton1 = $true )
& ( lamp = $true )
& ( symdiffI1 = $true )
& ( sepInPowerset = $true )
& ( subsetemptysetimpeq = $true )
& ( complementTcontraSubset = $true )
& ( binintersectSubset5 = $true )
& ( binintersectTELcontra = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( kfstsingleton = $true )
& ( theprop = $true )
& ( binunionE = $true )
& ( beta2 = $true )
& ( breln1SetBreln1 = $true )
& ( demorgan2a1 = $true )
& ( singletoninpowunion = $true )
& ( binintersectSubset3 = $true )
& ( descr__Cong = $true )
& ( in__Cong = $true )
& ( setminusER = $true )
& ( nonemptyI1 = $true )
& ( symdiffE = $true )
& ( ex1I2 = $true )
& ( doubleComplementSub2 = $true )
& ( funcextLem = $true )
& ( emptysetsubset = $true )
& ( infuncsetfunc = $true )
& ( vacuousDall = $true )
& ( dsetconstrEL = $true )
& ( binunionEcases = $true )
& ( notsubsetI = $true )
& ( iffalse = $true )
& ( cartprodmempaircEq = $true )
& ( contraSubsetComplement = $true )
& ( subsetRefl = $true )
& ( setext = $true )
& ( contrasubsetT = $true )
& ( iftrueProp1 = $true )
& ( setbeta = $true )
& ( foundationAx = $true )
& ( exuE2 = $true )
& ( ifSingleton = $true )
& ( quantDeMorgan4 = $true )
& ( setunionAx = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binunionT_lem = $true )
& ( exuE1 = $true )
& ( symdiffI2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setoftrueEq = $true )
& ( omegaIndAx = $true )
& ( setadjoinOr = $true )
& ( setextsub = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f874,f875]) ).
thf(f875,plain,
( ? [X0: $i > $i > $o,X1: $i] :
( ( breln1 @ X1
@ ( dpsetconstr @ X1 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) )
!= $true )
=> ( ( breln1 @ sK1
@ ( dpsetconstr @ sK1 @ sK1
@ ^ [Y0: $i,Y1: $i] : ( sK0 @ Y0 @ Y1 ) ) )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f874,plain,
( ( setunionI = $true )
& ( replAx = $true )
& ( complementTE1 = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( doubleComplementI1 = $true )
& ( kfstpairEq = $true )
& ( binintersectSubset1 = $true )
& ( funcext2 = $true )
& ( cartprodsndin = $true )
& ( powersetE1 = $true )
& ( binunionTE = $true )
& ( kpairp = $true )
& ( dpsetconstrERa = $true )
& ( uniqinunit = $true )
& ( upairequniteq = $true )
& ( dsetconstrI = $true )
& ( binintersectSubset2 = $true )
& ( cartprodmempair = $true )
& ( cartprodpairmemEL = $true )
& ( ap2apEq2 = $true )
& ( binintersectSubset4 = $true )
& ( descrp = $true )
& ( upairinpowunion = $true )
& ( kpairsurjEq = $true )
& ( subPowSU = $true )
& ( ubforcartprodlem2 = $true )
& ( singletonsubset = $true )
& ( powersetI = $true )
& ( inCongP = $true )
& ( dpsetconstrEL2 = $true )
& ( ifp = $true )
& ( powersetI1 = $true )
& ( choice2fnsingleton = $true )
& ( demorgan2a = $true )
& ( upairsetIR = $true )
& ( setadjoin__Cong = $true )
& ( app = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( singletonsswitch = $true )
& ( woz13rule2 = $true )
& ( ap2p = $true )
& ( exuE3e = $true )
& ( setminusIRneg = $true )
& ( inPowerset = $true )
& ( setminusERneg = $true )
& ( subsetTrans = $true )
& ( subsetI1 = $true )
& ( upairset2E = $true )
& ( setunionE = $true )
& ( setunion__Cong = $true )
& ( woz1_1 = $true )
& ( notdexE = $true )
& ( setukpairinjR2 = $true )
& ( subbreln = $true )
& ( setminusSubset1 = $true )
& ( beta1 = $true )
& ( setminusSubset2 = $true )
& ( powersetsubset = $true )
& ( ap2apEq1 = $true )
& ( binunionTEcontra = $true )
& ( prop2set2propI = $true )
& ( omegaSAx = $true )
& ( eta2 = $true )
& ? [X0: $i > $i > $o,X1: $i] :
( ( breln1 @ X1
@ ( dpsetconstr @ X1 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) )
!= $true )
& ( dsetconstrER = $true )
& ( powersetTE1 = $true )
& ( inIntersectImpInUnion = $true )
& ( notinsingleton = $true )
& ( exuE3u = $true )
& ( secondinupair = $true )
& ( quantDeMorgan3 = $true )
& ( doubleComplementE1 = $true )
& ( upairsetE = $true )
& ( binunionLsub = $true )
& ( dpsetconstrSub = $true )
& ( binintersectEL = $true )
& ( binintersectRsub = $true )
& ( setminusLsub = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( binunionRsub = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( cartprodfstin = $true )
& ( eqinunit = $true )
& ( demorgan2 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( upairset2IR = $true )
& ( contrasubsetT3 = $true )
& ( dpsetconstrEL1 = $true )
& ( contrasubsetT1 = $true )
& ( sepSubset = $true )
& ( theeq = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( binunionTILcontra = $true )
& ( singletoninpowerset = $true )
& ( woz1_2 = $true )
& ( woz1_4 = $true )
& ( demorgan2b2 = $true )
& ( setextT = $true )
& ( emptysetimpfalse = $true )
& ( funcinfuncset = $true )
& ( iffalseProp1 = $true )
& ( woz1_5 = $true )
& ( prop2setE = $true )
& ( setadjoinE = $true )
& ( ex1E1 = $true )
& ( setadjoinAx = $true )
& ( dsetconstr__Cong = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectLsub = $true )
& ( cartprodpairmemER = $true )
& ( powersetTI1 = $true )
& ( woz13rule4 = $true )
& ( eqbreln = $true )
& ( cartprodpairin = $true )
& ( emptyInPowerset = $true )
& ( singletonsuniq = $true )
& ( brelnall2 = $true )
& ( setminusILneg = $true )
& ( demorgan1b = $true )
& ( disjointsetsI1 = $true )
& ( setOfPairsIsBReln = $true )
& ( powersetAx = $true )
& ( setminusELneg = $true )
& ( demorgan1 = $true )
& ( demorgan1a = $true )
& ( iftrue = $true )
& ( inIntersectImpInUnion2 = $true )
& ( binintersectER = $true )
& ( nonemptyE1 = $true )
& ( upairsubunion = $true )
& ( setunionE2 = $true )
& ( setadjoinSub = $true )
& ( setunionsingleton2 = $true )
& ( binunionIL = $true )
& ( nonemptyImpWitness = $true )
& ( ksndpairEq = $true )
& ( powersetT_lem = $true )
& ( cartprodsndpairEq = $true )
& ( exuI1 = $true )
& ( subsetTI = $true )
& ( setadjoinIR = $true )
& ( woz13rule1 = $true )
& ( funcext = $true )
& ( ubforcartprodlem3 = $true )
& ( cartprodfstpairEq = $true )
& ( setminusEL = $true )
& ( subsetE = $true )
& ( dpsetconstrI = $true )
& ( setukpairIR = $true )
& ( funcGraphProp4 = $true )
& ( emptyI = $true )
& ( dpsetconstrER = $true )
& ( eqimpsubset2 = $true )
& ( funcGraphProp1 = $true )
& ( doubleComplementEq = $true )
& ( prop2setI = $true )
& ( powerset__Cong = $true )
& ( contrasubsetT2 = $true )
& ( setukpairinjL = $true )
& ( emptyinPowerset = $true )
& ( notdallE = $true )
& ( singletonprop = $true )
& ( setukpairinjL1 = $true )
& ( lam2lamEq = $true )
& ( emptyinunitempty = $true )
& ( subset2powerset = $true )
& ( symdiffIneg2 = $true )
& ( setadjoinSub2 = $true )
& ( complementT_lem = $true )
& ( woz13rule3 = $true )
& ( doubleComplementSub1 = $true )
& ( setadjoinIL = $true )
& ( setukpairIL = $true )
& ( emptyset__Cong = $true )
& ( binunionTIRcontra = $true )
& ( subsetI2 = $true )
& ( emptysetAx = $true )
& ( kpairiskpair = $true )
& ( setextAx = $true )
& ( setukpairinjL2 = $true )
& ( binintersectI = $true )
& ( binunionIR = $true )
& ( exuEu = $true )
& ( complementUnionInPowersetComplement = $true )
& ( complementImpComplementIntersect = $true )
& ( binintersectT_lem = $true )
& ( ex1E2 = $true )
& ( upairsetIL = $true )
& ( demorgan2a2 = $true )
& ( iftrueProp2 = $true )
& ( wellorderingAx = $true )
& ( quantDeMorgan2 = $true )
& ( powersetE = $true )
& ( cartprodmempair1 = $true )
& ( complementTnotintersectT = $true )
& ( notequalI2 = $true )
& ( exuI3 = $true )
& ( emptysetE = $true )
& ( setminusT_lem = $true )
& ( symdiffIneg1 = $true )
& ( demorgan2b = $true )
& ( complementTI1 = $true )
& ( nonemptyI = $true )
& ( apProp = $true )
& ( notinemptyset = $true )
& ( woz1_3 = $true )
& ( binintersectTERcontra = $true )
& ( woz13rule0 = $true )
& ( eqimpsubset1 = $true )
& ( emptyE1 = $true )
& ( notequalI1 = $true )
& ( subsetE2 = $true )
& ( setukpairinjR12 = $true )
& ( omega__Cong = $true )
& ( lamProp = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( setukpairinjR1 = $true )
& ( exu__Cong = $true )
& ( exuI2 = $true )
& ( ubforcartprodlem1 = $true )
& ( omega0Ax = $true )
& ( iffalseProp2 = $true )
& ( quantDeMorgan1 = $true )
& ( iftrueorfalse = $true )
& ( ex1I = $true )
& ( setunionsingleton = $true )
& ( setukpairinjR11 = $true )
& ( breln1all2 = $true )
& ( eta1 = $true )
& ( funcImageSingleton = $true )
& ( noeltsimpempty = $true )
& ( setunionsingleton1 = $true )
& ( lamp = $true )
& ( symdiffI1 = $true )
& ( sepInPowerset = $true )
& ( subsetemptysetimpeq = $true )
& ( complementTcontraSubset = $true )
& ( binintersectSubset5 = $true )
& ( binintersectTELcontra = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( kfstsingleton = $true )
& ( theprop = $true )
& ( binunionE = $true )
& ( beta2 = $true )
& ( breln1SetBreln1 = $true )
& ( demorgan2a1 = $true )
& ( singletoninpowunion = $true )
& ( binintersectSubset3 = $true )
& ( descr__Cong = $true )
& ( in__Cong = $true )
& ( setminusER = $true )
& ( nonemptyI1 = $true )
& ( symdiffE = $true )
& ( ex1I2 = $true )
& ( doubleComplementSub2 = $true )
& ( funcextLem = $true )
& ( emptysetsubset = $true )
& ( infuncsetfunc = $true )
& ( vacuousDall = $true )
& ( dsetconstrEL = $true )
& ( binunionEcases = $true )
& ( notsubsetI = $true )
& ( iffalse = $true )
& ( cartprodmempaircEq = $true )
& ( contraSubsetComplement = $true )
& ( subsetRefl = $true )
& ( setext = $true )
& ( contrasubsetT = $true )
& ( iftrueProp1 = $true )
& ( setbeta = $true )
& ( foundationAx = $true )
& ( exuE2 = $true )
& ( ifSingleton = $true )
& ( quantDeMorgan4 = $true )
& ( setunionAx = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binunionT_lem = $true )
& ( exuE1 = $true )
& ( symdiffI2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( setoftrueEq = $true )
& ( omegaIndAx = $true )
& ( setadjoinOr = $true )
& ( setextsub = $true ) ),
inference(flattening,[status(thm)],[f873]) ).
thf(f873,plain,
( ? [X0: $i > $i > $o,X1: $i] :
( ( breln1 @ X1
@ ( dpsetconstr @ X1 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) )
!= $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)],[f603]) ).
thf(f603,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 )
=> ! [X0: $i > $i > $o,X1: $i] :
( ( breln1 @ X1
@ ( dpsetconstr @ X1 @ X1
@ ^ [Y0: $i,Y1: $i] : ( X0 @ Y0 @ Y1 ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f602]) ).
thf(f602,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
=> ! [X0: $i > $i > $o,X1: $i] :
( breln1 @ X1
@ ( dpsetconstr @ X1 @ X1
@ ^ [X2: $i,X3: $i] : ( X0 @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f293]) ).
thf(f293,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
=> ! [X0: $i > $i > $o,X3: $i] :
( breln1 @ X3
@ ( dpsetconstr @ X3 @ X3
@ ^ [X1: $i,X2: $i] : ( X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f292]) ).
thf(f292,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
=> ! [X0: $i > $i > $o,X3: $i] :
( breln1 @ X3
@ ( dpsetconstr @ X3 @ X3
@ ^ [X1: $i,X2: $i] : ( X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.DOTu7T1G0A/DTF2THF_390.p',setOfPairsIsBReln1) ).
thf(f1452,plain,
( setOfPairsIsBReln
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $i > $o )
@ ^ [Y1: $i > $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( breln @ Y0 @ Y2
@ ( dpsetconstr @ Y0 @ Y2
@ ^ [Y3: $i,Y4: $i] : ( Y1 @ Y3 @ Y4 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f433]) ).
thf(f433,plain,
( setOfPairsIsBReln
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ ( $i > $i > $o )
@ ^ [Y1: $i > $i > $o] :
( !! @ $i
@ ^ [Y2: $i] :
( breln @ Y0 @ Y2
@ ( dpsetconstr @ Y0 @ Y2
@ ^ [Y3: $i,Y4: $i] : ( Y1 @ Y3 @ Y4 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f432]) ).
thf(f432,plain,
! [X0: $i,X1: $i > $i > $o,X2: $i] :
( ( breln @ X2 @ X0
@ ( dpsetconstr @ X2 @ X0
@ ^ [X3: $i,X4: $i] : ( X1 @ X3 @ X4 ) ) )
= setOfPairsIsBReln ),
inference(rectify,[status(thm)],[f187]) ).
thf(f187,axiom,
! [X4: $i,X0: $i > $i > $o,X3: $i] :
( ( breln @ X3 @ X4
@ ( dpsetconstr @ X3 @ X4
@ ^ [X1: $i,X2: $i] : ( X0 @ X1 @ X2 ) ) )
= setOfPairsIsBReln ),
file('/export/starexec/sandbox/tmp/tmp.DOTu7T1G0A/DTF2THF_390.p',setOfPairsIsBReln) ).
thf(f3803,plain,
( ( breln @ sK1 @ sK1 @ ( dpsetconstr @ sK1 @ sK1 @ sK0 ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f1485]) ).
thf(f1485,plain,
( ( ^ [Y0: $i,Y1: $i] : ( breln @ Y0 @ Y0 @ Y1 )
@ sK1
@ ( dpsetconstr @ sK1 @ sK1
@ ^ [Y0: $i,Y1: $i] : ( sK0 @ Y0 @ Y1 ) ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f1236,f1357]) ).
thf(f1357,plain,
( breln1
= ( ^ [Y0: $i,Y1: $i] : ( breln @ Y0 @ Y0 @ Y1 ) ) ),
inference(cnf_transformation,[status(thm)],[f633]) ).
thf(f633,plain,
( breln1
= ( ^ [Y0: $i,Y1: $i] : ( breln @ Y0 @ Y0 @ Y1 ) ) ),
inference(fool_elimination,[status(thm)],[f632]) ).
thf(f632,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.DOTu7T1G0A/DTF2THF_390.p',breln1) ).
thf(f1236,plain,
( ( breln1 @ sK1
@ ( dpsetconstr @ sK1 @ sK1
@ ^ [Y0: $i,Y1: $i] : ( sK0 @ Y0 @ Y1 ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f876]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU771^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n003.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:33:10 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.27/0.45 ---- Original DTF file ---
% 0.27/0.45 thf(spec,logic,$$dhol).
% 0.27/0.45 %------------------------------------------------------------------------------
% 0.27/0.45 % File : SEU771^1 : TPTP v9.2.1. Released v3.7.0.
% 0.27/0.45 % Domain : Set Theory
% 0.27/0.45 % Problem : Binary Relations on a Set
% 0.27/0.45 % Version : Especial.
% 0.27/0.45 % English : (! A:i.! phi:i>(i>o).breln1 A (dpsetconstr A A (^ x,y:i.phi x y)))
% 0.27/0.45
% 0.27/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.27/0.45 % Source : [Bro08]
% 0.27/0.45 % Names : ZFC273g [Bro08]
% 0.27/0.45
% 0.27/0.45 % Status : Theorem
% 0.27/0.45 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.27/0.45 % Syntax : Number of formulae : 625 ( 291 unt; 333 typ; 291 def)
% 0.27/0.45 % Number of atoms : 1746 ( 420 equ; 0 cnn)
% 0.27/0.45 % Maximal formula atoms : 288 ( 5 avg)
% 0.27/0.45 % Number of connectives : 3748 ( 73 ~; 12 |; 46 &;2716 @)
% 0.27/0.45 % ( 14 <=>; 887 =>; 0 <=; 0 <~>)
% 0.27/0.45 % Maximal formula depth : 296 ( 2 avg)
% 0.27/0.45 % Number of types : 2 ( 0 usr)
% 0.27/0.45 % Number of type conns : 151 ( 151 >; 0 *; 0 +; 0 <<)
% 0.27/0.45 % Number of symbols : 336 ( 333 usr; 291 con; 0-4 aty)
% 0.27/0.45 % Number of variables : 1003 ( 76 ^; 891 !; 36 ?;1003 :)
% 0.27/0.45 % SPC : TH0_THM_EQU_NAR
% 0.27/0.45
% 0.27/0.45 % Comments : http://mathgate.info/detsetitem.php?id=334
% 0.27/0.45 %------------------------------------------------------------------------------
% 0.27/0.45 thf(in_type,type,
% 0.27/0.45 in: $i > $i > $o ).
% 0.27/0.45
% 0.27/0.45 thf(exu_type,type,
% 0.27/0.45 exu: ( $i > $o ) > $o ).
% 0.27/0.45
% 0.27/0.45 thf(exu,definition,
% 0.27/0.45 ( exu
% 0.27/0.45 = ( ^ [Xphi: $i > $o] :
% 0.27/0.45 ? [Xx: $i] :
% 0.27/0.45 ( ( Xphi @ Xx )
% 0.27/0.45 & ! [Xy: $i] :
% 0.27/0.45 ( ( Xphi @ Xy )
% 0.27/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setextAx_type,type,
% 0.27/0.45 setextAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setextAx,definition,
% 0.27/0.45 ( setextAx
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 <=> ( in @ Xx @ B ) )
% 0.27/0.45 => ( A = B ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptyset_type,type,
% 0.27/0.45 emptyset: $i ).
% 0.27/0.45
% 0.27/0.45 thf(emptysetAx_type,type,
% 0.27/0.45 emptysetAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptysetAx,definition,
% 0.27/0.45 ( emptysetAx
% 0.27/0.45 = ( ! [Xx: $i] :
% 0.27/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoin_type,type,
% 0.27/0.45 setadjoin: $i > $i > $i ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinAx_type,type,
% 0.27/0.45 setadjoinAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinAx,definition,
% 0.27/0.45 ( setadjoinAx
% 0.27/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.45 <=> ( ( Xy = Xx )
% 0.27/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(powerset_type,type,
% 0.27/0.45 powerset: $i > $i ).
% 0.27/0.45
% 0.27/0.45 thf(powersetAx_type,type,
% 0.27/0.45 powersetAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(powersetAx,definition,
% 0.27/0.45 ( powersetAx
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.45 <=> ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ B )
% 0.27/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setunion_type,type,
% 0.27/0.45 setunion: $i > $i ).
% 0.27/0.45
% 0.27/0.45 thf(setunionAx_type,type,
% 0.27/0.45 setunionAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setunionAx,definition,
% 0.27/0.45 ( setunionAx
% 0.27/0.45 = ( ! [A: $i,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.45 <=> ? [B: $i] :
% 0.27/0.45 ( ( in @ Xx @ B )
% 0.27/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(omega_type,type,
% 0.27/0.45 omega: $i ).
% 0.27/0.45
% 0.27/0.45 thf(omega0Ax_type,type,
% 0.27/0.45 omega0Ax: $o ).
% 0.27/0.45
% 0.27/0.45 thf(omega0Ax,definition,
% 0.27/0.45 ( omega0Ax
% 0.27/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(omegaSAx_type,type,
% 0.27/0.45 omegaSAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(omegaSAx,definition,
% 0.27/0.45 ( omegaSAx
% 0.27/0.45 = ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ omega )
% 0.27/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(omegaIndAx_type,type,
% 0.27/0.45 omegaIndAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(omegaIndAx,definition,
% 0.27/0.45 ( omegaIndAx
% 0.27/0.45 = ( ! [A: $i] :
% 0.27/0.45 ( ( ( in @ emptyset @ A )
% 0.27/0.45 & ! [Xx: $i] :
% 0.27/0.45 ( ( ( in @ Xx @ omega )
% 0.27/0.45 & ( in @ Xx @ A ) )
% 0.27/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.27/0.45 => ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ omega )
% 0.27/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(replAx_type,type,
% 0.27/0.45 replAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(replAx,definition,
% 0.27/0.45 ( replAx
% 0.27/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( exu
% 0.27/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.27/0.45 => ? [B: $i] :
% 0.27/0.45 ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ B )
% 0.27/0.45 <=> ? [Xy: $i] :
% 0.27/0.45 ( ( in @ Xy @ A )
% 0.27/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(foundationAx_type,type,
% 0.27/0.45 foundationAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(foundationAx,definition,
% 0.27/0.45 ( foundationAx
% 0.27/0.45 = ( ! [A: $i] :
% 0.27/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.45 => ? [B: $i] :
% 0.27/0.45 ( ( in @ B @ A )
% 0.27/0.45 & ~ ? [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ B )
% 0.27/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(wellorderingAx_type,type,
% 0.27/0.45 wellorderingAx: $o ).
% 0.27/0.45
% 0.27/0.45 thf(wellorderingAx,definition,
% 0.27/0.45 ( wellorderingAx
% 0.27/0.45 = ( ! [A: $i] :
% 0.27/0.45 ? [B: $i] :
% 0.27/0.45 ( ! [C: $i] :
% 0.27/0.45 ( ( in @ C @ B )
% 0.27/0.45 => ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ C )
% 0.27/0.45 => ( in @ Xx @ A ) ) )
% 0.27/0.45 & ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( ( in @ Xx @ A )
% 0.27/0.45 & ( in @ Xy @ A ) )
% 0.27/0.45 => ( ! [C: $i] :
% 0.27/0.45 ( ( in @ C @ B )
% 0.27/0.45 => ( ( in @ Xx @ C )
% 0.27/0.45 <=> ( in @ Xy @ C ) ) )
% 0.27/0.45 => ( Xx = Xy ) ) )
% 0.27/0.45 & ! [C: $i,D: $i] :
% 0.27/0.45 ( ( ( in @ C @ B )
% 0.27/0.45 & ( in @ D @ B ) )
% 0.27/0.45 => ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ C )
% 0.27/0.45 => ( in @ Xx @ D ) )
% 0.27/0.45 | ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ D )
% 0.27/0.45 => ( in @ Xx @ C ) ) ) )
% 0.27/0.45 & ! [C: $i] :
% 0.27/0.45 ( ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ C )
% 0.27/0.45 => ( in @ Xx @ A ) )
% 0.27/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.27/0.45 => ? [D: $i,Xx: $i] :
% 0.27/0.45 ( ( in @ D @ B )
% 0.27/0.45 & ( in @ Xx @ C )
% 0.27/0.45 & ~ ? [Xy: $i] :
% 0.27/0.45 ( ( in @ Xy @ D )
% 0.27/0.45 & ( in @ Xy @ C ) )
% 0.27/0.45 & ! [E: $i] :
% 0.27/0.45 ( ( in @ E @ B )
% 0.27/0.45 => ( ! [Xy: $i] :
% 0.27/0.45 ( ( in @ Xy @ E )
% 0.27/0.45 => ( in @ Xy @ D ) )
% 0.27/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(descr_type,type,
% 0.27/0.45 descr: ( $i > $o ) > $i ).
% 0.27/0.45
% 0.27/0.45 thf(descrp_type,type,
% 0.27/0.45 descrp: $o ).
% 0.27/0.45
% 0.27/0.45 thf(descrp,definition,
% 0.27/0.45 ( descrp
% 0.27/0.45 = ( ! [Xphi: $i > $o] :
% 0.27/0.45 ( ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 => ( Xphi
% 0.27/0.45 @ ( descr
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(dsetconstr_type,type,
% 0.27/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.27/0.45
% 0.27/0.45 thf(dsetconstrI_type,type,
% 0.27/0.45 dsetconstrI: $o ).
% 0.27/0.45
% 0.27/0.45 thf(dsetconstrI,definition,
% 0.27/0.45 ( dsetconstrI
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( ( Xphi @ Xx )
% 0.27/0.45 => ( in @ Xx
% 0.27/0.45 @ ( dsetconstr @ A
% 0.27/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(dsetconstrEL_type,type,
% 0.27/0.45 dsetconstrEL: $o ).
% 0.27/0.45
% 0.27/0.45 thf(dsetconstrEL,definition,
% 0.27/0.45 ( dsetconstrEL
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx
% 0.27/0.45 @ ( dsetconstr @ A
% 0.27/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(dsetconstrER_type,type,
% 0.27/0.45 dsetconstrER: $o ).
% 0.27/0.45
% 0.27/0.45 thf(dsetconstrER,definition,
% 0.27/0.45 ( dsetconstrER
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx
% 0.27/0.45 @ ( dsetconstr @ A
% 0.27/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(exuE1_type,type,
% 0.27/0.45 exuE1: $o ).
% 0.27/0.45
% 0.27/0.45 thf(exuE1,definition,
% 0.27/0.45 ( exuE1
% 0.27/0.45 = ( ! [Xphi: $i > $o] :
% 0.27/0.45 ( ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 => ? [Xx: $i] :
% 0.27/0.45 ( ( Xphi @ Xx )
% 0.27/0.45 & ! [Xy: $i] :
% 0.27/0.45 ( ( Xphi @ Xy )
% 0.27/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(prop2set_type,type,
% 0.27/0.45 prop2set: $o > $i ).
% 0.27/0.45
% 0.27/0.45 thf(prop2setE_type,type,
% 0.27/0.45 prop2setE: $o ).
% 0.27/0.45
% 0.27/0.45 thf(prop2setE,definition,
% 0.27/0.45 ( prop2setE
% 0.27/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.27/0.45 => Xphi ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptysetE_type,type,
% 0.27/0.45 emptysetE: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptysetE,definition,
% 0.27/0.45 ( emptysetE
% 0.27/0.45 = ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ emptyset )
% 0.27/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptysetimpfalse_type,type,
% 0.27/0.45 emptysetimpfalse: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptysetimpfalse,definition,
% 0.27/0.45 ( emptysetimpfalse
% 0.27/0.45 = ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ emptyset )
% 0.27/0.45 => $false ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(notinemptyset_type,type,
% 0.27/0.45 notinemptyset: $o ).
% 0.27/0.45
% 0.27/0.45 thf(notinemptyset,definition,
% 0.27/0.45 ( notinemptyset
% 0.27/0.45 = ( ! [Xx: $i] :
% 0.27/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(exuE3e_type,type,
% 0.27/0.45 exuE3e: $o ).
% 0.27/0.45
% 0.27/0.45 thf(exuE3e,definition,
% 0.27/0.45 ( exuE3e
% 0.27/0.45 = ( ! [Xphi: $i > $o] :
% 0.27/0.45 ( ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setext_type,type,
% 0.27/0.45 setext: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setext,definition,
% 0.27/0.45 ( setext
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( in @ Xx @ B ) )
% 0.27/0.45 => ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ B )
% 0.27/0.45 => ( in @ Xx @ A ) )
% 0.27/0.45 => ( A = B ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptyI_type,type,
% 0.27/0.45 emptyI: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptyI,definition,
% 0.27/0.45 ( emptyI
% 0.27/0.45 = ( ! [A: $i] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ~ ( in @ Xx @ A )
% 0.27/0.45 => ( A = emptyset ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(noeltsimpempty_type,type,
% 0.27/0.45 noeltsimpempty: $o ).
% 0.27/0.45
% 0.27/0.45 thf(noeltsimpempty,definition,
% 0.27/0.45 ( noeltsimpempty
% 0.27/0.45 = ( ! [A: $i] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ~ ( in @ Xx @ A )
% 0.27/0.45 => ( A = emptyset ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setbeta_type,type,
% 0.27/0.45 setbeta: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setbeta,definition,
% 0.27/0.45 ( setbeta
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( ( in @ Xx
% 0.27/0.45 @ ( dsetconstr @ A
% 0.27/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.27/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(nonempty_type,type,
% 0.27/0.45 nonempty: $i > $o ).
% 0.27/0.45
% 0.27/0.45 thf(nonempty,definition,
% 0.27/0.45 ( nonempty
% 0.27/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(nonemptyE1_type,type,
% 0.27/0.45 nonemptyE1: $o ).
% 0.27/0.45
% 0.27/0.45 thf(nonemptyE1,definition,
% 0.27/0.45 ( nonemptyE1
% 0.27/0.45 = ( ! [A: $i] :
% 0.27/0.45 ( ( nonempty @ A )
% 0.27/0.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(nonemptyI_type,type,
% 0.27/0.45 nonemptyI: $o ).
% 0.27/0.45
% 0.27/0.45 thf(nonemptyI,definition,
% 0.27/0.45 ( nonemptyI
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( ( Xphi @ Xx )
% 0.27/0.45 => ( nonempty
% 0.27/0.45 @ ( dsetconstr @ A
% 0.27/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(nonemptyI1_type,type,
% 0.27/0.45 nonemptyI1: $o ).
% 0.27/0.45
% 0.27/0.45 thf(nonemptyI1,definition,
% 0.27/0.45 ( nonemptyI1
% 0.27/0.45 = ( ! [A: $i] :
% 0.27/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.27/0.45 => ( nonempty @ A ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinIL_type,type,
% 0.27/0.45 setadjoinIL: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinIL,definition,
% 0.27/0.45 ( setadjoinIL
% 0.27/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptyinunitempty_type,type,
% 0.27/0.45 emptyinunitempty: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptyinunitempty,definition,
% 0.27/0.45 ( emptyinunitempty
% 0.27/0.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinIR_type,type,
% 0.27/0.45 setadjoinIR: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinIR,definition,
% 0.27/0.45 ( setadjoinIR
% 0.27/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.45 ( ( in @ Xy @ A )
% 0.27/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinE_type,type,
% 0.27/0.45 setadjoinE: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinE,definition,
% 0.27/0.45 ( setadjoinE
% 0.27/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.45 => ! [Xphi: $o] :
% 0.27/0.45 ( ( ( Xy = Xx )
% 0.27/0.45 => Xphi )
% 0.27/0.45 => ( ( ( in @ Xy @ A )
% 0.27/0.45 => Xphi )
% 0.27/0.45 => Xphi ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinOr_type,type,
% 0.27/0.45 setadjoinOr: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinOr,definition,
% 0.27/0.45 ( setadjoinOr
% 0.27/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.27/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.27/0.45 => ( ( Xy = Xx )
% 0.27/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setoftrueEq_type,type,
% 0.27/0.45 setoftrueEq: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setoftrueEq,definition,
% 0.27/0.45 ( setoftrueEq
% 0.27/0.45 = ( ! [A: $i] :
% 0.27/0.45 ( ( dsetconstr @ A
% 0.27/0.45 @ ^ [Xx: $i] : $true )
% 0.27/0.45 = A ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(powersetI_type,type,
% 0.27/0.45 powersetI: $o ).
% 0.27/0.45
% 0.27/0.45 thf(powersetI,definition,
% 0.27/0.45 ( powersetI
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ B )
% 0.27/0.45 => ( in @ Xx @ A ) )
% 0.27/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptyinPowerset_type,type,
% 0.27/0.45 emptyinPowerset: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptyinPowerset,definition,
% 0.27/0.45 ( emptyinPowerset
% 0.27/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptyInPowerset_type,type,
% 0.27/0.45 emptyInPowerset: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptyInPowerset,definition,
% 0.27/0.45 ( emptyInPowerset
% 0.27/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(powersetE_type,type,
% 0.27/0.45 powersetE: $o ).
% 0.27/0.45
% 0.27/0.45 thf(powersetE,definition,
% 0.27/0.45 ( powersetE
% 0.27/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.45 => ( ( in @ Xx @ B )
% 0.27/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setunionI_type,type,
% 0.27/0.45 setunionI: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setunionI,definition,
% 0.27/0.45 ( setunionI
% 0.27/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.27/0.45 ( ( in @ Xx @ B )
% 0.27/0.45 => ( ( in @ B @ A )
% 0.27/0.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setunionE_type,type,
% 0.27/0.45 setunionE: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setunionE,definition,
% 0.27/0.45 ( setunionE
% 0.27/0.45 = ( ! [A: $i,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.45 => ! [Xphi: $o] :
% 0.27/0.45 ( ! [B: $i] :
% 0.27/0.45 ( ( in @ Xx @ B )
% 0.27/0.45 => ( ( in @ B @ A )
% 0.27/0.45 => Xphi ) )
% 0.27/0.45 => Xphi ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(subPowSU_type,type,
% 0.27/0.45 subPowSU: $o ).
% 0.27/0.45
% 0.27/0.45 thf(subPowSU,definition,
% 0.27/0.45 ( subPowSU
% 0.27/0.45 = ( ! [A: $i,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(exuE2_type,type,
% 0.27/0.45 exuE2: $o ).
% 0.27/0.45
% 0.27/0.45 thf(exuE2,definition,
% 0.27/0.45 ( exuE2
% 0.27/0.45 = ( ! [Xphi: $i > $o] :
% 0.27/0.45 ( ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 => ? [Xx: $i] :
% 0.27/0.45 ! [Xy: $i] :
% 0.27/0.45 ( ( Xphi @ Xy )
% 0.27/0.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(nonemptyImpWitness_type,type,
% 0.27/0.45 nonemptyImpWitness: $o ).
% 0.27/0.45
% 0.27/0.45 thf(nonemptyImpWitness,definition,
% 0.27/0.45 ( nonemptyImpWitness
% 0.27/0.45 = ( ! [A: $i] :
% 0.27/0.45 ( ( nonempty @ A )
% 0.27/0.45 => ? [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 & $true ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(uniqinunit_type,type,
% 0.27/0.45 uniqinunit: $o ).
% 0.27/0.45
% 0.27/0.45 thf(uniqinunit,definition,
% 0.27/0.45 ( uniqinunit
% 0.27/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(notinsingleton_type,type,
% 0.27/0.45 notinsingleton: $o ).
% 0.27/0.45
% 0.27/0.45 thf(notinsingleton,definition,
% 0.27/0.45 ( notinsingleton
% 0.27/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xx != Xy )
% 0.27/0.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(eqinunit_type,type,
% 0.27/0.45 eqinunit: $o ).
% 0.27/0.45
% 0.27/0.45 thf(eqinunit,definition,
% 0.27/0.45 ( eqinunit
% 0.27/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xx = Xy )
% 0.27/0.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(singletonsswitch_type,type,
% 0.27/0.45 singletonsswitch: $o ).
% 0.27/0.45
% 0.27/0.45 thf(singletonsswitch,definition,
% 0.27/0.45 ( singletonsswitch
% 0.27/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(upairsetE_type,type,
% 0.27/0.45 upairsetE: $o ).
% 0.27/0.45
% 0.27/0.45 thf(upairsetE,definition,
% 0.27/0.45 ( upairsetE
% 0.27/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.27/0.45 => ( ( Xz = Xx )
% 0.27/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(upairsetIL_type,type,
% 0.27/0.45 upairsetIL: $o ).
% 0.27/0.45
% 0.27/0.45 thf(upairsetIL,definition,
% 0.27/0.45 ( upairsetIL
% 0.27/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(upairsetIR_type,type,
% 0.27/0.45 upairsetIR: $o ).
% 0.27/0.45
% 0.27/0.45 thf(upairsetIR,definition,
% 0.27/0.45 ( upairsetIR
% 0.27/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptyE1_type,type,
% 0.27/0.45 emptyE1: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptyE1,definition,
% 0.27/0.45 ( emptyE1
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.45 ( ? [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 & ( Xphi @ Xx ) )
% 0.27/0.45 => ( ( ( dsetconstr @ A
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 = emptyset )
% 0.27/0.45 => $false ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(vacuousDall_type,type,
% 0.27/0.45 vacuousDall: $o ).
% 0.27/0.45
% 0.27/0.45 thf(vacuousDall,definition,
% 0.27/0.45 ( vacuousDall
% 0.27/0.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ emptyset )
% 0.27/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(quantDeMorgan1_type,type,
% 0.27/0.45 quantDeMorgan1: $o ).
% 0.27/0.45
% 0.27/0.45 thf(quantDeMorgan1,definition,
% 0.27/0.45 ( quantDeMorgan1
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.45 ( ~ ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( Xphi @ Xx ) )
% 0.27/0.45 => ? [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(quantDeMorgan2_type,type,
% 0.27/0.45 quantDeMorgan2: $o ).
% 0.27/0.45
% 0.27/0.45 thf(quantDeMorgan2,definition,
% 0.27/0.45 ( quantDeMorgan2
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ~ ( Xphi @ Xx ) )
% 0.27/0.45 => ~ ? [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(quantDeMorgan3_type,type,
% 0.27/0.45 quantDeMorgan3: $o ).
% 0.27/0.45
% 0.27/0.45 thf(quantDeMorgan3,definition,
% 0.27/0.45 ( quantDeMorgan3
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.45 ( ~ ? [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 & ( Xphi @ Xx ) )
% 0.27/0.45 => ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(quantDeMorgan4_type,type,
% 0.27/0.45 quantDeMorgan4: $o ).
% 0.27/0.45
% 0.27/0.45 thf(quantDeMorgan4,definition,
% 0.27/0.45 ( quantDeMorgan4
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.45 ( ? [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 & ~ ( Xphi @ Xx ) )
% 0.27/0.45 => ~ ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(prop2setI_type,type,
% 0.27/0.45 prop2setI: $o ).
% 0.27/0.45
% 0.27/0.45 thf(prop2setI,definition,
% 0.27/0.45 ( prop2setI
% 0.27/0.45 = ( ! [Xphi: $o] :
% 0.27/0.45 ( Xphi
% 0.27/0.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(set2prop_type,type,
% 0.27/0.45 set2prop: $i > $o ).
% 0.27/0.45
% 0.27/0.45 thf(prop2set2propI_type,type,
% 0.27/0.45 prop2set2propI: $o ).
% 0.27/0.45
% 0.27/0.45 thf(prop2set2propI,definition,
% 0.27/0.45 ( prop2set2propI
% 0.27/0.45 = ( ! [Xphi: $o] :
% 0.27/0.45 ( Xphi
% 0.27/0.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(notdexE_type,type,
% 0.27/0.45 notdexE: $o ).
% 0.27/0.45
% 0.27/0.45 thf(notdexE,definition,
% 0.27/0.45 ( notdexE
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.45 ( ~ ? [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 & ( Xphi @ Xx ) )
% 0.27/0.45 => ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(notdallE_type,type,
% 0.27/0.45 notdallE: $o ).
% 0.27/0.45
% 0.27/0.45 thf(notdallE,definition,
% 0.27/0.45 ( notdallE
% 0.27/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.45 ( ~ ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( Xphi @ Xx ) )
% 0.27/0.45 => ? [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(exuI1_type,type,
% 0.27/0.45 exuI1: $o ).
% 0.27/0.45
% 0.27/0.45 thf(exuI1,definition,
% 0.27/0.45 ( exuI1
% 0.27/0.45 = ( ! [Xphi: $i > $o] :
% 0.27/0.45 ( ? [Xx: $i] :
% 0.27/0.45 ( ( Xphi @ Xx )
% 0.27/0.45 & ! [Xy: $i] :
% 0.27/0.45 ( ( Xphi @ Xy )
% 0.27/0.45 => ( Xx = Xy ) ) )
% 0.27/0.45 => ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(exuI3_type,type,
% 0.27/0.45 exuI3: $o ).
% 0.27/0.45
% 0.27/0.45 thf(exuI3,definition,
% 0.27/0.45 ( exuI3
% 0.27/0.45 = ( ! [Xphi: $i > $o] :
% 0.27/0.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.27/0.45 => ( ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xphi @ Xx )
% 0.27/0.45 => ( ( Xphi @ Xy )
% 0.27/0.45 => ( Xx = Xy ) ) )
% 0.27/0.45 => ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(exuI2_type,type,
% 0.27/0.45 exuI2: $o ).
% 0.27/0.45
% 0.27/0.45 thf(exuI2,definition,
% 0.27/0.45 ( exuI2
% 0.27/0.45 = ( ! [Xphi: $i > $o] :
% 0.27/0.45 ( ? [Xx: $i] :
% 0.27/0.45 ! [Xy: $i] :
% 0.27/0.45 ( ( Xphi @ Xy )
% 0.27/0.45 <=> ( Xy = Xx ) )
% 0.27/0.45 => ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(inCongP_type,type,
% 0.27/0.45 inCongP: $o ).
% 0.27/0.45
% 0.27/0.45 thf(inCongP,definition,
% 0.27/0.45 ( inCongP
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ( A = B )
% 0.27/0.45 => ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xx = Xy )
% 0.27/0.45 => ( ( in @ Xx @ A )
% 0.27/0.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(in__Cong_type,type,
% 0.27/0.45 in__Cong: $o ).
% 0.27/0.45
% 0.27/0.45 thf(in__Cong,definition,
% 0.27/0.45 ( in__Cong
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ( A = B )
% 0.27/0.45 => ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xx = Xy )
% 0.27/0.45 => ( ( in @ Xx @ A )
% 0.27/0.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(exuE3u_type,type,
% 0.27/0.45 exuE3u: $o ).
% 0.27/0.45
% 0.27/0.45 thf(exuE3u,definition,
% 0.27/0.45 ( exuE3u
% 0.27/0.45 = ( ! [Xphi: $i > $o] :
% 0.27/0.45 ( ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 => ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xphi @ Xx )
% 0.27/0.45 => ( ( Xphi @ Xy )
% 0.27/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(exu__Cong_type,type,
% 0.27/0.45 exu__Cong: $o ).
% 0.27/0.45
% 0.27/0.45 thf(exu__Cong,definition,
% 0.27/0.45 ( exu__Cong
% 0.27/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xx = Xy )
% 0.27/0.45 => ( ( Xphi @ Xx )
% 0.27/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.27/0.45 => ( ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 <=> ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptyset__Cong_type,type,
% 0.27/0.45 emptyset__Cong: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptyset__Cong,definition,
% 0.27/0.45 ( emptyset__Cong
% 0.27/0.45 = ( emptyset = emptyset ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoin__Cong_type,type,
% 0.27/0.45 setadjoin__Cong: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoin__Cong,definition,
% 0.27/0.45 ( setadjoin__Cong
% 0.27/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xx = Xy )
% 0.27/0.45 => ! [Xz: $i,Xu: $i] :
% 0.27/0.45 ( ( Xz = Xu )
% 0.27/0.45 => ( ( setadjoin @ Xx @ Xz )
% 0.27/0.45 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(powerset__Cong_type,type,
% 0.27/0.45 powerset__Cong: $o ).
% 0.27/0.45
% 0.27/0.45 thf(powerset__Cong,definition,
% 0.27/0.45 ( powerset__Cong
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ( A = B )
% 0.27/0.45 => ( ( powerset @ A )
% 0.27/0.45 = ( powerset @ B ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setunion__Cong_type,type,
% 0.27/0.45 setunion__Cong: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setunion__Cong,definition,
% 0.27/0.45 ( setunion__Cong
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ( A = B )
% 0.27/0.45 => ( ( setunion @ A )
% 0.27/0.45 = ( setunion @ B ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(omega__Cong_type,type,
% 0.27/0.45 omega__Cong: $o ).
% 0.27/0.45
% 0.27/0.45 thf(omega__Cong,definition,
% 0.27/0.45 ( omega__Cong
% 0.27/0.45 = ( omega = omega ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(exuEu_type,type,
% 0.27/0.45 exuEu: $o ).
% 0.27/0.45
% 0.27/0.45 thf(exuEu,definition,
% 0.27/0.45 ( exuEu
% 0.27/0.45 = ( ! [Xphi: $i > $o] :
% 0.27/0.45 ( ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 => ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xphi @ Xx )
% 0.27/0.45 => ( ( Xphi @ Xy )
% 0.27/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(descr__Cong_type,type,
% 0.27/0.45 descr__Cong: $o ).
% 0.27/0.45
% 0.27/0.45 thf(descr__Cong,definition,
% 0.27/0.45 ( descr__Cong
% 0.27/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.27/0.45 ( ( Xx = Xy )
% 0.27/0.45 => ( ( Xphi @ Xx )
% 0.27/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.27/0.45 => ( ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 => ( ( exu
% 0.27/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.27/0.45 => ( ( descr
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 = ( descr
% 0.27/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(dsetconstr__Cong_type,type,
% 0.27/0.45 dsetconstr__Cong: $o ).
% 0.27/0.45
% 0.27/0.45 thf(dsetconstr__Cong,definition,
% 0.27/0.45 ( dsetconstr__Cong
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ( A = B )
% 0.27/0.45 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ! [Xy: $i] :
% 0.27/0.45 ( ( in @ Xy @ B )
% 0.27/0.45 => ( ( Xx = Xy )
% 0.27/0.45 => ( ( Xphi @ Xx )
% 0.27/0.45 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.27/0.45 => ( ( dsetconstr @ A
% 0.27/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.45 = ( dsetconstr @ B
% 0.27/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(subset_type,type,
% 0.27/0.45 subset: $i > $i > $o ).
% 0.27/0.45
% 0.27/0.45 thf(disjoint_type,type,
% 0.27/0.45 disjoint: $i > $i > $o ).
% 0.27/0.45
% 0.27/0.45 thf(setsmeet_type,type,
% 0.27/0.45 setsmeet: $i > $i > $o ).
% 0.27/0.45
% 0.27/0.45 thf(subsetI1_type,type,
% 0.27/0.45 subsetI1: $o ).
% 0.27/0.45
% 0.27/0.45 thf(subsetI1,definition,
% 0.27/0.45 ( subsetI1
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( in @ Xx @ B ) )
% 0.27/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(eqimpsubset2_type,type,
% 0.27/0.45 eqimpsubset2: $o ).
% 0.27/0.45
% 0.27/0.45 thf(eqimpsubset2,definition,
% 0.27/0.45 ( eqimpsubset2
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ( A = B )
% 0.27/0.45 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(eqimpsubset1_type,type,
% 0.27/0.45 eqimpsubset1: $o ).
% 0.27/0.45
% 0.27/0.45 thf(eqimpsubset1,definition,
% 0.27/0.45 ( eqimpsubset1
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ( A = B )
% 0.27/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(subsetI2_type,type,
% 0.27/0.45 subsetI2: $o ).
% 0.27/0.45
% 0.27/0.45 thf(subsetI2,definition,
% 0.27/0.45 ( subsetI2
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ! [Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( in @ Xx @ B ) )
% 0.27/0.45 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(emptysetsubset_type,type,
% 0.27/0.45 emptysetsubset: $o ).
% 0.27/0.45
% 0.27/0.45 thf(emptysetsubset,definition,
% 0.27/0.45 ( emptysetsubset
% 0.27/0.45 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(subsetE_type,type,
% 0.27/0.45 subsetE: $o ).
% 0.27/0.45
% 0.27/0.45 thf(subsetE,definition,
% 0.27/0.45 ( subsetE
% 0.27/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.45 ( ( subset @ A @ B )
% 0.27/0.45 => ( ( in @ Xx @ A )
% 0.27/0.45 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(subsetE2_type,type,
% 0.27/0.45 subsetE2: $o ).
% 0.27/0.45
% 0.27/0.45 thf(subsetE2,definition,
% 0.27/0.45 ( subsetE2
% 0.27/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.45 ( ( subset @ A @ B )
% 0.27/0.45 => ( ~ ( in @ Xx @ B )
% 0.27/0.45 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(notsubsetI_type,type,
% 0.27/0.45 notsubsetI: $o ).
% 0.27/0.45
% 0.27/0.45 thf(notsubsetI,definition,
% 0.27/0.45 ( notsubsetI
% 0.27/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( ~ ( in @ Xx @ B )
% 0.27/0.45 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(notequalI1_type,type,
% 0.27/0.45 notequalI1: $o ).
% 0.27/0.45
% 0.27/0.45 thf(notequalI1,definition,
% 0.27/0.45 ( notequalI1
% 0.27/0.45 = ( ! [A: $i,B: $i] :
% 0.27/0.45 ( ~ ( subset @ A @ B )
% 0.27/0.45 => ( A != B ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(notequalI2_type,type,
% 0.27/0.45 notequalI2: $o ).
% 0.27/0.45
% 0.27/0.45 thf(notequalI2,definition,
% 0.27/0.45 ( notequalI2
% 0.27/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.45 ( ( in @ Xx @ A )
% 0.27/0.45 => ( ~ ( in @ Xx @ B )
% 0.27/0.45 => ( A != B ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(subsetRefl_type,type,
% 0.27/0.45 subsetRefl: $o ).
% 0.27/0.45
% 0.27/0.45 thf(subsetRefl,definition,
% 0.27/0.45 ( subsetRefl
% 0.27/0.45 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(subsetTrans_type,type,
% 0.27/0.45 subsetTrans: $o ).
% 0.27/0.45
% 0.27/0.45 thf(subsetTrans,definition,
% 0.27/0.45 ( subsetTrans
% 0.27/0.45 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.45 ( ( subset @ A @ B )
% 0.27/0.45 => ( ( subset @ B @ C )
% 0.27/0.45 => ( subset @ A @ C ) ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinSub_type,type,
% 0.27/0.45 setadjoinSub: $o ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinSub,definition,
% 0.27/0.45 ( setadjoinSub
% 0.27/0.45 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.27/0.45
% 0.27/0.45 thf(setadjoinSub2_type,type,
% 0.27/0.45 setadjoinSub2: $o ).
% 0.27/0.45
% 0.27/0.46 thf(setadjoinSub2,definition,
% 0.27/0.46 ( setadjoinSub2
% 0.27/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.27/0.46 ( ( subset @ A @ B )
% 0.27/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(subset2powerset_type,type,
% 0.27/0.46 subset2powerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(subset2powerset,definition,
% 0.27/0.46 ( subset2powerset
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( subset @ A @ B )
% 0.27/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setextsub_type,type,
% 0.27/0.46 setextsub: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setextsub,definition,
% 0.27/0.46 ( setextsub
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( subset @ A @ B )
% 0.27/0.46 => ( ( subset @ B @ A )
% 0.27/0.46 => ( A = B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(subsetemptysetimpeq_type,type,
% 0.27/0.46 subsetemptysetimpeq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(subsetemptysetimpeq,definition,
% 0.27/0.46 ( subsetemptysetimpeq
% 0.27/0.46 = ( ! [A: $i] :
% 0.27/0.46 ( ( subset @ A @ emptyset )
% 0.27/0.46 => ( A = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetI1_type,type,
% 0.27/0.46 powersetI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetI1,definition,
% 0.27/0.46 ( powersetI1
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( subset @ B @ A )
% 0.27/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetE1_type,type,
% 0.27/0.46 powersetE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetE1,definition,
% 0.27/0.46 ( powersetE1
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.27/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(inPowerset_type,type,
% 0.27/0.46 inPowerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(inPowerset,definition,
% 0.27/0.46 ( inPowerset
% 0.27/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetsubset_type,type,
% 0.27/0.46 powersetsubset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetsubset,definition,
% 0.27/0.46 ( powersetsubset
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( subset @ A @ B )
% 0.27/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sepInPowerset_type,type,
% 0.27/0.46 sepInPowerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(sepInPowerset,definition,
% 0.27/0.46 ( sepInPowerset
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( in
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(sepSubset_type,type,
% 0.27/0.46 sepSubset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(sepSubset,definition,
% 0.27/0.46 ( sepSubset
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( subset
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunion_type,type,
% 0.27/0.46 binunion: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(binunionIL_type,type,
% 0.27/0.46 binunionIL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionIL,definition,
% 0.27/0.46 ( binunionIL
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairset2IR_type,type,
% 0.27/0.46 upairset2IR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairset2IR,definition,
% 0.27/0.46 ( upairset2IR
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionIR_type,type,
% 0.27/0.46 binunionIR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionIR,definition,
% 0.27/0.46 ( binunionIR
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionEcases_type,type,
% 0.27/0.46 binunionEcases: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionEcases,definition,
% 0.27/0.46 ( binunionEcases
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.27/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.27/0.46 => ( ( ( in @ Xx @ A )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => ( ( ( in @ Xx @ B )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => Xphi ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionE_type,type,
% 0.27/0.46 binunionE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionE,definition,
% 0.27/0.46 ( binunionE
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.27/0.46 => ( ( in @ Xx @ A )
% 0.27/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionLsub_type,type,
% 0.27/0.46 binunionLsub: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionLsub,definition,
% 0.27/0.46 ( binunionLsub
% 0.27/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionRsub_type,type,
% 0.27/0.46 binunionRsub: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionRsub,definition,
% 0.27/0.46 ( binunionRsub
% 0.27/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersect_type,type,
% 0.27/0.46 binintersect: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectI_type,type,
% 0.27/0.46 binintersectI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectI,definition,
% 0.27/0.46 ( binintersectI
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset5_type,type,
% 0.27/0.46 binintersectSubset5: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset5,definition,
% 0.27/0.46 ( binintersectSubset5
% 0.27/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.46 ( ( subset @ C @ A )
% 0.27/0.46 => ( ( subset @ C @ B )
% 0.27/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectEL_type,type,
% 0.27/0.46 binintersectEL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectEL,definition,
% 0.27/0.46 ( binintersectEL
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectLsub_type,type,
% 0.27/0.46 binintersectLsub: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectLsub,definition,
% 0.27/0.46 ( binintersectLsub
% 0.27/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset2_type,type,
% 0.27/0.46 binintersectSubset2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset2,definition,
% 0.27/0.46 ( binintersectSubset2
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( subset @ A @ B )
% 0.27/0.46 => ( ( binintersect @ A @ B )
% 0.27/0.46 = A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset3_type,type,
% 0.27/0.46 binintersectSubset3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset3,definition,
% 0.27/0.46 ( binintersectSubset3
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( ( binintersect @ A @ B )
% 0.27/0.46 = B )
% 0.27/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectER_type,type,
% 0.27/0.46 binintersectER: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectER,definition,
% 0.27/0.46 ( binintersectER
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.27/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(disjointsetsI1_type,type,
% 0.27/0.46 disjointsetsI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(disjointsetsI1,definition,
% 0.27/0.46 ( disjointsetsI1
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ~ ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( in @ Xx @ B ) )
% 0.27/0.46 => ( ( binintersect @ A @ B )
% 0.27/0.46 = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectRsub_type,type,
% 0.27/0.46 binintersectRsub: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectRsub,definition,
% 0.27/0.46 ( binintersectRsub
% 0.27/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset4_type,type,
% 0.27/0.46 binintersectSubset4: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset4,definition,
% 0.27/0.46 ( binintersectSubset4
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( subset @ B @ A )
% 0.27/0.46 => ( ( binintersect @ A @ B )
% 0.27/0.46 = B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset1_type,type,
% 0.27/0.46 binintersectSubset1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectSubset1,definition,
% 0.27/0.46 ( binintersectSubset1
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( ( binintersect @ A @ B )
% 0.27/0.46 = A )
% 0.27/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(bs114d_type,type,
% 0.27/0.46 bs114d: $o ).
% 0.27/0.46
% 0.27/0.46 thf(bs114d,definition,
% 0.27/0.46 ( bs114d
% 0.27/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.27/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.27/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(regular_type,type,
% 0.27/0.46 regular: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminus_type,type,
% 0.27/0.46 setminus: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(setminusI_type,type,
% 0.27/0.46 setminusI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusI,definition,
% 0.27/0.46 ( setminusI
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusEL_type,type,
% 0.27/0.46 setminusEL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusEL,definition,
% 0.27/0.46 ( setminusEL
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusER_type,type,
% 0.27/0.46 setminusER: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusER,definition,
% 0.27/0.46 ( setminusER
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusSubset2_type,type,
% 0.27/0.46 setminusSubset2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusSubset2,definition,
% 0.27/0.46 ( setminusSubset2
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( subset @ A @ B )
% 0.27/0.46 => ( ( setminus @ A @ B )
% 0.27/0.46 = emptyset ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusERneg_type,type,
% 0.27/0.46 setminusERneg: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusERneg,definition,
% 0.27/0.46 ( setminusERneg
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.46 => ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusELneg_type,type,
% 0.27/0.46 setminusELneg: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusELneg,definition,
% 0.27/0.46 ( setminusELneg
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.27/0.46 => ( ~ ( in @ Xx @ B )
% 0.27/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusILneg_type,type,
% 0.27/0.46 setminusILneg: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusILneg,definition,
% 0.27/0.46 ( setminusILneg
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ~ ( in @ Xx @ A )
% 0.27/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusIRneg_type,type,
% 0.27/0.46 setminusIRneg: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusIRneg,definition,
% 0.27/0.46 ( setminusIRneg
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ B )
% 0.27/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusLsub_type,type,
% 0.27/0.46 setminusLsub: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusLsub,definition,
% 0.27/0.46 ( setminusLsub
% 0.27/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusSubset1_type,type,
% 0.27/0.46 setminusSubset1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusSubset1,definition,
% 0.27/0.46 ( setminusSubset1
% 0.27/0.46 = ( ! [A: $i,B: $i] :
% 0.27/0.46 ( ( ( setminus @ A @ B )
% 0.27/0.46 = emptyset )
% 0.27/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(symdiff_type,type,
% 0.27/0.46 symdiff: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffE_type,type,
% 0.27/0.46 symdiffE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffE,definition,
% 0.27/0.46 ( symdiffE
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.27/0.46 => ! [Xphi: $o] :
% 0.27/0.46 ( ( ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ B )
% 0.27/0.46 => Xphi ) )
% 0.27/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ B )
% 0.27/0.46 => Xphi ) )
% 0.27/0.46 => Xphi ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffI1_type,type,
% 0.27/0.46 symdiffI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffI1,definition,
% 0.27/0.46 ( symdiffI1
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffI2_type,type,
% 0.27/0.46 symdiffI2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffI2,definition,
% 0.27/0.46 ( symdiffI2
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ~ ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ B )
% 0.27/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffIneg1_type,type,
% 0.27/0.46 symdiffIneg1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffIneg1,definition,
% 0.27/0.46 ( symdiffIneg1
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ B )
% 0.27/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffIneg2_type,type,
% 0.27/0.46 symdiffIneg2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(symdiffIneg2,definition,
% 0.27/0.46 ( symdiffIneg2
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ~ ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ B )
% 0.27/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(iskpair_type,type,
% 0.27/0.46 iskpair: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(secondinupair_type,type,
% 0.27/0.46 secondinupair: $o ).
% 0.27/0.46
% 0.27/0.46 thf(secondinupair,definition,
% 0.27/0.46 ( secondinupair
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairIL_type,type,
% 0.27/0.46 setukpairIL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairIL,definition,
% 0.27/0.46 ( setukpairIL
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairIR_type,type,
% 0.27/0.46 setukpairIR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairIR,definition,
% 0.27/0.46 ( setukpairIR
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(kpairiskpair_type,type,
% 0.27/0.46 kpairiskpair: $o ).
% 0.27/0.46
% 0.27/0.46 thf(kpairiskpair,definition,
% 0.27/0.46 ( kpairiskpair
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(kpair_type,type,
% 0.27/0.46 kpair: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(kpairp_type,type,
% 0.27/0.46 kpairp: $o ).
% 0.27/0.46
% 0.27/0.46 thf(kpairp,definition,
% 0.27/0.46 ( kpairp
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprod_type,type,
% 0.27/0.46 cartprod: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsubset_type,type,
% 0.27/0.46 singletonsubset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsubset,definition,
% 0.27/0.46 ( singletonsubset
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(singletoninpowerset_type,type,
% 0.27/0.46 singletoninpowerset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(singletoninpowerset,definition,
% 0.27/0.46 ( singletoninpowerset
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(singletoninpowunion_type,type,
% 0.27/0.46 singletoninpowunion: $o ).
% 0.27/0.46
% 0.27/0.46 thf(singletoninpowunion,definition,
% 0.27/0.46 ( singletoninpowunion
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairset2E_type,type,
% 0.27/0.46 upairset2E: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairset2E,definition,
% 0.27/0.46 ( upairset2E
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.27/0.46 => ( ( Xz = Xx )
% 0.27/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairsubunion_type,type,
% 0.27/0.46 upairsubunion: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsubunion,definition,
% 0.27/0.46 ( upairsubunion
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairinpowunion_type,type,
% 0.27/0.46 upairinpowunion: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairinpowunion,definition,
% 0.27/0.46 ( upairinpowunion
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ubforcartprodlem1_type,type,
% 0.27/0.46 ubforcartprodlem1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ubforcartprodlem1,definition,
% 0.27/0.46 ( ubforcartprodlem1
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ubforcartprodlem2_type,type,
% 0.27/0.46 ubforcartprodlem2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ubforcartprodlem2,definition,
% 0.27/0.46 ( ubforcartprodlem2
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ubforcartprodlem3_type,type,
% 0.27/0.46 ubforcartprodlem3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ubforcartprodlem3,definition,
% 0.27/0.46 ( ubforcartprodlem3
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodpairin_type,type,
% 0.27/0.46 cartprodpairin: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodpairin,definition,
% 0.27/0.46 ( cartprodpairin
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodmempair1_type,type,
% 0.27/0.46 cartprodmempair1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodmempair1,definition,
% 0.27/0.46 ( cartprodmempair1
% 0.27/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 & ( Xu
% 0.27/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodmempair_type,type,
% 0.27/0.46 cartprodmempair: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodmempair,definition,
% 0.27/0.46 ( cartprodmempair
% 0.27/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionE2_type,type,
% 0.27/0.46 setunionE2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionE2,definition,
% 0.27/0.46 ( setunionE2
% 0.27/0.46 = ( ! [A: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.27/0.46 => ? [X: $i] :
% 0.27/0.46 ( ( in @ X @ A )
% 0.27/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionsingleton1_type,type,
% 0.27/0.46 setunionsingleton1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionsingleton1,definition,
% 0.27/0.46 ( setunionsingleton1
% 0.27/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionsingleton2_type,type,
% 0.27/0.46 setunionsingleton2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionsingleton2,definition,
% 0.27/0.46 ( setunionsingleton2
% 0.27/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setunionsingleton_type,type,
% 0.27/0.46 setunionsingleton: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setunionsingleton,definition,
% 0.27/0.46 ( setunionsingleton
% 0.27/0.46 = ( ! [Xx: $i] :
% 0.27/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.27/0.46 = Xx ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(singleton_type,type,
% 0.27/0.46 singleton: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(singletonprop_type,type,
% 0.27/0.46 singletonprop: $o ).
% 0.27/0.46
% 0.27/0.46 thf(singletonprop,definition,
% 0.27/0.46 ( singletonprop
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) )
% 0.27/0.46 => ( ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) )
% 0.27/0.46 => ( singleton
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ex1_type,type,
% 0.27/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.27/0.46
% 0.27/0.46 thf(ex1E1_type,type,
% 0.27/0.46 ex1E1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ex1E1,definition,
% 0.27/0.46 ( ex1E1
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ( ex1 @ A
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ex1I_type,type,
% 0.27/0.46 ex1I: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ex1I,definition,
% 0.27/0.46 ( ex1I
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xy = Xx ) ) )
% 0.27/0.46 => ( ex1 @ A
% 0.27/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ex1I2_type,type,
% 0.27/0.46 ex1I2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ex1I2,definition,
% 0.27/0.46 ( ex1I2
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) )
% 0.27/0.46 => ( ? [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 & ( Xphi @ Xx ) )
% 0.27/0.46 => ( ex1 @ A
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsuniq_type,type,
% 0.27/0.46 singletonsuniq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(singletonsuniq,definition,
% 0.27/0.46 ( singletonsuniq
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.27/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(atmost1p_type,type,
% 0.27/0.46 atmost1p: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(atleast2p_type,type,
% 0.27/0.46 atleast2p: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(atmost2p_type,type,
% 0.27/0.46 atmost2p: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairsetp_type,type,
% 0.27/0.46 upairsetp: $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjL1_type,type,
% 0.27/0.46 setukpairinjL1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjL1,definition,
% 0.27/0.46 ( setukpairinjL1
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(kfstsingleton_type,type,
% 0.27/0.46 kfstsingleton: $o ).
% 0.27/0.46
% 0.27/0.46 thf(kfstsingleton,definition,
% 0.27/0.46 ( kfstsingleton
% 0.27/0.46 = ( ! [Xu: $i] :
% 0.27/0.46 ( ( iskpair @ Xu )
% 0.27/0.46 => ( singleton
% 0.27/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.27/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(theprop_type,type,
% 0.27/0.46 theprop: $o ).
% 0.27/0.46
% 0.27/0.46 thf(theprop,definition,
% 0.27/0.46 ( theprop
% 0.27/0.46 = ( ! [X: $i] :
% 0.27/0.46 ( ( singleton @ X )
% 0.27/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(kfst_type,type,
% 0.27/0.46 kfst: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(kfstpairEq_type,type,
% 0.27/0.46 kfstpairEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(kfstpairEq,definition,
% 0.27/0.46 ( kfstpairEq
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.27/0.46 = Xx ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodfstin_type,type,
% 0.27/0.46 cartprodfstin: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodfstin,definition,
% 0.27/0.46 ( cartprodfstin
% 0.27/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjL2_type,type,
% 0.27/0.46 setukpairinjL2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjL2,definition,
% 0.27/0.46 ( setukpairinjL2
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjL_type,type,
% 0.27/0.46 setukpairinjL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjL,definition,
% 0.27/0.46 ( setukpairinjL
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.27/0.46 = ( kpair @ Xz @ Xu ) )
% 0.27/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR11_type,type,
% 0.27/0.46 setukpairinjR11: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR11,definition,
% 0.27/0.46 ( setukpairinjR11
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR12_type,type,
% 0.27/0.46 setukpairinjR12: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR12,definition,
% 0.27/0.46 ( setukpairinjR12
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xx = Xy )
% 0.27/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR1_type,type,
% 0.27/0.46 setukpairinjR1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR1,definition,
% 0.27/0.46 ( setukpairinjR1
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.46 => ( ( Xz = Xu )
% 0.27/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(upairequniteq_type,type,
% 0.27/0.46 upairequniteq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(upairequniteq,definition,
% 0.27/0.46 ( upairequniteq
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.27/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.27/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR2_type,type,
% 0.27/0.46 setukpairinjR2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR2,definition,
% 0.27/0.46 ( setukpairinjR2
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.27/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.27/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR_type,type,
% 0.27/0.46 setukpairinjR: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setukpairinjR,definition,
% 0.27/0.46 ( setukpairinjR
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.27/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.27/0.46 = ( kpair @ Xz @ Xu ) )
% 0.27/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ksndsingleton_type,type,
% 0.27/0.46 ksndsingleton: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ksndsingleton,definition,
% 0.27/0.46 ( ksndsingleton
% 0.27/0.46 = ( ! [Xu: $i] :
% 0.27/0.46 ( ( iskpair @ Xu )
% 0.27/0.46 => ( singleton
% 0.27/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.27/0.46 @ ^ [Xx: $i] :
% 0.27/0.46 ( Xu
% 0.27/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ksnd_type,type,
% 0.27/0.46 ksnd: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(ksndpairEq_type,type,
% 0.27/0.46 ksndpairEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ksndpairEq,definition,
% 0.27/0.46 ( ksndpairEq
% 0.27/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.27/0.46 = Xy ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(kpairsurjEq_type,type,
% 0.27/0.46 kpairsurjEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(kpairsurjEq,definition,
% 0.27/0.46 ( kpairsurjEq
% 0.27/0.46 = ( ! [Xu: $i] :
% 0.27/0.46 ( ( iskpair @ Xu )
% 0.27/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.27/0.46 = Xu ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodsndin_type,type,
% 0.27/0.46 cartprodsndin: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodsndin,definition,
% 0.27/0.46 ( cartprodsndin
% 0.27/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodpairmemEL_type,type,
% 0.27/0.46 cartprodpairmemEL: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodpairmemEL,definition,
% 0.27/0.46 ( cartprodpairmemEL
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodpairmemER_type,type,
% 0.27/0.46 cartprodpairmemER: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodpairmemER,definition,
% 0.27/0.46 ( cartprodpairmemER
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.27/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodmempaircEq_type,type,
% 0.27/0.46 cartprodmempaircEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodmempaircEq,definition,
% 0.27/0.46 ( cartprodmempaircEq
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( kpair @ Xx @ Xy )
% 0.27/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodfstpairEq_type,type,
% 0.27/0.46 cartprodfstpairEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodfstpairEq,definition,
% 0.27/0.46 ( cartprodfstpairEq
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.27/0.46 = Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodsndpairEq_type,type,
% 0.27/0.46 cartprodsndpairEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodsndpairEq,definition,
% 0.27/0.46 ( cartprodsndpairEq
% 0.27/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.27/0.46 = Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodpairsurjEq_type,type,
% 0.27/0.46 cartprodpairsurjEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(cartprodpairsurjEq,definition,
% 0.27/0.46 ( cartprodpairsurjEq
% 0.27/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.27/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.27/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.27/0.46 = Xu ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(breln_type,type,
% 0.27/0.46 breln: $i > $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(breln,definition,
% 0.27/0.46 ( breln
% 0.27/0.46 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstr_type,type,
% 0.27/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrI_type,type,
% 0.27/0.46 dpsetconstrI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrI,definition,
% 0.27/0.46 ( dpsetconstrI
% 0.27/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.27/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.46 @ ( dpsetconstr @ A @ B
% 0.27/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrSub_type,type,
% 0.27/0.46 dpsetconstrSub: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrSub,definition,
% 0.27/0.46 ( dpsetconstrSub
% 0.27/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.46 ( subset
% 0.27/0.46 @ ( dpsetconstr @ A @ B
% 0.27/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.27/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setOfPairsIsBReln_type,type,
% 0.27/0.46 setOfPairsIsBReln: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setOfPairsIsBReln,definition,
% 0.27/0.46 ( setOfPairsIsBReln
% 0.27/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.46 ( breln @ A @ B
% 0.27/0.46 @ ( dpsetconstr @ A @ B
% 0.27/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrERa_type,type,
% 0.27/0.46 dpsetconstrERa: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrERa,definition,
% 0.27/0.46 ( dpsetconstrERa
% 0.27/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.46 @ ( dpsetconstr @ A @ B
% 0.27/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrEL1_type,type,
% 0.27/0.46 dpsetconstrEL1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrEL1,definition,
% 0.27/0.46 ( dpsetconstrEL1
% 0.27/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.46 @ ( dpsetconstr @ A @ B
% 0.27/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrEL2_type,type,
% 0.27/0.46 dpsetconstrEL2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrEL2,definition,
% 0.27/0.46 ( dpsetconstrEL2
% 0.27/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.46 @ ( dpsetconstr @ A @ B
% 0.27/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrER_type,type,
% 0.27/0.46 dpsetconstrER: $o ).
% 0.27/0.46
% 0.27/0.46 thf(dpsetconstrER,definition,
% 0.27/0.46 ( dpsetconstrER
% 0.27/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.27/0.46 @ ( dpsetconstr @ A @ B
% 0.27/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.27/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(func_type,type,
% 0.27/0.46 func: $i > $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcSet_type,type,
% 0.27/0.46 funcSet: $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(funcImageSingleton_type,type,
% 0.27/0.46 funcImageSingleton: $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcImageSingleton,definition,
% 0.27/0.46 ( funcImageSingleton
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( singleton
% 0.27/0.46 @ ( dsetconstr @ B
% 0.27/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(apProp_type,type,
% 0.27/0.46 apProp: $o ).
% 0.27/0.46
% 0.27/0.46 thf(apProp,definition,
% 0.27/0.46 ( apProp
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in
% 0.27/0.46 @ ( setunion
% 0.27/0.46 @ ( dsetconstr @ B
% 0.27/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.27/0.46 @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ap_type,type,
% 0.27/0.46 ap: $i > $i > $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(app_type,type,
% 0.27/0.46 app: $o ).
% 0.27/0.46
% 0.27/0.46 thf(app,definition,
% 0.27/0.46 ( app
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(infuncsetfunc_type,type,
% 0.27/0.46 infuncsetfunc: $o ).
% 0.27/0.46
% 0.27/0.46 thf(infuncsetfunc,definition,
% 0.27/0.46 ( infuncsetfunc
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ap2p_type,type,
% 0.27/0.46 ap2p: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ap2p,definition,
% 0.27/0.46 ( ap2p
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(funcinfuncset_type,type,
% 0.27/0.46 funcinfuncset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcinfuncset,definition,
% 0.27/0.46 ( funcinfuncset
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(lamProp_type,type,
% 0.27/0.46 lamProp: $o ).
% 0.27/0.46
% 0.27/0.46 thf(lamProp,definition,
% 0.27/0.46 ( lamProp
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.46 => ( func @ A @ B
% 0.27/0.46 @ ( dpsetconstr @ A @ B
% 0.27/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.27/0.46 ( ( Xf @ Xx )
% 0.27/0.46 = Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(lam_type,type,
% 0.27/0.46 lam: $i > $i > ( $i > $i ) > $i ).
% 0.27/0.46
% 0.27/0.46 thf(lamp_type,type,
% 0.27/0.46 lamp: $o ).
% 0.27/0.46
% 0.27/0.46 thf(lamp,definition,
% 0.27/0.46 ( lamp
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.46 => ( func @ A @ B
% 0.27/0.46 @ ( lam @ A @ B
% 0.27/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(lam2p_type,type,
% 0.27/0.46 lam2p: $o ).
% 0.27/0.46
% 0.27/0.46 thf(lam2p,definition,
% 0.27/0.46 ( lam2p
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.46 => ( in
% 0.27/0.46 @ ( lam @ A @ B
% 0.27/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.27/0.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(brelnall1_type,type,
% 0.27/0.46 brelnall1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(brelnall1,definition,
% 0.27/0.46 ( brelnall1
% 0.27/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.46 ( ( breln @ A @ B @ R )
% 0.27/0.46 => ! [Xphi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ R )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(brelnall2_type,type,
% 0.27/0.46 brelnall2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(brelnall2,definition,
% 0.27/0.46 ( brelnall2
% 0.27/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.46 ( ( breln @ A @ B @ R )
% 0.27/0.46 => ! [Xphi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ R )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ex1E2_type,type,
% 0.27/0.46 ex1E2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ex1E2,definition,
% 0.27/0.46 ( ex1E2
% 0.27/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.27/0.46 ( ( ex1 @ A
% 0.27/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( ( Xphi @ Xx )
% 0.27/0.46 => ( ( Xphi @ Xy )
% 0.27/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(funcGraphProp1_type,type,
% 0.27/0.46 funcGraphProp1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcGraphProp1,definition,
% 0.27/0.46 ( funcGraphProp1
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(funcGraphProp3_type,type,
% 0.27/0.46 funcGraphProp3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcGraphProp3,definition,
% 0.27/0.46 ( funcGraphProp3
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(funcGraphProp2_type,type,
% 0.27/0.46 funcGraphProp2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcGraphProp2,definition,
% 0.27/0.46 ( funcGraphProp2
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(funcextLem_type,type,
% 0.27/0.46 funcextLem: $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcextLem,definition,
% 0.27/0.46 ( funcextLem
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ! [Xg: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xg )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.27/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(funcGraphProp4_type,type,
% 0.27/0.46 funcGraphProp4: $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcGraphProp4,definition,
% 0.27/0.46 ( funcGraphProp4
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.27/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(subbreln_type,type,
% 0.27/0.46 subbreln: $o ).
% 0.27/0.46
% 0.27/0.46 thf(subbreln,definition,
% 0.27/0.46 ( subbreln
% 0.27/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.46 ( ( breln @ A @ B @ R )
% 0.27/0.46 => ! [S: $i] :
% 0.27/0.46 ( ( breln @ A @ B @ S )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(eqbreln_type,type,
% 0.27/0.46 eqbreln: $o ).
% 0.27/0.46
% 0.27/0.46 thf(eqbreln,definition,
% 0.27/0.46 ( eqbreln
% 0.27/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.27/0.46 ( ( breln @ A @ B @ R )
% 0.27/0.46 => ! [S: $i] :
% 0.27/0.46 ( ( breln @ A @ B @ S )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.27/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.27/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(funcext_type,type,
% 0.27/0.46 funcext: $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcext,definition,
% 0.27/0.46 ( funcext
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ! [Xg: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xg )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(funcext2_type,type,
% 0.27/0.46 funcext2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(funcext2,definition,
% 0.27/0.46 ( funcext2
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.46 => ! [Xg: $i] :
% 0.27/0.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.27/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ap2apEq1_type,type,
% 0.27/0.46 ap2apEq1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ap2apEq1,definition,
% 0.27/0.46 ( ap2apEq1
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ap2apEq2_type,type,
% 0.27/0.46 ap2apEq2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ap2apEq2,definition,
% 0.27/0.46 ( ap2apEq2
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.27/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(beta1_type,type,
% 0.27/0.46 beta1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(beta1,definition,
% 0.27/0.46 ( beta1
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( ap @ A @ B
% 0.27/0.46 @ ( lam @ A @ B
% 0.27/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.27/0.46 @ Xx )
% 0.27/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(eta1_type,type,
% 0.27/0.46 eta1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(eta1,definition,
% 0.27/0.46 ( eta1
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( func @ A @ B @ Xf )
% 0.27/0.46 => ( ( lam @ A @ B
% 0.27/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.27/0.46 = Xf ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(lam2lamEq_type,type,
% 0.27/0.46 lam2lamEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(lam2lamEq,definition,
% 0.27/0.46 ( lam2lamEq
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.46 => ( ( lam @ A @ B
% 0.27/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.27/0.46 = ( lam @ A @ B
% 0.27/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(beta2_type,type,
% 0.27/0.46 beta2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(beta2,definition,
% 0.27/0.46 ( beta2
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( ap @ A @ B
% 0.27/0.46 @ ( lam @ A @ B
% 0.27/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.27/0.46 @ Xx )
% 0.27/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(eta2_type,type,
% 0.27/0.46 eta2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(eta2,definition,
% 0.27/0.46 ( eta2
% 0.27/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.27/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.27/0.46 => ( ( lam @ A @ B
% 0.27/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.27/0.46 = Xf ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(iffalseProp1_type,type,
% 0.27/0.46 iffalseProp1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(iffalseProp1,definition,
% 0.27/0.46 ( iffalseProp1
% 0.27/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( ~ Xphi
% 0.27/0.46 => ( in @ Xy
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xz: $i] :
% 0.27/0.46 ( ( Xphi
% 0.27/0.46 & ( Xz = Xx ) )
% 0.27/0.46 | ( ~ Xphi
% 0.27/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(iffalseProp2_type,type,
% 0.27/0.46 iffalseProp2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(iffalseProp2,definition,
% 0.27/0.46 ( iffalseProp2
% 0.27/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( ~ Xphi
% 0.27/0.46 => ( ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xz: $i] :
% 0.27/0.46 ( ( Xphi
% 0.27/0.46 & ( Xz = Xx ) )
% 0.27/0.46 | ( ~ Xphi
% 0.27/0.46 & ( Xz = Xy ) ) ) )
% 0.27/0.46 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(iftrueProp1_type,type,
% 0.27/0.46 iftrueProp1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(iftrueProp1,definition,
% 0.27/0.46 ( iftrueProp1
% 0.27/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( Xphi
% 0.27/0.46 => ( in @ Xx
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xz: $i] :
% 0.27/0.46 ( ( Xphi
% 0.27/0.46 & ( Xz = Xx ) )
% 0.27/0.46 | ( ~ Xphi
% 0.27/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(iftrueProp2_type,type,
% 0.27/0.46 iftrueProp2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(iftrueProp2,definition,
% 0.27/0.46 ( iftrueProp2
% 0.27/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( Xphi
% 0.27/0.46 => ( ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xz: $i] :
% 0.27/0.46 ( ( Xphi
% 0.27/0.46 & ( Xz = Xx ) )
% 0.27/0.46 | ( ~ Xphi
% 0.27/0.46 & ( Xz = Xy ) ) ) )
% 0.27/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(ifSingleton_type,type,
% 0.27/0.46 ifSingleton: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ifSingleton,definition,
% 0.27/0.46 ( ifSingleton
% 0.27/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( singleton
% 0.27/0.46 @ ( dsetconstr @ A
% 0.27/0.46 @ ^ [Xz: $i] :
% 0.27/0.46 ( ( Xphi
% 0.27/0.46 & ( Xz = Xx ) )
% 0.27/0.46 | ( ~ Xphi
% 0.27/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(if_type,type,
% 0.27/0.46 if: $i > $o > $i > $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(ifp_type,type,
% 0.27/0.46 ifp: $o ).
% 0.27/0.46
% 0.27/0.46 thf(ifp,definition,
% 0.27/0.46 ( ifp
% 0.27/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(theeq_type,type,
% 0.27/0.46 theeq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(theeq,definition,
% 0.27/0.46 ( theeq
% 0.27/0.46 = ( ! [X: $i] :
% 0.27/0.46 ( ( singleton @ X )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ X )
% 0.27/0.46 => ( ( setunion @ X )
% 0.27/0.46 = Xx ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(iftrue_type,type,
% 0.27/0.46 iftrue: $o ).
% 0.27/0.46
% 0.27/0.46 thf(iftrue,definition,
% 0.27/0.46 ( iftrue
% 0.27/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( Xphi
% 0.27/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.27/0.46 = Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(iffalse_type,type,
% 0.27/0.46 iffalse: $o ).
% 0.27/0.46
% 0.27/0.46 thf(iffalse,definition,
% 0.27/0.46 ( iffalse
% 0.27/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( ~ Xphi
% 0.27/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.27/0.46 = Xy ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(iftrueorfalse_type,type,
% 0.27/0.46 iftrueorfalse: $o ).
% 0.27/0.46
% 0.27/0.46 thf(iftrueorfalse,definition,
% 0.27/0.46 ( iftrueorfalse
% 0.27/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectT_lem_type,type,
% 0.27/0.46 binintersectT_lem: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectT_lem,definition,
% 0.27/0.46 ( binintersectT_lem
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionT_lem_type,type,
% 0.27/0.46 binunionT_lem: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionT_lem,definition,
% 0.27/0.46 ( binunionT_lem
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetT_lem_type,type,
% 0.27/0.46 powersetT_lem: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetT_lem,definition,
% 0.27/0.46 ( powersetT_lem
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setminusT_lem_type,type,
% 0.27/0.46 setminusT_lem: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setminusT_lem,definition,
% 0.27/0.46 ( setminusT_lem
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(complementT_lem_type,type,
% 0.27/0.46 complementT_lem: $o ).
% 0.27/0.46
% 0.27/0.46 thf(complementT_lem,definition,
% 0.27/0.46 ( complementT_lem
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setextT_type,type,
% 0.27/0.46 setextT: $o ).
% 0.27/0.46
% 0.27/0.46 thf(setextT,definition,
% 0.27/0.46 ( setextT
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ X )
% 0.27/0.46 => ( in @ Xx @ Y ) ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ Y )
% 0.27/0.46 => ( in @ Xx @ X ) ) )
% 0.27/0.46 => ( X = Y ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(subsetTI_type,type,
% 0.27/0.46 subsetTI: $o ).
% 0.27/0.46
% 0.27/0.46 thf(subsetTI,definition,
% 0.27/0.46 ( subsetTI
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ X )
% 0.27/0.46 => ( in @ Xx @ Y ) ) )
% 0.27/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetTI1_type,type,
% 0.27/0.46 powersetTI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetTI1,definition,
% 0.27/0.46 ( powersetTI1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ X )
% 0.27/0.46 => ( in @ Xx @ Y ) ) )
% 0.27/0.46 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(powersetTE1_type,type,
% 0.27/0.46 powersetTE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(powersetTE1,definition,
% 0.27/0.46 ( powersetTE1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ X @ ( powerset @ Y ) )
% 0.27/0.46 => ( ( in @ Xx @ X )
% 0.27/0.46 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(complementTI1_type,type,
% 0.27/0.46 complementTI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(complementTI1,definition,
% 0.27/0.46 ( complementTI1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ X )
% 0.27/0.46 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(complementTE1_type,type,
% 0.27/0.46 complementTE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(complementTE1,definition,
% 0.27/0.46 ( complementTE1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.27/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectTELcontra_type,type,
% 0.27/0.46 binintersectTELcontra: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectTELcontra,definition,
% 0.27/0.46 ( binintersectTELcontra
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ X )
% 0.27/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectTERcontra_type,type,
% 0.27/0.46 binintersectTERcontra: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binintersectTERcontra,definition,
% 0.27/0.46 ( binintersectTERcontra
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ Y )
% 0.27/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(contrasubsetT_type,type,
% 0.27/0.46 contrasubsetT: $o ).
% 0.27/0.46
% 0.27/0.46 thf(contrasubsetT,definition,
% 0.27/0.46 ( contrasubsetT
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.27/0.46 => ( ( in @ Xx @ Y )
% 0.27/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(contrasubsetT1_type,type,
% 0.27/0.46 contrasubsetT1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(contrasubsetT1,definition,
% 0.27/0.46 ( contrasubsetT1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( subset @ X @ Y )
% 0.27/0.46 => ( ~ ( in @ Xx @ Y )
% 0.27/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(contrasubsetT2_type,type,
% 0.27/0.46 contrasubsetT2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(contrasubsetT2,definition,
% 0.27/0.46 ( contrasubsetT2
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ( subset @ X @ Y )
% 0.27/0.46 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(contrasubsetT3_type,type,
% 0.27/0.46 contrasubsetT3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(contrasubsetT3,definition,
% 0.27/0.46 ( contrasubsetT3
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.27/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementI1_type,type,
% 0.27/0.46 doubleComplementI1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementI1,definition,
% 0.27/0.46 ( doubleComplementI1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ X )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementE1_type,type,
% 0.27/0.46 doubleComplementE1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementE1,definition,
% 0.27/0.46 ( doubleComplementE1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.27/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementSub1_type,type,
% 0.27/0.46 doubleComplementSub1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementSub1,definition,
% 0.27/0.46 ( doubleComplementSub1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementSub2_type,type,
% 0.27/0.46 doubleComplementSub2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementSub2,definition,
% 0.27/0.46 ( doubleComplementSub2
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementEq_type,type,
% 0.27/0.46 doubleComplementEq: $o ).
% 0.27/0.46
% 0.27/0.46 thf(doubleComplementEq,definition,
% 0.27/0.46 ( doubleComplementEq
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ( X
% 0.27/0.46 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(complementTnotintersectT_type,type,
% 0.27/0.46 complementTnotintersectT: $o ).
% 0.27/0.46
% 0.27/0.46 thf(complementTnotintersectT,definition,
% 0.27/0.46 ( complementTnotintersectT
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.27/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(complementImpComplementIntersect_type,type,
% 0.27/0.46 complementImpComplementIntersect: $o ).
% 0.27/0.46
% 0.27/0.46 thf(complementImpComplementIntersect,definition,
% 0.27/0.46 ( complementImpComplementIntersect
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(complementSubsetComplementIntersect_type,type,
% 0.27/0.46 complementSubsetComplementIntersect: $o ).
% 0.27/0.46
% 0.27/0.46 thf(complementSubsetComplementIntersect,definition,
% 0.27/0.46 ( complementSubsetComplementIntersect
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(complementInPowersetComplementIntersect_type,type,
% 0.27/0.46 complementInPowersetComplementIntersect: $o ).
% 0.27/0.46
% 0.27/0.46 thf(complementInPowersetComplementIntersect,definition,
% 0.27/0.46 ( complementInPowersetComplementIntersect
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(contraSubsetComplement_type,type,
% 0.27/0.46 contraSubsetComplement: $o ).
% 0.27/0.46
% 0.27/0.46 thf(contraSubsetComplement,definition,
% 0.27/0.46 ( contraSubsetComplement
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ Y )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(complementTcontraSubset_type,type,
% 0.27/0.46 complementTcontraSubset: $o ).
% 0.27/0.46
% 0.27/0.46 thf(complementTcontraSubset,definition,
% 0.27/0.46 ( complementTcontraSubset
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.27/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionTILcontra_type,type,
% 0.27/0.46 binunionTILcontra: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionTILcontra,definition,
% 0.27/0.46 ( binunionTILcontra
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.27/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionTIRcontra_type,type,
% 0.27/0.46 binunionTIRcontra: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionTIRcontra,definition,
% 0.27/0.46 ( binunionTIRcontra
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.27/0.46 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(inIntersectImpInUnion_type,type,
% 0.27/0.46 inIntersectImpInUnion: $o ).
% 0.27/0.46
% 0.27/0.46 thf(inIntersectImpInUnion,definition,
% 0.27/0.46 ( inIntersectImpInUnion
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.27/0.46 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(inIntersectImpInUnion2_type,type,
% 0.27/0.46 inIntersectImpInUnion2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(inIntersectImpInUnion2,definition,
% 0.27/0.46 ( inIntersectImpInUnion2
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.27/0.46 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(inIntersectImpInIntersectUnions_type,type,
% 0.27/0.46 inIntersectImpInIntersectUnions: $o ).
% 0.27/0.46
% 0.27/0.46 thf(inIntersectImpInIntersectUnions,definition,
% 0.27/0.46 ( inIntersectImpInIntersectUnions
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.27/0.46 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(intersectInPowersetIntersectUnions_type,type,
% 0.27/0.46 intersectInPowersetIntersectUnions: $o ).
% 0.27/0.46
% 0.27/0.46 thf(intersectInPowersetIntersectUnions,definition,
% 0.27/0.46 ( intersectInPowersetIntersectUnions
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(inComplementUnionImpNotIn1_type,type,
% 0.27/0.46 inComplementUnionImpNotIn1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(inComplementUnionImpNotIn1,definition,
% 0.27/0.46 ( inComplementUnionImpNotIn1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(inComplementUnionImpInComplement1_type,type,
% 0.27/0.46 inComplementUnionImpInComplement1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(inComplementUnionImpInComplement1,definition,
% 0.27/0.46 ( inComplementUnionImpInComplement1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionTE_type,type,
% 0.27/0.46 binunionTE: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionTE,definition,
% 0.27/0.46 ( binunionTE
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xphi: $o,Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.27/0.46 => ( ( ( in @ Xx @ X )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => ( ( ( in @ Xx @ Y )
% 0.27/0.46 => Xphi )
% 0.27/0.46 => Xphi ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(binunionTEcontra_type,type,
% 0.27/0.46 binunionTEcontra: $o ).
% 0.27/0.46
% 0.27/0.46 thf(binunionTEcontra,definition,
% 0.27/0.46 ( binunionTEcontra
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ~ ( in @ Xx @ X )
% 0.27/0.46 => ( ~ ( in @ Xx @ Y )
% 0.27/0.46 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2a1_type,type,
% 0.27/0.46 demorgan2a1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2a1,definition,
% 0.27/0.46 ( demorgan2a1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(complementUnionInPowersetComplement_type,type,
% 0.27/0.46 complementUnionInPowersetComplement: $o ).
% 0.27/0.46
% 0.27/0.46 thf(complementUnionInPowersetComplement,definition,
% 0.27/0.46 ( complementUnionInPowersetComplement
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2a2_type,type,
% 0.27/0.46 demorgan2a2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2a2,definition,
% 0.27/0.46 ( demorgan2a2
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan1a_type,type,
% 0.27/0.46 demorgan1a: $o ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan1a,definition,
% 0.27/0.46 ( demorgan1a
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.27/0.46 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan1b_type,type,
% 0.27/0.46 demorgan1b: $o ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan1b,definition,
% 0.27/0.46 ( demorgan1b
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan1_type,type,
% 0.27/0.46 demorgan1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan1,definition,
% 0.27/0.46 ( demorgan1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.27/0.46 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2a_type,type,
% 0.27/0.46 demorgan2a: $o ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2a,definition,
% 0.27/0.46 ( demorgan2a
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.27/0.46 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2b2_type,type,
% 0.27/0.46 demorgan2b2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2b2,definition,
% 0.27/0.46 ( demorgan2b2
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.27/0.46 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2b_type,type,
% 0.27/0.46 demorgan2b: $o ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2b,definition,
% 0.27/0.46 ( demorgan2b
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.27/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2_type,type,
% 0.27/0.46 demorgan2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(demorgan2,definition,
% 0.27/0.46 ( demorgan2
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.27/0.46 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule0_type,type,
% 0.27/0.46 woz13rule0: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule0,definition,
% 0.27/0.46 ( woz13rule0
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.27/0.46 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule1_type,type,
% 0.27/0.46 woz13rule1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule1,definition,
% 0.27/0.46 ( woz13rule1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ( ( subset @ X @ Z )
% 0.27/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule2_type,type,
% 0.27/0.46 woz13rule2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule2,definition,
% 0.27/0.46 ( woz13rule2
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ( ( subset @ Y @ Z )
% 0.27/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule3_type,type,
% 0.27/0.46 woz13rule3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule3,definition,
% 0.27/0.46 ( woz13rule3
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ( ( subset @ X @ Y )
% 0.27/0.46 => ( ( subset @ X @ Z )
% 0.27/0.46 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule4_type,type,
% 0.27/0.46 woz13rule4: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz13rule4,definition,
% 0.27/0.46 ( woz13rule4
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ! [W: $i] :
% 0.27/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.27/0.46 => ( ( subset @ X @ Z )
% 0.27/0.46 => ( ( subset @ Y @ W )
% 0.27/0.46 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_1_type,type,
% 0.27/0.46 woz1_1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_1,definition,
% 0.27/0.46 ( woz1_1
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_2_type,type,
% 0.27/0.46 woz1_2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_2,definition,
% 0.27/0.46 ( woz1_2
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ! [W: $i] :
% 0.27/0.46 ( ( in @ W @ ( powerset @ A ) )
% 0.27/0.46 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.27/0.46 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_3_type,type,
% 0.27/0.46 woz1_3: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_3,definition,
% 0.27/0.46 ( woz1_3
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ! [Z: $i] :
% 0.27/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_4_type,type,
% 0.27/0.46 woz1_4: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_4,definition,
% 0.27/0.46 ( woz1_4
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.27/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_5_type,type,
% 0.27/0.46 woz1_5: $o ).
% 0.27/0.46
% 0.27/0.46 thf(woz1_5,definition,
% 0.27/0.46 ( woz1_5
% 0.27/0.46 = ( ! [A: $i,X: $i] :
% 0.27/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.27/0.46 => ! [Y: $i] :
% 0.27/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.27/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(breln1_type,type,
% 0.27/0.46 breln1: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(breln1,definition,
% 0.27/0.46 ( breln1
% 0.27/0.46 = ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(breln1all2_type,type,
% 0.27/0.46 breln1all2: $o ).
% 0.27/0.46
% 0.27/0.46 thf(breln1all2,definition,
% 0.27/0.46 ( breln1all2
% 0.27/0.46 = ( ! [A: $i,R: $i] :
% 0.27/0.46 ( ( breln1 @ A @ R )
% 0.27/0.46 => ! [Xphi: $i > $o] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ! [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ A )
% 0.27/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.27/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ R )
% 0.27/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(breln1Set_type,type,
% 0.27/0.46 breln1Set: $i > $i ).
% 0.27/0.46
% 0.27/0.46 thf(breln1SetBreln1_type,type,
% 0.27/0.46 breln1SetBreln1: $o ).
% 0.27/0.46
% 0.27/0.46 thf(breln1SetBreln1,definition,
% 0.27/0.46 ( breln1SetBreln1
% 0.27/0.46 = ( ! [A: $i,R: $i] :
% 0.27/0.46 ( ( in @ R @ ( breln1Set @ A ) )
% 0.27/0.46 => ( breln1 @ A @ R ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(transitive_type,type,
% 0.27/0.46 transitive: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(antisymmetric_type,type,
% 0.27/0.46 antisymmetric: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(reflexive_type,type,
% 0.27/0.46 reflexive: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(refltransitive_type,type,
% 0.27/0.46 refltransitive: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(refllinearorder_type,type,
% 0.27/0.46 refllinearorder: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(reflwellordering_type,type,
% 0.27/0.46 reflwellordering: $i > $i > $o ).
% 0.27/0.46
% 0.27/0.46 thf(choice2fnsingleton_type,type,
% 0.27/0.46 choice2fnsingleton: $o ).
% 0.27/0.46
% 0.27/0.46 thf(choice2fnsingleton,definition,
% 0.27/0.46 ( choice2fnsingleton
% 0.27/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.27/0.46 ( ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ? [Xy: $i] :
% 0.27/0.46 ( ( in @ Xy @ B )
% 0.27/0.46 & ( Xphi @ Xx @ Xy ) ) )
% 0.27/0.46 => ! [R: $i] :
% 0.27/0.46 ( ( in @ R @ ( breln1Set @ B ) )
% 0.27/0.46 => ( ( reflwellordering @ B @ R )
% 0.27/0.46 => ! [Xx: $i] :
% 0.27/0.46 ( ( in @ Xx @ A )
% 0.27/0.46 => ( singleton
% 0.27/0.46 @ ( dsetconstr @ B
% 0.27/0.46 @ ^ [Xy: $i] :
% 0.27/0.46 ( ( Xphi @ Xx @ Xy )
% 0.27/0.46 & ! [Xz: $i] :
% 0.27/0.46 ( ( in @ Xz @ B )
% 0.27/0.46 => ( ( Xphi @ Xx @ Xz )
% 0.27/0.46 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 thf(setOfPairsIsBReln1,conjecture,
% 0.27/0.46 ( setextAx
% 0.27/0.46 => ( emptysetAx
% 0.27/0.46 => ( setadjoinAx
% 0.27/0.46 => ( powersetAx
% 0.27/0.46 => ( setunionAx
% 0.27/0.46 => ( omega0Ax
% 0.27/0.46 => ( omegaSAx
% 0.27/0.46 => ( omegaIndAx
% 0.27/0.46 => ( replAx
% 0.27/0.46 => ( foundationAx
% 0.27/0.46 => ( wellorderingAx
% 0.27/0.46 => ( descrp
% 0.27/0.46 => ( dsetconstrI
% 0.27/0.46 => ( dsetconstrEL
% 0.27/0.46 => ( dsetconstrER
% 0.27/0.46 => ( exuE1
% 0.27/0.46 => ( prop2setE
% 0.27/0.46 => ( emptysetE
% 0.27/0.46 => ( emptysetimpfalse
% 0.27/0.46 => ( notinemptyset
% 0.27/0.46 => ( exuE3e
% 0.27/0.46 => ( setext
% 0.27/0.46 => ( emptyI
% 0.27/0.46 => ( noeltsimpempty
% 0.27/0.46 => ( setbeta
% 0.27/0.46 => ( nonemptyE1
% 0.27/0.46 => ( nonemptyI
% 0.27/0.46 => ( nonemptyI1
% 0.27/0.46 => ( setadjoinIL
% 0.27/0.46 => ( emptyinunitempty
% 0.27/0.46 => ( setadjoinIR
% 0.27/0.46 => ( setadjoinE
% 0.27/0.46 => ( setadjoinOr
% 0.27/0.46 => ( setoftrueEq
% 0.27/0.46 => ( powersetI
% 0.27/0.46 => ( emptyinPowerset
% 0.27/0.46 => ( emptyInPowerset
% 0.27/0.46 => ( powersetE
% 0.27/0.46 => ( setunionI
% 0.27/0.46 => ( setunionE
% 0.27/0.46 => ( subPowSU
% 0.27/0.46 => ( exuE2
% 0.27/0.46 => ( nonemptyImpWitness
% 0.27/0.46 => ( uniqinunit
% 0.27/0.46 => ( notinsingleton
% 0.27/0.46 => ( eqinunit
% 0.27/0.46 => ( singletonsswitch
% 0.27/0.46 => ( upairsetE
% 0.27/0.46 => ( upairsetIL
% 0.27/0.46 => ( upairsetIR
% 0.27/0.46 => ( emptyE1
% 0.27/0.46 => ( vacuousDall
% 0.27/0.46 => ( quantDeMorgan1
% 0.27/0.46 => ( quantDeMorgan2
% 0.27/0.46 => ( quantDeMorgan3
% 0.27/0.46 => ( quantDeMorgan4
% 0.27/0.46 => ( prop2setI
% 0.27/0.46 => ( prop2set2propI
% 0.27/0.46 => ( notdexE
% 0.27/0.46 => ( notdallE
% 0.27/0.46 => ( exuI1
% 0.27/0.46 => ( exuI3
% 0.27/0.46 => ( exuI2
% 0.27/0.46 => ( inCongP
% 0.27/0.46 => ( in__Cong
% 0.27/0.46 => ( exuE3u
% 0.27/0.46 => ( exu__Cong
% 0.27/0.46 => ( emptyset__Cong
% 0.27/0.46 => ( setadjoin__Cong
% 0.27/0.46 => ( powerset__Cong
% 0.27/0.46 => ( setunion__Cong
% 0.27/0.46 => ( omega__Cong
% 0.27/0.46 => ( exuEu
% 0.27/0.46 => ( descr__Cong
% 0.27/0.46 => ( dsetconstr__Cong
% 0.27/0.46 => ( subsetI1
% 0.27/0.46 => ( eqimpsubset2
% 0.27/0.46 => ( eqimpsubset1
% 0.27/0.46 => ( subsetI2
% 0.27/0.46 => ( emptysetsubset
% 0.27/0.46 => ( subsetE
% 0.27/0.46 => ( subsetE2
% 0.27/0.46 => ( notsubsetI
% 0.27/0.46 => ( notequalI1
% 0.27/0.46 => ( notequalI2
% 0.27/0.46 => ( subsetRefl
% 0.27/0.46 => ( subsetTrans
% 0.27/0.46 => ( setadjoinSub
% 0.27/0.46 => ( setadjoinSub2
% 0.27/0.46 => ( subset2powerset
% 0.27/0.46 => ( setextsub
% 0.27/0.46 => ( subsetemptysetimpeq
% 0.27/0.46 => ( powersetI1
% 0.27/0.46 => ( powersetE1
% 0.27/0.46 => ( inPowerset
% 0.27/0.46 => ( powersetsubset
% 0.27/0.46 => ( sepInPowerset
% 0.27/0.46 => ( sepSubset
% 0.27/0.46 => ( binunionIL
% 0.27/0.46 => ( upairset2IR
% 0.27/0.46 => ( binunionIR
% 0.27/0.46 => ( binunionEcases
% 0.27/0.46 => ( binunionE
% 0.27/0.46 => ( binunionLsub
% 0.27/0.46 => ( binunionRsub
% 0.27/0.46 => ( binintersectI
% 0.27/0.46 => ( binintersectSubset5
% 0.27/0.46 => ( binintersectEL
% 0.27/0.46 => ( binintersectLsub
% 0.27/0.46 => ( binintersectSubset2
% 0.27/0.46 => ( binintersectSubset3
% 0.27/0.46 => ( binintersectER
% 0.27/0.46 => ( disjointsetsI1
% 0.27/0.46 => ( binintersectRsub
% 0.27/0.46 => ( binintersectSubset4
% 0.27/0.46 => ( binintersectSubset1
% 0.27/0.46 => ( bs114d
% 0.27/0.46 => ( setminusI
% 0.27/0.46 => ( setminusEL
% 0.27/0.46 => ( setminusER
% 0.27/0.46 => ( setminusSubset2
% 0.27/0.46 => ( setminusERneg
% 0.27/0.46 => ( setminusELneg
% 0.27/0.46 => ( setminusILneg
% 0.27/0.46 => ( setminusIRneg
% 0.27/0.46 => ( setminusLsub
% 0.27/0.46 => ( setminusSubset1
% 0.27/0.46 => ( symdiffE
% 0.27/0.46 => ( symdiffI1
% 0.27/0.46 => ( symdiffI2
% 0.27/0.46 => ( symdiffIneg1
% 0.27/0.46 => ( symdiffIneg2
% 0.27/0.46 => ( secondinupair
% 0.27/0.46 => ( setukpairIL
% 0.27/0.46 => ( setukpairIR
% 0.27/0.46 => ( kpairiskpair
% 0.27/0.46 => ( kpairp
% 0.27/0.46 => ( singletonsubset
% 0.27/0.46 => ( singletoninpowerset
% 0.27/0.46 => ( singletoninpowunion
% 0.27/0.46 => ( upairset2E
% 0.27/0.46 => ( upairsubunion
% 0.27/0.46 => ( upairinpowunion
% 0.27/0.46 => ( ubforcartprodlem1
% 0.27/0.46 => ( ubforcartprodlem2
% 0.27/0.46 => ( ubforcartprodlem3
% 0.27/0.46 => ( cartprodpairin
% 0.27/0.46 => ( cartprodmempair1
% 0.27/0.46 => ( cartprodmempair
% 0.27/0.46 => ( setunionE2
% 0.27/0.46 => ( setunionsingleton1
% 0.27/0.46 => ( setunionsingleton2
% 0.27/0.46 => ( setunionsingleton
% 0.27/0.46 => ( singletonprop
% 0.27/0.46 => ( ex1E1
% 0.27/0.46 => ( ex1I
% 0.27/0.46 => ( ex1I2
% 0.27/0.46 => ( singletonsuniq
% 0.27/0.46 => ( setukpairinjL1
% 0.27/0.46 => ( kfstsingleton
% 0.27/0.46 => ( theprop
% 0.27/0.46 => ( kfstpairEq
% 0.27/0.46 => ( cartprodfstin
% 0.27/0.46 => ( setukpairinjL2
% 0.27/0.46 => ( setukpairinjL
% 0.27/0.46 => ( setukpairinjR11
% 0.27/0.46 => ( setukpairinjR12
% 0.27/0.46 => ( setukpairinjR1
% 0.27/0.46 => ( upairequniteq
% 0.27/0.46 => ( setukpairinjR2
% 0.27/0.46 => ( setukpairinjR
% 0.27/0.46 => ( ksndsingleton
% 0.27/0.46 => ( ksndpairEq
% 0.27/0.46 => ( kpairsurjEq
% 0.27/0.46 => ( cartprodsndin
% 0.27/0.46 => ( cartprodpairmemEL
% 0.27/0.46 => ( cartprodpairmemER
% 0.27/0.46 => ( cartprodmempaircEq
% 0.27/0.46 => ( cartprodfstpairEq
% 0.27/0.46 => ( cartprodsndpairEq
% 0.27/0.46 => ( cartprodpairsurjEq
% 0.27/0.46 => ( dpsetconstrI
% 0.27/0.46 => ( dpsetconstrSub
% 0.27/0.46 => ( setOfPairsIsBReln
% 0.27/0.46 => ( dpsetconstrERa
% 0.27/0.46 => ( dpsetconstrEL1
% 0.27/0.46 => ( dpsetconstrEL2
% 0.27/0.46 => ( dpsetconstrER
% 0.27/0.46 => ( funcImageSingleton
% 0.27/0.46 => ( apProp
% 0.27/0.46 => ( app
% 0.27/0.46 => ( infuncsetfunc
% 0.27/0.46 => ( ap2p
% 0.27/0.46 => ( funcinfuncset
% 0.27/0.46 => ( lamProp
% 0.27/0.46 => ( lamp
% 0.27/0.46 => ( lam2p
% 0.27/0.46 => ( brelnall1
% 0.27/0.46 => ( brelnall2
% 0.27/0.46 => ( ex1E2
% 0.27/0.46 => ( funcGraphProp1
% 0.27/0.46 => ( funcGraphProp3
% 0.27/0.46 => ( funcGraphProp2
% 0.27/0.46 => ( funcextLem
% 0.27/0.46 => ( funcGraphProp4
% 0.27/0.46 => ( subbreln
% 0.27/0.46 => ( eqbreln
% 0.27/0.46 => ( funcext
% 0.27/0.46 => ( funcext2
% 0.27/0.46 => ( ap2apEq1
% 0.27/0.46 => ( ap2apEq2
% 0.27/0.46 => ( beta1
% 0.27/0.46 => ( eta1
% 0.27/0.46 => ( lam2lamEq
% 0.27/0.46 => ( beta2
% 0.27/0.46 => ( eta2
% 0.27/0.46 => ( iffalseProp1
% 0.27/0.46 => ( iffalseProp2
% 0.27/0.46 => ( iftrueProp1
% 0.27/0.46 => ( iftrueProp2
% 0.27/0.46 => ( ifSingleton
% 0.27/0.46 => ( ifp
% 0.27/0.46 => ( theeq
% 0.27/0.46 => ( iftrue
% 0.27/0.46 => ( iffalse
% 0.27/0.46 => ( iftrueorfalse
% 0.27/0.46 => ( binintersectT_lem
% 0.27/0.46 => ( binunionT_lem
% 0.27/0.46 => ( powersetT_lem
% 0.27/0.46 => ( setminusT_lem
% 0.27/0.46 => ( complementT_lem
% 0.27/0.46 => ( setextT
% 0.27/0.46 => ( subsetTI
% 0.27/0.46 => ( powersetTI1
% 0.27/0.46 => ( powersetTE1
% 0.27/0.46 => ( complementTI1
% 0.27/0.46 => ( complementTE1
% 0.27/0.46 => ( binintersectTELcontra
% 0.27/0.46 => ( binintersectTERcontra
% 0.27/0.46 => ( contrasubsetT
% 0.27/0.46 => ( contrasubsetT1
% 0.27/0.46 => ( contrasubsetT2
% 0.27/0.46 => ( contrasubsetT3
% 0.27/0.46 => ( doubleComplementI1
% 0.27/0.46 => ( doubleComplementE1
% 0.27/0.46 => ( doubleComplementSub1
% 0.27/0.46 => ( doubleComplementSub2
% 0.27/0.46 => ( doubleComplementEq
% 0.27/0.46 => ( complementTnotintersectT
% 0.27/0.46 => ( complementImpComplementIntersect
% 0.27/0.46 => ( complementSubsetComplementIntersect
% 0.27/0.46 => ( complementInPowersetComplementIntersect
% 0.27/0.46 => ( contraSubsetComplement
% 0.27/0.46 => ( complementTcontraSubset
% 0.27/0.46 => ( binunionTILcontra
% 0.27/0.46 => ( binunionTIRcontra
% 0.27/0.46 => ( inIntersectImpInUnion
% 0.27/0.46 => ( inIntersectImpInUnion2
% 0.27/0.46 => ( inIntersectImpInIntersectUnions
% 0.27/0.46 => ( intersectInPowersetIntersectUnions
% 0.27/0.46 => ( inComplementUnionImpNotIn1
% 0.27/0.46 => ( inComplementUnionImpInComplement1
% 0.27/0.46 => ( binunionTE
% 0.27/0.46 => ( binunionTEcontra
% 0.27/0.46 => ( demorgan2a1
% 0.27/0.46 => ( complementUnionInPowersetComplement
% 0.27/0.46 => ( demorgan2a2
% 0.27/0.46 => ( demorgan1a
% 0.27/0.46 => ( demorgan1b
% 0.27/0.46 => ( demorgan1
% 0.27/0.46 => ( demorgan2a
% 0.27/0.46 => ( demorgan2b2
% 0.27/0.46 => ( demorgan2b
% 0.27/0.46 => ( demorgan2
% 0.27/0.46 => ( woz13rule0
% 0.27/0.46 => ( woz13rule1
% 0.27/0.46 => ( woz13rule2
% 0.27/0.46 => ( woz13rule3
% 0.27/0.46 => ( woz13rule4
% 0.27/0.46 => ( woz1_1
% 0.27/0.46 => ( woz1_2
% 0.27/0.46 => ( woz1_3
% 0.27/0.46 => ( woz1_4
% 0.27/0.46 => ( woz1_5
% 0.27/0.46 => ( breln1all2
% 0.27/0.46 => ( breln1SetBreln1
% 0.27/0.46 => ( choice2fnsingleton
% 0.27/0.46 => ! [A: $i,Xphi: $i > $i > $o] :
% 0.27/0.46 ( breln1 @ A
% 0.27/0.46 @ ( dpsetconstr @ A @ A
% 0.27/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.27/0.46
% 0.27/0.46 %------------------------------------------------------------------------------
% 0.27/0.46 ------------------------
% 2.68/1.47 ---- Embedded in THF ---
% 2.68/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.68/1.47 %%% Generated on Tue Feb 24 12:33:11 EST 2026
% 2.68/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.68/1.47 %%% Logic specification used:
% 2.68/1.47 %%% thf(spec, logic, $$dhol).
% 2.68/1.47
% 2.68/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.DOTu7T1G0A/DTF2DTF390.p
% See solution above
% 2.68/1.48 ------------------------
% 2.93/1.51 ---- Cleaned THF ---
% 2.93/1.51 thf(in_type,type,
% 2.93/1.51 in: $i > $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(exu_type,type,
% 2.93/1.51 exu: ( $i > $o ) > $o ).
% 2.93/1.51
% 2.93/1.51 thf(exu,definition,
% 2.93/1.51 ( exu
% 2.93/1.51 = ( ^ [Xphi: $i > $o] :
% 2.93/1.51 ? [Xx: $i] :
% 2.93/1.51 ( ( Xphi @ Xx )
% 2.93/1.51 & ! [Xy: $i] :
% 2.93/1.51 ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setextAx_type,type,
% 2.93/1.51 setextAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setextAx,definition,
% 2.93/1.51 ( setextAx
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 <=> ( in @ Xx @ B ) )
% 2.93/1.51 => ( A = B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptyset_type,type,
% 2.93/1.51 emptyset: $i ).
% 2.93/1.51
% 2.93/1.51 thf(emptysetAx_type,type,
% 2.93/1.51 emptysetAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptysetAx,definition,
% 2.93/1.51 ( emptysetAx
% 2.93/1.51 = ( ! [Xx: $i] :
% 2.93/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoin_type,type,
% 2.93/1.51 setadjoin: $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinAx_type,type,
% 2.93/1.51 setadjoinAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinAx,definition,
% 2.93/1.51 ( setadjoinAx
% 2.93/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.93/1.51 <=> ( ( Xy = Xx )
% 2.93/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(powerset_type,type,
% 2.93/1.51 powerset: $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(powersetAx_type,type,
% 2.93/1.51 powersetAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(powersetAx,definition,
% 2.93/1.51 ( powersetAx
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.93/1.51 <=> ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setunion_type,type,
% 2.93/1.51 setunion: $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(setunionAx_type,type,
% 2.93/1.51 setunionAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setunionAx,definition,
% 2.93/1.51 ( setunionAx
% 2.93/1.51 = ( ! [A: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.93/1.51 <=> ? [B: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 & ( in @ B @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(omega_type,type,
% 2.93/1.51 omega: $i ).
% 2.93/1.51
% 2.93/1.51 thf(omega0Ax_type,type,
% 2.93/1.51 omega0Ax: $o ).
% 2.93/1.51
% 2.93/1.51 thf(omega0Ax,definition,
% 2.93/1.51 ( omega0Ax
% 2.93/1.51 = ( in @ emptyset @ omega ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(omegaSAx_type,type,
% 2.93/1.51 omegaSAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(omegaSAx,definition,
% 2.93/1.51 ( omegaSAx
% 2.93/1.51 = ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ omega )
% 2.93/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(omegaIndAx_type,type,
% 2.93/1.51 omegaIndAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(omegaIndAx,definition,
% 2.93/1.51 ( omegaIndAx
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ( ( ( in @ emptyset @ A )
% 2.93/1.51 & ! [Xx: $i] :
% 2.93/1.51 ( ( ( in @ Xx @ omega )
% 2.93/1.51 & ( in @ Xx @ A ) )
% 2.93/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ omega )
% 2.93/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(replAx_type,type,
% 2.93/1.51 replAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(replAx,definition,
% 2.93/1.51 ( replAx
% 2.93/1.51 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( exu
% 2.93/1.51 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.93/1.51 => ? [B: $i] :
% 2.93/1.51 ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 <=> ? [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(foundationAx_type,type,
% 2.93/1.51 foundationAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(foundationAx,definition,
% 2.93/1.51 ( foundationAx
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.93/1.51 => ? [B: $i] :
% 2.93/1.51 ( ( in @ B @ A )
% 2.93/1.51 & ~ ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(wellorderingAx_type,type,
% 2.93/1.51 wellorderingAx: $o ).
% 2.93/1.51
% 2.93/1.51 thf(wellorderingAx,definition,
% 2.93/1.51 ( wellorderingAx
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ? [B: $i] :
% 2.93/1.51 ( ! [C: $i] :
% 2.93/1.51 ( ( in @ C @ B )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ C )
% 2.93/1.51 => ( in @ Xx @ A ) ) )
% 2.93/1.51 & ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( ( in @ Xx @ A )
% 2.93/1.51 & ( in @ Xy @ A ) )
% 2.93/1.51 => ( ! [C: $i] :
% 2.93/1.51 ( ( in @ C @ B )
% 2.93/1.51 => ( ( in @ Xx @ C )
% 2.93/1.51 <=> ( in @ Xy @ C ) ) )
% 2.93/1.51 => ( Xx = Xy ) ) )
% 2.93/1.51 & ! [C: $i,D: $i] :
% 2.93/1.51 ( ( ( in @ C @ B )
% 2.93/1.51 & ( in @ D @ B ) )
% 2.93/1.51 => ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ C )
% 2.93/1.51 => ( in @ Xx @ D ) )
% 2.93/1.51 | ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ D )
% 2.93/1.51 => ( in @ Xx @ C ) ) ) )
% 2.93/1.51 & ! [C: $i] :
% 2.93/1.51 ( ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ C )
% 2.93/1.51 => ( in @ Xx @ A ) )
% 2.93/1.51 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.93/1.51 => ? [D: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ D @ B )
% 2.93/1.51 & ( in @ Xx @ C )
% 2.93/1.51 & ~ ? [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ D )
% 2.93/1.51 & ( in @ Xy @ C ) )
% 2.93/1.51 & ! [E: $i] :
% 2.93/1.51 ( ( in @ E @ B )
% 2.93/1.51 => ( ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ E )
% 2.93/1.51 => ( in @ Xy @ D ) )
% 2.93/1.51 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(descr_type,type,
% 2.93/1.51 descr: ( $i > $o ) > $i ).
% 2.93/1.51
% 2.93/1.51 thf(descrp_type,type,
% 2.93/1.51 descrp: $o ).
% 2.93/1.51
% 2.93/1.51 thf(descrp,definition,
% 2.93/1.51 ( descrp
% 2.93/1.51 = ( ! [Xphi: $i > $o] :
% 2.93/1.51 ( ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 => ( Xphi
% 2.93/1.51 @ ( descr
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dsetconstr_type,type,
% 2.93/1.51 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.93/1.51
% 2.93/1.51 thf(dsetconstrI_type,type,
% 2.93/1.51 dsetconstrI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dsetconstrI,definition,
% 2.93/1.51 ( dsetconstrI
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( Xphi @ Xx )
% 2.93/1.51 => ( in @ Xx
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dsetconstrEL_type,type,
% 2.93/1.51 dsetconstrEL: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dsetconstrEL,definition,
% 2.93/1.51 ( dsetconstrEL
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.93/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dsetconstrER_type,type,
% 2.93/1.51 dsetconstrER: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dsetconstrER,definition,
% 2.93/1.51 ( dsetconstrER
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.93/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(exuE1_type,type,
% 2.93/1.51 exuE1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(exuE1,definition,
% 2.93/1.51 ( exuE1
% 2.93/1.51 = ( ! [Xphi: $i > $o] :
% 2.93/1.51 ( ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 => ? [Xx: $i] :
% 2.93/1.51 ( ( Xphi @ Xx )
% 2.93/1.51 & ! [Xy: $i] :
% 2.93/1.51 ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(prop2set_type,type,
% 2.93/1.51 prop2set: $o > $i ).
% 2.93/1.51
% 2.93/1.51 thf(prop2setE_type,type,
% 2.93/1.51 prop2setE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(prop2setE,definition,
% 2.93/1.51 ( prop2setE
% 2.93/1.51 = ( ! [Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.93/1.51 => Xphi ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptysetE_type,type,
% 2.93/1.51 emptysetE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptysetE,definition,
% 2.93/1.51 ( emptysetE
% 2.93/1.51 = ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ emptyset )
% 2.93/1.51 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptysetimpfalse_type,type,
% 2.93/1.51 emptysetimpfalse: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptysetimpfalse,definition,
% 2.93/1.51 ( emptysetimpfalse
% 2.93/1.51 = ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ emptyset )
% 2.93/1.51 => $false ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(notinemptyset_type,type,
% 2.93/1.51 notinemptyset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(notinemptyset,definition,
% 2.93/1.51 ( notinemptyset
% 2.93/1.51 = ( ! [Xx: $i] :
% 2.93/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(exuE3e_type,type,
% 2.93/1.51 exuE3e: $o ).
% 2.93/1.51
% 2.93/1.51 thf(exuE3e,definition,
% 2.93/1.51 ( exuE3e
% 2.93/1.51 = ( ! [Xphi: $i > $o] :
% 2.93/1.51 ( ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setext_type,type,
% 2.93/1.51 setext: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setext,definition,
% 2.93/1.51 ( setext
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ Xx @ B ) )
% 2.93/1.51 => ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 => ( in @ Xx @ A ) )
% 2.93/1.51 => ( A = B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptyI_type,type,
% 2.93/1.51 emptyI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptyI,definition,
% 2.93/1.51 ( emptyI
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ~ ( in @ Xx @ A )
% 2.93/1.51 => ( A = emptyset ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(noeltsimpempty_type,type,
% 2.93/1.51 noeltsimpempty: $o ).
% 2.93/1.51
% 2.93/1.51 thf(noeltsimpempty,definition,
% 2.93/1.51 ( noeltsimpempty
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ~ ( in @ Xx @ A )
% 2.93/1.51 => ( A = emptyset ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setbeta_type,type,
% 2.93/1.51 setbeta: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setbeta,definition,
% 2.93/1.51 ( setbeta
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( in @ Xx
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.93/1.51 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(nonempty_type,type,
% 2.93/1.51 nonempty: $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(nonempty,definition,
% 2.93/1.51 ( nonempty
% 2.93/1.51 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(nonemptyE1_type,type,
% 2.93/1.51 nonemptyE1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(nonemptyE1,definition,
% 2.93/1.51 ( nonemptyE1
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ( ( nonempty @ A )
% 2.93/1.51 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(nonemptyI_type,type,
% 2.93/1.51 nonemptyI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(nonemptyI,definition,
% 2.93/1.51 ( nonemptyI
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( Xphi @ Xx )
% 2.93/1.51 => ( nonempty
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(nonemptyI1_type,type,
% 2.93/1.51 nonemptyI1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(nonemptyI1,definition,
% 2.93/1.51 ( nonemptyI1
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.93/1.51 => ( nonempty @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinIL_type,type,
% 2.93/1.51 setadjoinIL: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinIL,definition,
% 2.93/1.51 ( setadjoinIL
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptyinunitempty_type,type,
% 2.93/1.51 emptyinunitempty: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptyinunitempty,definition,
% 2.93/1.51 ( emptyinunitempty
% 2.93/1.51 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinIR_type,type,
% 2.93/1.51 setadjoinIR: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinIR,definition,
% 2.93/1.51 ( setadjoinIR
% 2.93/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinE_type,type,
% 2.93/1.51 setadjoinE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinE,definition,
% 2.93/1.51 ( setadjoinE
% 2.93/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.93/1.51 => ! [Xphi: $o] :
% 2.93/1.51 ( ( ( Xy = Xx )
% 2.93/1.51 => Xphi )
% 2.93/1.51 => ( ( ( in @ Xy @ A )
% 2.93/1.51 => Xphi )
% 2.93/1.51 => Xphi ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinOr_type,type,
% 2.93/1.51 setadjoinOr: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinOr,definition,
% 2.93/1.51 ( setadjoinOr
% 2.93/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.93/1.51 => ( ( Xy = Xx )
% 2.93/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setoftrueEq_type,type,
% 2.93/1.51 setoftrueEq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setoftrueEq,definition,
% 2.93/1.51 ( setoftrueEq
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ( ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xx: $i] : $true )
% 2.93/1.51 = A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(powersetI_type,type,
% 2.93/1.51 powersetI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(powersetI,definition,
% 2.93/1.51 ( powersetI
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 => ( in @ Xx @ A ) )
% 2.93/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptyinPowerset_type,type,
% 2.93/1.51 emptyinPowerset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptyinPowerset,definition,
% 2.93/1.51 ( emptyinPowerset
% 2.93/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptyInPowerset_type,type,
% 2.93/1.51 emptyInPowerset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptyInPowerset,definition,
% 2.93/1.51 ( emptyInPowerset
% 2.93/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(powersetE_type,type,
% 2.93/1.51 powersetE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(powersetE,definition,
% 2.93/1.51 ( powersetE
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.93/1.51 => ( ( in @ Xx @ B )
% 2.93/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setunionI_type,type,
% 2.93/1.51 setunionI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setunionI,definition,
% 2.93/1.51 ( setunionI
% 2.93/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 => ( ( in @ B @ A )
% 2.93/1.51 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setunionE_type,type,
% 2.93/1.51 setunionE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setunionE,definition,
% 2.93/1.51 ( setunionE
% 2.93/1.51 = ( ! [A: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.93/1.51 => ! [Xphi: $o] :
% 2.93/1.51 ( ! [B: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 => ( ( in @ B @ A )
% 2.93/1.51 => Xphi ) )
% 2.93/1.51 => Xphi ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subPowSU_type,type,
% 2.93/1.51 subPowSU: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subPowSU,definition,
% 2.93/1.51 ( subPowSU
% 2.93/1.51 = ( ! [A: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(exuE2_type,type,
% 2.93/1.51 exuE2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(exuE2,definition,
% 2.93/1.51 ( exuE2
% 2.93/1.51 = ( ! [Xphi: $i > $o] :
% 2.93/1.51 ( ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 => ? [Xx: $i] :
% 2.93/1.51 ! [Xy: $i] :
% 2.93/1.51 ( ( Xphi @ Xy )
% 2.93/1.51 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(nonemptyImpWitness_type,type,
% 2.93/1.51 nonemptyImpWitness: $o ).
% 2.93/1.51
% 2.93/1.51 thf(nonemptyImpWitness,definition,
% 2.93/1.51 ( nonemptyImpWitness
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ( ( nonempty @ A )
% 2.93/1.51 => ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & $true ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(uniqinunit_type,type,
% 2.93/1.51 uniqinunit: $o ).
% 2.93/1.51
% 2.93/1.51 thf(uniqinunit,definition,
% 2.93/1.51 ( uniqinunit
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(notinsingleton_type,type,
% 2.93/1.51 notinsingleton: $o ).
% 2.93/1.51
% 2.93/1.51 thf(notinsingleton,definition,
% 2.93/1.51 ( notinsingleton
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xx != Xy )
% 2.93/1.51 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(eqinunit_type,type,
% 2.93/1.51 eqinunit: $o ).
% 2.93/1.51
% 2.93/1.51 thf(eqinunit,definition,
% 2.93/1.51 ( eqinunit
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xx = Xy )
% 2.93/1.51 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(singletonsswitch_type,type,
% 2.93/1.51 singletonsswitch: $o ).
% 2.93/1.51
% 2.93/1.51 thf(singletonsswitch,definition,
% 2.93/1.51 ( singletonsswitch
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(upairsetE_type,type,
% 2.93/1.51 upairsetE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(upairsetE,definition,
% 2.93/1.51 ( upairsetE
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.93/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.93/1.51 => ( ( Xz = Xx )
% 2.93/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(upairsetIL_type,type,
% 2.93/1.51 upairsetIL: $o ).
% 2.93/1.51
% 2.93/1.51 thf(upairsetIL,definition,
% 2.93/1.51 ( upairsetIL
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(upairsetIR_type,type,
% 2.93/1.51 upairsetIR: $o ).
% 2.93/1.51
% 2.93/1.51 thf(upairsetIR,definition,
% 2.93/1.51 ( upairsetIR
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptyE1_type,type,
% 2.93/1.51 emptyE1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptyE1,definition,
% 2.93/1.51 ( emptyE1
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ( Xphi @ Xx ) )
% 2.93/1.51 => ( ( ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 = emptyset )
% 2.93/1.51 => $false ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(vacuousDall_type,type,
% 2.93/1.51 vacuousDall: $o ).
% 2.93/1.51
% 2.93/1.51 thf(vacuousDall,definition,
% 2.93/1.51 ( vacuousDall
% 2.93/1.51 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ emptyset )
% 2.93/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(quantDeMorgan1_type,type,
% 2.93/1.51 quantDeMorgan1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(quantDeMorgan1,definition,
% 2.93/1.51 ( quantDeMorgan1
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ~ ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( Xphi @ Xx ) )
% 2.93/1.51 => ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(quantDeMorgan2_type,type,
% 2.93/1.51 quantDeMorgan2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(quantDeMorgan2,definition,
% 2.93/1.51 ( quantDeMorgan2
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ~ ( Xphi @ Xx ) )
% 2.93/1.51 => ~ ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(quantDeMorgan3_type,type,
% 2.93/1.51 quantDeMorgan3: $o ).
% 2.93/1.51
% 2.93/1.51 thf(quantDeMorgan3,definition,
% 2.93/1.51 ( quantDeMorgan3
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ~ ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ( Xphi @ Xx ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(quantDeMorgan4_type,type,
% 2.93/1.51 quantDeMorgan4: $o ).
% 2.93/1.51
% 2.93/1.51 thf(quantDeMorgan4,definition,
% 2.93/1.51 ( quantDeMorgan4
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ~ ( Xphi @ Xx ) )
% 2.93/1.51 => ~ ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(prop2setI_type,type,
% 2.93/1.51 prop2setI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(prop2setI,definition,
% 2.93/1.51 ( prop2setI
% 2.93/1.51 = ( ! [Xphi: $o] :
% 2.93/1.51 ( Xphi
% 2.93/1.51 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(set2prop_type,type,
% 2.93/1.51 set2prop: $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(prop2set2propI_type,type,
% 2.93/1.51 prop2set2propI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(prop2set2propI,definition,
% 2.93/1.51 ( prop2set2propI
% 2.93/1.51 = ( ! [Xphi: $o] :
% 2.93/1.51 ( Xphi
% 2.93/1.51 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(notdexE_type,type,
% 2.93/1.51 notdexE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(notdexE,definition,
% 2.93/1.51 ( notdexE
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ~ ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ( Xphi @ Xx ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(notdallE_type,type,
% 2.93/1.51 notdallE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(notdallE,definition,
% 2.93/1.51 ( notdallE
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ~ ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( Xphi @ Xx ) )
% 2.93/1.51 => ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(exuI1_type,type,
% 2.93/1.51 exuI1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(exuI1,definition,
% 2.93/1.51 ( exuI1
% 2.93/1.51 = ( ! [Xphi: $i > $o] :
% 2.93/1.51 ( ? [Xx: $i] :
% 2.93/1.51 ( ( Xphi @ Xx )
% 2.93/1.51 & ! [Xy: $i] :
% 2.93/1.51 ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xx = Xy ) ) )
% 2.93/1.51 => ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(exuI3_type,type,
% 2.93/1.51 exuI3: $o ).
% 2.93/1.51
% 2.93/1.51 thf(exuI3,definition,
% 2.93/1.51 ( exuI3
% 2.93/1.51 = ( ! [Xphi: $i > $o] :
% 2.93/1.51 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.93/1.51 => ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xphi @ Xx )
% 2.93/1.51 => ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xx = Xy ) ) )
% 2.93/1.51 => ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(exuI2_type,type,
% 2.93/1.51 exuI2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(exuI2,definition,
% 2.93/1.51 ( exuI2
% 2.93/1.51 = ( ! [Xphi: $i > $o] :
% 2.93/1.51 ( ? [Xx: $i] :
% 2.93/1.51 ! [Xy: $i] :
% 2.93/1.51 ( ( Xphi @ Xy )
% 2.93/1.51 <=> ( Xy = Xx ) )
% 2.93/1.51 => ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(inCongP_type,type,
% 2.93/1.51 inCongP: $o ).
% 2.93/1.51
% 2.93/1.51 thf(inCongP,definition,
% 2.93/1.51 ( inCongP
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( A = B )
% 2.93/1.51 => ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xx = Xy )
% 2.93/1.51 => ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(in__Cong_type,type,
% 2.93/1.51 in__Cong: $o ).
% 2.93/1.51
% 2.93/1.51 thf(in__Cong,definition,
% 2.93/1.51 ( in__Cong
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( A = B )
% 2.93/1.51 => ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xx = Xy )
% 2.93/1.51 => ( ( in @ Xx @ A )
% 2.93/1.51 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(exuE3u_type,type,
% 2.93/1.51 exuE3u: $o ).
% 2.93/1.51
% 2.93/1.51 thf(exuE3u,definition,
% 2.93/1.51 ( exuE3u
% 2.93/1.51 = ( ! [Xphi: $i > $o] :
% 2.93/1.51 ( ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 => ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xphi @ Xx )
% 2.93/1.51 => ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(exu__Cong_type,type,
% 2.93/1.51 exu__Cong: $o ).
% 2.93/1.51
% 2.93/1.51 thf(exu__Cong,definition,
% 2.93/1.51 ( exu__Cong
% 2.93/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.93/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xx = Xy )
% 2.93/1.51 => ( ( Xphi @ Xx )
% 2.93/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.93/1.51 => ( ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 <=> ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptyset__Cong_type,type,
% 2.93/1.51 emptyset__Cong: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptyset__Cong,definition,
% 2.93/1.51 ( emptyset__Cong
% 2.93/1.51 = ( emptyset = emptyset ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoin__Cong_type,type,
% 2.93/1.51 setadjoin__Cong: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoin__Cong,definition,
% 2.93/1.51 ( setadjoin__Cong
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xx = Xy )
% 2.93/1.51 => ! [Xz: $i,Xu: $i] :
% 2.93/1.51 ( ( Xz = Xu )
% 2.93/1.51 => ( ( setadjoin @ Xx @ Xz )
% 2.93/1.51 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(powerset__Cong_type,type,
% 2.93/1.51 powerset__Cong: $o ).
% 2.93/1.51
% 2.93/1.51 thf(powerset__Cong,definition,
% 2.93/1.51 ( powerset__Cong
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( A = B )
% 2.93/1.51 => ( ( powerset @ A )
% 2.93/1.51 = ( powerset @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setunion__Cong_type,type,
% 2.93/1.51 setunion__Cong: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setunion__Cong,definition,
% 2.93/1.51 ( setunion__Cong
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( A = B )
% 2.93/1.51 => ( ( setunion @ A )
% 2.93/1.51 = ( setunion @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(omega__Cong_type,type,
% 2.93/1.51 omega__Cong: $o ).
% 2.93/1.51
% 2.93/1.51 thf(omega__Cong,definition,
% 2.93/1.51 ( omega__Cong
% 2.93/1.51 = ( omega = omega ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(exuEu_type,type,
% 2.93/1.51 exuEu: $o ).
% 2.93/1.51
% 2.93/1.51 thf(exuEu,definition,
% 2.93/1.51 ( exuEu
% 2.93/1.51 = ( ! [Xphi: $i > $o] :
% 2.93/1.51 ( ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 => ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xphi @ Xx )
% 2.93/1.51 => ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(descr__Cong_type,type,
% 2.93/1.51 descr__Cong: $o ).
% 2.93/1.51
% 2.93/1.51 thf(descr__Cong,definition,
% 2.93/1.51 ( descr__Cong
% 2.93/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.93/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xx = Xy )
% 2.93/1.51 => ( ( Xphi @ Xx )
% 2.93/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.93/1.51 => ( ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 => ( ( exu
% 2.93/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.93/1.51 => ( ( descr
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 = ( descr
% 2.93/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dsetconstr__Cong_type,type,
% 2.93/1.51 dsetconstr__Cong: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dsetconstr__Cong,definition,
% 2.93/1.51 ( dsetconstr__Cong
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( A = B )
% 2.93/1.51 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( Xx = Xy )
% 2.93/1.51 => ( ( Xphi @ Xx )
% 2.93/1.51 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.93/1.51 => ( ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 = ( dsetconstr @ B
% 2.93/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subset_type,type,
% 2.93/1.51 subset: $i > $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(disjoint_type,type,
% 2.93/1.51 disjoint: $i > $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(setsmeet_type,type,
% 2.93/1.51 setsmeet: $i > $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(subsetI1_type,type,
% 2.93/1.51 subsetI1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subsetI1,definition,
% 2.93/1.51 ( subsetI1
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ Xx @ B ) )
% 2.93/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(eqimpsubset2_type,type,
% 2.93/1.51 eqimpsubset2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(eqimpsubset2,definition,
% 2.93/1.51 ( eqimpsubset2
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( A = B )
% 2.93/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(eqimpsubset1_type,type,
% 2.93/1.51 eqimpsubset1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(eqimpsubset1,definition,
% 2.93/1.51 ( eqimpsubset1
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( A = B )
% 2.93/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subsetI2_type,type,
% 2.93/1.51 subsetI2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subsetI2,definition,
% 2.93/1.51 ( subsetI2
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ Xx @ B ) )
% 2.93/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(emptysetsubset_type,type,
% 2.93/1.51 emptysetsubset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(emptysetsubset,definition,
% 2.93/1.51 ( emptysetsubset
% 2.93/1.51 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subsetE_type,type,
% 2.93/1.51 subsetE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subsetE,definition,
% 2.93/1.51 ( subsetE
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( subset @ A @ B )
% 2.93/1.51 => ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subsetE2_type,type,
% 2.93/1.51 subsetE2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subsetE2,definition,
% 2.93/1.51 ( subsetE2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( subset @ A @ B )
% 2.93/1.51 => ( ~ ( in @ Xx @ B )
% 2.93/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(notsubsetI_type,type,
% 2.93/1.51 notsubsetI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(notsubsetI,definition,
% 2.93/1.51 ( notsubsetI
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ~ ( in @ Xx @ B )
% 2.93/1.51 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(notequalI1_type,type,
% 2.93/1.51 notequalI1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(notequalI1,definition,
% 2.93/1.51 ( notequalI1
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ~ ( subset @ A @ B )
% 2.93/1.51 => ( A != B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(notequalI2_type,type,
% 2.93/1.51 notequalI2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(notequalI2,definition,
% 2.93/1.51 ( notequalI2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ~ ( in @ Xx @ B )
% 2.93/1.51 => ( A != B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subsetRefl_type,type,
% 2.93/1.51 subsetRefl: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subsetRefl,definition,
% 2.93/1.51 ( subsetRefl
% 2.93/1.51 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subsetTrans_type,type,
% 2.93/1.51 subsetTrans: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subsetTrans,definition,
% 2.93/1.51 ( subsetTrans
% 2.93/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.93/1.51 ( ( subset @ A @ B )
% 2.93/1.51 => ( ( subset @ B @ C )
% 2.93/1.51 => ( subset @ A @ C ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinSub_type,type,
% 2.93/1.51 setadjoinSub: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinSub,definition,
% 2.93/1.51 ( setadjoinSub
% 2.93/1.51 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinSub2_type,type,
% 2.93/1.51 setadjoinSub2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setadjoinSub2,definition,
% 2.93/1.51 ( setadjoinSub2
% 2.93/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.93/1.51 ( ( subset @ A @ B )
% 2.93/1.51 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subset2powerset_type,type,
% 2.93/1.51 subset2powerset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subset2powerset,definition,
% 2.93/1.51 ( subset2powerset
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( subset @ A @ B )
% 2.93/1.51 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setextsub_type,type,
% 2.93/1.51 setextsub: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setextsub,definition,
% 2.93/1.51 ( setextsub
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( subset @ A @ B )
% 2.93/1.51 => ( ( subset @ B @ A )
% 2.93/1.51 => ( A = B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subsetemptysetimpeq_type,type,
% 2.93/1.51 subsetemptysetimpeq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subsetemptysetimpeq,definition,
% 2.93/1.51 ( subsetemptysetimpeq
% 2.93/1.51 = ( ! [A: $i] :
% 2.93/1.51 ( ( subset @ A @ emptyset )
% 2.93/1.51 => ( A = emptyset ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(powersetI1_type,type,
% 2.93/1.51 powersetI1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(powersetI1,definition,
% 2.93/1.51 ( powersetI1
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( subset @ B @ A )
% 2.93/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(powersetE1_type,type,
% 2.93/1.51 powersetE1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(powersetE1,definition,
% 2.93/1.51 ( powersetE1
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.93/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(inPowerset_type,type,
% 2.93/1.51 inPowerset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(inPowerset,definition,
% 2.93/1.51 ( inPowerset
% 2.93/1.51 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(powersetsubset_type,type,
% 2.93/1.51 powersetsubset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(powersetsubset,definition,
% 2.93/1.51 ( powersetsubset
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( subset @ A @ B )
% 2.93/1.51 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sepInPowerset_type,type,
% 2.93/1.51 sepInPowerset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(sepInPowerset,definition,
% 2.93/1.51 ( sepInPowerset
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( in
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 @ ( powerset @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(sepSubset_type,type,
% 2.93/1.51 sepSubset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(sepSubset,definition,
% 2.93/1.51 ( sepSubset
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( subset
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binunion_type,type,
% 2.93/1.51 binunion: $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(binunionIL_type,type,
% 2.93/1.51 binunionIL: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binunionIL,definition,
% 2.93/1.51 ( binunionIL
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(upairset2IR_type,type,
% 2.93/1.51 upairset2IR: $o ).
% 2.93/1.51
% 2.93/1.51 thf(upairset2IR,definition,
% 2.93/1.51 ( upairset2IR
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binunionIR_type,type,
% 2.93/1.51 binunionIR: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binunionIR,definition,
% 2.93/1.51 ( binunionIR
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binunionEcases_type,type,
% 2.93/1.51 binunionEcases: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binunionEcases,definition,
% 2.93/1.51 ( binunionEcases
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.93/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.93/1.51 => ( ( ( in @ Xx @ A )
% 2.93/1.51 => Xphi )
% 2.93/1.51 => ( ( ( in @ Xx @ B )
% 2.93/1.51 => Xphi )
% 2.93/1.51 => Xphi ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binunionE_type,type,
% 2.93/1.51 binunionE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binunionE,definition,
% 2.93/1.51 ( binunionE
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.93/1.51 => ( ( in @ Xx @ A )
% 2.93/1.51 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binunionLsub_type,type,
% 2.93/1.51 binunionLsub: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binunionLsub,definition,
% 2.93/1.51 ( binunionLsub
% 2.93/1.51 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binunionRsub_type,type,
% 2.93/1.51 binunionRsub: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binunionRsub,definition,
% 2.93/1.51 ( binunionRsub
% 2.93/1.51 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersect_type,type,
% 2.93/1.51 binintersect: $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectI_type,type,
% 2.93/1.51 binintersectI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectI,definition,
% 2.93/1.51 ( binintersectI
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( in @ Xx @ B )
% 2.93/1.51 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset5_type,type,
% 2.93/1.51 binintersectSubset5: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset5,definition,
% 2.93/1.51 ( binintersectSubset5
% 2.93/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.93/1.51 ( ( subset @ C @ A )
% 2.93/1.51 => ( ( subset @ C @ B )
% 2.93/1.51 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectEL_type,type,
% 2.93/1.51 binintersectEL: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectEL,definition,
% 2.93/1.51 ( binintersectEL
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.93/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectLsub_type,type,
% 2.93/1.51 binintersectLsub: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectLsub,definition,
% 2.93/1.51 ( binintersectLsub
% 2.93/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset2_type,type,
% 2.93/1.51 binintersectSubset2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset2,definition,
% 2.93/1.51 ( binintersectSubset2
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( subset @ A @ B )
% 2.93/1.51 => ( ( binintersect @ A @ B )
% 2.93/1.51 = A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset3_type,type,
% 2.93/1.51 binintersectSubset3: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset3,definition,
% 2.93/1.51 ( binintersectSubset3
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( ( binintersect @ A @ B )
% 2.93/1.51 = B )
% 2.93/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectER_type,type,
% 2.93/1.51 binintersectER: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectER,definition,
% 2.93/1.51 ( binintersectER
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.93/1.51 => ( in @ Xx @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(disjointsetsI1_type,type,
% 2.93/1.51 disjointsetsI1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(disjointsetsI1,definition,
% 2.93/1.51 ( disjointsetsI1
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ~ ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ( in @ Xx @ B ) )
% 2.93/1.51 => ( ( binintersect @ A @ B )
% 2.93/1.51 = emptyset ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectRsub_type,type,
% 2.93/1.51 binintersectRsub: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectRsub,definition,
% 2.93/1.51 ( binintersectRsub
% 2.93/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset4_type,type,
% 2.93/1.51 binintersectSubset4: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset4,definition,
% 2.93/1.51 ( binintersectSubset4
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( subset @ B @ A )
% 2.93/1.51 => ( ( binintersect @ A @ B )
% 2.93/1.51 = B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset1_type,type,
% 2.93/1.51 binintersectSubset1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(binintersectSubset1,definition,
% 2.93/1.51 ( binintersectSubset1
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( ( binintersect @ A @ B )
% 2.93/1.51 = A )
% 2.93/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(bs114d_type,type,
% 2.93/1.51 bs114d: $o ).
% 2.93/1.51
% 2.93/1.51 thf(bs114d,definition,
% 2.93/1.51 ( bs114d
% 2.93/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.93/1.51 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.93/1.51 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(regular_type,type,
% 2.93/1.51 regular: $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminus_type,type,
% 2.93/1.51 setminus: $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(setminusI_type,type,
% 2.93/1.51 setminusI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusI,definition,
% 2.93/1.51 ( setminusI
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ~ ( in @ Xx @ B )
% 2.93/1.51 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setminusEL_type,type,
% 2.93/1.51 setminusEL: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusEL,definition,
% 2.93/1.51 ( setminusEL
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.93/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setminusER_type,type,
% 2.93/1.51 setminusER: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusER,definition,
% 2.93/1.51 ( setminusER
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.93/1.51 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setminusSubset2_type,type,
% 2.93/1.51 setminusSubset2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusSubset2,definition,
% 2.93/1.51 ( setminusSubset2
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( subset @ A @ B )
% 2.93/1.51 => ( ( setminus @ A @ B )
% 2.93/1.51 = emptyset ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setminusERneg_type,type,
% 2.93/1.51 setminusERneg: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusERneg,definition,
% 2.93/1.51 ( setminusERneg
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.93/1.51 => ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setminusELneg_type,type,
% 2.93/1.51 setminusELneg: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusELneg,definition,
% 2.93/1.51 ( setminusELneg
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.93/1.51 => ( ~ ( in @ Xx @ B )
% 2.93/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setminusILneg_type,type,
% 2.93/1.51 setminusILneg: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusILneg,definition,
% 2.93/1.51 ( setminusILneg
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ~ ( in @ Xx @ A )
% 2.93/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setminusIRneg_type,type,
% 2.93/1.51 setminusIRneg: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusIRneg,definition,
% 2.93/1.51 ( setminusIRneg
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ B )
% 2.93/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setminusLsub_type,type,
% 2.93/1.51 setminusLsub: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusLsub,definition,
% 2.93/1.51 ( setminusLsub
% 2.93/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setminusSubset1_type,type,
% 2.93/1.51 setminusSubset1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setminusSubset1,definition,
% 2.93/1.51 ( setminusSubset1
% 2.93/1.51 = ( ! [A: $i,B: $i] :
% 2.93/1.51 ( ( ( setminus @ A @ B )
% 2.93/1.51 = emptyset )
% 2.93/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(symdiff_type,type,
% 2.93/1.51 symdiff: $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffE_type,type,
% 2.93/1.51 symdiffE: $o ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffE,definition,
% 2.93/1.51 ( symdiffE
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.93/1.51 => ! [Xphi: $o] :
% 2.93/1.51 ( ( ( in @ Xx @ A )
% 2.93/1.51 => ( ~ ( in @ Xx @ B )
% 2.93/1.51 => Xphi ) )
% 2.93/1.51 => ( ( ~ ( in @ Xx @ A )
% 2.93/1.51 => ( ( in @ Xx @ B )
% 2.93/1.51 => Xphi ) )
% 2.93/1.51 => Xphi ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffI1_type,type,
% 2.93/1.51 symdiffI1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffI1,definition,
% 2.93/1.51 ( symdiffI1
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ~ ( in @ Xx @ B )
% 2.93/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffI2_type,type,
% 2.93/1.51 symdiffI2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffI2,definition,
% 2.93/1.51 ( symdiffI2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ~ ( in @ Xx @ A )
% 2.93/1.51 => ( ( in @ Xx @ B )
% 2.93/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffIneg1_type,type,
% 2.93/1.51 symdiffIneg1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffIneg1,definition,
% 2.93/1.51 ( symdiffIneg1
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( in @ Xx @ B )
% 2.93/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffIneg2_type,type,
% 2.93/1.51 symdiffIneg2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(symdiffIneg2,definition,
% 2.93/1.51 ( symdiffIneg2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ~ ( in @ Xx @ A )
% 2.93/1.51 => ( ~ ( in @ Xx @ B )
% 2.93/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(iskpair_type,type,
% 2.93/1.51 iskpair: $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(secondinupair_type,type,
% 2.93/1.51 secondinupair: $o ).
% 2.93/1.51
% 2.93/1.51 thf(secondinupair,definition,
% 2.93/1.51 ( secondinupair
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairIL_type,type,
% 2.93/1.51 setukpairIL: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairIL,definition,
% 2.93/1.51 ( setukpairIL
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairIR_type,type,
% 2.93/1.51 setukpairIR: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairIR,definition,
% 2.93/1.51 ( setukpairIR
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(kpairiskpair_type,type,
% 2.93/1.51 kpairiskpair: $o ).
% 2.93/1.51
% 2.93/1.51 thf(kpairiskpair,definition,
% 2.93/1.51 ( kpairiskpair
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(kpair_type,type,
% 2.93/1.51 kpair: $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(kpairp_type,type,
% 2.93/1.51 kpairp: $o ).
% 2.93/1.51
% 2.93/1.51 thf(kpairp,definition,
% 2.93/1.51 ( kpairp
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprod_type,type,
% 2.93/1.51 cartprod: $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(singletonsubset_type,type,
% 2.93/1.51 singletonsubset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(singletonsubset,definition,
% 2.93/1.51 ( singletonsubset
% 2.93/1.51 = ( ! [A: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(singletoninpowerset_type,type,
% 2.93/1.51 singletoninpowerset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(singletoninpowerset,definition,
% 2.93/1.51 ( singletoninpowerset
% 2.93/1.51 = ( ! [A: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(singletoninpowunion_type,type,
% 2.93/1.51 singletoninpowunion: $o ).
% 2.93/1.51
% 2.93/1.51 thf(singletoninpowunion,definition,
% 2.93/1.51 ( singletoninpowunion
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(upairset2E_type,type,
% 2.93/1.51 upairset2E: $o ).
% 2.93/1.51
% 2.93/1.51 thf(upairset2E,definition,
% 2.93/1.51 ( upairset2E
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.93/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.93/1.51 => ( ( Xz = Xx )
% 2.93/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(upairsubunion_type,type,
% 2.93/1.51 upairsubunion: $o ).
% 2.93/1.51
% 2.93/1.51 thf(upairsubunion,definition,
% 2.93/1.51 ( upairsubunion
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(upairinpowunion_type,type,
% 2.93/1.51 upairinpowunion: $o ).
% 2.93/1.51
% 2.93/1.51 thf(upairinpowunion,definition,
% 2.93/1.51 ( upairinpowunion
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ubforcartprodlem1_type,type,
% 2.93/1.51 ubforcartprodlem1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ubforcartprodlem1,definition,
% 2.93/1.51 ( ubforcartprodlem1
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ubforcartprodlem2_type,type,
% 2.93/1.51 ubforcartprodlem2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ubforcartprodlem2,definition,
% 2.93/1.51 ( ubforcartprodlem2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ubforcartprodlem3_type,type,
% 2.93/1.51 ubforcartprodlem3: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ubforcartprodlem3,definition,
% 2.93/1.51 ( ubforcartprodlem3
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodpairin_type,type,
% 2.93/1.51 cartprodpairin: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodpairin,definition,
% 2.93/1.51 ( cartprodpairin
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodmempair1_type,type,
% 2.93/1.51 cartprodmempair1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodmempair1,definition,
% 2.93/1.51 ( cartprodmempair1
% 2.93/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.51 => ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ? [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 & ( Xu
% 2.93/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodmempair_type,type,
% 2.93/1.51 cartprodmempair: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodmempair,definition,
% 2.93/1.51 ( cartprodmempair
% 2.93/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.51 => ( iskpair @ Xu ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setunionE2_type,type,
% 2.93/1.51 setunionE2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setunionE2,definition,
% 2.93/1.51 ( setunionE2
% 2.93/1.51 = ( ! [A: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.93/1.51 => ? [X: $i] :
% 2.93/1.51 ( ( in @ X @ A )
% 2.93/1.51 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setunionsingleton1_type,type,
% 2.93/1.51 setunionsingleton1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setunionsingleton1,definition,
% 2.93/1.51 ( setunionsingleton1
% 2.93/1.51 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setunionsingleton2_type,type,
% 2.93/1.51 setunionsingleton2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setunionsingleton2,definition,
% 2.93/1.51 ( setunionsingleton2
% 2.93/1.51 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setunionsingleton_type,type,
% 2.93/1.51 setunionsingleton: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setunionsingleton,definition,
% 2.93/1.51 ( setunionsingleton
% 2.93/1.51 = ( ! [Xx: $i] :
% 2.93/1.51 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.93/1.51 = Xx ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(singleton_type,type,
% 2.93/1.51 singleton: $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(singletonprop_type,type,
% 2.93/1.51 singletonprop: $o ).
% 2.93/1.51
% 2.93/1.51 thf(singletonprop,definition,
% 2.93/1.51 ( singletonprop
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( ( Xphi @ Xx )
% 2.93/1.51 => ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) )
% 2.93/1.51 => ( ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ( Xphi @ Xx ) )
% 2.93/1.51 => ( singleton
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ex1_type,type,
% 2.93/1.51 ex1: $i > ( $i > $o ) > $o ).
% 2.93/1.51
% 2.93/1.51 thf(ex1E1_type,type,
% 2.93/1.51 ex1E1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ex1E1,definition,
% 2.93/1.51 ( ex1E1
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ( ex1 @ A
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 => ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ex1I_type,type,
% 2.93/1.51 ex1I: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ex1I,definition,
% 2.93/1.51 ( ex1I
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( Xphi @ Xx )
% 2.93/1.51 => ( ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xy = Xx ) ) )
% 2.93/1.51 => ( ex1 @ A
% 2.93/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ex1I2_type,type,
% 2.93/1.51 ex1I2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ex1I2,definition,
% 2.93/1.51 ( ex1I2
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( ( Xphi @ Xx )
% 2.93/1.51 => ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) )
% 2.93/1.51 => ( ? [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 & ( Xphi @ Xx ) )
% 2.93/1.51 => ( ex1 @ A
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(singletonsuniq_type,type,
% 2.93/1.51 singletonsuniq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(singletonsuniq,definition,
% 2.93/1.51 ( singletonsuniq
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( ( setadjoin @ Xx @ emptyset )
% 2.93/1.51 = ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(atmost1p_type,type,
% 2.93/1.51 atmost1p: $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(atleast2p_type,type,
% 2.93/1.51 atleast2p: $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(atmost2p_type,type,
% 2.93/1.51 atmost2p: $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(upairsetp_type,type,
% 2.93/1.51 upairsetp: $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjL1_type,type,
% 2.93/1.51 setukpairinjL1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjL1,definition,
% 2.93/1.51 ( setukpairinjL1
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.93/1.51 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.93/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(kfstsingleton_type,type,
% 2.93/1.51 kfstsingleton: $o ).
% 2.93/1.51
% 2.93/1.51 thf(kfstsingleton,definition,
% 2.93/1.51 ( kfstsingleton
% 2.93/1.51 = ( ! [Xu: $i] :
% 2.93/1.51 ( ( iskpair @ Xu )
% 2.93/1.51 => ( singleton
% 2.93/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.93/1.51 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(theprop_type,type,
% 2.93/1.51 theprop: $o ).
% 2.93/1.51
% 2.93/1.51 thf(theprop,definition,
% 2.93/1.51 ( theprop
% 2.93/1.51 = ( ! [X: $i] :
% 2.93/1.51 ( ( singleton @ X )
% 2.93/1.51 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(kfst_type,type,
% 2.93/1.51 kfst: $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(kfstpairEq_type,type,
% 2.93/1.51 kfstpairEq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(kfstpairEq,definition,
% 2.93/1.51 ( kfstpairEq
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.93/1.51 = Xx ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodfstin_type,type,
% 2.93/1.51 cartprodfstin: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodfstin,definition,
% 2.93/1.51 ( cartprodfstin
% 2.93/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.51 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjL2_type,type,
% 2.93/1.51 setukpairinjL2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjL2,definition,
% 2.93/1.51 ( setukpairinjL2
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.93/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.93/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjL_type,type,
% 2.93/1.51 setukpairinjL: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjL,definition,
% 2.93/1.51 ( setukpairinjL
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.93/1.51 = ( kpair @ Xz @ Xu ) )
% 2.93/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR11_type,type,
% 2.93/1.51 setukpairinjR11: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR11,definition,
% 2.93/1.51 ( setukpairinjR11
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xx = Xy )
% 2.93/1.51 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR12_type,type,
% 2.93/1.51 setukpairinjR12: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR12,definition,
% 2.93/1.51 ( setukpairinjR12
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xx = Xy )
% 2.93/1.51 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.93/1.51 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR1_type,type,
% 2.93/1.51 setukpairinjR1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR1,definition,
% 2.93/1.51 ( setukpairinjR1
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.93/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.93/1.51 => ( ( Xz = Xu )
% 2.93/1.51 => ( Xy = Xu ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(upairequniteq_type,type,
% 2.93/1.51 upairequniteq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(upairequniteq,definition,
% 2.93/1.51 ( upairequniteq
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.93/1.51 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.93/1.51 = ( setadjoin @ Xz @ emptyset ) )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR2_type,type,
% 2.93/1.51 setukpairinjR2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR2,definition,
% 2.93/1.51 ( setukpairinjR2
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.93/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.93/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR_type,type,
% 2.93/1.51 setukpairinjR: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setukpairinjR,definition,
% 2.93/1.51 ( setukpairinjR
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.93/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.93/1.51 = ( kpair @ Xz @ Xu ) )
% 2.93/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ksndsingleton_type,type,
% 2.93/1.51 ksndsingleton: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ksndsingleton,definition,
% 2.93/1.51 ( ksndsingleton
% 2.93/1.51 = ( ! [Xu: $i] :
% 2.93/1.51 ( ( iskpair @ Xu )
% 2.93/1.51 => ( singleton
% 2.93/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.93/1.51 @ ^ [Xx: $i] :
% 2.93/1.51 ( Xu
% 2.93/1.51 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ksnd_type,type,
% 2.93/1.51 ksnd: $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(ksndpairEq_type,type,
% 2.93/1.51 ksndpairEq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ksndpairEq,definition,
% 2.93/1.51 ( ksndpairEq
% 2.93/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.93/1.51 = Xy ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(kpairsurjEq_type,type,
% 2.93/1.51 kpairsurjEq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(kpairsurjEq,definition,
% 2.93/1.51 ( kpairsurjEq
% 2.93/1.51 = ( ! [Xu: $i] :
% 2.93/1.51 ( ( iskpair @ Xu )
% 2.93/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.93/1.51 = Xu ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodsndin_type,type,
% 2.93/1.51 cartprodsndin: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodsndin,definition,
% 2.93/1.51 ( cartprodsndin
% 2.93/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.51 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodpairmemEL_type,type,
% 2.93/1.51 cartprodpairmemEL: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodpairmemEL,definition,
% 2.93/1.51 ( cartprodpairmemEL
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.93/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodpairmemER_type,type,
% 2.93/1.51 cartprodpairmemER: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodpairmemER,definition,
% 2.93/1.51 ( cartprodpairmemER
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.93/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodmempaircEq_type,type,
% 2.93/1.51 cartprodmempaircEq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodmempaircEq,definition,
% 2.93/1.51 ( cartprodmempaircEq
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( kpair @ Xx @ Xy )
% 2.93/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodfstpairEq_type,type,
% 2.93/1.51 cartprodfstpairEq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodfstpairEq,definition,
% 2.93/1.51 ( cartprodfstpairEq
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.93/1.51 = Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodsndpairEq_type,type,
% 2.93/1.51 cartprodsndpairEq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodsndpairEq,definition,
% 2.93/1.51 ( cartprodsndpairEq
% 2.93/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.93/1.51 = Xy ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodpairsurjEq_type,type,
% 2.93/1.51 cartprodpairsurjEq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(cartprodpairsurjEq,definition,
% 2.93/1.51 ( cartprodpairsurjEq
% 2.93/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.93/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.93/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.93/1.51 = Xu ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(breln_type,type,
% 2.93/1.51 breln: $i > $i > $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(breln,definition,
% 2.93/1.51 ( breln
% 2.93/1.51 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstr_type,type,
% 2.93/1.51 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrI_type,type,
% 2.93/1.51 dpsetconstrI: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrI,definition,
% 2.93/1.51 ( dpsetconstrI
% 2.93/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( Xphi @ Xx @ Xy )
% 2.93/1.51 => ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.51 @ ( dpsetconstr @ A @ B
% 2.93/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrSub_type,type,
% 2.93/1.51 dpsetconstrSub: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrSub,definition,
% 2.93/1.51 ( dpsetconstrSub
% 2.93/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.93/1.51 ( subset
% 2.93/1.51 @ ( dpsetconstr @ A @ B
% 2.93/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.93/1.51 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(setOfPairsIsBReln_type,type,
% 2.93/1.51 setOfPairsIsBReln: $o ).
% 2.93/1.51
% 2.93/1.51 thf(setOfPairsIsBReln,definition,
% 2.93/1.51 ( setOfPairsIsBReln
% 2.93/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.93/1.51 ( breln @ A @ B
% 2.93/1.51 @ ( dpsetconstr @ A @ B
% 2.93/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrERa_type,type,
% 2.93/1.51 dpsetconstrERa: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrERa,definition,
% 2.93/1.51 ( dpsetconstrERa
% 2.93/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.51 @ ( dpsetconstr @ A @ B
% 2.93/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.93/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrEL1_type,type,
% 2.93/1.51 dpsetconstrEL1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrEL1,definition,
% 2.93/1.51 ( dpsetconstrEL1
% 2.93/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.51 @ ( dpsetconstr @ A @ B
% 2.93/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.93/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrEL2_type,type,
% 2.93/1.51 dpsetconstrEL2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrEL2,definition,
% 2.93/1.51 ( dpsetconstrEL2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.51 @ ( dpsetconstr @ A @ B
% 2.93/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.93/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrER_type,type,
% 2.93/1.51 dpsetconstrER: $o ).
% 2.93/1.51
% 2.93/1.51 thf(dpsetconstrER,definition,
% 2.93/1.51 ( dpsetconstrER
% 2.93/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.93/1.51 @ ( dpsetconstr @ A @ B
% 2.93/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.93/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(func_type,type,
% 2.93/1.51 func: $i > $i > $i > $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcSet_type,type,
% 2.93/1.51 funcSet: $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(funcImageSingleton_type,type,
% 2.93/1.51 funcImageSingleton: $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcImageSingleton,definition,
% 2.93/1.51 ( funcImageSingleton
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( singleton
% 2.93/1.51 @ ( dsetconstr @ B
% 2.93/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(apProp_type,type,
% 2.93/1.51 apProp: $o ).
% 2.93/1.51
% 2.93/1.51 thf(apProp,definition,
% 2.93/1.51 ( apProp
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in
% 2.93/1.51 @ ( setunion
% 2.93/1.51 @ ( dsetconstr @ B
% 2.93/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.93/1.51 @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ap_type,type,
% 2.93/1.51 ap: $i > $i > $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(app_type,type,
% 2.93/1.51 app: $o ).
% 2.93/1.51
% 2.93/1.51 thf(app,definition,
% 2.93/1.51 ( app
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(infuncsetfunc_type,type,
% 2.93/1.51 infuncsetfunc: $o ).
% 2.93/1.51
% 2.93/1.51 thf(infuncsetfunc,definition,
% 2.93/1.51 ( infuncsetfunc
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.51 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ap2p_type,type,
% 2.93/1.51 ap2p: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ap2p,definition,
% 2.93/1.51 ( ap2p
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(funcinfuncset_type,type,
% 2.93/1.51 funcinfuncset: $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcinfuncset,definition,
% 2.93/1.51 ( funcinfuncset
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(lamProp_type,type,
% 2.93/1.51 lamProp: $o ).
% 2.93/1.51
% 2.93/1.51 thf(lamProp,definition,
% 2.93/1.51 ( lamProp
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.51 => ( func @ A @ B
% 2.93/1.51 @ ( dpsetconstr @ A @ B
% 2.93/1.51 @ ^ [Xx: $i,Xy: $i] :
% 2.93/1.51 ( ( Xf @ Xx )
% 2.93/1.51 = Xy ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(lam_type,type,
% 2.93/1.51 lam: $i > $i > ( $i > $i ) > $i ).
% 2.93/1.51
% 2.93/1.51 thf(lamp_type,type,
% 2.93/1.51 lamp: $o ).
% 2.93/1.51
% 2.93/1.51 thf(lamp,definition,
% 2.93/1.51 ( lamp
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.51 => ( func @ A @ B
% 2.93/1.51 @ ( lam @ A @ B
% 2.93/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(lam2p_type,type,
% 2.93/1.51 lam2p: $o ).
% 2.93/1.51
% 2.93/1.51 thf(lam2p,definition,
% 2.93/1.51 ( lam2p
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.51 => ( in
% 2.93/1.51 @ ( lam @ A @ B
% 2.93/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.93/1.51 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(brelnall1_type,type,
% 2.93/1.51 brelnall1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(brelnall1,definition,
% 2.93/1.51 ( brelnall1
% 2.93/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.93/1.51 ( ( breln @ A @ B @ R )
% 2.93/1.51 => ! [Xphi: $i > $o] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ R )
% 2.93/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(brelnall2_type,type,
% 2.93/1.51 brelnall2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(brelnall2,definition,
% 2.93/1.51 ( brelnall2
% 2.93/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.93/1.51 ( ( breln @ A @ B @ R )
% 2.93/1.51 => ! [Xphi: $i > $o] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ R )
% 2.93/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ex1E2_type,type,
% 2.93/1.51 ex1E2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ex1E2,definition,
% 2.93/1.51 ( ex1E2
% 2.93/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.93/1.51 ( ( ex1 @ A
% 2.93/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( ( Xphi @ Xx )
% 2.93/1.51 => ( ( Xphi @ Xy )
% 2.93/1.51 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(funcGraphProp1_type,type,
% 2.93/1.51 funcGraphProp1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcGraphProp1,definition,
% 2.93/1.51 ( funcGraphProp1
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(funcGraphProp3_type,type,
% 2.93/1.51 funcGraphProp3: $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcGraphProp3,definition,
% 2.93/1.51 ( funcGraphProp3
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(funcGraphProp2_type,type,
% 2.93/1.51 funcGraphProp2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcGraphProp2,definition,
% 2.93/1.51 ( funcGraphProp2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.93/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(funcextLem_type,type,
% 2.93/1.51 funcextLem: $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcextLem,definition,
% 2.93/1.51 ( funcextLem
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ! [Xg: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xg )
% 2.93/1.51 => ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.93/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(funcGraphProp4_type,type,
% 2.93/1.51 funcGraphProp4: $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcGraphProp4,definition,
% 2.93/1.51 ( funcGraphProp4
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.93/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(subbreln_type,type,
% 2.93/1.51 subbreln: $o ).
% 2.93/1.51
% 2.93/1.51 thf(subbreln,definition,
% 2.93/1.51 ( subbreln
% 2.93/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.93/1.51 ( ( breln @ A @ B @ R )
% 2.93/1.51 => ! [S: $i] :
% 2.93/1.51 ( ( breln @ A @ B @ S )
% 2.93/1.51 => ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.93/1.51 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(eqbreln_type,type,
% 2.93/1.51 eqbreln: $o ).
% 2.93/1.51
% 2.93/1.51 thf(eqbreln,definition,
% 2.93/1.51 ( eqbreln
% 2.93/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.93/1.51 ( ( breln @ A @ B @ R )
% 2.93/1.51 => ! [S: $i] :
% 2.93/1.51 ( ( breln @ A @ B @ S )
% 2.93/1.51 => ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.93/1.51 => ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ B )
% 2.93/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.93/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.93/1.51 => ( R = S ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(funcext_type,type,
% 2.93/1.51 funcext: $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcext,definition,
% 2.93/1.51 ( funcext
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ! [Xg: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xg )
% 2.93/1.51 => ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.93/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(funcext2_type,type,
% 2.93/1.51 funcext2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(funcext2,definition,
% 2.93/1.51 ( funcext2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.51 => ! [Xg: $i] :
% 2.93/1.51 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.93/1.51 => ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.93/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ap2apEq1_type,type,
% 2.93/1.51 ap2apEq1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ap2apEq1,definition,
% 2.93/1.51 ( ap2apEq1
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ap2apEq2_type,type,
% 2.93/1.51 ap2apEq2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ap2apEq2,definition,
% 2.93/1.51 ( ap2apEq2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.93/1.51 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(beta1_type,type,
% 2.93/1.51 beta1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(beta1,definition,
% 2.93/1.51 ( beta1
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( ap @ A @ B
% 2.93/1.51 @ ( lam @ A @ B
% 2.93/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.93/1.51 @ Xx )
% 2.93/1.51 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(eta1_type,type,
% 2.93/1.51 eta1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(eta1,definition,
% 2.93/1.51 ( eta1
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( func @ A @ B @ Xf )
% 2.93/1.51 => ( ( lam @ A @ B
% 2.93/1.51 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.93/1.51 = Xf ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(lam2lamEq_type,type,
% 2.93/1.51 lam2lamEq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(lam2lamEq,definition,
% 2.93/1.51 ( lam2lamEq
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.51 => ( ( lam @ A @ B
% 2.93/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.93/1.51 = ( lam @ A @ B
% 2.93/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(beta2_type,type,
% 2.93/1.51 beta2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(beta2,definition,
% 2.93/1.51 ( beta2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.93/1.51 ( ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ( ( ap @ A @ B
% 2.93/1.51 @ ( lam @ A @ B
% 2.93/1.51 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.93/1.51 @ Xx )
% 2.93/1.51 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(eta2_type,type,
% 2.93/1.51 eta2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(eta2,definition,
% 2.93/1.51 ( eta2
% 2.93/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.93/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.93/1.51 => ( ( lam @ A @ B
% 2.93/1.51 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.93/1.51 = Xf ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(iffalseProp1_type,type,
% 2.93/1.51 iffalseProp1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(iffalseProp1,definition,
% 2.93/1.51 ( iffalseProp1
% 2.93/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( ~ Xphi
% 2.93/1.51 => ( in @ Xy
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xz: $i] :
% 2.93/1.51 ( ( Xphi
% 2.93/1.51 & ( Xz = Xx ) )
% 2.93/1.51 | ( ~ Xphi
% 2.93/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(iffalseProp2_type,type,
% 2.93/1.51 iffalseProp2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(iffalseProp2,definition,
% 2.93/1.51 ( iffalseProp2
% 2.93/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( ~ Xphi
% 2.93/1.51 => ( ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xz: $i] :
% 2.93/1.51 ( ( Xphi
% 2.93/1.51 & ( Xz = Xx ) )
% 2.93/1.51 | ( ~ Xphi
% 2.93/1.51 & ( Xz = Xy ) ) ) )
% 2.93/1.51 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(iftrueProp1_type,type,
% 2.93/1.51 iftrueProp1: $o ).
% 2.93/1.51
% 2.93/1.51 thf(iftrueProp1,definition,
% 2.93/1.51 ( iftrueProp1
% 2.93/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( Xphi
% 2.93/1.51 => ( in @ Xx
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xz: $i] :
% 2.93/1.51 ( ( Xphi
% 2.93/1.51 & ( Xz = Xx ) )
% 2.93/1.51 | ( ~ Xphi
% 2.93/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(iftrueProp2_type,type,
% 2.93/1.51 iftrueProp2: $o ).
% 2.93/1.51
% 2.93/1.51 thf(iftrueProp2,definition,
% 2.93/1.51 ( iftrueProp2
% 2.93/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( Xphi
% 2.93/1.51 => ( ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xz: $i] :
% 2.93/1.51 ( ( Xphi
% 2.93/1.51 & ( Xz = Xx ) )
% 2.93/1.51 | ( ~ Xphi
% 2.93/1.51 & ( Xz = Xy ) ) ) )
% 2.93/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(ifSingleton_type,type,
% 2.93/1.51 ifSingleton: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ifSingleton,definition,
% 2.93/1.51 ( ifSingleton
% 2.93/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( singleton
% 2.93/1.51 @ ( dsetconstr @ A
% 2.93/1.51 @ ^ [Xz: $i] :
% 2.93/1.51 ( ( Xphi
% 2.93/1.51 & ( Xz = Xx ) )
% 2.93/1.51 | ( ~ Xphi
% 2.93/1.51 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(if_type,type,
% 2.93/1.51 if: $i > $o > $i > $i > $i ).
% 2.93/1.51
% 2.93/1.51 thf(ifp_type,type,
% 2.93/1.51 ifp: $o ).
% 2.93/1.51
% 2.93/1.51 thf(ifp,definition,
% 2.93/1.51 ( ifp
% 2.93/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(theeq_type,type,
% 2.93/1.51 theeq: $o ).
% 2.93/1.51
% 2.93/1.51 thf(theeq,definition,
% 2.93/1.51 ( theeq
% 2.93/1.51 = ( ! [X: $i] :
% 2.93/1.51 ( ( singleton @ X )
% 2.93/1.51 => ! [Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ X )
% 2.93/1.51 => ( ( setunion @ X )
% 2.93/1.51 = Xx ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(iftrue_type,type,
% 2.93/1.51 iftrue: $o ).
% 2.93/1.51
% 2.93/1.51 thf(iftrue,definition,
% 2.93/1.51 ( iftrue
% 2.93/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( Xphi
% 2.93/1.51 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.93/1.51 = Xx ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(iffalse_type,type,
% 2.93/1.51 iffalse: $o ).
% 2.93/1.51
% 2.93/1.51 thf(iffalse,definition,
% 2.93/1.51 ( iffalse
% 2.93/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( ~ Xphi
% 2.93/1.51 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.93/1.51 = Xy ) ) ) ) ) ) ).
% 2.93/1.51
% 2.93/1.51 thf(iftrueorfalse_type,type,
% 2.93/1.51 iftrueorfalse: $o ).
% 2.93/1.51
% 2.93/1.51 thf(iftrueorfalse,definition,
% 2.93/1.51 ( iftrueorfalse
% 2.93/1.51 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.93/1.51 ( ( in @ Xx @ A )
% 2.93/1.51 => ! [Xy: $i] :
% 2.93/1.51 ( ( in @ Xy @ A )
% 2.93/1.51 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(binintersectT_lem_type,type,
% 2.93/1.52 binintersectT_lem: $o ).
% 2.93/1.52
% 2.93/1.52 thf(binintersectT_lem,definition,
% 2.93/1.52 ( binintersectT_lem
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(binunionT_lem_type,type,
% 2.93/1.52 binunionT_lem: $o ).
% 2.93/1.52
% 2.93/1.52 thf(binunionT_lem,definition,
% 2.93/1.52 ( binunionT_lem
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(powersetT_lem_type,type,
% 2.93/1.52 powersetT_lem: $o ).
% 2.93/1.52
% 2.93/1.52 thf(powersetT_lem,definition,
% 2.93/1.52 ( powersetT_lem
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(setminusT_lem_type,type,
% 2.93/1.52 setminusT_lem: $o ).
% 2.93/1.52
% 2.93/1.52 thf(setminusT_lem,definition,
% 2.93/1.52 ( setminusT_lem
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(complementT_lem_type,type,
% 2.93/1.52 complementT_lem: $o ).
% 2.93/1.52
% 2.93/1.52 thf(complementT_lem,definition,
% 2.93/1.52 ( complementT_lem
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(setextT_type,type,
% 2.93/1.52 setextT: $o ).
% 2.93/1.52
% 2.93/1.52 thf(setextT,definition,
% 2.93/1.52 ( setextT
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ X )
% 2.93/1.52 => ( in @ Xx @ Y ) ) )
% 2.93/1.52 => ( ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ Y )
% 2.93/1.52 => ( in @ Xx @ X ) ) )
% 2.93/1.52 => ( X = Y ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(subsetTI_type,type,
% 2.93/1.52 subsetTI: $o ).
% 2.93/1.52
% 2.93/1.52 thf(subsetTI,definition,
% 2.93/1.52 ( subsetTI
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ X )
% 2.93/1.52 => ( in @ Xx @ Y ) ) )
% 2.93/1.52 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(powersetTI1_type,type,
% 2.93/1.52 powersetTI1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(powersetTI1,definition,
% 2.93/1.52 ( powersetTI1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ X )
% 2.93/1.52 => ( in @ Xx @ Y ) ) )
% 2.93/1.52 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(powersetTE1_type,type,
% 2.93/1.52 powersetTE1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(powersetTE1,definition,
% 2.93/1.52 ( powersetTE1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ X @ ( powerset @ Y ) )
% 2.93/1.52 => ( ( in @ Xx @ X )
% 2.93/1.52 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(complementTI1_type,type,
% 2.93/1.52 complementTI1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(complementTI1,definition,
% 2.93/1.52 ( complementTI1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ X )
% 2.93/1.52 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(complementTE1_type,type,
% 2.93/1.52 complementTE1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(complementTE1,definition,
% 2.93/1.52 ( complementTE1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.52 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(binintersectTELcontra_type,type,
% 2.93/1.52 binintersectTELcontra: $o ).
% 2.93/1.52
% 2.93/1.52 thf(binintersectTELcontra,definition,
% 2.93/1.52 ( binintersectTELcontra
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ~ ( in @ Xx @ X )
% 2.93/1.52 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(binintersectTERcontra_type,type,
% 2.93/1.52 binintersectTERcontra: $o ).
% 2.93/1.52
% 2.93/1.52 thf(binintersectTERcontra,definition,
% 2.93/1.52 ( binintersectTERcontra
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ~ ( in @ Xx @ Y )
% 2.93/1.52 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(contrasubsetT_type,type,
% 2.93/1.52 contrasubsetT: $o ).
% 2.93/1.52
% 2.93/1.52 thf(contrasubsetT,definition,
% 2.93/1.52 ( contrasubsetT
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.52 => ( ( in @ Xx @ Y )
% 2.93/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(contrasubsetT1_type,type,
% 2.93/1.52 contrasubsetT1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(contrasubsetT1,definition,
% 2.93/1.52 ( contrasubsetT1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( subset @ X @ Y )
% 2.93/1.52 => ( ~ ( in @ Xx @ Y )
% 2.93/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(contrasubsetT2_type,type,
% 2.93/1.52 contrasubsetT2: $o ).
% 2.93/1.52
% 2.93/1.52 thf(contrasubsetT2,definition,
% 2.93/1.52 ( contrasubsetT2
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ( subset @ X @ Y )
% 2.93/1.52 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(contrasubsetT3_type,type,
% 2.93/1.52 contrasubsetT3: $o ).
% 2.93/1.52
% 2.93/1.52 thf(contrasubsetT3,definition,
% 2.93/1.52 ( contrasubsetT3
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.93/1.52 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementI1_type,type,
% 2.93/1.52 doubleComplementI1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementI1,definition,
% 2.93/1.52 ( doubleComplementI1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ X )
% 2.93/1.52 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementE1_type,type,
% 2.93/1.52 doubleComplementE1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementE1,definition,
% 2.93/1.52 ( doubleComplementE1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.93/1.52 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementSub1_type,type,
% 2.93/1.52 doubleComplementSub1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementSub1,definition,
% 2.93/1.52 ( doubleComplementSub1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementSub2_type,type,
% 2.93/1.52 doubleComplementSub2: $o ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementSub2,definition,
% 2.93/1.52 ( doubleComplementSub2
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementEq_type,type,
% 2.93/1.52 doubleComplementEq: $o ).
% 2.93/1.52
% 2.93/1.52 thf(doubleComplementEq,definition,
% 2.93/1.52 ( doubleComplementEq
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ( X
% 2.93/1.52 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(complementTnotintersectT_type,type,
% 2.93/1.52 complementTnotintersectT: $o ).
% 2.93/1.52
% 2.93/1.52 thf(complementTnotintersectT,definition,
% 2.93/1.52 ( complementTnotintersectT
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.52 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(complementImpComplementIntersect_type,type,
% 2.93/1.52 complementImpComplementIntersect: $o ).
% 2.93/1.52
% 2.93/1.52 thf(complementImpComplementIntersect,definition,
% 2.93/1.52 ( complementImpComplementIntersect
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.52 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(complementSubsetComplementIntersect_type,type,
% 2.93/1.52 complementSubsetComplementIntersect: $o ).
% 2.93/1.52
% 2.93/1.52 thf(complementSubsetComplementIntersect,definition,
% 2.93/1.52 ( complementSubsetComplementIntersect
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(complementInPowersetComplementIntersect_type,type,
% 2.93/1.52 complementInPowersetComplementIntersect: $o ).
% 2.93/1.52
% 2.93/1.52 thf(complementInPowersetComplementIntersect,definition,
% 2.93/1.52 ( complementInPowersetComplementIntersect
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(contraSubsetComplement_type,type,
% 2.93/1.52 contraSubsetComplement: $o ).
% 2.93/1.52
% 2.93/1.52 thf(contraSubsetComplement,definition,
% 2.93/1.52 ( contraSubsetComplement
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ Y )
% 2.93/1.52 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(complementTcontraSubset_type,type,
% 2.93/1.52 complementTcontraSubset: $o ).
% 2.93/1.52
% 2.93/1.52 thf(complementTcontraSubset,definition,
% 2.93/1.52 ( complementTcontraSubset
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.52 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(binunionTILcontra_type,type,
% 2.93/1.52 binunionTILcontra: $o ).
% 2.93/1.52
% 2.93/1.52 thf(binunionTILcontra,definition,
% 2.93/1.52 ( binunionTILcontra
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.93/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(binunionTIRcontra_type,type,
% 2.93/1.52 binunionTIRcontra: $o ).
% 2.93/1.52
% 2.93/1.52 thf(binunionTIRcontra,definition,
% 2.93/1.52 ( binunionTIRcontra
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.93/1.52 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(inIntersectImpInUnion_type,type,
% 2.93/1.52 inIntersectImpInUnion: $o ).
% 2.93/1.52
% 2.93/1.52 thf(inIntersectImpInUnion,definition,
% 2.93/1.52 ( inIntersectImpInUnion
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.93/1.52 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(inIntersectImpInUnion2_type,type,
% 2.93/1.52 inIntersectImpInUnion2: $o ).
% 2.93/1.52
% 2.93/1.52 thf(inIntersectImpInUnion2,definition,
% 2.93/1.52 ( inIntersectImpInUnion2
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.93/1.52 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(inIntersectImpInIntersectUnions_type,type,
% 2.93/1.52 inIntersectImpInIntersectUnions: $o ).
% 2.93/1.52
% 2.93/1.52 thf(inIntersectImpInIntersectUnions,definition,
% 2.93/1.52 ( inIntersectImpInIntersectUnions
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.93/1.52 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(intersectInPowersetIntersectUnions_type,type,
% 2.93/1.52 intersectInPowersetIntersectUnions: $o ).
% 2.93/1.52
% 2.93/1.52 thf(intersectInPowersetIntersectUnions,definition,
% 2.93/1.52 ( intersectInPowersetIntersectUnions
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(inComplementUnionImpNotIn1_type,type,
% 2.93/1.52 inComplementUnionImpNotIn1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(inComplementUnionImpNotIn1,definition,
% 2.93/1.52 ( inComplementUnionImpNotIn1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.52 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(inComplementUnionImpInComplement1_type,type,
% 2.93/1.52 inComplementUnionImpInComplement1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(inComplementUnionImpInComplement1,definition,
% 2.93/1.52 ( inComplementUnionImpInComplement1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.52 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(binunionTE_type,type,
% 2.93/1.52 binunionTE: $o ).
% 2.93/1.52
% 2.93/1.52 thf(binunionTE,definition,
% 2.93/1.52 ( binunionTE
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xphi: $o,Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.93/1.52 => ( ( ( in @ Xx @ X )
% 2.93/1.52 => Xphi )
% 2.93/1.52 => ( ( ( in @ Xx @ Y )
% 2.93/1.52 => Xphi )
% 2.93/1.52 => Xphi ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(binunionTEcontra_type,type,
% 2.93/1.52 binunionTEcontra: $o ).
% 2.93/1.52
% 2.93/1.52 thf(binunionTEcontra,definition,
% 2.93/1.52 ( binunionTEcontra
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ~ ( in @ Xx @ X )
% 2.93/1.52 => ( ~ ( in @ Xx @ Y )
% 2.93/1.52 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2a1_type,type,
% 2.93/1.52 demorgan2a1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2a1,definition,
% 2.93/1.52 ( demorgan2a1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.52 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(complementUnionInPowersetComplement_type,type,
% 2.93/1.52 complementUnionInPowersetComplement: $o ).
% 2.93/1.52
% 2.93/1.52 thf(complementUnionInPowersetComplement,definition,
% 2.93/1.52 ( complementUnionInPowersetComplement
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2a2_type,type,
% 2.93/1.52 demorgan2a2: $o ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2a2,definition,
% 2.93/1.52 ( demorgan2a2
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.52 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan1a_type,type,
% 2.93/1.52 demorgan1a: $o ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan1a,definition,
% 2.93/1.52 ( demorgan1a
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.93/1.52 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan1b_type,type,
% 2.93/1.52 demorgan1b: $o ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan1b,definition,
% 2.93/1.52 ( demorgan1b
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.93/1.52 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan1_type,type,
% 2.93/1.52 demorgan1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan1,definition,
% 2.93/1.52 ( demorgan1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.93/1.52 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2a_type,type,
% 2.93/1.52 demorgan2a: $o ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2a,definition,
% 2.93/1.52 ( demorgan2a
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.93/1.52 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2b2_type,type,
% 2.93/1.52 demorgan2b2: $o ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2b2,definition,
% 2.93/1.52 ( demorgan2b2
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.93/1.52 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.93/1.52 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2b_type,type,
% 2.93/1.52 demorgan2b: $o ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2b,definition,
% 2.93/1.52 ( demorgan2b
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.93/1.52 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2_type,type,
% 2.93/1.52 demorgan2: $o ).
% 2.93/1.52
% 2.93/1.52 thf(demorgan2,definition,
% 2.93/1.52 ( demorgan2
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.93/1.52 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule0_type,type,
% 2.93/1.52 woz13rule0: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule0,definition,
% 2.93/1.52 ( woz13rule0
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.93/1.52 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule1_type,type,
% 2.93/1.52 woz13rule1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule1,definition,
% 2.93/1.52 ( woz13rule1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ( ( subset @ X @ Z )
% 2.93/1.52 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule2_type,type,
% 2.93/1.52 woz13rule2: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule2,definition,
% 2.93/1.52 ( woz13rule2
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ( ( subset @ Y @ Z )
% 2.93/1.52 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule3_type,type,
% 2.93/1.52 woz13rule3: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule3,definition,
% 2.93/1.52 ( woz13rule3
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ( ( subset @ X @ Y )
% 2.93/1.52 => ( ( subset @ X @ Z )
% 2.93/1.52 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule4_type,type,
% 2.93/1.52 woz13rule4: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz13rule4,definition,
% 2.93/1.52 ( woz13rule4
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ! [W: $i] :
% 2.93/1.52 ( ( in @ W @ ( powerset @ A ) )
% 2.93/1.52 => ( ( subset @ X @ Z )
% 2.93/1.52 => ( ( subset @ Y @ W )
% 2.93/1.52 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_1_type,type,
% 2.93/1.52 woz1_1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_1,definition,
% 2.93/1.52 ( woz1_1
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_2_type,type,
% 2.93/1.52 woz1_2: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_2,definition,
% 2.93/1.52 ( woz1_2
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ! [W: $i] :
% 2.93/1.52 ( ( in @ W @ ( powerset @ A ) )
% 2.93/1.52 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 2.93/1.52 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_3_type,type,
% 2.93/1.52 woz1_3: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_3,definition,
% 2.93/1.52 ( woz1_3
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ! [Z: $i] :
% 2.93/1.52 ( ( in @ Z @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_4_type,type,
% 2.93/1.52 woz1_4: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_4,definition,
% 2.93/1.52 ( woz1_4
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.93/1.52 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_5_type,type,
% 2.93/1.52 woz1_5: $o ).
% 2.93/1.52
% 2.93/1.52 thf(woz1_5,definition,
% 2.93/1.52 ( woz1_5
% 2.93/1.52 = ( ! [A: $i,X: $i] :
% 2.93/1.52 ( ( in @ X @ ( powerset @ A ) )
% 2.93/1.52 => ! [Y: $i] :
% 2.93/1.52 ( ( in @ Y @ ( powerset @ A ) )
% 2.93/1.52 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(breln1_type,type,
% 2.93/1.52 breln1: $i > $i > $o ).
% 2.93/1.52
% 2.93/1.52 thf(breln1,definition,
% 2.93/1.52 ( breln1
% 2.93/1.52 = ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(breln1all2_type,type,
% 2.93/1.52 breln1all2: $o ).
% 2.93/1.52
% 2.93/1.52 thf(breln1all2,definition,
% 2.93/1.52 ( breln1all2
% 2.93/1.52 = ( ! [A: $i,R: $i] :
% 2.93/1.52 ( ( breln1 @ A @ R )
% 2.93/1.52 => ! [Xphi: $i > $o] :
% 2.93/1.52 ( ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ! [Xy: $i] :
% 2.93/1.52 ( ( in @ Xy @ A )
% 2.93/1.52 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.93/1.52 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ R )
% 2.93/1.52 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(breln1Set_type,type,
% 2.93/1.52 breln1Set: $i > $i ).
% 2.93/1.52
% 2.93/1.52 thf(breln1SetBreln1_type,type,
% 2.93/1.52 breln1SetBreln1: $o ).
% 2.93/1.52
% 2.93/1.52 thf(breln1SetBreln1,definition,
% 2.93/1.52 ( breln1SetBreln1
% 2.93/1.52 = ( ! [A: $i,R: $i] :
% 2.93/1.52 ( ( in @ R @ ( breln1Set @ A ) )
% 2.93/1.52 => ( breln1 @ A @ R ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(transitive_type,type,
% 2.93/1.52 transitive: $i > $i > $o ).
% 2.93/1.52
% 2.93/1.52 thf(antisymmetric_type,type,
% 2.93/1.52 antisymmetric: $i > $i > $o ).
% 2.93/1.52
% 2.93/1.52 thf(reflexive_type,type,
% 2.93/1.52 reflexive: $i > $i > $o ).
% 2.93/1.52
% 2.93/1.52 thf(refltransitive_type,type,
% 2.93/1.52 refltransitive: $i > $i > $o ).
% 2.93/1.52
% 2.93/1.52 thf(refllinearorder_type,type,
% 2.93/1.52 refllinearorder: $i > $i > $o ).
% 2.93/1.52
% 2.93/1.52 thf(reflwellordering_type,type,
% 2.93/1.52 reflwellordering: $i > $i > $o ).
% 2.93/1.52
% 2.93/1.52 thf(choice2fnsingleton_type,type,
% 2.93/1.52 choice2fnsingleton: $o ).
% 2.93/1.52
% 2.93/1.52 thf(choice2fnsingleton,definition,
% 2.93/1.52 ( choice2fnsingleton
% 2.93/1.52 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.93/1.52 ( ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ? [Xy: $i] :
% 2.93/1.52 ( ( in @ Xy @ B )
% 2.93/1.52 & ( Xphi @ Xx @ Xy ) ) )
% 2.93/1.52 => ! [R: $i] :
% 2.93/1.52 ( ( in @ R @ ( breln1Set @ B ) )
% 2.93/1.52 => ( ( reflwellordering @ B @ R )
% 2.93/1.52 => ! [Xx: $i] :
% 2.93/1.52 ( ( in @ Xx @ A )
% 2.93/1.52 => ( singleton
% 2.93/1.52 @ ( dsetconstr @ B
% 2.93/1.52 @ ^ [Xy: $i] :
% 2.93/1.52 ( ( Xphi @ Xx @ Xy )
% 2.93/1.52 & ! [Xz: $i] :
% 2.93/1.52 ( ( in @ Xz @ B )
% 2.93/1.52 => ( ( Xphi @ Xx @ Xz )
% 2.93/1.52 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52
% 2.93/1.52 thf(setOfPairsIsBReln1,conjecture,
% 2.93/1.52 ( setextAx
% 2.93/1.52 => ( emptysetAx
% 2.93/1.52 => ( setadjoinAx
% 2.93/1.52 => ( powersetAx
% 2.93/1.52 => ( setunionAx
% 2.93/1.52 => ( omega0Ax
% 2.93/1.52 => ( omegaSAx
% 2.93/1.52 => ( omegaIndAx
% 2.93/1.52 => ( replAx
% 2.93/1.52 => ( foundationAx
% 2.93/1.52 => ( wellorderingAx
% 2.93/1.52 => ( descrp
% 2.93/1.52 => ( dsetconstrI
% 2.93/1.52 => ( dsetconstrEL
% 2.93/1.52 => ( dsetconstrER
% 2.93/1.52 => ( exuE1
% 2.93/1.52 => ( prop2setE
% 2.93/1.52 => ( emptysetE
% 2.93/1.52 => ( emptysetimpfalse
% 2.93/1.52 => ( notinemptyset
% 2.93/1.52 => ( exuE3e
% 2.93/1.52 => ( setext
% 2.93/1.52 => ( emptyI
% 2.93/1.52 => ( noeltsimpempty
% 2.93/1.52 => ( setbeta
% 2.93/1.52 => ( nonemptyE1
% 2.93/1.52 => ( nonemptyI
% 2.93/1.52 => ( nonemptyI1
% 2.93/1.52 => ( setadjoinIL
% 2.93/1.52 => ( emptyinunitempty
% 2.93/1.52 => ( setadjoinIR
% 2.93/1.52 => ( setadjoinE
% 2.93/1.52 => ( setadjoinOr
% 2.93/1.52 => ( setoftrueEq
% 2.93/1.52 => ( powersetI
% 2.93/1.52 => ( emptyinPowerset
% 2.93/1.52 => ( emptyInPowerset
% 2.93/1.52 => ( powersetE
% 2.93/1.52 => ( setunionI
% 2.93/1.52 => ( setunionE
% 2.93/1.52 => ( subPowSU
% 2.93/1.52 => ( exuE2
% 2.93/1.52 => ( nonemptyImpWitness
% 2.93/1.52 => ( uniqinunit
% 2.93/1.52 => ( notinsingleton
% 2.93/1.52 => ( eqinunit
% 2.93/1.52 => ( singletonsswitch
% 2.93/1.52 => ( upairsetE
% 2.93/1.52 => ( upairsetIL
% 2.93/1.52 => ( upairsetIR
% 2.93/1.52 => ( emptyE1
% 2.93/1.52 => ( vacuousDall
% 2.93/1.52 => ( quantDeMorgan1
% 2.93/1.52 => ( quantDeMorgan2
% 2.93/1.52 => ( quantDeMorgan3
% 2.93/1.52 => ( quantDeMorgan4
% 2.93/1.52 => ( prop2setI
% 2.93/1.52 => ( prop2set2propI
% 2.93/1.52 => ( notdexE
% 2.93/1.52 => ( notdallE
% 2.93/1.52 => ( exuI1
% 2.93/1.52 => ( exuI3
% 2.93/1.52 => ( exuI2
% 2.93/1.52 => ( inCongP
% 2.93/1.52 => ( in__Cong
% 2.93/1.52 => ( exuE3u
% 2.93/1.52 => ( exu__Cong
% 2.93/1.52 => ( emptyset__Cong
% 2.93/1.52 => ( setadjoin__Cong
% 2.93/1.52 => ( powerset__Cong
% 2.93/1.52 => ( setunion__Cong
% 2.93/1.52 => ( omega__Cong
% 2.93/1.52 => ( exuEu
% 2.93/1.52 => ( descr__Cong
% 2.93/1.52 => ( dsetconstr__Cong
% 2.93/1.52 => ( subsetI1
% 2.93/1.52 => ( eqimpsubset2
% 2.93/1.52 => ( eqimpsubset1
% 2.93/1.52 => ( subsetI2
% 2.93/1.52 => ( emptysetsubset
% 2.93/1.52 => ( subsetE
% 2.93/1.52 => ( subsetE2
% 2.93/1.52 => ( notsubsetI
% 2.93/1.52 => ( notequalI1
% 2.93/1.52 => ( notequalI2
% 2.93/1.52 => ( subsetRefl
% 2.93/1.52 => ( subsetTrans
% 2.93/1.52 => ( setadjoinSub
% 2.93/1.52 => ( setadjoinSub2
% 2.93/1.52 => ( subset2powerset
% 2.93/1.52 => ( setextsub
% 2.93/1.52 => ( subsetemptysetimpeq
% 2.93/1.52 => ( powersetI1
% 2.93/1.52 => ( powersetE1
% 2.93/1.52 => ( inPowerset
% 2.93/1.52 => ( powersetsubset
% 2.93/1.52 => ( sepInPowerset
% 2.93/1.52 => ( sepSubset
% 2.93/1.52 => ( binunionIL
% 2.93/1.52 => ( upairset2IR
% 2.93/1.52 => ( binunionIR
% 2.93/1.52 => ( binunionEcases
% 2.93/1.52 => ( binunionE
% 2.93/1.52 => ( binunionLsub
% 2.93/1.52 => ( binunionRsub
% 2.93/1.52 => ( binintersectI
% 2.93/1.52 => ( binintersectSubset5
% 2.93/1.52 => ( binintersectEL
% 2.93/1.52 => ( binintersectLsub
% 2.93/1.52 => ( binintersectSubset2
% 2.93/1.52 => ( binintersectSubset3
% 2.93/1.52 => ( binintersectER
% 2.93/1.52 => ( disjointsetsI1
% 2.93/1.52 => ( binintersectRsub
% 2.93/1.52 => ( binintersectSubset4
% 2.93/1.52 => ( binintersectSubset1
% 2.93/1.52 => ( bs114d
% 2.93/1.52 => ( setminusI
% 2.93/1.52 => ( setminusEL
% 2.93/1.52 => ( setminusER
% 2.93/1.52 => ( setminusSubset2
% 2.93/1.52 => ( setminusERneg
% 2.93/1.52 => ( setminusELneg
% 2.93/1.52 => ( setminusILneg
% 2.93/1.52 => ( setminusIRneg
% 2.93/1.52 => ( setminusLsub
% 2.93/1.52 => ( setminusSubset1
% 2.93/1.52 => ( symdiffE
% 2.93/1.52 => ( symdiffI1
% 2.93/1.52 => ( symdiffI2
% 2.93/1.52 => ( symdiffIneg1
% 2.93/1.52 => ( symdiffIneg2
% 2.93/1.52 => ( secondinupair
% 2.93/1.52 => ( setukpairIL
% 2.93/1.52 => ( setukpairIR
% 2.93/1.52 => ( kpairiskpair
% 2.93/1.52 => ( kpairp
% 2.93/1.52 => ( singletonsubset
% 2.93/1.52 => ( singletoninpowerset
% 2.93/1.52 => ( singletoninpowunion
% 2.93/1.52 => ( upairset2E
% 2.93/1.52 => ( upairsubunion
% 2.93/1.52 => ( upairinpowunion
% 2.93/1.52 => ( ubforcartprodlem1
% 2.93/1.52 => ( ubforcartprodlem2
% 2.93/1.52 => ( ubforcartprodlem3
% 2.93/1.52 => ( cartprodpairin
% 2.93/1.52 => ( cartprodmempair1
% 2.93/1.52 => ( cartprodmempair
% 2.93/1.52 => ( setunionE2
% 2.93/1.52 => ( setunionsingleton1
% 2.93/1.52 => ( setunionsingleton2
% 2.93/1.52 => ( setunionsingleton
% 2.93/1.52 => ( singletonprop
% 2.93/1.52 => ( ex1E1
% 2.93/1.52 => ( ex1I
% 2.93/1.52 => ( ex1I2
% 2.93/1.52 => ( singletonsuniq
% 2.93/1.52 => ( setukpairinjL1
% 2.93/1.52 => ( kfstsingleton
% 2.93/1.52 => ( theprop
% 2.93/1.52 => ( kfstpairEq
% 2.93/1.52 => ( cartprodfstin
% 2.93/1.52 => ( setukpairinjL2
% 2.93/1.52 => ( setukpairinjL
% 2.93/1.52 => ( setukpairinjR11
% 2.93/1.52 => ( setukpairinjR12
% 2.93/1.52 => ( setukpairinjR1
% 2.93/1.52 => ( upairequniteq
% 2.93/1.52 => ( setukpairinjR2
% 2.93/1.52 => ( setukpairinjR
% 2.93/1.52 => ( ksndsingleton
% 2.93/1.52 => ( ksndpairEq
% 2.93/1.52 => ( kpairsurjEq
% 2.93/1.52 => ( cartprodsndin
% 2.93/1.52 => ( cartprodpairmemEL
% 2.93/1.52 => ( cartprodpairmemER
% 2.93/1.52 => ( cartprodmempaircEq
% 2.93/1.52 => ( cartprodfstpairEq
% 2.93/1.52 => ( cartprodsndpairEq
% 2.93/1.52 => ( cartprodpairsurjEq
% 2.93/1.52 => ( dpsetconstrI
% 2.93/1.52 => ( dpsetconstrSub
% 2.93/1.52 => ( setOfPairsIsBReln
% 2.93/1.52 => ( dpsetconstrERa
% 2.93/1.52 => ( dpsetconstrEL1
% 2.93/1.52 => ( dpsetconstrEL2
% 2.93/1.52 => ( dpsetconstrER
% 2.93/1.52 => ( funcImageSingleton
% 2.93/1.52 => ( apProp
% 2.93/1.52 => ( app
% 2.93/1.52 => ( infuncsetfunc
% 2.93/1.52 => ( ap2p
% 2.93/1.52 => ( funcinfuncset
% 2.93/1.52 => ( lamProp
% 2.93/1.52 => ( lamp
% 2.93/1.52 => ( lam2p
% 2.93/1.52 => ( brelnall1
% 2.93/1.52 => ( brelnall2
% 2.93/1.52 => ( ex1E2
% 2.93/1.52 => ( funcGraphProp1
% 2.93/1.52 => ( funcGraphProp3
% 2.93/1.52 => ( funcGraphProp2
% 2.93/1.52 => ( funcextLem
% 2.93/1.52 => ( funcGraphProp4
% 2.93/1.52 => ( subbreln
% 2.93/1.52 => ( eqbreln
% 2.93/1.52 => ( funcext
% 2.93/1.52 => ( funcext2
% 2.93/1.52 => ( ap2apEq1
% 2.93/1.52 => ( ap2apEq2
% 2.93/1.52 => ( beta1
% 2.93/1.52 => ( eta1
% 2.93/1.52 => ( lam2lamEq
% 2.93/1.52 => ( beta2
% 2.93/1.52 => ( eta2
% 2.93/1.52 => ( iffalseProp1
% 2.93/1.52 => ( iffalseProp2
% 2.93/1.52 => ( iftrueProp1
% 2.93/1.52 => ( iftrueProp2
% 2.93/1.52 => ( ifSingleton
% 2.93/1.52 => ( ifp
% 2.93/1.52 => ( theeq
% 2.93/1.52 => ( iftrue
% 2.93/1.52 => ( iffalse
% 2.93/1.52 => ( iftrueorfalse
% 2.93/1.52 => ( binintersectT_lem
% 2.93/1.52 => ( binunionT_lem
% 2.93/1.52 => ( powersetT_lem
% 2.93/1.52 => ( setminusT_lem
% 2.93/1.52 => ( complementT_lem
% 2.93/1.52 => ( setextT
% 2.93/1.52 => ( subsetTI
% 2.93/1.52 => ( powersetTI1
% 2.93/1.52 => ( powersetTE1
% 2.93/1.52 => ( complementTI1
% 2.93/1.52 => ( complementTE1
% 2.93/1.52 => ( binintersectTELcontra
% 2.93/1.52 => ( binintersectTERcontra
% 2.93/1.52 => ( contrasubsetT
% 2.93/1.52 => ( contrasubsetT1
% 2.93/1.52 => ( contrasubsetT2
% 2.93/1.52 => ( contrasubsetT3
% 2.93/1.52 => ( doubleComplementI1
% 2.93/1.52 => ( doubleComplementE1
% 2.93/1.52 => ( doubleComplementSub1
% 2.93/1.52 => ( doubleComplementSub2
% 2.93/1.52 => ( doubleComplementEq
% 2.93/1.52 => ( complementTnotintersectT
% 2.93/1.52 => ( complementImpComplementIntersect
% 2.93/1.52 => ( complementSubsetComplementIntersect
% 2.93/1.52 => ( complementInPowersetComplementIntersect
% 2.93/1.52 => ( contraSubsetComplement
% 2.93/1.52 => ( complementTcontraSubset
% 2.93/1.52 => ( binunionTILcontra
% 2.93/1.52 => ( binunionTIRcontra
% 2.93/1.52 => ( inIntersectImpInUnion
% 2.93/1.52 => ( inIntersectImpInUnion2
% 2.93/1.52 => ( inIntersectImpInIntersectUnions
% 2.93/1.52 => ( intersectInPowersetIntersectUnions
% 2.93/1.52 => ( inComplementUnionImpNotIn1
% 2.93/1.52 => ( inComplementUnionImpInComplement1
% 2.93/1.52 => ( binunionTE
% 2.93/1.52 => ( binunionTEcontra
% 2.93/1.52 => ( demorgan2a1
% 2.93/1.52 => ( complementUnionInPowersetComplement
% 2.93/1.52 => ( demorgan2a2
% 2.93/1.52 => ( demorgan1a
% 2.93/1.52 => ( demorgan1b
% 2.93/1.52 => ( demorgan1
% 2.93/1.52 => ( demorgan2a
% 2.93/1.52 => ( demorgan2b2
% 2.93/1.52 => ( demorgan2b
% 2.93/1.52 => ( demorgan2
% 2.93/1.52 => ( woz13rule0
% 2.93/1.52 => ( woz13rule1
% 2.93/1.52 => ( woz13rule2
% 2.93/1.52 => ( woz13rule3
% 2.93/1.52 => ( woz13rule4
% 2.93/1.52 => ( woz1_1
% 2.93/1.52 => ( woz1_2
% 2.93/1.52 => ( woz1_3
% 2.93/1.52 => ( woz1_4
% 2.93/1.52 => ( woz1_5
% 2.93/1.52 => ( breln1all2
% 2.93/1.52 => ( breln1SetBreln1
% 2.93/1.52 => ( choice2fnsingleton
% 2.93/1.52 => ! [A: $i,Xphi: $i > $i > $o] :
% 2.93/1.52 ( breln1 @ A
% 2.93/1.52 @ ( dpsetconstr @ A @ A
% 2.93/1.52 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.93/1.52 ------------------------
% 2.93/1.56 % (512)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_390 for (2999ds/4Mi)
% 2.93/1.56 % (511)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/183Mi)
% 2.93/1.56 % (514)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/2Mi)
% 2.93/1.56 % (512)Instruction limit reached!
% 2.93/1.56 % (512)------------------------------
% 2.93/1.56 % (512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.56 % (512)Termination reason: Unknown
% 2.93/1.56 % (512)Termination phase: shuffling
% 2.93/1.56
% 2.93/1.56 % (512)Memory used [KB]: 1535
% 2.93/1.56 % (512)Time elapsed: 0.003 s
% 2.93/1.56 % (512)Instructions burned: 6 (million)
% 2.93/1.56 % (512)------------------------------
% 2.93/1.56 % (512)------------------------------
% 2.93/1.56 % (514)Instruction limit reached!
% 2.93/1.56 % (514)------------------------------
% 2.93/1.56 % (514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.56 % (514)Termination reason: Unknown
% 2.93/1.56 % (514)Termination phase: shuffling
% 2.93/1.56
% 2.93/1.56 % (514)Memory used [KB]: 1535
% 2.93/1.56 % (514)Time elapsed: 0.002 s
% 2.93/1.56 % (514)Instructions burned: 3 (million)
% 2.93/1.56 % (514)------------------------------
% 2.93/1.56 % (514)------------------------------
% 2.93/1.56 % (520)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/18Mi)
% 2.93/1.56 % (513)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_390 for (2999ds/27Mi)
% 2.93/1.56 % (519)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_390 for (2999ds/275Mi)
% 2.93/1.57 % (518)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_390 for (2999ds/2Mi)
% 2.93/1.57 % (520)Instruction limit reached!
% 2.93/1.57 % (520)------------------------------
% 2.93/1.57 % (520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.57 % (520)Termination reason: Unknown
% 2.93/1.57 % (520)Termination phase: shuffling
% 2.93/1.57
% 2.93/1.57 % (520)Memory used [KB]: 1791
% 2.93/1.57 % (520)Time elapsed: 0.008 s
% 2.93/1.57 % (520)Instructions burned: 20 (million)
% 2.93/1.57 % (520)------------------------------
% 2.93/1.57 % (520)------------------------------
% 2.93/1.57 % (518)Instruction limit reached!
% 2.93/1.57 % (518)------------------------------
% 2.93/1.57 % (518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.57 % (518)Termination reason: Unknown
% 2.93/1.57 % (518)Termination phase: shuffling
% 2.93/1.57
% 2.93/1.57 % (518)Memory used [KB]: 1535
% 2.93/1.57 % (518)Time elapsed: 0.004 s
% 2.93/1.57 % (518)Instructions burned: 3 (million)
% 2.93/1.57 % (518)------------------------------
% 2.93/1.57 % (518)------------------------------
% 2.93/1.57 % (521)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/3Mi)
% 2.93/1.57 % (522)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_390 for (2999ds/37Mi)
% 2.93/1.57 % (521)Instruction limit reached!
% 2.93/1.57 % (521)------------------------------
% 2.93/1.57 % (521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.57 % (521)Termination reason: Unknown
% 2.93/1.57 % (521)Termination phase: shuffling
% 2.93/1.57
% 2.93/1.57 % (521)Memory used [KB]: 1535
% 2.93/1.57 % (521)Time elapsed: 0.002 s
% 2.93/1.57 % (521)Instructions burned: 3 (million)
% 2.93/1.57 % (521)------------------------------
% 2.93/1.57 % (521)------------------------------
% 2.93/1.58 % (523)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_390 for (2999ds/15Mi)
% 2.93/1.58 % (522)Instruction limit reached!
% 2.93/1.58 % (522)------------------------------
% 2.93/1.58 % (522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.58 % (522)Termination reason: Unknown
% 2.93/1.58 % (522)Termination phase: shuffling
% 2.93/1.58
% 2.93/1.58 % (522)Memory used [KB]: 2302
% 2.93/1.58 % (522)Time elapsed: 0.011 s
% 2.93/1.58 % (522)Instructions burned: 38 (million)
% 2.93/1.58 % (522)------------------------------
% 2.93/1.58 % (522)------------------------------
% 2.93/1.58 % (525)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_390 for (2999ds/1041Mi)
% 2.93/1.58 % (513)Instruction limit reached!
% 2.93/1.58 % (513)------------------------------
% 2.93/1.58 % (513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.58 % (513)Termination reason: Unknown
% 2.93/1.58 % (513)Termination phase: shuffling
% 2.93/1.58
% 2.93/1.58 % (513)Memory used [KB]: 2046
% 2.93/1.58 % (513)Time elapsed: 0.017 s
% 2.93/1.58 % (513)Instructions burned: 28 (million)
% 2.93/1.58 % (513)------------------------------
% 2.93/1.58 % (513)------------------------------
% 2.93/1.58 % (523)Instruction limit reached!
% 2.93/1.58 % (523)------------------------------
% 2.93/1.58 % (523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.58 % (523)Termination reason: Unknown
% 2.93/1.58 % (523)Termination phase: shuffling
% 2.93/1.58
% 2.93/1.58 % (523)Memory used [KB]: 1791
% 2.93/1.58 % (523)Time elapsed: 0.006 s
% 2.93/1.58 % (523)Instructions burned: 16 (million)
% 2.93/1.58 % (523)------------------------------
% 2.93/1.58 % (523)------------------------------
% 2.93/1.58 % (524)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_390 for (2999ds/3Mi)
% 2.93/1.59 % (524)Instruction limit reached!
% 2.93/1.59 % (524)------------------------------
% 2.93/1.59 % (524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (524)Termination reason: Unknown
% 2.93/1.59 % (524)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (524)Memory used [KB]: 1535
% 2.93/1.59 % (524)Time elapsed: 0.004 s
% 2.93/1.59 % (524)Instructions burned: 5 (million)
% 2.93/1.59 % (524)------------------------------
% 2.93/1.59 % (524)------------------------------
% 2.93/1.59 % (526)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/7Mi)
% 2.93/1.59 % (526)Instruction limit reached!
% 2.93/1.59 % (526)------------------------------
% 2.93/1.59 % (526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (526)Termination reason: Unknown
% 2.93/1.59 % (526)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (526)Memory used [KB]: 1535
% 2.93/1.59 % (526)Time elapsed: 0.003 s
% 2.93/1.59 % (526)Instructions burned: 7 (million)
% 2.93/1.59 % (526)------------------------------
% 2.93/1.59 % (526)------------------------------
% 2.93/1.59 % (528)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_390 for (2999ds/3Mi)
% 2.93/1.59 % (528)Instruction limit reached!
% 2.93/1.59 % (528)------------------------------
% 2.93/1.59 % (528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.93/1.59 % (528)Termination reason: Unknown
% 2.93/1.59 % (528)Termination phase: shuffling
% 2.93/1.59
% 2.93/1.59 % (528)Memory used [KB]: 1535
% 2.93/1.59 % (528)Time elapsed: 0.002 s
% 2.93/1.59 % (528)Instructions burned: 3 (million)
% 2.93/1.59 % (528)------------------------------
% 2.93/1.59 % (528)------------------------------
% 2.96/1.60 % (527)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/16Mi)
% 2.96/1.60 % (529)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_390 for (2999ds/3Mi)
% 2.96/1.60 % (530)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/7Mi)
% 2.96/1.60 % (529)Instruction limit reached!
% 2.96/1.60 % (529)------------------------------
% 2.96/1.60 % (529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.60 % (529)Termination reason: Unknown
% 2.96/1.60 % (529)Termination phase: shuffling
% 2.96/1.60
% 2.96/1.60 % (529)Memory used [KB]: 1535
% 2.96/1.60 % (529)Time elapsed: 0.003 s
% 2.96/1.60 % (529)Instructions burned: 6 (million)
% 2.96/1.60 % (529)------------------------------
% 2.96/1.60 % (529)------------------------------
% 2.96/1.60 % (531)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/3Mi)
% 2.96/1.60 % (530)Instruction limit reached!
% 2.96/1.60 % (530)------------------------------
% 2.96/1.60 % (530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.60 % (530)Termination reason: Unknown
% 2.96/1.60 % (530)Termination phase: shuffling
% 2.96/1.60
% 2.96/1.60 % (530)Memory used [KB]: 1535
% 2.96/1.60 % (530)Time elapsed: 0.003 s
% 2.96/1.60 % (530)Instructions burned: 7 (million)
% 2.96/1.60 % (530)------------------------------
% 2.96/1.60 % (530)------------------------------
% 2.96/1.60 % (531)Instruction limit reached!
% 2.96/1.60 % (531)------------------------------
% 2.96/1.60 % (531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.60 % (531)Termination reason: Unknown
% 2.96/1.60 % (531)Termination phase: shuffling
% 2.96/1.60
% 2.96/1.60 % (531)Memory used [KB]: 1535
% 2.96/1.60 % (531)Time elapsed: 0.002 s
% 2.96/1.60 % (531)Instructions burned: 4 (million)
% 2.96/1.60 % (531)------------------------------
% 2.96/1.60 % (531)------------------------------
% 2.96/1.60 % (527)Instruction limit reached!
% 2.96/1.60 % (527)------------------------------
% 2.96/1.60 % (527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.60 % (511)Instruction limit reached!
% 2.96/1.60 % (511)------------------------------
% 2.96/1.60 % (511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.60 % (511)Termination reason: Unknown
% 2.96/1.60 % (511)Termination phase: Saturation
% 2.96/1.60
% 2.96/1.60 % (511)Memory used [KB]: 7419
% 2.96/1.60 % (511)Time elapsed: 0.047 s
% 2.96/1.60 % (511)Instructions burned: 183 (million)
% 2.96/1.60 % (511)------------------------------
% 2.96/1.60 % (511)------------------------------
% 2.96/1.60 % (527)Termination reason: Unknown
% 2.96/1.60 % (527)Termination phase: shuffling
% 2.96/1.60
% 2.96/1.60 % (527)Memory used [KB]: 1918
% 2.96/1.60 % (527)Time elapsed: 0.007 s
% 2.96/1.60 % (527)Instructions burned: 16 (million)
% 2.96/1.60 % (527)------------------------------
% 2.96/1.60 % (527)------------------------------
% 2.96/1.61 % (532)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/4Mi)
% 2.96/1.61 % (533)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/18Mi)
% 2.96/1.61 % (532)Instruction limit reached!
% 2.96/1.61 % (532)------------------------------
% 2.96/1.61 % (532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.61 % (532)Termination reason: Unknown
% 2.96/1.61 % (532)Termination phase: shuffling
% 2.96/1.61
% 2.96/1.61 % (532)Memory used [KB]: 1535
% 2.96/1.61 % (534)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/710Mi)
% 2.96/1.61 % (532)Time elapsed: 0.002 s
% 2.96/1.61 % (532)Instructions burned: 4 (million)
% 2.96/1.61 % (532)------------------------------
% 2.96/1.61 % (532)------------------------------
% 2.96/1.61 % (535)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/6Mi)
% 2.96/1.61 % (536)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_390 for (2999ds/902Mi)
% 2.96/1.61 % (535)Instruction limit reached!
% 2.96/1.61 % (535)------------------------------
% 2.96/1.61 % (535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.61 % (535)Termination reason: Unknown
% 2.96/1.61 % (535)Termination phase: shuffling
% 2.96/1.61
% 2.96/1.61 % (535)Memory used [KB]: 1535
% 2.96/1.61 % (535)Time elapsed: 0.003 s
% 2.96/1.61 % (535)Instructions burned: 7 (million)
% 2.96/1.61 % (535)------------------------------
% 2.96/1.61 % (535)------------------------------
% 2.96/1.61 % (533)Instruction limit reached!
% 2.96/1.61 % (533)------------------------------
% 2.96/1.61 % (533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.61 % (533)Termination reason: Unknown
% 2.96/1.61 % (533)Termination phase: shuffling
% 2.96/1.61
% 2.96/1.62 % (533)Memory used [KB]: 1791
% 2.96/1.62 % (533)Time elapsed: 0.006 s
% 2.96/1.62 % (533)Instructions burned: 19 (million)
% 2.96/1.62 % (533)------------------------------
% 2.96/1.62 % (533)------------------------------
% 2.96/1.62 % (537)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_390 for (2999ds/21Mi)
% 2.96/1.62 % (538)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_390 for (2999ds/5Mi)
% 2.96/1.62 % (539)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_390 for (2999ds/6Mi)
% 2.96/1.62 % (538)Instruction limit reached!
% 2.96/1.62 % (538)------------------------------
% 2.96/1.62 % (538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.62 % (538)Termination reason: Unknown
% 2.96/1.62 % (538)Termination phase: shuffling
% 2.96/1.63
% 2.96/1.63 % (538)Memory used [KB]: 1663
% 2.96/1.63 % (538)Time elapsed: 0.003 s
% 2.96/1.63 % (538)Instructions burned: 7 (million)
% 2.96/1.63 % (538)------------------------------
% 2.96/1.63 % (538)------------------------------
% 2.96/1.63 % (539)Instruction limit reached!
% 2.96/1.63 % (539)------------------------------
% 2.96/1.63 % (539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.63 % (539)Termination reason: Unknown
% 2.96/1.63 % (539)Termination phase: shuffling
% 2.96/1.63
% 2.96/1.63 % (539)Memory used [KB]: 1535
% 2.96/1.63 % (539)Time elapsed: 0.003 s
% 2.96/1.63 % (539)Instructions burned: 7 (million)
% 2.96/1.63 % (539)------------------------------
% 2.96/1.63 % (539)------------------------------
% 2.96/1.63 % (537)Instruction limit reached!
% 2.96/1.63 % (537)------------------------------
% 2.96/1.63 % (537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.63 % (537)Termination reason: Unknown
% 2.96/1.63 % (537)Termination phase: shuffling
% 2.96/1.63
% 2.96/1.63 % (537)Memory used [KB]: 1918
% 2.96/1.63 % (537)Time elapsed: 0.007 s
% 2.96/1.63 % (537)Instructions burned: 21 (million)
% 2.96/1.63 % (537)------------------------------
% 2.96/1.63 % (537)------------------------------
% 2.96/1.63 % (540)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_390 for (2998ds/377Mi)
% 2.96/1.64 % (541)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_390 for (2998ds/779Mi)
% 2.96/1.64 % (542)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_390 for (2998ds/19Mi)
% 2.96/1.64 % (542)Instruction limit reached!
% 2.96/1.64 % (542)------------------------------
% 2.96/1.64 % (542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.64 % (542)Termination reason: Unknown
% 2.96/1.64 % (542)Termination phase: shuffling
% 2.96/1.64
% 2.96/1.64 % (542)Memory used [KB]: 1918
% 2.96/1.64 % (542)Time elapsed: 0.007 s
% 2.96/1.64 % (542)Instructions burned: 20 (million)
% 2.96/1.64 % (542)------------------------------
% 2.96/1.64 % (542)------------------------------
% 2.96/1.65 % (543)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_390 for (2998ds/879Mi)
% 2.96/1.69 % (519)Instruction limit reached!
% 2.96/1.69 % (519)------------------------------
% 2.96/1.69 % (519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.69 % (519)Termination reason: Unknown
% 2.96/1.69 % (519)Termination phase: Saturation
% 2.96/1.69
% 2.96/1.69 % (519)Memory used [KB]: 10106
% 2.96/1.69 % (519)Time elapsed: 0.132 s
% 2.96/1.69 % (519)Instructions burned: 277 (million)
% 2.96/1.69 % (519)------------------------------
% 2.96/1.69 % (519)------------------------------
% 2.96/1.71 % (544)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_390 for (2998ds/17Mi)
% 2.96/1.71 % (544)Instruction limit reached!
% 2.96/1.71 % (544)------------------------------
% 2.96/1.71 % (544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.71 % (544)Termination reason: Unknown
% 2.96/1.71 % (544)Termination phase: shuffling
% 2.96/1.71
% 2.96/1.71 % (544)Memory used [KB]: 1918
% 2.96/1.71 % (544)Time elapsed: 0.007 s
% 2.96/1.71 % (544)Instructions burned: 19 (million)
% 2.96/1.71 % (544)------------------------------
% 2.96/1.71 % (544)------------------------------
% 2.96/1.72 % (545)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_390 for (2998ds/3Mi)
% 2.96/1.72 % (545)Instruction limit reached!
% 2.96/1.72 % (545)------------------------------
% 2.96/1.72 % (545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.72 % (545)Termination reason: Unknown
% 2.96/1.72 % (545)Termination phase: shuffling
% 2.96/1.72
% 2.96/1.72 % (545)Memory used [KB]: 1535
% 2.96/1.72 % (545)Time elapsed: 0.002 s
% 2.96/1.72 % (545)Instructions burned: 4 (million)
% 2.96/1.72 % (545)------------------------------
% 2.96/1.72 % (545)------------------------------
% 2.96/1.73 % (546)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_390 for (2997ds/30Mi)
% 2.96/1.73 % (540)Instruction limit reached!
% 2.96/1.73 % (540)------------------------------
% 2.96/1.73 % (540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.73 % (540)Termination reason: Unknown
% 2.96/1.73 % (540)Termination phase: Saturation
% 2.96/1.73
% 2.96/1.73 % (540)Memory used [KB]: 9466
% 2.96/1.73 % (540)Time elapsed: 0.100 s
% 2.96/1.73 % (540)Instructions burned: 379 (million)
% 2.96/1.73 % (540)------------------------------
% 2.96/1.73 % (540)------------------------------
% 2.96/1.74 % (536)Refutation not found, incomplete strategy
% 2.96/1.74 % (536)------------------------------
% 2.96/1.74 % (536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.74 % (536)Termination reason: Refutation not found, incomplete strategy
% 2.96/1.74
% 2.96/1.74
% 2.96/1.74 % (536)Memory used [KB]: 10746
% 2.96/1.74 % (536)Time elapsed: 0.152 s
% 2.96/1.74 % (536)Instructions burned: 401 (million)
% 2.96/1.74 % (536)------------------------------
% 2.96/1.74 % (536)------------------------------
% 2.96/1.74 % (546)Instruction limit reached!
% 2.96/1.74 % (546)------------------------------
% 2.96/1.74 % (546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.74 % (546)Termination reason: Unknown
% 2.96/1.74 % (546)Termination phase: shuffling
% 2.96/1.74
% 2.96/1.74 % (546)Memory used [KB]: 2174
% 2.96/1.74 % (546)Time elapsed: 0.010 s
% 2.96/1.74 % (546)Instructions burned: 31 (million)
% 2.96/1.74 % (546)------------------------------
% 2.96/1.74 % (546)------------------------------
% 2.96/1.74 % (547)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_390 for (2997ds/127Mi)
% 2.96/1.75 % (548)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_390 for (2997ds/100Mi)
% 2.96/1.75 % (549)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_390 for (2997ds/3Mi)
% 2.96/1.75 % (549)Instruction limit reached!
% 2.96/1.75 % (549)------------------------------
% 2.96/1.75 % (549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.75 % (549)Termination reason: Unknown
% 2.96/1.75 % (549)Termination phase: shuffling
% 2.96/1.75
% 2.96/1.75 % (549)Memory used [KB]: 1535
% 2.96/1.75 % (549)Time elapsed: 0.002 s
% 2.96/1.75 % (549)Instructions burned: 3 (million)
% 2.96/1.75 % (549)------------------------------
% 2.96/1.75 % (549)------------------------------
% 2.96/1.76 % (550)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_390 for (2997ds/20Mi)
% 2.96/1.77 % (550)Instruction limit reached!
% 2.96/1.77 % (550)------------------------------
% 2.96/1.77 % (550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.77 % (550)Termination reason: Unknown
% 2.96/1.77 % (550)Termination phase: shuffling
% 2.96/1.77
% 2.96/1.77 % (550)Memory used [KB]: 1791
% 2.96/1.77 % (550)Time elapsed: 0.008 s
% 2.96/1.77 % (550)Instructions burned: 20 (million)
% 2.96/1.77 % (550)------------------------------
% 2.96/1.77 % (550)------------------------------
% 2.96/1.78 % (547)Instruction limit reached!
% 2.96/1.78 % (547)------------------------------
% 2.96/1.78 % (547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.78 % (547)Termination reason: Unknown
% 2.96/1.78 % (547)Termination phase: Property scanning
% 2.96/1.78
% 2.96/1.78 % (547)Memory used [KB]: 2686
% 2.96/1.78 % (547)Time elapsed: 0.033 s
% 2.96/1.78 % (547)Instructions burned: 127 (million)
% 2.96/1.78 % (547)------------------------------
% 2.96/1.78 % (547)------------------------------
% 2.96/1.78 % (543)First to succeed.
% 2.96/1.78 % (548)Instruction limit reached!
% 2.96/1.78 % (548)------------------------------
% 2.96/1.78 % (548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.78 % (548)Termination reason: Unknown
% 2.96/1.78 % (548)Termination phase: Property scanning
% 2.96/1.78
% 2.96/1.78 % (548)Memory used [KB]: 2686
% 2.96/1.78 % (548)Time elapsed: 0.029 s
% 2.96/1.78 % (548)Instructions burned: 101 (million)
% 2.96/1.78 % (548)------------------------------
% 2.96/1.78 % (548)------------------------------
% 2.96/1.78 % (543)Refutation found. Thanks to Tanya!
% 2.96/1.78 % SZS status Theorem for DTF2THF_390
% 2.96/1.78 % SZS output start Proof for DTF2THF_390
% 2.96/1.78 thf(func_def_0, type, in: $i > $i > $o).
% 2.96/1.78 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.96/1.78 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.96/1.78 thf(func_def_8, type, powerset: $i > $i).
% 2.96/1.78 thf(func_def_10, type, setunion: $i > $i).
% 2.96/1.78 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.96/1.78 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_26, type, prop2set: $o > $i).
% 2.96/1.78 thf(func_def_36, type, nonempty: $i > $o).
% 2.96/1.78 thf(func_def_69, type, set2prop: $i > $o).
% 2.96/1.78 thf(func_def_88, type, subset: $i > $i > $o).
% 2.96/1.78 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.96/1.78 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.96/1.78 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.96/1.78 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.96/1.78 thf(func_def_135, type, regular: $i > $o).
% 2.96/1.78 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.96/1.78 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.96/1.78 thf(func_def_153, type, iskpair: $i > $o).
% 2.96/1.78 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.96/1.78 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.96/1.78 thf(func_def_177, type, singleton: $i > $o).
% 2.96/1.78 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.96/1.78 thf(func_def_184, type, atmost1p: $i > $o).
% 2.96/1.78 thf(func_def_185, type, atleast2p: $i > $o).
% 2.96/1.78 thf(func_def_186, type, atmost2p: $i > $o).
% 2.96/1.78 thf(func_def_187, type, upairsetp: $i > $o).
% 2.96/1.78 thf(func_def_191, type, kfst: $i > $i).
% 2.96/1.78 thf(func_def_203, type, ksnd: $i > $i).
% 2.96/1.78 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.96/1.78 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.96/1.78 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.96/1.78 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.96/1.78 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.96/1.78 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.96/1.78 thf(func_def_323, type, breln1: $i > $i > $o).
% 2.96/1.78 thf(func_def_325, type, breln1Set: $i > $i).
% 2.96/1.78 thf(func_def_327, type, transitive: $i > $i > $o).
% 2.96/1.78 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 2.96/1.78 thf(func_def_329, type, reflexive: $i > $i > $o).
% 2.96/1.78 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 2.96/1.78 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 2.96/1.78 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 2.96/1.78 thf(func_def_352, type, sK0: $i > $i > $o).
% 2.96/1.78 thf(func_def_355, type, sK3: $i > $i > $i).
% 2.96/1.78 thf(func_def_356, type, sK4: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_357, type, sK5: $i > $i > ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_358, type, sK6: $i > $i > $i > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_359, type, sK7: $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_360, type, sK8: $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_361, type, sK9: $i > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_362, type, sK10: $i > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_363, type, sK11: $i > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_364, type, sK12: $i > ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_365, type, sK13: $i > ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_366, type, sK14: ($i > $i > $o) > $i > $i > $i).
% 2.96/1.78 thf(func_def_367, type, sK15: ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_368, type, sK16: $i > $i > ($i > $o) > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_369, type, sK17: $i > ($i > $o) > ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_370, type, sK18: $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_371, type, sK19: $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_372, type, sK20: $i > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_373, type, sK21: $i > $i).
% 2.96/1.78 thf(func_def_374, type, sK22: ($i > $o) > $i).
% 2.96/1.78 thf(func_def_375, type, sK23: $i > $i).
% 2.96/1.78 thf(func_def_376, type, sK24: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_377, type, sK25: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_378, type, sK26: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_379, type, sK27: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_380, type, sK28: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_381, type, sK29: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_382, type, sK30: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_383, type, sK31: ($i > $o) > $i).
% 2.96/1.78 thf(func_def_384, type, sK32: ($i > $o) > $i).
% 2.96/1.78 thf(func_def_385, type, sK33: ($i > $i) > $i > $i > $i).
% 2.96/1.78 thf(func_def_386, type, sK34: $i > $i > $i).
% 2.96/1.78 thf(func_def_387, type, sK35: $i > $i).
% 2.96/1.78 thf(func_def_388, type, sK36: $i > $i > $i > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_389, type, sK37: $i > $i > ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_390, type, sK38: $i > $i > $i).
% 2.96/1.78 thf(func_def_391, type, sK39: $i > $i > $i).
% 2.96/1.78 thf(func_def_392, type, sK40: ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_393, type, sK41: $i > $i > ($i > $i) > $i).
% 2.96/1.78 thf(func_def_394, type, sK42: $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_395, type, sK43: ($i > $o) > $i).
% 2.96/1.78 thf(func_def_396, type, sK44: $i > $i > $i).
% 2.96/1.78 thf(func_def_397, type, sK45: $i > $i > ($i > $i) > $i).
% 2.96/1.78 thf(func_def_398, type, sK46: ($i > $o) > $i > $i > $i).
% 2.96/1.78 thf(func_def_399, type, sK47: ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_400, type, sK48: $i > $i).
% 2.96/1.78 thf(func_def_401, type, sK49: $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_402, type, sK50: $i > $i > $i).
% 2.96/1.78 thf(func_def_403, type, sK51: $i > $i > $i).
% 2.96/1.78 thf(func_def_404, type, sK52: $i > $i > $i).
% 2.96/1.78 thf(func_def_405, type, sK53: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_406, type, sK54: ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_407, type, sK55: ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_408, type, sK56: ($i > $o) > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_409, type, sK57: ($i > $o) > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_410, type, sK58: $i > $i).
% 2.96/1.78 thf(func_def_411, type, sK59: $i > $i > $i).
% 2.96/1.78 thf(func_def_412, type, sK60: $i > $i > $i).
% 2.96/1.78 thf(func_def_413, type, sK61: $i > ($i > $i) > $i > $i).
% 2.96/1.78 thf(func_def_414, type, sK62: $i > $i > $i).
% 2.96/1.78 thf(func_def_415, type, sK63: $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_416, type, sK64: $i > $i > $i).
% 2.96/1.78 thf(func_def_417, type, sK65: $i > $i > $o > $i).
% 2.96/1.78 thf(func_def_418, type, sK66: ($i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_419, type, sK67: ($i > $o) > $i).
% 2.96/1.78 thf(func_def_420, type, sK68: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_421, type, sK69: $i > $i).
% 2.96/1.78 thf(func_def_422, type, sK70: $i > $i).
% 2.96/1.78 thf(func_def_423, type, sK71: $i > $i > ($i > $i) > $i).
% 2.96/1.78 thf(func_def_424, type, sK72: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_425, type, sK73: $i > $i > $i > $i > $i).
% 2.96/1.78 thf(func_def_426, type, sK74: $i > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_427, type, sK75: ($i > $o) > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_428, type, sK76: ($i > $o) > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_429, type, sK77: $i > $i > $i).
% 2.96/1.78 thf(func_def_430, type, sK78: $i > ($i > $i > $o) > $i).
% 2.96/1.78 thf(func_def_431, type, sK79: $i > ($i > $i > $o) > $i).
% 2.96/1.78 thf(func_def_432, type, sK80: $i > ($i > $i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_433, type, sK81: $i > ($i > $i > $o) > $i > $i).
% 2.96/1.78 thf(func_def_434, type, sK82: $i > ($i > $o) > $i).
% 2.96/1.78 thf(func_def_435, type, sK83: ($i > $o) > $i > $i).
% 2.96/1.78 thf(f4777,plain,(
% 2.96/1.78 $false),
% 2.96/1.78 inference(subsumption_resolution,[status(thm)],[f3803,f4656])).
% 2.96/1.78 thf(f4656,plain,(
% 2.96/1.78 ( ! [X2 : $i > $i > $o,X3 : $i,X1 : $i] : (((breln @ X1 @ X3 @ (dpsetconstr @ X1 @ X3 @ X2)) = $true)) )),
% 2.96/1.78 inference(beta_eta_normalization,[status(thm)],[f4655])).
% 2.96/1.78 thf(f4655,plain,(
% 2.96/1.78 ( ! [X2 : $i > $i > $o,X3 : $i,X1 : $i] : ((((^[Y0 : $i]: (breln @ X1 @ Y0 @ (dpsetconstr @ X1 @ Y0 @ X2))) @ X3) = $true)) )),
% 2.96/1.78 inference(pi_clausification,[status(thm)],[f4654])).
% 2.96/1.78 thf(f4654,plain,(
% 2.96/1.78 ( ! [X2 : $i > $i > $o,X1 : $i] : (($true = (!! @ $i @ (^[Y0 : $i]: (breln @ X1 @ Y0 @ (dpsetconstr @ X1 @ Y0 @ X2)))))) )),
% 2.96/1.78 inference(beta_eta_normalization,[status(thm)],[f4653])).
% 2.96/1.78 thf(f4653,plain,(
% 2.96/1.78 ( ! [X2 : $i > $i > $o,X1 : $i] : ((((^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (breln @ X1 @ Y1 @ (dpsetconstr @ X1 @ Y1 @ Y0))))) @ X2) = $true)) )),
% 2.96/1.78 inference(pi_clausification,[status(thm)],[f4652])).
% 2.96/1.78 thf(f4652,plain,(
% 2.96/1.78 ( ! [X1 : $i] : (((!! @ ($i > $i > $o) @ (^[Y0 : $i > $i > $o]: (!! @ $i @ (^[Y1 : $i]: (breln @ X1 @ Y1 @ (dpsetconstr @ X1 @ Y1 @ Y0)))))) = $true)) )),
% 2.96/1.78 inference(beta_eta_normalization,[status(thm)],[f4651])).
% 2.96/1.78 thf(f4651,plain,(
% 2.96/1.78 ( ! [X1 : $i] : (($true = ((^[Y0 : $i]: (!! @ ($i > $i > $o) @ (^[Y1 : $i > $i > $o]: (!! @ $i @ (^[Y2 : $i]: (breln @ Y0 @ Y2 @ (dpsetconstr @ Y0 @ Y2 @ Y1))))))) @ X1))) )),
% 2.96/1.78 inference(pi_clausification,[status(thm)],[f4650])).
% 2.96/1.78 thf(f4650,plain,(
% 2.96/1.78 ($true = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $i > $o) @ (^[Y1 : $i > $i > $o]: (!! @ $i @ (^[Y2 : $i]: (breln @ Y0 @ Y2 @ (dpsetconstr @ Y0 @ Y2 @ Y1)))))))))),
% 2.96/1.78 inference(beta_eta_normalization,[status(thm)],[f1742])).
% 2.96/1.78 thf(f1742,plain,(
% 2.96/1.78 ((!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $i > $o) @ (^[Y1 : $i > $i > $o]: (!! @ $i @ (^[Y2 : $i]: (breln @ Y0 @ Y2 @ (dpsetconstr @ Y0 @ Y2 @ (^[Y3 : $i]: ((^[Y4 : $i]: (Y1 @ Y3 @ Y4)))))))))))) = $true)),
% 2.96/1.78 inference(definition_unfolding,[status(thm)],[f1452,f1178])).
% 2.96/1.78 thf(f1178,plain,(
% 2.96/1.78 (setOfPairsIsBReln = $true)),
% 2.96/1.78 inference(cnf_transformation,[status(thm)],[f876])).
% 2.96/1.78 thf(f876,plain,(
% 2.96/1.78 (setunionI = $true) & (replAx = $true) & (complementTE1 = $true) & (ksndsingleton = $true) & (setukpairinjR = $true) & (doubleComplementI1 = $true) & (kfstpairEq = $true) & (binintersectSubset1 = $true) & (funcext2 = $true) & (cartprodsndin = $true) & (powersetE1 = $true) & (binunionTE = $true) & (kpairp = $true) & (dpsetconstrERa = $true) & (uniqinunit = $true) & (upairequniteq = $true) & (dsetconstrI = $true) & (binintersectSubset2 = $true) & (cartprodmempair = $true) & (cartprodpairmemEL = $true) & (ap2apEq2 = $true) & (binintersectSubset4 = $true) & (descrp = $true) & (upairinpowunion = $true) & (kpairsurjEq = $true) & (subPowSU = $true) & (ubforcartprodlem2 = $true) & (singletonsubset = $true) & (powersetI = $true) & (inCongP = $true) & (dpsetconstrEL2 = $true) & (ifp = $true) & (powersetI1 = $true) & (choice2fnsingleton = $true) & (demorgan2a = $true) & (upairsetIR = $true) & (setadjoin__Cong = $true) & (app = $true) & (inIntersectImpInIntersectUnions = $true) & (singletonsswitch = $true) & (woz13rule2 = $true) & (ap2p = $true) & (exuE3e = $true) & (setminusIRneg = $true) & (inPowerset = $true) & (setminusERneg = $true) & (subsetTrans = $true) & (subsetI1 = $true) & (upairset2E = $true) & (setunionE = $true) & (setunion__Cong = $true) & (woz1_1 = $true) & (notdexE = $true) & (setukpairinjR2 = $true) & (subbreln = $true) & (setminusSubset1 = $true) & (beta1 = $true) & (setminusSubset2 = $true) & (powersetsubset = $true) & (ap2apEq1 = $true) & (binunionTEcontra = $true) & (prop2set2propI = $true) & (omegaSAx = $true) & (eta2 = $true) & ((breln1 @ sK1 @ (dpsetconstr @ sK1 @ sK1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK0 @ Y0 @ Y1)))))) != $true) & (dsetconstrER = $true) & (powersetTE1 = $true) & (inIntersectImpInUnion = $true) & (notinsingleton = $true) & (exuE3u = $true) & (secondinupair = $true) & (quantDeMorgan3 = $true) & (doubleComplementE1 = $true) & (upairsetE = $true) & (binunionLsub = $true) & (dpsetconstrSub = $true) & (binintersectEL = $true) & (binintersectRsub = $true) & (setminusLsub = $true) & (brelnall1 = $true) & (lam2p = $true) & (binunionRsub = $true) & (inComplementUnionImpNotIn1 = $true) & (cartprodfstin = $true) & (eqinunit = $true) & (demorgan2 = $true) & (inComplementUnionImpInComplement1 = $true) & (upairset2IR = $true) & (contrasubsetT3 = $true) & (dpsetconstrEL1 = $true) & (contrasubsetT1 = $true) & (sepSubset = $true) & (theeq = $true) & (intersectInPowersetIntersectUnions = $true) & (binunionTILcontra = $true) & (singletoninpowerset = $true) & (woz1_2 = $true) & (woz1_4 = $true) & (demorgan2b2 = $true) & (setextT = $true) & (emptysetimpfalse = $true) & (funcinfuncset = $true) & (iffalseProp1 = $true) & (woz1_5 = $true) & (prop2setE = $true) & (setadjoinE = $true) & (ex1E1 = $true) & (setadjoinAx = $true) & (dsetconstr__Cong = $true) & (cartprodpairsurjEq = $true) & (binintersectLsub = $true) & (cartprodpairmemER = $true) & (powersetTI1 = $true) & (woz13rule4 = $true) & (eqbreln = $true) & (cartprodpairin = $true) & (emptyInPowerset = $true) & (singletonsuniq = $true) & (brelnall2 = $true) & (setminusILneg = $true) & (demorgan1b = $true) & (disjointsetsI1 = $true) & (setOfPairsIsBReln = $true) & (powersetAx = $true) & (setminusELneg = $true) & (demorgan1 = $true) & (demorgan1a = $true) & (iftrue = $true) & (inIntersectImpInUnion2 = $true) & (binintersectER = $true) & (nonemptyE1 = $true) & (upairsubunion = $true) & (setunionE2 = $true) & (setadjoinSub = $true) & (setunionsingleton2 = $true) & (binunionIL = $true) & (nonemptyImpWitness = $true) & (ksndpairEq = $true) & (powersetT_lem = $true) & (cartprodsndpairEq = $true) & (exuI1 = $true) & (subsetTI = $true) & (setadjoinIR = $true) & (woz13rule1 = $true) & (funcext = $true) & (ubforcartprodlem3 = $true) & (cartprodfstpairEq = $true) & (setminusEL = $true) & (subsetE = $true) & (dpsetconstrI = $true) & (setukpairIR = $true) & (funcGraphProp4 = $true) & (emptyI = $true) & (dpsetconstrER = $true) & (eqimpsubset2 = $true) & (funcGraphProp1 = $true) & (doubleComplementEq = $true) & (prop2setI = $true) & (powerset__Cong = $true) & (contrasubsetT2 = $true) & (setukpairinjL = $true) & (emptyinPowerset = $true) & (notdallE = $true) & (singletonprop = $true) & (setukpairinjL1 = $true) & (lam2lamEq = $true) & (emptyinunitempty = $true) & (subset2powerset = $true) & (symdiffIneg2 = $true) & (setadjoinSub2 = $true) & (complementT_lem = $true) & (woz13rule3 = $true) & (doubleComplementSub1 = $true) & (setadjoinIL = $true) & (setukpairIL = $true) & (emptyset__Cong = $true) & (binunionTIRcontra = $true) & (subsetI2 = $true) & (emptysetAx = $true) & (kpairiskpair = $true) & (setextAx = $true) & (setukpairinjL2 = $true) & (binintersectI = $true) & (binunionIR = $true) & (exuEu = $true) & (complementUnionInPowersetComplement = $true) & (complementImpComplementIntersect = $true) & (binintersectT_lem = $true) & (ex1E2 = $true) & (upairsetIL = $true) & (demorgan2a2 = $true) & (iftrueProp2 = $true) & (wellorderingAx = $true) & (quantDeMorgan2 = $true) & (powersetE = $true) & (cartprodmempair1 = $true) & (complementTnotintersectT = $true) & (notequalI2 = $true) & (exuI3 = $true) & (emptysetE = $true) & (setminusT_lem = $true) & (symdiffIneg1 = $true) & (demorgan2b = $true) & (complementTI1 = $true) & (nonemptyI = $true) & (apProp = $true) & (notinemptyset = $true) & (woz1_3 = $true) & (binintersectTERcontra = $true) & (woz13rule0 = $true) & (eqimpsubset1 = $true) & (emptyE1 = $true) & (notequalI1 = $true) & (subsetE2 = $true) & (setukpairinjR12 = $true) & (omega__Cong = $true) & (lamProp = $true) & (funcGraphProp2 = $true) & (funcGraphProp3 = $true) & (setukpairinjR1 = $true) & (exu__Cong = $true) & (exuI2 = $true) & (ubforcartprodlem1 = $true) & (omega0Ax = $true) & (iffalseProp2 = $true) & (quantDeMorgan1 = $true) & (iftrueorfalse = $true) & (ex1I = $true) & (setunionsingleton = $true) & (setukpairinjR11 = $true) & (breln1all2 = $true) & (eta1 = $true) & (funcImageSingleton = $true) & (noeltsimpempty = $true) & (setunionsingleton1 = $true) & (lamp = $true) & (symdiffI1 = $true) & (sepInPowerset = $true) & (subsetemptysetimpeq = $true) & (complementTcontraSubset = $true) & (binintersectSubset5 = $true) & (binintersectTELcontra = $true) & (complementInPowersetComplementIntersect = $true) & (kfstsingleton = $true) & (theprop = $true) & (binunionE = $true) & (beta2 = $true) & (breln1SetBreln1 = $true) & (demorgan2a1 = $true) & (singletoninpowunion = $true) & (binintersectSubset3 = $true) & (descr__Cong = $true) & (in__Cong = $true) & (setminusER = $true) & (nonemptyI1 = $true) & (symdiffE = $true) & (ex1I2 = $true) & (doubleComplementSub2 = $true) & (funcextLem = $true) & (emptysetsubset = $true) & (infuncsetfunc = $true) & (vacuousDall = $true) & (dsetconstrEL = $true) & (binunionEcases = $true) & (notsubsetI = $true) & (iffalse = $true) & (cartprodmempaircEq = $true) & (contraSubsetComplement = $true) & (subsetRefl = $true) & (setext = $true) & (contrasubsetT = $true) & (iftrueProp1 = $true) & (setbeta = $true) & (foundationAx = $true) & (exuE2 = $true) & (ifSingleton = $true) & (quantDeMorgan4 = $true) & (setunionAx = $true) & (setminusI = $true) & (bs114d = $true) & (binunionT_lem = $true) & (exuE1 = $true) & (symdiffI2 = $true) & (complementSubsetComplementIntersect = $true) & (setoftrueEq = $true) & (omegaIndAx = $true) & (setadjoinOr = $true) & (setextsub = $true)),
% 2.96/1.78 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1])],[f874,f875])).
% 2.96/1.78 thf(f875,plain,(
% 2.96/1.78 ? [X0 : $i > $i > $o,X1 : $i] : ((breln1 @ X1 @ (dpsetconstr @ X1 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1)))))) != $true) => ((breln1 @ sK1 @ (dpsetconstr @ sK1 @ sK1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK0 @ Y0 @ Y1)))))) != $true)),
% 2.96/1.78 introduced(definition,[],[choice_axiom])).
% 2.96/1.78 thf(f874,plain,(
% 2.96/1.78 (setunionI = $true) & (replAx = $true) & (complementTE1 = $true) & (ksndsingleton = $true) & (setukpairinjR = $true) & (doubleComplementI1 = $true) & (kfstpairEq = $true) & (binintersectSubset1 = $true) & (funcext2 = $true) & (cartprodsndin = $true) & (powersetE1 = $true) & (binunionTE = $true) & (kpairp = $true) & (dpsetconstrERa = $true) & (uniqinunit = $true) & (upairequniteq = $true) & (dsetconstrI = $true) & (binintersectSubset2 = $true) & (cartprodmempair = $true) & (cartprodpairmemEL = $true) & (ap2apEq2 = $true) & (binintersectSubset4 = $true) & (descrp = $true) & (upairinpowunion = $true) & (kpairsurjEq = $true) & (subPowSU = $true) & (ubforcartprodlem2 = $true) & (singletonsubset = $true) & (powersetI = $true) & (inCongP = $true) & (dpsetconstrEL2 = $true) & (ifp = $true) & (powersetI1 = $true) & (choice2fnsingleton = $true) & (demorgan2a = $true) & (upairsetIR = $true) & (setadjoin__Cong = $true) & (app = $true) & (inIntersectImpInIntersectUnions = $true) & (singletonsswitch = $true) & (woz13rule2 = $true) & (ap2p = $true) & (exuE3e = $true) & (setminusIRneg = $true) & (inPowerset = $true) & (setminusERneg = $true) & (subsetTrans = $true) & (subsetI1 = $true) & (upairset2E = $true) & (setunionE = $true) & (setunion__Cong = $true) & (woz1_1 = $true) & (notdexE = $true) & (setukpairinjR2 = $true) & (subbreln = $true) & (setminusSubset1 = $true) & (beta1 = $true) & (setminusSubset2 = $true) & (powersetsubset = $true) & (ap2apEq1 = $true) & (binunionTEcontra = $true) & (prop2set2propI = $true) & (omegaSAx = $true) & (eta2 = $true) & ? [X0 : $i > $i > $o,X1 : $i] : ((breln1 @ X1 @ (dpsetconstr @ X1 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1)))))) != $true) & (dsetconstrER = $true) & (powersetTE1 = $true) & (inIntersectImpInUnion = $true) & (notinsingleton = $true) & (exuE3u = $true) & (secondinupair = $true) & (quantDeMorgan3 = $true) & (doubleComplementE1 = $true) & (upairsetE = $true) & (binunionLsub = $true) & (dpsetconstrSub = $true) & (binintersectEL = $true) & (binintersectRsub = $true) & (setminusLsub = $true) & (brelnall1 = $true) & (lam2p = $true) & (binunionRsub = $true) & (inComplementUnionImpNotIn1 = $true) & (cartprodfstin = $true) & (eqinunit = $true) & (demorgan2 = $true) & (inComplementUnionImpInComplement1 = $true) & (upairset2IR = $true) & (contrasubsetT3 = $true) & (dpsetconstrEL1 = $true) & (contrasubsetT1 = $true) & (sepSubset = $true) & (theeq = $true) & (intersectInPowersetIntersectUnions = $true) & (binunionTILcontra = $true) & (singletoninpowerset = $true) & (woz1_2 = $true) & (woz1_4 = $true) & (demorgan2b2 = $true) & (setextT = $true) & (emptysetimpfalse = $true) & (funcinfuncset = $true) & (iffalseProp1 = $true) & (woz1_5 = $true) & (prop2setE = $true) & (setadjoinE = $true) & (ex1E1 = $true) & (setadjoinAx = $true) & (dsetconstr__Cong = $true) & (cartprodpairsurjEq = $true) & (binintersectLsub = $true) & (cartprodpairmemER = $true) & (powersetTI1 = $true) & (woz13rule4 = $true) & (eqbreln = $true) & (cartprodpairin = $true) & (emptyInPowerset = $true) & (singletonsuniq = $true) & (brelnall2 = $true) & (setminusILneg = $true) & (demorgan1b = $true) & (disjointsetsI1 = $true) & (setOfPairsIsBReln = $true) & (powersetAx = $true) & (setminusELneg = $true) & (demorgan1 = $true) & (demorgan1a = $true) & (iftrue = $true) & (inIntersectImpInUnion2 = $true) & (binintersectER = $true) & (nonemptyE1 = $true) & (upairsubunion = $true) & (setunionE2 = $true) & (setadjoinSub = $true) & (setunionsingleton2 = $true) & (binunionIL = $true) & (nonemptyImpWitness = $true) & (ksndpairEq = $true) & (powersetT_lem = $true) & (cartprodsndpairEq = $true) & (exuI1 = $true) & (subsetTI = $true) & (setadjoinIR = $true) & (woz13rule1 = $true) & (funcext = $true) & (ubforcartprodlem3 = $true) & (cartprodfstpairEq = $true) & (setminusEL = $true) & (subsetE = $true) & (dpsetconstrI = $true) & (setukpairIR = $true) & (funcGraphProp4 = $true) & (emptyI = $true) & (dpsetconstrER = $true) & (eqimpsubset2 = $true) & (funcGraphProp1 = $true) & (doubleComplementEq = $true) & (prop2setI = $true) & (powerset__Cong = $true) & (contrasubsetT2 = $true) & (setukpairinjL = $true) & (emptyinPowerset = $true) & (notdallE = $true) & (singletonprop = $true) & (setukpairinjL1 = $true) & (lam2lamEq = $true) & (emptyinunitempty = $true) & (subset2powerset = $true) & (symdiffIneg2 = $true) & (setadjoinSub2 = $true) & (complementT_lem = $true) & (woz13rule3 = $true) & (doubleComplementSub1 = $true) & (setadjoinIL = $true) & (setukpairIL = $true) & (emptyset__Cong = $true) & (binunionTIRcontra = $true) & (subsetI2 = $true) & (emptysetAx = $true) & (kpairiskpair = $true) & (setextAx = $true) & (setukpairinjL2 = $true) & (binintersectI = $true) & (binunionIR = $true) & (exuEu = $true) & (complementUnionInPowersetComplement = $true) & (complementImpComplementIntersect = $true) & (binintersectT_lem = $true) & (ex1E2 = $true) & (upairsetIL = $true) & (demorgan2a2 = $true) & (iftrueProp2 = $true) & (wellorderingAx = $true) & (quantDeMorgan2 = $true) & (powersetE = $true) & (cartprodmempair1 = $true) & (complementTnotintersectT = $true) & (notequalI2 = $true) & (exuI3 = $true) & (emptysetE = $true) & (setminusT_lem = $true) & (symdiffIneg1 = $true) & (demorgan2b = $true) & (complementTI1 = $true) & (nonemptyI = $true) & (apProp = $true) & (notinemptyset = $true) & (woz1_3 = $true) & (binintersectTERcontra = $true) & (woz13rule0 = $true) & (eqimpsubset1 = $true) & (emptyE1 = $true) & (notequalI1 = $true) & (subsetE2 = $true) & (setukpairinjR12 = $true) & (omega__Cong = $true) & (lamProp = $true) & (funcGraphProp2 = $true) & (funcGraphProp3 = $true) & (setukpairinjR1 = $true) & (exu__Cong = $true) & (exuI2 = $true) & (ubforcartprodlem1 = $true) & (omega0Ax = $true) & (iffalseProp2 = $true) & (quantDeMorgan1 = $true) & (iftrueorfalse = $true) & (ex1I = $true) & (setunionsingleton = $true) & (setukpairinjR11 = $true) & (breln1all2 = $true) & (eta1 = $true) & (funcImageSingleton = $true) & (noeltsimpempty = $true) & (setunionsingleton1 = $true) & (lamp = $true) & (symdiffI1 = $true) & (sepInPowerset = $true) & (subsetemptysetimpeq = $true) & (complementTcontraSubset = $true) & (binintersectSubset5 = $true) & (binintersectTELcontra = $true) & (complementInPowersetComplementIntersect = $true) & (kfstsingleton = $true) & (theprop = $true) & (binunionE = $true) & (beta2 = $true) & (breln1SetBreln1 = $true) & (demorgan2a1 = $true) & (singletoninpowunion = $true) & (binintersectSubset3 = $true) & (descr__Cong = $true) & (in__Cong = $true) & (setminusER = $true) & (nonemptyI1 = $true) & (symdiffE = $true) & (ex1I2 = $true) & (doubleComplementSub2 = $true) & (funcextLem = $true) & (emptysetsubset = $true) & (infuncsetfunc = $true) & (vacuousDall = $true) & (dsetconstrEL = $true) & (binunionEcases = $true) & (notsubsetI = $true) & (iffalse = $true) & (cartprodmempaircEq = $true) & (contraSubsetComplement = $true) & (subsetRefl = $true) & (setext = $true) & (contrasubsetT = $true) & (iftrueProp1 = $true) & (setbeta = $true) & (foundationAx = $true) & (exuE2 = $true) & (ifSingleton = $true) & (quantDeMorgan4 = $true) & (setunionAx = $true) & (setminusI = $true) & (bs114d = $true) & (binunionT_lem = $true) & (exuE1 = $true) & (symdiffI2 = $true) & (complementSubsetComplementIntersect = $true) & (setoftrueEq = $true) & (omegaIndAx = $true) & (setadjoinOr = $true) & (setextsub = $true)),
% 2.96/1.78 inference(flattening,[status(thm)],[f873])).
% 2.96/1.78 thf(f873,plain,(
% 2.96/1.78 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i > $i > $o,X1 : $i] : ((breln1 @ X1 @ (dpsetconstr @ X1 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1)))))) != $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)),
% 2.96/1.78 inference(ennf_transformation,[status(thm)],[f603])).
% 2.96/1.78 thf(f603,plain,(
% 2.96/1.78 ~((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) => ! [X0 : $i > $i > $o,X1 : $i] : ((breln1 @ X1 @ (dpsetconstr @ X1 @ X1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (X0 @ Y0 @ Y1)))))) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.96/1.78 inference(fool_elimination,[status(thm)],[f602])).
% 2.96/1.78 thf(f602,plain,(
% 2.96/1.78 ~(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 => ! [X0 : $i > $i > $o,X1 : $i] : (breln1 @ X1 @ (dpsetconstr @ X1 @ X1 @ (^[X2 : $i, X3 : $i] : (X0 @ X2 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.96/1.78 inference(rectify,[status(thm)],[f293])).
% 2.96/1.78 thf(f293,negated_conjecture,(
% 2.96/1.78 ~(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 => ! [X0 : $i > $i > $o,X3 : $i] : (breln1 @ X3 @ (dpsetconstr @ X3 @ X3 @ (^[X1 : $i, X2 : $i] : (X0 @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.96/1.78 inference(negated_conjecture,[status(cth)],[f292])).
% 2.96/1.78 thf(f292,conjecture,(
% 2.96/1.78 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 => ! [X0 : $i > $i > $o,X3 : $i] : (breln1 @ X3 @ (dpsetconstr @ X3 @ X3 @ (^[X1 : $i, X2 : $i] : (X0 @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.96/1.78 file('/export/starexec/sandbox/tmp/tmp.DOTu7T1G0A/DTF2THF_390.p',setOfPairsIsBReln1)).
% 2.96/1.78 thf(f1452,plain,(
% 2.96/1.78 (setOfPairsIsBReln = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $i > $o) @ (^[Y1 : $i > $i > $o]: (!! @ $i @ (^[Y2 : $i]: (breln @ Y0 @ Y2 @ (dpsetconstr @ Y0 @ Y2 @ (^[Y3 : $i]: ((^[Y4 : $i]: (Y1 @ Y3 @ Y4)))))))))))))),
% 2.96/1.78 inference(cnf_transformation,[status(thm)],[f433])).
% 2.96/1.78 thf(f433,plain,(
% 2.96/1.78 (setOfPairsIsBReln = (!! @ $i @ (^[Y0 : $i]: (!! @ ($i > $i > $o) @ (^[Y1 : $i > $i > $o]: (!! @ $i @ (^[Y2 : $i]: (breln @ Y0 @ Y2 @ (dpsetconstr @ Y0 @ Y2 @ (^[Y3 : $i]: ((^[Y4 : $i]: (Y1 @ Y3 @ Y4)))))))))))))),
% 2.96/1.78 inference(fool_elimination,[status(thm)],[f432])).
% 2.96/1.78 thf(f432,plain,(
% 2.96/1.78 (! [X0 : $i,X1 : $i > $i > $o,X2 : $i] : (breln @ X2 @ X0 @ (dpsetconstr @ X2 @ X0 @ (^[X3 : $i, X4 : $i] : (X1 @ X3 @ X4)))) = setOfPairsIsBReln)),
% 2.96/1.78 inference(rectify,[status(thm)],[f187])).
% 2.96/1.78 thf(f187,axiom,(
% 2.96/1.78 (! [X4 : $i,X0 : $i > $i > $o,X3 : $i] : (breln @ X3 @ X4 @ (dpsetconstr @ X3 @ X4 @ (^[X1 : $i, X2 : $i] : (X0 @ X1 @ X2)))) = setOfPairsIsBReln)),
% 2.96/1.78 file('/export/starexec/sandbox/tmp/tmp.DOTu7T1G0A/DTF2THF_390.p',setOfPairsIsBReln)).
% 2.96/1.78 thf(f3803,plain,(
% 2.96/1.78 ((breln @ sK1 @ sK1 @ (dpsetconstr @ sK1 @ sK1 @ sK0)) != $true)),
% 2.96/1.78 inference(beta_eta_normalization,[status(thm)],[f1485])).
% 2.96/1.78 thf(f1485,plain,(
% 2.96/1.78 (((^[Y0 : $i]: ((^[Y1 : $i]: (breln @ Y0 @ Y0 @ Y1)))) @ sK1 @ (dpsetconstr @ sK1 @ sK1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK0 @ Y0 @ Y1)))))) != $true)),
% 2.96/1.78 inference(definition_unfolding,[status(thm)],[f1236,f1357])).
% 2.96/1.78 thf(f1357,plain,(
% 2.96/1.78 (breln1 = (^[Y0 : $i]: ((^[Y1 : $i]: (breln @ Y0 @ Y0 @ Y1)))))),
% 2.96/1.78 inference(cnf_transformation,[status(thm)],[f633])).
% 2.96/1.78 thf(f633,plain,(
% 2.96/1.78 (breln1 = (^[Y0 : $i]: ((^[Y1 : $i]: (breln @ Y0 @ Y0 @ Y1)))))),
% 2.96/1.78 inference(fool_elimination,[status(thm)],[f632])).
% 2.96/1.78 thf(f632,plain,(
% 2.96/1.78 ((^[X0 : $i, X1 : $i] : (breln @ X0 @ X0 @ X1)) = breln1)),
% 2.96/1.78 inference(rectify,[status(thm)],[f288])).
% 2.96/1.78 thf(f288,axiom,(
% 2.96/1.78 ((^[X3 : $i, X13 : $i] : (breln @ X3 @ X3 @ X13)) = breln1)),
% 2.96/1.78 file('/export/starexec/sandbox/tmp/tmp.DOTu7T1G0A/DTF2THF_390.p',breln1)).
% 2.96/1.78 thf(f1236,plain,(
% 2.96/1.78 ((breln1 @ sK1 @ (dpsetconstr @ sK1 @ sK1 @ (^[Y0 : $i]: ((^[Y1 : $i]: (sK0 @ Y0 @ Y1)))))) != $true)),
% 2.96/1.78 inference(cnf_transformation,[status(thm)],[f876])).
% 2.96/1.78 % SZS output end Proof for DTF2THF_390
% 2.96/1.78 % (543)------------------------------
% 2.96/1.78 % (543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.96/1.78 % (543)Termination reason: Refutation
% 2.96/1.78
% 2.96/1.78 % (543)Memory used [KB]: 9850
% 2.96/1.78 % (543)Time elapsed: 0.126 s
% 2.96/1.78 % (543)Instructions burned: 421 (million)
% 2.96/1.78 % (543)------------------------------
% 2.96/1.78 % (543)------------------------------
% 2.96/1.78 % (510)Success in time 0.257 s
%------------------------------------------------------------------------------