%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU812^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n013.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 09:37:24 AM UTC 2026
% Result : Theorem 5.44s 2.09s
% Output : Refutation 5.44s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 6
% Syntax : Number of formulae : 28 ( 14 unt; 0 typ; 0 def)
% Number of atoms : 5117 (1359 equ; 0 cnn)
% Maximal formula atoms : 331 ( 182 avg)
% Number of connectives : 2468 ( 11 ~; 7 |; 996 &; 116 @)
% ( 0 <=>;1333 =>; 0 <=; 0 <~>)
% Maximal formula depth : 336 ( 85 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 1 ( 1 >; 0 *; 0 +; 0 <<)
% Number of symbols : 394 ( 390 usr; 333 con; 0-4 aty)
% ( 5 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 37 ( 11 ^; 18 !; 8 ?; 37 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_323,type,
breln1: $i > $i > $o ).
thf(func_def_325,type,
breln1Set: $i > $i ).
thf(func_def_327,type,
transitive: $i > $i > $o ).
thf(func_def_328,type,
antisymmetric: $i > $i > $o ).
thf(func_def_329,type,
reflexive: $i > $i > $o ).
thf(func_def_330,type,
refltransitive: $i > $i > $o ).
thf(func_def_331,type,
refllinearorder: $i > $i > $o ).
thf(func_def_332,type,
reflwellordering: $i > $i > $o ).
thf(func_def_338,type,
breln1invset: $i > $i > $i ).
thf(func_def_342,type,
breln1compset: $i > $i > $i > $i ).
thf(func_def_360,type,
image1: $i > ( $i > $i ) > $i ).
thf(func_def_364,type,
injective: $i > $i > $i > $o ).
thf(func_def_365,type,
injFuncSet: $i > $i > $i ).
thf(func_def_369,type,
surjective: $i > $i > $i > $o ).
thf(func_def_370,type,
surjFuncSet: $i > $i > $i ).
thf(func_def_378,type,
omegaS: $i > $i ).
thf(func_def_382,type,
peano3: $i > $i > $i > $o ).
thf(func_def_383,type,
peano4: $i > $i > $i > $o ).
thf(func_def_384,type,
peano5: $i > $i > $i > $o ).
thf(func_def_386,type,
peano: $i > $i > $i > $o ).
thf(func_def_387,type,
transitiveset: $i > $o ).
thf(func_def_406,type,
sK0:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_407,type,
sK1:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_411,type,
ph5:
!>[X0: $tType] : X0 ).
thf(f2422,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f2421,f1483]) ).
thf(f1483,plain,
( ( subset @ sK3 @ sK2 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1016]) ).
thf(f1016,plain,
( ( foundation2 = $true )
& ( notinself2 = $true )
& ( setextT = $true )
& ( cartprodfstin = $true )
& ( emptyset__Cong = $true )
& ( woz1_1 = $true )
& ( breln1compprop = $true )
& ( setadjoinE = $true )
& ( breln1unionIL = $true )
& ( notinsingleton = $true )
& ( lam2p = $true )
& ( binunionTIRcontra = $true )
& ( infuncsetfunc = $true )
& ( demorgan2a2 = $true )
& ( binunionT_lem = $true )
& ( setOfPairsIsBReln = $true )
& ( setminusSubset2 = $true )
& ( setext = $true )
& ( symdiffI2 = $true )
& ( omegaIndAx = $true )
& ( setukpairIR = $true )
& ( eta2 = $true )
& ( injFuncSetFuncInj = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR2 = $true )
& ( setminusELneg = $true )
& ( beta2 = $true )
& ( emptysetimpfalse = $true )
& ( eta1 = $true )
& ( contraSubsetComplement = $true )
& ( nonemptyE1 = $true )
& ( singletonsubset = $true )
& ( exuI1 = $true )
& ( ( transitiveset @ sK2 )
= $true )
& ( ( in @ sK3 @ sK2 )
= $true )
& ( ( subset @ sK3 @ sK2 )
!= $true )
& ( breln1unionE = $true )
& ( powersetTE1 = $true )
& ( woz1_3 = $true )
& ( cartprodpairsurjEq = $true )
& ( setadjoinSub2 = $true )
& ( funcGraphProp1 = $true )
& ( sepInPowerset = $true )
& ( cartprodmempair1 = $true )
& ( brelnall2 = $true )
& ( ap2p = $true )
& ( powerset__Cong = $true )
& ( ex1E2 = $true )
& ( in__Cong = $true )
& ( binunionTE = $true )
& ( setunionsingleton2 = $true )
& ( app = $true )
& ( setukpairinjR = $true )
& ( emptyI = $true )
& ( ex1I = $true )
& ( setminusSubset1 = $true )
& ( demorgan1 = $true )
& ( woz2A = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjL1 = $true )
& ( binintersectI = $true )
& ( powersetT_lem = $true )
& ( ap2apEq1 = $true )
& ( demorgan2a = $true )
& ( binintersectTERcontra = $true )
& ( eqinunit = $true )
& ( subsetI2 = $true )
& ( lamProp = $true )
& ( setadjoinIR = $true )
& ( omegaSp = $true )
& ( setadjoinOr = $true )
& ( sepSubset = $true )
& ( dsetconstrER = $true )
& ( setunionAx = $true )
& ( upairequniteq = $true )
& ( powersetAx = $true )
& ( apProp = $true )
& ( ex1I2 = $true )
& ( contrasubsetT1 = $true )
& ( wellorderingAx = $true )
& ( setukpairinjR12 = $true )
& ( powersetE1 = $true )
& ( dpsetconstrSub = $true )
& ( iftrueProp2 = $true )
& ( exu__Cong = $true )
& ( symdiffIneg1 = $true )
& ( setunionsingleton1 = $true )
& ( powersetsubset = $true )
& ( singletonsuniq = $true )
& ( woz13rule3 = $true )
& ( setextAx = $true )
& ( uniqinunit = $true )
& ( powersetE = $true )
& ( breln1unionIR = $true )
& ( exuE2 = $true )
& ( singletonsswitch = $true )
& ( upairsubunion = $true )
& ( image1E = $true )
& ( demorgan1a = $true )
& ( dsetconstrI = $true )
& ( breln1unionI = $true )
& ( complementTnotintersectT = $true )
& ( subsetE2 = $true )
& ( binunionIR = $true )
& ( upairset2E = $true )
& ( iffalseProp1 = $true )
& ( doubleComplementE1 = $true )
& ( quantDeMorgan3 = $true )
& ( setminusILneg = $true )
& ( breln1all2 = $true )
& ( iftrueorfalse = $true )
& ( omegaSAx = $true )
& ( binintersectSubset2 = $true )
& ( emptyinunitempty = $true )
& ( doubleComplementI1 = $true )
& ( setminusIRneg = $true )
& ( complementT_lem = $true )
& ( funcGraphProp2 = $true )
& ( dsetconstr__Cong = $true )
& ( singletoninpowunion = $true )
& ( singletonprop = $true )
& ( setminusT_lem = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( complementUnionInPowersetComplement = $true )
& ( doubleComplementEq = $true )
& ( replAx = $true )
& ( exuE1 = $true )
& ( descr__Cong = $true )
& ( woz13rule1 = $true )
& ( ifp = $true )
& ( vacuousDall = $true )
& ( binintersectLsub = $true )
& ( breln1invI = $true )
& ( setminusI = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( cartprodfstpairEq = $true )
& ( eqbreln = $true )
& ( setunionI = $true )
& ( binintersectTELcontra = $true )
& ( contrasubsetT = $true )
& ( inIntersectImpInUnion2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( powersetI = $true )
& ( singletoninpowerset = $true )
& ( setextsub = $true )
& ( iftrue = $true )
& ( cartprodsndin = $true )
& ( brelnall1 = $true )
& ( setunionE2 = $true )
& ( binintersectSubset3 = $true )
& ( binunionTILcontra = $true )
& ( subsetTI = $true )
& ( peano0notS = $true )
& ( lam2lamEq = $true )
& ( funcGraphProp4 = $true )
& ( woz2B = $true )
& ( inPowerset = $true )
& ( inCongP = $true )
& ( funcextLem = $true )
& ( nonemptyI = $true )
& ( binunionEcases = $true )
& ( complementTcontraSubset = $true )
& ( binunionLsub = $true )
& ( upairsetIR = $true )
& ( exuE3e = $true )
& ( surjCantorThm = $true )
& ( setukpairinjL2 = $true )
& ( notinself = $true )
& ( notdexE = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setadjoin__Cong = $true )
& ( ubforcartprodlem1 = $true )
& ( ap2apEq2 = $true )
& ( funcext2 = $true )
& ( woz2W = $true )
& ( choice2fnsingleton = $true )
& ( dpsetconstrI = $true )
& ( setminusEL = $true )
& ( subPowSU = $true )
& ( binunionIL = $true )
& ( eqbreln1 = $true )
& ( image1Ex1 = $true )
& ( surjFuncSetFuncSurj = $true )
& ( image1Ex = $true )
& ( subsetI1 = $true )
& ( cartprodmempair = $true )
& ( breln1SetBreln1 = $true )
& ( exuI3 = $true )
& ( breln1invE = $true )
& ( notequalI1 = $true )
& ( kfstpairEq = $true )
& ( subsetRefl = $true )
& ( ksndsingleton = $true )
& ( emptyinPowerset = $true )
& ( injFuncSetFuncIn = $true )
& ( kpairp = $true )
& ( dsetconstrEL = $true )
& ( quantDeMorgan1 = $true )
& ( binintersectSubset1 = $true )
& ( binintersectER = $true )
& ( funcImageSingleton = $true )
& ( woz13rule2 = $true )
& ( breln1invprop = $true )
& ( demorgan2b2 = $true )
& ( setukpairinjL = $true )
& ( foundationAx = $true )
& ( iftrueProp1 = $true )
& ( complementTI1 = $true )
& ( inIntersectImpInUnion = $true )
& ( image1I = $true )
& ( kfstsingleton = $true )
& ( setadjoinIL = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( omegaSclos = $true )
& ( upairinpowunion = $true )
& ( woz2Ex = $true )
& ( demorgan2a1 = $true )
& ( doubleComplementSub1 = $true )
& ( prop2set2propI = $true )
& ( contrasubsetT3 = $true )
& ( nonemptyI1 = $true )
& ( upairsetE = $true )
& ( breln1unionprop = $true )
& ( demorgan2b = $true )
& ( binintersectRsub = $true )
& ( contrasubsetT2 = $true )
& ( kpairiskpair = $true )
& ( woz1_5 = $true )
& ( cartprodpairmemER = $true )
& ( setunionE = $true )
& ( notsubsetI = $true )
& ( setunion__Cong = $true )
& ( dpsetconstrERa = $true )
& ( breln1all1 = $true )
& ( breln1compEex = $true )
& ( nonemptyImpWitness = $true )
& ( powersetI1 = $true )
& ( ifSingleton = $true )
& ( complementTE1 = $true )
& ( binintersectSubset4 = $true )
& ( surjFuncSetFuncIn = $true )
& ( setminusLsub = $true )
& ( breln1compE = $true )
& ( upairset2IR = $true )
& ( omega0Ax = $true )
& ( funcGraphProp3 = $true )
& ( quantDeMorgan2 = $true )
& ( subsetE = $true )
& ( noeltsimpempty = $true )
& ( ubforcartprodlem3 = $true )
& ( leftInvIsSurj = $true )
& ( demorgan1b = $true )
& ( lamp = $true )
& ( theeq = $true )
& ( setoftrueEq = $true )
& ( dpsetconstrEL1 = $true )
& ( upairsetIL = $true )
& ( prop2setE = $true )
& ( complementImpComplementIntersect = $true )
& ( emptysetAx = $true )
& ( dpsetconstrEL2 = $true )
& ( binintersectT_lem = $true )
& ( demorgan2 = $true )
& ( dpsetconstrER = $true )
& ( woz13rule4 = $true )
& ( funcinfuncset = $true )
& ( notdallE = $true )
& ( image1Equiv = $true )
& ( emptysetsubset = $true )
& ( subsetemptysetimpeq = $true )
& ( woz1_4 = $true )
& ( binunionTEcontra = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( iffalse = $true )
& ( descrp = $true )
& ( emptyE1 = $true )
& ( subset2powerset = $true )
& ( cartprodpairmemEL = $true )
& ( subbreln1 = $true )
& ( binunionRsub = $true )
& ( theprop = $true )
& ( setminusER = $true )
& ( notinemptyset = $true )
& ( exuE3u = $true )
& ( cartprodsndpairEq = $true )
& ( binintersectEL = $true )
& ( symdiffI1 = $true )
& ( breln1unionEcases = $true )
& ( injFuncInInjFuncSet = $true )
& ( setminusERneg = $true )
& ( ex1E1 = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjR11 = $true )
& ( woz1_2 = $true )
& ( subsetTrans = $true )
& ( setadjoinSub = $true )
& ( cartprodpairin = $true )
& ( woz13rule0 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( breln1unionCommutes = $true )
& ( prop2setI = $true )
& ( breln1compI = $true )
& ( exuI2 = $true )
& ( emptyInPowerset = $true )
& ( funcext = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( ubforcartprodlem2 = $true )
& ( quantDeMorgan4 = $true )
& ( bs114d = $true )
& ( powersetTI1 = $true )
& ( beta1 = $true )
& ( binunionE = $true )
& ( setunionsingleton = $true )
& ( iffalseProp2 = $true )
& ( disjointsetsI1 = $true )
& ( eqimpsubset2 = $true )
& ( eqimpsubset1 = $true )
& ( symdiffE = $true )
& ( setadjoinAx = $true )
& ( omega__Cong = $true )
& ( kpairsurjEq = $true )
& ( emptysetE = $true )
& ( ksndpairEq = $true )
& ( subbreln = $true )
& ( setbeta = $true )
& ( notequalI2 = $true )
& ( exuEu = $true )
& ( doubleComplementSub2 = $true )
& ( peanoSinj = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3])],[f1013,f1015,f1014]) ).
thf(f1014,plain,
( ? [X0: $i] :
( ( $true
= ( transitiveset @ X0 ) )
& ? [X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ( ( subset @ X1 @ X0 )
!= $true ) ) )
=> ( ( ( transitiveset @ sK2 )
= $true )
& ? [X1: $i] :
( ( ( in @ X1 @ sK2 )
= $true )
& ( ( subset @ X1 @ sK2 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1015,plain,
( ? [X1: $i] :
( ( ( in @ X1 @ sK2 )
= $true )
& ( ( subset @ X1 @ sK2 )
!= $true ) )
=> ( ( ( in @ sK3 @ sK2 )
= $true )
& ( ( subset @ sK3 @ sK2 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1013,plain,
( ( foundation2 = $true )
& ( notinself2 = $true )
& ( setextT = $true )
& ( cartprodfstin = $true )
& ( emptyset__Cong = $true )
& ( woz1_1 = $true )
& ( breln1compprop = $true )
& ( setadjoinE = $true )
& ( breln1unionIL = $true )
& ( notinsingleton = $true )
& ( lam2p = $true )
& ( binunionTIRcontra = $true )
& ( infuncsetfunc = $true )
& ( demorgan2a2 = $true )
& ( binunionT_lem = $true )
& ( setOfPairsIsBReln = $true )
& ( setminusSubset2 = $true )
& ( setext = $true )
& ( symdiffI2 = $true )
& ( omegaIndAx = $true )
& ( setukpairIR = $true )
& ( eta2 = $true )
& ( injFuncSetFuncInj = $true )
& ( binintersectSubset5 = $true )
& ( setukpairinjR2 = $true )
& ( setminusELneg = $true )
& ( beta2 = $true )
& ( emptysetimpfalse = $true )
& ( eta1 = $true )
& ( contraSubsetComplement = $true )
& ( nonemptyE1 = $true )
& ( singletonsubset = $true )
& ( exuI1 = $true )
& ? [X0: $i] :
( ( $true
= ( transitiveset @ X0 ) )
& ? [X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ( ( subset @ X1 @ X0 )
!= $true ) ) )
& ( breln1unionE = $true )
& ( powersetTE1 = $true )
& ( woz1_3 = $true )
& ( cartprodpairsurjEq = $true )
& ( setadjoinSub2 = $true )
& ( funcGraphProp1 = $true )
& ( sepInPowerset = $true )
& ( cartprodmempair1 = $true )
& ( brelnall2 = $true )
& ( ap2p = $true )
& ( powerset__Cong = $true )
& ( ex1E2 = $true )
& ( in__Cong = $true )
& ( binunionTE = $true )
& ( setunionsingleton2 = $true )
& ( app = $true )
& ( setukpairinjR = $true )
& ( emptyI = $true )
& ( ex1I = $true )
& ( setminusSubset1 = $true )
& ( demorgan1 = $true )
& ( woz2A = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjL1 = $true )
& ( binintersectI = $true )
& ( powersetT_lem = $true )
& ( ap2apEq1 = $true )
& ( demorgan2a = $true )
& ( binintersectTERcontra = $true )
& ( eqinunit = $true )
& ( subsetI2 = $true )
& ( lamProp = $true )
& ( setadjoinIR = $true )
& ( omegaSp = $true )
& ( setadjoinOr = $true )
& ( sepSubset = $true )
& ( dsetconstrER = $true )
& ( setunionAx = $true )
& ( upairequniteq = $true )
& ( powersetAx = $true )
& ( apProp = $true )
& ( ex1I2 = $true )
& ( contrasubsetT1 = $true )
& ( wellorderingAx = $true )
& ( setukpairinjR12 = $true )
& ( powersetE1 = $true )
& ( dpsetconstrSub = $true )
& ( iftrueProp2 = $true )
& ( exu__Cong = $true )
& ( symdiffIneg1 = $true )
& ( setunionsingleton1 = $true )
& ( powersetsubset = $true )
& ( singletonsuniq = $true )
& ( woz13rule3 = $true )
& ( setextAx = $true )
& ( uniqinunit = $true )
& ( powersetE = $true )
& ( breln1unionIR = $true )
& ( exuE2 = $true )
& ( singletonsswitch = $true )
& ( upairsubunion = $true )
& ( image1E = $true )
& ( demorgan1a = $true )
& ( dsetconstrI = $true )
& ( breln1unionI = $true )
& ( complementTnotintersectT = $true )
& ( subsetE2 = $true )
& ( binunionIR = $true )
& ( upairset2E = $true )
& ( iffalseProp1 = $true )
& ( doubleComplementE1 = $true )
& ( quantDeMorgan3 = $true )
& ( setminusILneg = $true )
& ( breln1all2 = $true )
& ( iftrueorfalse = $true )
& ( omegaSAx = $true )
& ( binintersectSubset2 = $true )
& ( emptyinunitempty = $true )
& ( doubleComplementI1 = $true )
& ( setminusIRneg = $true )
& ( complementT_lem = $true )
& ( funcGraphProp2 = $true )
& ( dsetconstr__Cong = $true )
& ( singletoninpowunion = $true )
& ( singletonprop = $true )
& ( setminusT_lem = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( complementUnionInPowersetComplement = $true )
& ( doubleComplementEq = $true )
& ( replAx = $true )
& ( exuE1 = $true )
& ( descr__Cong = $true )
& ( woz13rule1 = $true )
& ( ifp = $true )
& ( vacuousDall = $true )
& ( binintersectLsub = $true )
& ( breln1invI = $true )
& ( setminusI = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( cartprodfstpairEq = $true )
& ( eqbreln = $true )
& ( setunionI = $true )
& ( binintersectTELcontra = $true )
& ( contrasubsetT = $true )
& ( inIntersectImpInUnion2 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( powersetI = $true )
& ( singletoninpowerset = $true )
& ( setextsub = $true )
& ( iftrue = $true )
& ( cartprodsndin = $true )
& ( brelnall1 = $true )
& ( setunionE2 = $true )
& ( binintersectSubset3 = $true )
& ( binunionTILcontra = $true )
& ( subsetTI = $true )
& ( peano0notS = $true )
& ( lam2lamEq = $true )
& ( funcGraphProp4 = $true )
& ( woz2B = $true )
& ( inPowerset = $true )
& ( inCongP = $true )
& ( funcextLem = $true )
& ( nonemptyI = $true )
& ( binunionEcases = $true )
& ( complementTcontraSubset = $true )
& ( binunionLsub = $true )
& ( upairsetIR = $true )
& ( exuE3e = $true )
& ( surjCantorThm = $true )
& ( setukpairinjL2 = $true )
& ( notinself = $true )
& ( notdexE = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setadjoin__Cong = $true )
& ( ubforcartprodlem1 = $true )
& ( ap2apEq2 = $true )
& ( funcext2 = $true )
& ( woz2W = $true )
& ( choice2fnsingleton = $true )
& ( dpsetconstrI = $true )
& ( setminusEL = $true )
& ( subPowSU = $true )
& ( binunionIL = $true )
& ( eqbreln1 = $true )
& ( image1Ex1 = $true )
& ( surjFuncSetFuncSurj = $true )
& ( image1Ex = $true )
& ( subsetI1 = $true )
& ( cartprodmempair = $true )
& ( breln1SetBreln1 = $true )
& ( exuI3 = $true )
& ( breln1invE = $true )
& ( notequalI1 = $true )
& ( kfstpairEq = $true )
& ( subsetRefl = $true )
& ( ksndsingleton = $true )
& ( emptyinPowerset = $true )
& ( injFuncSetFuncIn = $true )
& ( kpairp = $true )
& ( dsetconstrEL = $true )
& ( quantDeMorgan1 = $true )
& ( binintersectSubset1 = $true )
& ( binintersectER = $true )
& ( funcImageSingleton = $true )
& ( woz13rule2 = $true )
& ( breln1invprop = $true )
& ( demorgan2b2 = $true )
& ( setukpairinjL = $true )
& ( foundationAx = $true )
& ( iftrueProp1 = $true )
& ( complementTI1 = $true )
& ( inIntersectImpInUnion = $true )
& ( image1I = $true )
& ( kfstsingleton = $true )
& ( setadjoinIL = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( omegaSclos = $true )
& ( upairinpowunion = $true )
& ( woz2Ex = $true )
& ( demorgan2a1 = $true )
& ( doubleComplementSub1 = $true )
& ( prop2set2propI = $true )
& ( contrasubsetT3 = $true )
& ( nonemptyI1 = $true )
& ( upairsetE = $true )
& ( breln1unionprop = $true )
& ( demorgan2b = $true )
& ( binintersectRsub = $true )
& ( contrasubsetT2 = $true )
& ( kpairiskpair = $true )
& ( woz1_5 = $true )
& ( cartprodpairmemER = $true )
& ( setunionE = $true )
& ( notsubsetI = $true )
& ( setunion__Cong = $true )
& ( dpsetconstrERa = $true )
& ( breln1all1 = $true )
& ( breln1compEex = $true )
& ( nonemptyImpWitness = $true )
& ( powersetI1 = $true )
& ( ifSingleton = $true )
& ( complementTE1 = $true )
& ( binintersectSubset4 = $true )
& ( surjFuncSetFuncIn = $true )
& ( setminusLsub = $true )
& ( breln1compE = $true )
& ( upairset2IR = $true )
& ( omega0Ax = $true )
& ( funcGraphProp3 = $true )
& ( quantDeMorgan2 = $true )
& ( subsetE = $true )
& ( noeltsimpempty = $true )
& ( ubforcartprodlem3 = $true )
& ( leftInvIsSurj = $true )
& ( demorgan1b = $true )
& ( lamp = $true )
& ( theeq = $true )
& ( setoftrueEq = $true )
& ( dpsetconstrEL1 = $true )
& ( upairsetIL = $true )
& ( prop2setE = $true )
& ( complementImpComplementIntersect = $true )
& ( emptysetAx = $true )
& ( dpsetconstrEL2 = $true )
& ( binintersectT_lem = $true )
& ( demorgan2 = $true )
& ( dpsetconstrER = $true )
& ( woz13rule4 = $true )
& ( funcinfuncset = $true )
& ( notdallE = $true )
& ( image1Equiv = $true )
& ( emptysetsubset = $true )
& ( subsetemptysetimpeq = $true )
& ( woz1_4 = $true )
& ( binunionTEcontra = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( iffalse = $true )
& ( descrp = $true )
& ( emptyE1 = $true )
& ( subset2powerset = $true )
& ( cartprodpairmemEL = $true )
& ( subbreln1 = $true )
& ( binunionRsub = $true )
& ( theprop = $true )
& ( setminusER = $true )
& ( notinemptyset = $true )
& ( exuE3u = $true )
& ( cartprodsndpairEq = $true )
& ( binintersectEL = $true )
& ( symdiffI1 = $true )
& ( breln1unionEcases = $true )
& ( injFuncInInjFuncSet = $true )
& ( setminusERneg = $true )
& ( ex1E1 = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjR11 = $true )
& ( woz1_2 = $true )
& ( subsetTrans = $true )
& ( setadjoinSub = $true )
& ( cartprodpairin = $true )
& ( woz13rule0 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( breln1unionCommutes = $true )
& ( prop2setI = $true )
& ( breln1compI = $true )
& ( exuI2 = $true )
& ( emptyInPowerset = $true )
& ( funcext = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( ubforcartprodlem2 = $true )
& ( quantDeMorgan4 = $true )
& ( bs114d = $true )
& ( powersetTI1 = $true )
& ( beta1 = $true )
& ( binunionE = $true )
& ( setunionsingleton = $true )
& ( iffalseProp2 = $true )
& ( disjointsetsI1 = $true )
& ( eqimpsubset2 = $true )
& ( eqimpsubset1 = $true )
& ( symdiffE = $true )
& ( setadjoinAx = $true )
& ( omega__Cong = $true )
& ( kpairsurjEq = $true )
& ( emptysetE = $true )
& ( ksndpairEq = $true )
& ( subbreln = $true )
& ( setbeta = $true )
& ( notequalI2 = $true )
& ( exuEu = $true )
& ( doubleComplementSub2 = $true )
& ( peanoSinj = $true ) ),
inference(flattening,[status(thm)],[f1012]) ).
thf(f1012,plain,
( ? [X0: $i] :
( ( $true
= ( transitiveset @ X0 ) )
& ? [X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
& ( ( subset @ X1 @ X0 )
!= $true ) ) )
& ( peanoSinj = $true )
& ( peano0notS = $true )
& ( omegaSclos = $true )
& ( omegaSp = $true )
& ( notinself2 = $true )
& ( notinself = $true )
& ( foundation2 = $true )
& ( surjCantorThm = $true )
& ( leftInvIsSurj = $true )
& ( surjFuncSetFuncSurj = $true )
& ( surjFuncSetFuncIn = $true )
& ( injFuncSetFuncInj = $true )
& ( injFuncSetFuncIn = $true )
& ( injFuncInInjFuncSet = $true )
& ( image1I = $true )
& ( image1E = $true )
& ( image1Equiv = $true )
& ( image1Ex1 = $true )
& ( image1Ex = $true )
& ( woz2B = $true )
& ( woz2A = $true )
& ( woz2W = $true )
& ( woz2Ex = $true )
& ( breln1unionCommutes = $true )
& ( breln1unionEcases = $true )
& ( breln1unionE = $true )
& ( breln1unionI = $true )
& ( breln1unionIR = $true )
& ( breln1unionIL = $true )
& ( breln1unionprop = $true )
& ( breln1compEex = $true )
& ( breln1compE = $true )
& ( breln1compI = $true )
& ( breln1compprop = $true )
& ( breln1invE = $true )
& ( breln1invI = $true )
& ( breln1invprop = $true )
& ( eqbreln1 = $true )
& ( subbreln1 = $true )
& ( breln1all1 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( choice2fnsingleton = $true )
& ( breln1SetBreln1 = $true )
& ( breln1all2 = $true )
& ( woz1_5 = $true )
& ( woz1_4 = $true )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $true )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f910]) ).
thf(f910,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ( ( woz1_4 = $true )
=> ( ( woz1_5 = $true )
=> ( ( breln1all2 = $true )
=> ( ( breln1SetBreln1 = $true )
=> ( ( choice2fnsingleton = $true )
=> ( ( setOfPairsIsBReln1 = $true )
=> ( ( breln1all1 = $true )
=> ( ( subbreln1 = $true )
=> ( ( eqbreln1 = $true )
=> ( ( breln1invprop = $true )
=> ( ( breln1invI = $true )
=> ( ( breln1invE = $true )
=> ( ( breln1compprop = $true )
=> ( ( breln1compI = $true )
=> ( ( breln1compE = $true )
=> ( ( breln1compEex = $true )
=> ( ( breln1unionprop = $true )
=> ( ( breln1unionIL = $true )
=> ( ( breln1unionIR = $true )
=> ( ( breln1unionI = $true )
=> ( ( breln1unionE = $true )
=> ( ( breln1unionEcases = $true )
=> ( ( breln1unionCommutes = $true )
=> ( ( woz2Ex = $true )
=> ( ( woz2W = $true )
=> ( ( woz2A = $true )
=> ( ( woz2B = $true )
=> ( ( image1Ex = $true )
=> ( ( image1Ex1 = $true )
=> ( ( image1Equiv = $true )
=> ( ( image1E = $true )
=> ( ( image1I = $true )
=> ( ( injFuncInInjFuncSet = $true )
=> ( ( injFuncSetFuncIn = $true )
=> ( ( injFuncSetFuncInj = $true )
=> ( ( surjFuncSetFuncIn = $true )
=> ( ( surjFuncSetFuncSurj = $true )
=> ( ( leftInvIsSurj = $true )
=> ( ( surjCantorThm = $true )
=> ( ( foundation2 = $true )
=> ( ( notinself = $true )
=> ( ( notinself2 = $true )
=> ( ( omegaSp = $true )
=> ( ( omegaSclos = $true )
=> ( ( peano0notS = $true )
=> ( ( peanoSinj = $true )
=> ! [X0: $i] :
( ( $true
= ( transitiveset @ X0 ) )
=> ! [X1: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
=> ( ( subset @ X1 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f909]) ).
thf(f909,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ! [X0: $i] :
( ( transitiveset @ X0 )
=> ! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( subset @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f333]) ).
thf(f333,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ! [X11: $i] :
( ( transitiveset @ X11 )
=> ! [X3: $i] :
( ( in @ X3 @ X11 )
=> ( subset @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f332]) ).
thf(f332,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ( breln1unionCommutes
=> ( woz2Ex
=> ( woz2W
=> ( woz2A
=> ( woz2B
=> ( image1Ex
=> ( image1Ex1
=> ( image1Equiv
=> ( image1E
=> ( image1I
=> ( injFuncInInjFuncSet
=> ( injFuncSetFuncIn
=> ( injFuncSetFuncInj
=> ( surjFuncSetFuncIn
=> ( surjFuncSetFuncSurj
=> ( leftInvIsSurj
=> ( surjCantorThm
=> ( foundation2
=> ( notinself
=> ( notinself2
=> ( omegaSp
=> ( omegaSclos
=> ( peano0notS
=> ( peanoSinj
=> ! [X11: $i] :
( ( transitiveset @ X11 )
=> ! [X3: $i] :
( ( in @ X3 @ X11 )
=> ( subset @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.4NXBcYPEpZ/DTF2THF_16671.p',transitivesetOp1) ).
thf(f2421,plain,
( ( subset @ sK3 @ sK2 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f2420]) ).
thf(f2420,plain,
( ( ( subset @ sK3 @ sK2 )
= $true )
| ( $false = $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f2417]) ).
thf(f2417,plain,
( ( $false = $true )
| ( $false = $true )
| ( ( subset @ sK3 @ sK2 )
= $true ) ),
inference(superposition,[status(thm)],[f1005,f2378]) ).
thf(f2378,plain,
( ( $true
=> ( subset @ sK3 @ sK2 ) )
= $true ),
inference(superposition,[status(thm)],[f2206,f1484]) ).
thf(f1484,plain,
( ( in @ sK3 @ sK2 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1016]) ).
thf(f2206,plain,
! [X0: $i] :
( $true
= ( ( in @ X0 @ sK2 )
=> ( subset @ X0 @ sK2 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2205]) ).
thf(f2205,plain,
! [X0: $i] :
( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( subset @ Y0 @ sK2 ) )
@ X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2202]) ).
thf(f2202,plain,
! [X0: $i] :
( ( $true
= ( ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( subset @ Y0 @ sK2 ) )
@ X0 ) )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f1000,f2061]) ).
thf(f2061,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( ( in @ Y0 @ sK2 )
=> ( subset @ Y0 @ sK2 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1707]) ).
thf(f1707,plain,
( ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( subset @ Y1 @ Y0 ) ) )
@ sK2 )
= $true ),
inference(definition_unfolding,[status(thm)],[f1485,f1677]) ).
thf(f1677,plain,
( transitiveset
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( subset @ Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f522]) ).
thf(f522,plain,
( transitiveset
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( subset @ Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f521]) ).
thf(f521,plain,
( ( ^ [X0: $i] :
! [X1: $i] :
( ( in @ X1 @ X0 )
=> ( subset @ X1 @ X0 ) ) )
= transitiveset ),
inference(rectify,[status(thm)],[f331]) ).
thf(f331,axiom,
( ( ^ [X3: $i] :
! [X11: $i] :
( ( in @ X11 @ X3 )
=> ( subset @ X11 @ X3 ) ) )
= transitiveset ),
file('/export/starexec/sandbox2/tmp/tmp.4NXBcYPEpZ/DTF2THF_16671.p',transitiveset) ).
thf(f1485,plain,
( ( transitiveset @ sK2 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1016]) ).
thf(f1000,plain,
! [X0: $tType,X2: X0,X1: X0 > $o] :
( ( $false
= ( !! @ X0 @ X1 ) )
| ( ( X1 @ X2 )
= $true ) ),
introduced(definition,[],[pi_proxy_axiom]) ).
thf(f1005,plain,
! [X2: $o,X1: $o] :
( ( $false
= ( X1
=> X2 ) )
| ( $true = X2 )
| ( $false = X1 ) ),
introduced(definition,[],[implies_proxy_axiom]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU812^1 : TPTP v9.2.1. Released v3.7.0.
% 0.13/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n013.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:34:38 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SEU812^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.46 % Domain : Set Theory
% 0.30/0.46 % Problem : Transitive Sets
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English : (! X:i.transitiveset X -> (! A:i.in A X -> subset A X))
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.46 % Source : [Bro08]
% 0.30/0.46 % Names : ZFC314g [Bro08]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v4.0.1, 1.00 v4.0.0, 0.67 v3.7.0
% 0.30/0.46 % Syntax : Number of formulae : 719 ( 331 unt; 387 typ; 331 def)
% 0.30/0.46 % Number of atoms : 2035 ( 476 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 331 ( 6 avg)
% 0.30/0.46 % Number of connectives : 4438 ( 77 ~; 14 |; 54 &;3227 @)
% 0.30/0.46 % ( 17 <=>;1049 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 335 ( 2 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 190 ( 190 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 390 ( 387 usr; 332 con; 0-4 aty)
% 0.30/0.46 % Number of variables : 1155 ( 76 ^;1035 !; 44 ?;1155 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : http://mathgate.info/detsetitem.php?id=500
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(in_type,type,
% 0.30/0.46 in: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu_type,type,
% 0.30/0.46 exu: ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu,definition,
% 0.30/0.46 ( exu
% 0.30/0.46 = ( ^ [Xphi: $i > $o] :
% 0.30/0.46 ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx_type,type,
% 0.30/0.46 setextAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx,definition,
% 0.30/0.46 ( setextAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xx @ B ) )
% 0.30/0.46 => ( A = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset_type,type,
% 0.30/0.46 emptyset: $i ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx_type,type,
% 0.30/0.46 emptysetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx,definition,
% 0.30/0.46 ( emptysetAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin_type,type,
% 0.30/0.46 setadjoin: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx_type,type,
% 0.30/0.46 setadjoinAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx,definition,
% 0.30/0.46 ( setadjoinAx
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 <=> ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset_type,type,
% 0.30/0.46 powerset: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx_type,type,
% 0.30/0.46 powersetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx,definition,
% 0.30/0.46 ( powersetAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 <=> ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion_type,type,
% 0.30/0.46 setunion: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx_type,type,
% 0.30/0.46 setunionAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx,definition,
% 0.30/0.46 ( setunionAx
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 <=> ? [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega_type,type,
% 0.30/0.46 omega: $i ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax_type,type,
% 0.30/0.46 omega0Ax: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax,definition,
% 0.30/0.46 ( omega0Ax
% 0.30/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx_type,type,
% 0.30/0.46 omegaSAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx,definition,
% 0.30/0.46 ( omegaSAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx_type,type,
% 0.30/0.46 omegaIndAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx,definition,
% 0.30/0.46 ( omegaIndAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( ( in @ emptyset @ A )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ omega )
% 0.30/0.46 & ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(replAx_type,type,
% 0.30/0.46 replAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(replAx,definition,
% 0.30/0.46 ( replAx
% 0.30/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx_type,type,
% 0.30/0.46 foundationAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx,definition,
% 0.30/0.46 ( foundationAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ( ( in @ B @ A )
% 0.30/0.46 & ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx_type,type,
% 0.30/0.46 wellorderingAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx,definition,
% 0.30/0.46 ( wellorderingAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ? [B: $i] :
% 0.30/0.46 ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) ) )
% 0.30/0.46 & ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xy @ A ) )
% 0.30/0.46 => ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ( ( in @ Xx @ C )
% 0.30/0.46 <=> ( in @ Xy @ C ) ) )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 & ! [C: $i,D: $i] :
% 0.30/0.46 ( ( ( in @ C @ B )
% 0.30/0.46 & ( in @ D @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ D ) )
% 0.30/0.46 | ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ D )
% 0.30/0.46 => ( in @ Xx @ C ) ) ) )
% 0.30/0.46 & ! [C: $i] :
% 0.30/0.46 ( ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.46 => ? [D: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ D @ B )
% 0.30/0.46 & ( in @ Xx @ C )
% 0.30/0.46 & ~ ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ D )
% 0.30/0.46 & ( in @ Xy @ C ) )
% 0.30/0.46 & ! [E: $i] :
% 0.30/0.46 ( ( in @ E @ B )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ E )
% 0.30/0.46 => ( in @ Xy @ D ) )
% 0.30/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr_type,type,
% 0.30/0.46 descr: ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(descrp_type,type,
% 0.30/0.46 descrp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descrp,definition,
% 0.30/0.46 ( descrp
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 @ ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr_type,type,
% 0.30/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI_type,type,
% 0.30/0.46 dsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI,definition,
% 0.30/0.46 ( dsetconstrI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL_type,type,
% 0.30/0.46 dsetconstrEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL,definition,
% 0.30/0.46 ( dsetconstrEL
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER_type,type,
% 0.30/0.46 dsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER,definition,
% 0.30/0.46 ( dsetconstrER
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1_type,type,
% 0.30/0.46 exuE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1,definition,
% 0.30/0.46 ( exuE1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set_type,type,
% 0.30/0.46 prop2set: $o > $i ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE_type,type,
% 0.30/0.46 prop2setE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE,definition,
% 0.30/0.46 ( prop2setE
% 0.30/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE_type,type,
% 0.30/0.46 emptysetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE,definition,
% 0.30/0.46 ( emptysetE
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse_type,type,
% 0.30/0.46 emptysetimpfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse,definition,
% 0.30/0.46 ( emptysetimpfalse
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => $false ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset_type,type,
% 0.30/0.46 notinemptyset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset,definition,
% 0.30/0.46 ( notinemptyset
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e_type,type,
% 0.30/0.46 exuE3e: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e,definition,
% 0.30/0.46 ( exuE3e
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setext_type,type,
% 0.30/0.46 setext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setext,definition,
% 0.30/0.46 ( setext
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI_type,type,
% 0.30/0.46 emptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI,definition,
% 0.30/0.46 ( emptyI
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty_type,type,
% 0.30/0.46 noeltsimpempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty,definition,
% 0.30/0.46 ( noeltsimpempty
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta_type,type,
% 0.30/0.46 setbeta: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta,definition,
% 0.30/0.46 ( setbeta
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty_type,type,
% 0.30/0.46 nonempty: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty,definition,
% 0.30/0.46 ( nonempty
% 0.30/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1_type,type,
% 0.30/0.46 nonemptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1,definition,
% 0.30/0.46 ( nonemptyE1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI_type,type,
% 0.30/0.46 nonemptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI,definition,
% 0.30/0.46 ( nonemptyI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( nonempty
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1_type,type,
% 0.30/0.46 nonemptyI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1,definition,
% 0.30/0.46 ( nonemptyI1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL_type,type,
% 0.30/0.46 setadjoinIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL,definition,
% 0.30/0.46 ( setadjoinIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty_type,type,
% 0.30/0.46 emptyinunitempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty,definition,
% 0.30/0.46 ( emptyinunitempty
% 0.30/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR_type,type,
% 0.30/0.46 setadjoinIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR,definition,
% 0.30/0.46 ( setadjoinIR
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE_type,type,
% 0.30/0.46 setadjoinE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE,definition,
% 0.30/0.46 ( setadjoinE
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( Xy = Xx )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xy @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr_type,type,
% 0.30/0.46 setadjoinOr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr,definition,
% 0.30/0.46 ( setadjoinOr
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq_type,type,
% 0.30/0.46 setoftrueEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq,definition,
% 0.30/0.46 ( setoftrueEq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : $true )
% 0.30/0.46 = A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI_type,type,
% 0.30/0.46 powersetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI,definition,
% 0.30/0.46 ( powersetI
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset_type,type,
% 0.30/0.46 emptyinPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset,definition,
% 0.30/0.46 ( emptyinPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset_type,type,
% 0.30/0.46 emptyInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset,definition,
% 0.30/0.46 ( emptyInPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE_type,type,
% 0.30/0.46 powersetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE,definition,
% 0.30/0.46 ( powersetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI_type,type,
% 0.30/0.46 setunionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI,definition,
% 0.30/0.46 ( setunionI
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE_type,type,
% 0.30/0.46 setunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE,definition,
% 0.30/0.46 ( setunionE
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU_type,type,
% 0.30/0.46 subPowSU: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU,definition,
% 0.30/0.46 ( subPowSU
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2_type,type,
% 0.30/0.46 exuE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2,definition,
% 0.30/0.46 ( exuE2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness_type,type,
% 0.30/0.46 nonemptyImpWitness: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness,definition,
% 0.30/0.46 ( nonemptyImpWitness
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & $true ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit_type,type,
% 0.30/0.46 uniqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit,definition,
% 0.30/0.46 ( uniqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton_type,type,
% 0.30/0.46 notinsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton,definition,
% 0.30/0.46 ( notinsingleton
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx != Xy )
% 0.30/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit_type,type,
% 0.30/0.46 eqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit,definition,
% 0.30/0.46 ( eqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch_type,type,
% 0.30/0.46 singletonsswitch: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch,definition,
% 0.30/0.46 ( singletonsswitch
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE_type,type,
% 0.30/0.46 upairsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE,definition,
% 0.30/0.46 ( upairsetE
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL_type,type,
% 0.30/0.46 upairsetIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL,definition,
% 0.30/0.46 ( upairsetIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR_type,type,
% 0.30/0.46 upairsetIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR,definition,
% 0.30/0.46 ( upairsetIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1_type,type,
% 0.30/0.46 emptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1,definition,
% 0.30/0.46 ( emptyE1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => $false ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall_type,type,
% 0.30/0.46 vacuousDall: $o ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall,definition,
% 0.30/0.46 ( vacuousDall
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1_type,type,
% 0.30/0.46 quantDeMorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1,definition,
% 0.30/0.46 ( quantDeMorgan1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2_type,type,
% 0.30/0.46 quantDeMorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2,definition,
% 0.30/0.46 ( quantDeMorgan2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3_type,type,
% 0.30/0.46 quantDeMorgan3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3,definition,
% 0.30/0.46 ( quantDeMorgan3
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4_type,type,
% 0.30/0.46 quantDeMorgan4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4,definition,
% 0.30/0.46 ( quantDeMorgan4
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI_type,type,
% 0.30/0.46 prop2setI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI,definition,
% 0.30/0.46 ( prop2setI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(set2prop_type,type,
% 0.30/0.46 set2prop: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI_type,type,
% 0.30/0.46 prop2set2propI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI,definition,
% 0.30/0.46 ( prop2set2propI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE_type,type,
% 0.30/0.46 notdexE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE,definition,
% 0.30/0.46 ( notdexE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE_type,type,
% 0.30/0.46 notdallE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE,definition,
% 0.30/0.46 ( notdallE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1_type,type,
% 0.30/0.46 exuI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1,definition,
% 0.30/0.46 ( exuI1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3_type,type,
% 0.30/0.46 exuI3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3,definition,
% 0.30/0.46 ( exuI3
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2_type,type,
% 0.30/0.46 exuI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2,definition,
% 0.30/0.46 ( exuI2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP_type,type,
% 0.30/0.46 inCongP: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP,definition,
% 0.30/0.46 ( inCongP
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong_type,type,
% 0.30/0.46 in__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong,definition,
% 0.30/0.46 ( in__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u_type,type,
% 0.30/0.46 exuE3u: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u,definition,
% 0.30/0.46 ( exuE3u
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong_type,type,
% 0.30/0.46 exu__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong,definition,
% 0.30/0.46 ( exu__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 <=> ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong_type,type,
% 0.30/0.46 emptyset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong,definition,
% 0.30/0.46 ( emptyset__Cong
% 0.30/0.46 = ( emptyset = emptyset ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong_type,type,
% 0.30/0.46 setadjoin__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong,definition,
% 0.30/0.46 ( setadjoin__Cong
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ! [Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( Xz = Xu )
% 0.30/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong_type,type,
% 0.30/0.46 powerset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong,definition,
% 0.30/0.46 ( powerset__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( powerset @ A )
% 0.30/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong_type,type,
% 0.30/0.46 setunion__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong,definition,
% 0.30/0.46 ( setunion__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( setunion @ A )
% 0.30/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong_type,type,
% 0.30/0.46 omega__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong,definition,
% 0.30/0.46 ( omega__Cong
% 0.30/0.46 = ( omega = omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu_type,type,
% 0.30/0.46 exuEu: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu,definition,
% 0.30/0.46 ( exuEu
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong_type,type,
% 0.30/0.46 descr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong,definition,
% 0.30/0.46 ( descr__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.46 => ( ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong_type,type,
% 0.30/0.46 dsetconstr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong,definition,
% 0.30/0.46 ( dsetconstr__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset_type,type,
% 0.30/0.46 subset: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjoint_type,type,
% 0.30/0.46 disjoint: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setsmeet_type,type,
% 0.30/0.46 setsmeet: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1_type,type,
% 0.30/0.46 subsetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1,definition,
% 0.30/0.46 ( subsetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2_type,type,
% 0.30/0.46 eqimpsubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2,definition,
% 0.30/0.46 ( eqimpsubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1_type,type,
% 0.30/0.46 eqimpsubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1,definition,
% 0.30/0.46 ( eqimpsubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2_type,type,
% 0.30/0.46 subsetI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2,definition,
% 0.30/0.46 ( subsetI2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset_type,type,
% 0.30/0.46 emptysetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset,definition,
% 0.30/0.46 ( emptysetsubset
% 0.30/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE_type,type,
% 0.30/0.46 subsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE,definition,
% 0.30/0.46 ( subsetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2_type,type,
% 0.30/0.46 subsetE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2,definition,
% 0.30/0.46 ( subsetE2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI_type,type,
% 0.30/0.46 notsubsetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI,definition,
% 0.30/0.46 ( notsubsetI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1_type,type,
% 0.30/0.46 notequalI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1,definition,
% 0.30/0.46 ( notequalI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ( subset @ A @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2_type,type,
% 0.30/0.46 notequalI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2,definition,
% 0.30/0.46 ( notequalI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl_type,type,
% 0.30/0.46 subsetRefl: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl,definition,
% 0.30/0.46 ( subsetRefl
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans_type,type,
% 0.30/0.46 subsetTrans: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans,definition,
% 0.30/0.46 ( subsetTrans
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ C )
% 0.30/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub_type,type,
% 0.30/0.46 setadjoinSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub,definition,
% 0.30/0.46 ( setadjoinSub
% 0.30/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2_type,type,
% 0.30/0.46 setadjoinSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2,definition,
% 0.30/0.46 ( setadjoinSub2
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset_type,type,
% 0.30/0.46 subset2powerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset,definition,
% 0.30/0.46 ( subset2powerset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub_type,type,
% 0.30/0.46 setextsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub,definition,
% 0.30/0.46 ( setextsub
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ A )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq_type,type,
% 0.30/0.46 subsetemptysetimpeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq,definition,
% 0.30/0.46 ( subsetemptysetimpeq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( subset @ A @ emptyset )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1_type,type,
% 0.30/0.46 powersetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1,definition,
% 0.30/0.46 ( powersetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1_type,type,
% 0.30/0.46 powersetE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1,definition,
% 0.30/0.46 ( powersetE1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset_type,type,
% 0.30/0.46 inPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset,definition,
% 0.30/0.46 ( inPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset_type,type,
% 0.30/0.46 powersetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset,definition,
% 0.30/0.46 ( powersetsubset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset_type,type,
% 0.30/0.46 sepInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset,definition,
% 0.30/0.46 ( sepInPowerset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( in
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset_type,type,
% 0.30/0.46 sepSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset,definition,
% 0.30/0.46 ( sepSubset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunion_type,type,
% 0.30/0.46 binunion: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL_type,type,
% 0.30/0.46 binunionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL,definition,
% 0.30/0.46 ( binunionIL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR_type,type,
% 0.30/0.46 upairset2IR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR,definition,
% 0.30/0.46 ( upairset2IR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR_type,type,
% 0.30/0.46 binunionIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR,definition,
% 0.30/0.46 ( binunionIR
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases_type,type,
% 0.30/0.46 binunionEcases: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases,definition,
% 0.30/0.46 ( binunionEcases
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( ( in @ Xx @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE_type,type,
% 0.30/0.46 binunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE,definition,
% 0.30/0.46 ( binunionE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub_type,type,
% 0.30/0.46 binunionLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub,definition,
% 0.30/0.46 ( binunionLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub_type,type,
% 0.30/0.46 binunionRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub,definition,
% 0.30/0.46 ( binunionRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersect_type,type,
% 0.30/0.46 binintersect: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI_type,type,
% 0.30/0.46 binintersectI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI,definition,
% 0.30/0.46 ( binintersectI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5_type,type,
% 0.30/0.46 binintersectSubset5: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5,definition,
% 0.30/0.46 ( binintersectSubset5
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ C @ A )
% 0.30/0.46 => ( ( subset @ C @ B )
% 0.30/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL_type,type,
% 0.30/0.46 binintersectEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL,definition,
% 0.30/0.46 ( binintersectEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub_type,type,
% 0.30/0.46 binintersectLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub,definition,
% 0.30/0.46 ( binintersectLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2_type,type,
% 0.30/0.46 binintersectSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2,definition,
% 0.30/0.46 ( binintersectSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3_type,type,
% 0.30/0.46 binintersectSubset3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3,definition,
% 0.30/0.46 ( binintersectSubset3
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER_type,type,
% 0.30/0.46 binintersectER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER,definition,
% 0.30/0.46 ( binintersectER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1_type,type,
% 0.30/0.46 disjointsetsI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1,definition,
% 0.30/0.46 ( disjointsetsI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xx @ B ) )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub_type,type,
% 0.30/0.46 binintersectRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub,definition,
% 0.30/0.46 ( binintersectRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4_type,type,
% 0.30/0.46 binintersectSubset4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4,definition,
% 0.30/0.46 ( binintersectSubset4
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1_type,type,
% 0.30/0.46 binintersectSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1,definition,
% 0.30/0.46 ( binintersectSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = A )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d_type,type,
% 0.30/0.46 bs114d: $o ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d,definition,
% 0.30/0.46 ( bs114d
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.30/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(regular_type,type,
% 0.30/0.46 regular: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminus_type,type,
% 0.30/0.46 setminus: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI_type,type,
% 0.30/0.46 setminusI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI,definition,
% 0.30/0.46 ( setminusI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL_type,type,
% 0.30/0.46 setminusEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL,definition,
% 0.30/0.46 ( setminusEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER_type,type,
% 0.30/0.46 setminusER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER,definition,
% 0.30/0.46 ( setminusER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2_type,type,
% 0.30/0.46 setminusSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2,definition,
% 0.30/0.46 ( setminusSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg_type,type,
% 0.30/0.46 setminusERneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg,definition,
% 0.30/0.46 ( setminusERneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg_type,type,
% 0.30/0.46 setminusELneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg,definition,
% 0.30/0.46 ( setminusELneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg_type,type,
% 0.30/0.46 setminusILneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg,definition,
% 0.30/0.46 ( setminusILneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg_type,type,
% 0.30/0.46 setminusIRneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg,definition,
% 0.30/0.46 ( setminusIRneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub_type,type,
% 0.30/0.46 setminusLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub,definition,
% 0.30/0.46 ( setminusLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1_type,type,
% 0.30/0.46 setminusSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1,definition,
% 0.30/0.46 ( setminusSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiff_type,type,
% 0.30/0.46 symdiff: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE_type,type,
% 0.30/0.46 symdiffE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE,definition,
% 0.30/0.46 ( symdiffE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1_type,type,
% 0.30/0.46 symdiffI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1,definition,
% 0.30/0.46 ( symdiffI1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2_type,type,
% 0.30/0.46 symdiffI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2,definition,
% 0.30/0.46 ( symdiffI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1_type,type,
% 0.30/0.46 symdiffIneg1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1,definition,
% 0.30/0.46 ( symdiffIneg1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2_type,type,
% 0.30/0.46 symdiffIneg2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2,definition,
% 0.30/0.46 ( symdiffIneg2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iskpair_type,type,
% 0.30/0.46 iskpair: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair_type,type,
% 0.30/0.46 secondinupair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair,definition,
% 0.30/0.46 ( secondinupair
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL_type,type,
% 0.30/0.46 setukpairIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL,definition,
% 0.30/0.46 ( setukpairIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIR_type,type,
% 0.30/0.46 setukpairIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIR,definition,
% 0.30/0.46 ( setukpairIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpairiskpair_type,type,
% 0.30/0.46 kpairiskpair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairiskpair,definition,
% 0.30/0.46 ( kpairiskpair
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kpair_type,type,
% 0.30/0.46 kpair: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(kpairp_type,type,
% 0.30/0.46 kpairp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kpairp,definition,
% 0.30/0.46 ( kpairp
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprod_type,type,
% 0.30/0.46 cartprod: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsubset_type,type,
% 0.30/0.46 singletonsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsubset,definition,
% 0.30/0.46 ( singletonsubset
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowerset_type,type,
% 0.30/0.46 singletoninpowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowerset,definition,
% 0.30/0.46 ( singletoninpowerset
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowunion_type,type,
% 0.30/0.46 singletoninpowunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletoninpowunion,definition,
% 0.30/0.46 ( singletoninpowunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2E_type,type,
% 0.30/0.46 upairset2E: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2E,definition,
% 0.30/0.46 ( upairset2E
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsubunion_type,type,
% 0.30/0.46 upairsubunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsubunion,definition,
% 0.30/0.46 ( upairsubunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairinpowunion_type,type,
% 0.30/0.46 upairinpowunion: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairinpowunion,definition,
% 0.30/0.46 ( upairinpowunion
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem1_type,type,
% 0.30/0.46 ubforcartprodlem1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem1,definition,
% 0.30/0.46 ( ubforcartprodlem1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem2_type,type,
% 0.30/0.46 ubforcartprodlem2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem2,definition,
% 0.30/0.46 ( ubforcartprodlem2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem3_type,type,
% 0.30/0.46 ubforcartprodlem3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ubforcartprodlem3,definition,
% 0.30/0.46 ( ubforcartprodlem3
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairin_type,type,
% 0.30/0.46 cartprodpairin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodpairin,definition,
% 0.30/0.46 ( cartprodpairin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair1_type,type,
% 0.30/0.46 cartprodmempair1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair1,definition,
% 0.30/0.46 ( cartprodmempair1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 & ( Xu
% 0.30/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair_type,type,
% 0.30/0.46 cartprodmempair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodmempair,definition,
% 0.30/0.46 ( cartprodmempair
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE2_type,type,
% 0.30/0.46 setunionE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE2,definition,
% 0.30/0.46 ( setunionE2
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ? [X: $i] :
% 0.30/0.46 ( ( in @ X @ A )
% 0.30/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton1_type,type,
% 0.30/0.46 setunionsingleton1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton1,definition,
% 0.30/0.46 ( setunionsingleton1
% 0.30/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton2_type,type,
% 0.30/0.46 setunionsingleton2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton2,definition,
% 0.30/0.46 ( setunionsingleton2
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton_type,type,
% 0.30/0.46 setunionsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionsingleton,definition,
% 0.30/0.46 ( setunionsingleton
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.30/0.46 = Xx ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singleton_type,type,
% 0.30/0.46 singleton: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonprop_type,type,
% 0.30/0.46 singletonprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonprop,definition,
% 0.30/0.46 ( singletonprop
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) )
% 0.30/0.46 => ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1_type,type,
% 0.30/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E1_type,type,
% 0.30/0.46 ex1E1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1E1,definition,
% 0.30/0.46 ( ex1E1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I_type,type,
% 0.30/0.46 ex1I: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I,definition,
% 0.30/0.46 ( ex1I
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xy = Xx ) ) )
% 0.30/0.46 => ( ex1 @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I2_type,type,
% 0.30/0.46 ex1I2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(ex1I2,definition,
% 0.30/0.46 ( ex1I2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) )
% 0.30/0.46 => ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ex1 @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsuniq_type,type,
% 0.30/0.46 singletonsuniq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsuniq,definition,
% 0.30/0.46 ( singletonsuniq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.30/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(atmost1p_type,type,
% 0.30/0.46 atmost1p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(atleast2p_type,type,
% 0.30/0.46 atleast2p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(atmost2p_type,type,
% 0.30/0.46 atmost2p: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetp_type,type,
% 0.30/0.46 upairsetp: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL1_type,type,
% 0.30/0.46 setukpairinjL1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL1,definition,
% 0.30/0.46 ( setukpairinjL1
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kfstsingleton_type,type,
% 0.30/0.46 kfstsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kfstsingleton,definition,
% 0.30/0.46 ( kfstsingleton
% 0.30/0.46 = ( ! [Xu: $i] :
% 0.30/0.46 ( ( iskpair @ Xu )
% 0.30/0.46 => ( singleton
% 0.30/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(theprop_type,type,
% 0.30/0.46 theprop: $o ).
% 0.30/0.46
% 0.30/0.46 thf(theprop,definition,
% 0.30/0.46 ( theprop
% 0.30/0.46 = ( ! [X: $i] :
% 0.30/0.46 ( ( singleton @ X )
% 0.30/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(kfst_type,type,
% 0.30/0.46 kfst: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(kfstpairEq_type,type,
% 0.30/0.46 kfstpairEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(kfstpairEq,definition,
% 0.30/0.46 ( kfstpairEq
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.46 = Xx ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstin_type,type,
% 0.30/0.46 cartprodfstin: $o ).
% 0.30/0.46
% 0.30/0.46 thf(cartprodfstin,definition,
% 0.30/0.46 ( cartprodfstin
% 0.30/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL2_type,type,
% 0.30/0.46 setukpairinjL2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL2,definition,
% 0.30/0.46 ( setukpairinjL2
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL_type,type,
% 0.30/0.46 setukpairinjL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjL,definition,
% 0.30/0.46 ( setukpairinjL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.46 = ( kpair @ Xz @ Xu ) )
% 0.30/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR11_type,type,
% 0.30/0.46 setukpairinjR11: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairinjR11,definition,
% 0.30/0.46 ( setukpairinjR11
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR12_type,type,
% 0.30/0.47 setukpairinjR12: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR12,definition,
% 0.30/0.47 ( setukpairinjR12
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR1_type,type,
% 0.30/0.47 setukpairinjR1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR1,definition,
% 0.30/0.47 ( setukpairinjR1
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( ( Xz = Xu )
% 0.30/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairequniteq_type,type,
% 0.30/0.47 upairequniteq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairequniteq,definition,
% 0.30/0.47 ( upairequniteq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR2_type,type,
% 0.30/0.47 setukpairinjR2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR2,definition,
% 0.30/0.47 ( setukpairinjR2
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR_type,type,
% 0.30/0.47 setukpairinjR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR,definition,
% 0.30/0.47 ( setukpairinjR
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.47 = ( kpair @ Xz @ Xu ) )
% 0.30/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ksndsingleton_type,type,
% 0.30/0.47 ksndsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ksndsingleton,definition,
% 0.30/0.47 ( ksndsingleton
% 0.30/0.47 = ( ! [Xu: $i] :
% 0.30/0.47 ( ( iskpair @ Xu )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.47 @ ^ [Xx: $i] :
% 0.30/0.47 ( Xu
% 0.30/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ksnd_type,type,
% 0.30/0.47 ksnd: $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(ksndpairEq_type,type,
% 0.30/0.47 ksndpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ksndpairEq,definition,
% 0.30/0.47 ( ksndpairEq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xy ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kpairsurjEq_type,type,
% 0.30/0.47 kpairsurjEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kpairsurjEq,definition,
% 0.30/0.47 ( kpairsurjEq
% 0.30/0.47 = ( ! [Xu: $i] :
% 0.30/0.47 ( ( iskpair @ Xu )
% 0.30/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.47 = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndin_type,type,
% 0.30/0.47 cartprodsndin: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndin,definition,
% 0.30/0.47 ( cartprodsndin
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemEL_type,type,
% 0.30/0.47 cartprodpairmemEL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemEL,definition,
% 0.30/0.47 ( cartprodpairmemEL
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemER_type,type,
% 0.30/0.47 cartprodpairmemER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemER,definition,
% 0.30/0.47 ( cartprodpairmemER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempaircEq_type,type,
% 0.30/0.47 cartprodmempaircEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempaircEq,definition,
% 0.30/0.47 ( cartprodmempaircEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( kpair @ Xx @ Xy )
% 0.30/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstpairEq_type,type,
% 0.30/0.47 cartprodfstpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstpairEq,definition,
% 0.30/0.47 ( cartprodfstpairEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndpairEq_type,type,
% 0.30/0.47 cartprodsndpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndpairEq,definition,
% 0.30/0.47 ( cartprodsndpairEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairsurjEq_type,type,
% 0.30/0.47 cartprodpairsurjEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairsurjEq,definition,
% 0.30/0.47 ( cartprodpairsurjEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.47 = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln_type,type,
% 0.30/0.47 breln: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstr_type,type,
% 0.30/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrI_type,type,
% 0.30/0.47 dpsetconstrI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrI,definition,
% 0.30/0.47 ( dpsetconstrI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrSub_type,type,
% 0.30/0.47 dpsetconstrSub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrSub,definition,
% 0.30/0.47 ( dpsetconstrSub
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.47 ( subset
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.30/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setOfPairsIsBReln_type,type,
% 0.30/0.47 setOfPairsIsBReln: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setOfPairsIsBReln,definition,
% 0.30/0.47 ( setOfPairsIsBReln
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.47 ( breln @ A @ B
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrERa_type,type,
% 0.30/0.47 dpsetconstrERa: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrERa,definition,
% 0.30/0.47 ( dpsetconstrERa
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL1_type,type,
% 0.30/0.47 dpsetconstrEL1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL1,definition,
% 0.30/0.47 ( dpsetconstrEL1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL2_type,type,
% 0.30/0.47 dpsetconstrEL2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL2,definition,
% 0.30/0.47 ( dpsetconstrEL2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrER_type,type,
% 0.30/0.47 dpsetconstrER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrER,definition,
% 0.30/0.47 ( dpsetconstrER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(func_type,type,
% 0.30/0.47 func: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcSet_type,type,
% 0.30/0.47 funcSet: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(funcImageSingleton_type,type,
% 0.30/0.47 funcImageSingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcImageSingleton,definition,
% 0.30/0.47 ( funcImageSingleton
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(apProp_type,type,
% 0.30/0.47 apProp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(apProp,definition,
% 0.30/0.47 ( apProp
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in
% 0.30/0.47 @ ( setunion
% 0.30/0.47 @ ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.30/0.47 @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap_type,type,
% 0.30/0.47 ap: $i > $i > $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(app_type,type,
% 0.30/0.47 app: $o ).
% 0.30/0.47
% 0.30/0.47 thf(app,definition,
% 0.30/0.47 ( app
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(infuncsetfunc_type,type,
% 0.30/0.47 infuncsetfunc: $o ).
% 0.30/0.47
% 0.30/0.47 thf(infuncsetfunc,definition,
% 0.30/0.47 ( infuncsetfunc
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap2p_type,type,
% 0.30/0.47 ap2p: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ap2p,definition,
% 0.30/0.47 ( ap2p
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcinfuncset_type,type,
% 0.30/0.47 funcinfuncset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcinfuncset,definition,
% 0.30/0.47 ( funcinfuncset
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lamProp_type,type,
% 0.30/0.47 lamProp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lamProp,definition,
% 0.30/0.47 ( lamProp
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( func @ A @ B
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xf @ Xx )
% 0.30/0.47 = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lam_type,type,
% 0.30/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(lamp_type,type,
% 0.30/0.47 lamp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lamp,definition,
% 0.30/0.47 ( lamp
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( func @ A @ B
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lam2p_type,type,
% 0.30/0.47 lam2p: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lam2p,definition,
% 0.30/0.47 ( lam2p
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( in
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall1_type,type,
% 0.30/0.47 brelnall1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall1,definition,
% 0.30/0.47 ( brelnall1
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ R )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall2_type,type,
% 0.30/0.47 brelnall2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall2,definition,
% 0.30/0.47 ( brelnall2
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ R )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E2_type,type,
% 0.30/0.47 ex1E2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E2,definition,
% 0.30/0.47 ( ex1E2
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ( ex1 @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp1_type,type,
% 0.30/0.47 funcGraphProp1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp1,definition,
% 0.30/0.47 ( funcGraphProp1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp3_type,type,
% 0.30/0.47 funcGraphProp3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp3,definition,
% 0.30/0.47 ( funcGraphProp3
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp2_type,type,
% 0.30/0.47 funcGraphProp2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp2,definition,
% 0.30/0.47 ( funcGraphProp2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcextLem_type,type,
% 0.30/0.47 funcextLem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcextLem,definition,
% 0.30/0.47 ( funcextLem
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xg )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp4_type,type,
% 0.30/0.47 funcGraphProp4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp4,definition,
% 0.30/0.47 ( funcGraphProp4
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subbreln_type,type,
% 0.30/0.47 subbreln: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subbreln,definition,
% 0.30/0.47 ( subbreln
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ S )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqbreln_type,type,
% 0.30/0.47 eqbreln: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqbreln,definition,
% 0.30/0.47 ( eqbreln
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ S )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcext_type,type,
% 0.30/0.47 funcext: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcext,definition,
% 0.30/0.47 ( funcext
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xg )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcext2_type,type,
% 0.30/0.47 funcext2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcext2,definition,
% 0.30/0.47 ( funcext2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq1_type,type,
% 0.30/0.47 ap2apEq1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq1,definition,
% 0.30/0.47 ( ap2apEq1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq2_type,type,
% 0.30/0.47 ap2apEq2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq2,definition,
% 0.30/0.47 ( ap2apEq2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(beta1_type,type,
% 0.30/0.47 beta1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(beta1,definition,
% 0.30/0.47 ( beta1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.47 @ Xx )
% 0.30/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eta1_type,type,
% 0.30/0.47 eta1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eta1,definition,
% 0.30/0.47 ( eta1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ( ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.47 = Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lam2lamEq_type,type,
% 0.30/0.47 lam2lamEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lam2lamEq,definition,
% 0.30/0.47 ( lam2lamEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.47 = ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(beta2_type,type,
% 0.30/0.47 beta2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(beta2,definition,
% 0.30/0.47 ( beta2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.47 @ Xx )
% 0.30/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eta2_type,type,
% 0.30/0.47 eta2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eta2,definition,
% 0.30/0.47 ( eta2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.47 = Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp1_type,type,
% 0.30/0.47 iffalseProp1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp1,definition,
% 0.30/0.47 ( iffalseProp1
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ~ Xphi
% 0.30/0.47 => ( in @ Xy
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp2_type,type,
% 0.30/0.47 iffalseProp2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp2,definition,
% 0.30/0.47 ( iffalseProp2
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ~ Xphi
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) )
% 0.30/0.47 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp1_type,type,
% 0.30/0.47 iftrueProp1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp1,definition,
% 0.30/0.47 ( iftrueProp1
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( Xphi
% 0.30/0.47 => ( in @ Xx
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp2_type,type,
% 0.30/0.47 iftrueProp2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp2,definition,
% 0.30/0.47 ( iftrueProp2
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( Xphi
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) )
% 0.30/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ifSingleton_type,type,
% 0.30/0.47 ifSingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ifSingleton,definition,
% 0.30/0.47 ( ifSingleton
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(if_type,type,
% 0.30/0.47 if: $i > $o > $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(ifp_type,type,
% 0.30/0.47 ifp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ifp,definition,
% 0.30/0.47 ( ifp
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(theeq_type,type,
% 0.30/0.47 theeq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(theeq,definition,
% 0.30/0.47 ( theeq
% 0.30/0.47 = ( ! [X: $i] :
% 0.30/0.47 ( ( singleton @ X )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ X )
% 0.30/0.47 => ( ( setunion @ X )
% 0.30/0.47 = Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrue_type,type,
% 0.30/0.47 iftrue: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrue,definition,
% 0.30/0.47 ( iftrue
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( Xphi
% 0.30/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.47 = Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iffalse_type,type,
% 0.30/0.47 iffalse: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iffalse,definition,
% 0.30/0.47 ( iffalse
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ~ Xphi
% 0.30/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.47 = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueorfalse_type,type,
% 0.30/0.47 iftrueorfalse: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueorfalse,definition,
% 0.30/0.47 ( iftrueorfalse
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectT_lem_type,type,
% 0.30/0.47 binintersectT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectT_lem,definition,
% 0.30/0.47 ( binintersectT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionT_lem_type,type,
% 0.30/0.47 binunionT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionT_lem,definition,
% 0.30/0.47 ( binunionT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetT_lem_type,type,
% 0.30/0.47 powersetT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetT_lem,definition,
% 0.30/0.47 ( powersetT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusT_lem_type,type,
% 0.30/0.47 setminusT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusT_lem,definition,
% 0.30/0.47 ( setminusT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementT_lem_type,type,
% 0.30/0.47 complementT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementT_lem,definition,
% 0.30/0.47 ( complementT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setextT_type,type,
% 0.30/0.47 setextT: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setextT,definition,
% 0.30/0.47 ( setextT
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ( in @ Xx @ X ) ) )
% 0.30/0.47 => ( X = Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTI_type,type,
% 0.30/0.47 subsetTI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTI,definition,
% 0.30/0.47 ( subsetTI
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) )
% 0.30/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTI1_type,type,
% 0.30/0.47 powersetTI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTI1,definition,
% 0.30/0.47 ( powersetTI1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) )
% 0.30/0.47 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTE1_type,type,
% 0.30/0.47 powersetTE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTE1,definition,
% 0.30/0.47 ( powersetTE1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ X @ ( powerset @ Y ) )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTI1_type,type,
% 0.30/0.47 complementTI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTI1,definition,
% 0.30/0.47 ( complementTI1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTE1_type,type,
% 0.30/0.47 complementTE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTE1,definition,
% 0.30/0.47 ( complementTE1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTELcontra_type,type,
% 0.30/0.47 binintersectTELcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTELcontra,definition,
% 0.30/0.47 ( binintersectTELcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ X )
% 0.30/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTERcontra_type,type,
% 0.30/0.47 binintersectTERcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTERcontra,definition,
% 0.30/0.47 ( binintersectTERcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT_type,type,
% 0.30/0.47 contrasubsetT: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT,definition,
% 0.30/0.47 ( contrasubsetT
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT1_type,type,
% 0.30/0.47 contrasubsetT1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT1,definition,
% 0.30/0.47 ( contrasubsetT1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( subset @ X @ Y )
% 0.30/0.47 => ( ~ ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT2_type,type,
% 0.30/0.47 contrasubsetT2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT2,definition,
% 0.30/0.47 ( contrasubsetT2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ Y )
% 0.30/0.47 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT3_type,type,
% 0.30/0.47 contrasubsetT3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT3,definition,
% 0.30/0.47 ( contrasubsetT3
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementI1_type,type,
% 0.30/0.47 doubleComplementI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementI1,definition,
% 0.30/0.47 ( doubleComplementI1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementE1_type,type,
% 0.30/0.47 doubleComplementE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementE1,definition,
% 0.30/0.47 ( doubleComplementE1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.30/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub1_type,type,
% 0.30/0.47 doubleComplementSub1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub1,definition,
% 0.30/0.47 ( doubleComplementSub1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub2_type,type,
% 0.30/0.47 doubleComplementSub2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub2,definition,
% 0.30/0.47 ( doubleComplementSub2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementEq_type,type,
% 0.30/0.47 doubleComplementEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementEq,definition,
% 0.30/0.47 ( doubleComplementEq
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( X
% 0.30/0.47 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTnotintersectT_type,type,
% 0.30/0.47 complementTnotintersectT: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTnotintersectT,definition,
% 0.30/0.47 ( complementTnotintersectT
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementImpComplementIntersect_type,type,
% 0.30/0.47 complementImpComplementIntersect: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementImpComplementIntersect,definition,
% 0.30/0.47 ( complementImpComplementIntersect
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementSubsetComplementIntersect_type,type,
% 0.30/0.47 complementSubsetComplementIntersect: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementSubsetComplementIntersect,definition,
% 0.30/0.47 ( complementSubsetComplementIntersect
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementInPowersetComplementIntersect_type,type,
% 0.30/0.47 complementInPowersetComplementIntersect: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementInPowersetComplementIntersect,definition,
% 0.30/0.47 ( complementInPowersetComplementIntersect
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contraSubsetComplement_type,type,
% 0.30/0.47 contraSubsetComplement: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contraSubsetComplement,definition,
% 0.30/0.47 ( contraSubsetComplement
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTcontraSubset_type,type,
% 0.30/0.47 complementTcontraSubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTcontraSubset,definition,
% 0.30/0.47 ( complementTcontraSubset
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTILcontra_type,type,
% 0.30/0.47 binunionTILcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTILcontra,definition,
% 0.30/0.47 ( binunionTILcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTIRcontra_type,type,
% 0.30/0.47 binunionTIRcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTIRcontra,definition,
% 0.30/0.47 ( binunionTIRcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.47 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInUnion_type,type,
% 0.30/0.47 inIntersectImpInUnion: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInUnion,definition,
% 0.30/0.47 ( inIntersectImpInUnion
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInUnion2_type,type,
% 0.30/0.47 inIntersectImpInUnion2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInUnion2,definition,
% 0.30/0.47 ( inIntersectImpInUnion2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInIntersectUnions_type,type,
% 0.30/0.47 inIntersectImpInIntersectUnions: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInIntersectUnions,definition,
% 0.30/0.47 ( inIntersectImpInIntersectUnions
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.47 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(intersectInPowersetIntersectUnions_type,type,
% 0.30/0.47 intersectInPowersetIntersectUnions: $o ).
% 0.30/0.47
% 0.30/0.47 thf(intersectInPowersetIntersectUnions,definition,
% 0.30/0.47 ( intersectInPowersetIntersectUnions
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inComplementUnionImpNotIn1_type,type,
% 0.30/0.47 inComplementUnionImpNotIn1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inComplementUnionImpNotIn1,definition,
% 0.30/0.47 ( inComplementUnionImpNotIn1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inComplementUnionImpInComplement1_type,type,
% 0.30/0.47 inComplementUnionImpInComplement1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inComplementUnionImpInComplement1,definition,
% 0.30/0.47 ( inComplementUnionImpInComplement1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTE_type,type,
% 0.30/0.47 binunionTE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTE,definition,
% 0.30/0.47 ( binunionTE
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.47 => ( ( ( in @ Xx @ X )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => ( ( ( in @ Xx @ Y )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTEcontra_type,type,
% 0.30/0.47 binunionTEcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTEcontra,definition,
% 0.30/0.47 ( binunionTEcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ X )
% 0.30/0.47 => ( ~ ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a1_type,type,
% 0.30/0.47 demorgan2a1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a1,definition,
% 0.30/0.47 ( demorgan2a1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementUnionInPowersetComplement_type,type,
% 0.30/0.47 complementUnionInPowersetComplement: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementUnionInPowersetComplement,definition,
% 0.30/0.47 ( complementUnionInPowersetComplement
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a2_type,type,
% 0.30/0.47 demorgan2a2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a2,definition,
% 0.30/0.47 ( demorgan2a2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1a_type,type,
% 0.30/0.47 demorgan1a: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1a,definition,
% 0.30/0.47 ( demorgan1a
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1b_type,type,
% 0.30/0.47 demorgan1b: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1b,definition,
% 0.30/0.47 ( demorgan1b
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1_type,type,
% 0.30/0.47 demorgan1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1,definition,
% 0.30/0.47 ( demorgan1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.30/0.47 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a_type,type,
% 0.30/0.47 demorgan2a: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a,definition,
% 0.30/0.47 ( demorgan2a
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2b2_type,type,
% 0.30/0.47 demorgan2b2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2b2,definition,
% 0.30/0.47 ( demorgan2b2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2b_type,type,
% 0.30/0.47 demorgan2b: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2b,definition,
% 0.30/0.47 ( demorgan2b
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2_type,type,
% 0.30/0.47 demorgan2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2,definition,
% 0.30/0.47 ( demorgan2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.30/0.47 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule0_type,type,
% 0.30/0.47 woz13rule0: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule0,definition,
% 0.30/0.47 ( woz13rule0
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule1_type,type,
% 0.30/0.47 woz13rule1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule1,definition,
% 0.30/0.47 ( woz13rule1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ Z )
% 0.30/0.47 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule2_type,type,
% 0.30/0.47 woz13rule2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule2,definition,
% 0.30/0.47 ( woz13rule2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ Y @ Z )
% 0.30/0.47 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule3_type,type,
% 0.30/0.47 woz13rule3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule3,definition,
% 0.30/0.47 ( woz13rule3
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ Y )
% 0.30/0.47 => ( ( subset @ X @ Z )
% 0.30/0.47 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule4_type,type,
% 0.30/0.47 woz13rule4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule4,definition,
% 0.30/0.47 ( woz13rule4
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [W: $i] :
% 0.30/0.47 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ Z )
% 0.30/0.47 => ( ( subset @ Y @ W )
% 0.30/0.47 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_1_type,type,
% 0.30/0.47 woz1_1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_1,definition,
% 0.30/0.47 ( woz1_1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_2_type,type,
% 0.30/0.47 woz1_2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_2,definition,
% 0.30/0.47 ( woz1_2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [W: $i] :
% 0.30/0.47 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.47 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.30/0.47 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_3_type,type,
% 0.30/0.47 woz1_3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_3,definition,
% 0.30/0.47 ( woz1_3
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_4_type,type,
% 0.30/0.47 woz1_4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_4,definition,
% 0.30/0.47 ( woz1_4
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_5_type,type,
% 0.30/0.47 woz1_5: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_5,definition,
% 0.30/0.47 ( woz1_5
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1_type,type,
% 0.30/0.47 breln1: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1all2_type,type,
% 0.30/0.47 breln1all2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1all2,definition,
% 0.30/0.47 ( breln1all2
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ R )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1Set_type,type,
% 0.30/0.47 breln1Set: $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(breln1SetBreln1_type,type,
% 0.30/0.47 breln1SetBreln1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1SetBreln1,definition,
% 0.30/0.47 ( breln1SetBreln1
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( in @ R @ ( breln1Set @ A ) )
% 0.30/0.47 => ( breln1 @ A @ R ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(transitive_type,type,
% 0.30/0.47 transitive: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(antisymmetric_type,type,
% 0.30/0.47 antisymmetric: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(reflexive_type,type,
% 0.30/0.47 reflexive: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(refltransitive_type,type,
% 0.30/0.47 refltransitive: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(refllinearorder_type,type,
% 0.30/0.47 refllinearorder: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(reflwellordering_type,type,
% 0.30/0.47 reflwellordering: $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(choice2fnsingleton_type,type,
% 0.30/0.47 choice2fnsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(choice2fnsingleton,definition,
% 0.30/0.47 ( choice2fnsingleton
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 & ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.47 => ! [R: $i] :
% 0.30/0.47 ( ( in @ R @ ( breln1Set @ B ) )
% 0.30/0.47 => ( ( reflwellordering @ B @ R )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xy: $i] :
% 0.30/0.47 ( ( Xphi @ Xx @ Xy )
% 0.30/0.47 & ! [Xz: $i] :
% 0.30/0.47 ( ( in @ Xz @ B )
% 0.30/0.47 => ( ( Xphi @ Xx @ Xz )
% 0.30/0.47 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setOfPairsIsBReln1_type,type,
% 0.30/0.47 setOfPairsIsBReln1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setOfPairsIsBReln1,definition,
% 0.30/0.47 ( setOfPairsIsBReln1
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.30/0.47 ( breln1 @ A
% 0.30/0.47 @ ( dpsetconstr @ A @ A
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1all1_type,type,
% 0.30/0.47 breln1all1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1all1,definition,
% 0.30/0.47 ( breln1all1
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ R )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subbreln1_type,type,
% 0.30/0.47 subbreln1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subbreln1,definition,
% 0.30/0.47 ( subbreln1
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqbreln1_type,type,
% 0.30/0.47 eqbreln1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqbreln1,definition,
% 0.30/0.47 ( eqbreln1
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1invset_type,type,
% 0.30/0.47 breln1invset: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(breln1invprop_type,type,
% 0.30/0.47 breln1invprop: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1invprop,definition,
% 0.30/0.47 ( breln1invprop
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1invI_type,type,
% 0.30/0.47 breln1invI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1invI,definition,
% 0.30/0.47 ( breln1invI
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1invE_type,type,
% 0.30/0.47 breln1invE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1invE,definition,
% 0.30/0.47 ( breln1invE
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1compset_type,type,
% 0.30/0.47 breln1compset: $i > $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(breln1compprop_type,type,
% 0.30/0.47 breln1compprop: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1compprop,definition,
% 0.30/0.47 ( breln1compprop
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1compI_type,type,
% 0.30/0.47 breln1compI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1compI,definition,
% 0.30/0.47 ( breln1compI
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ! [Xz: $i] :
% 0.30/0.47 ( ( in @ Xz @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.47 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1compE_type,type,
% 0.30/0.47 breln1compE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1compE,definition,
% 0.30/0.47 ( breln1compE
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.47 => ? [Xz: $i] :
% 0.30/0.47 ( ( in @ Xz @ A )
% 0.30/0.47 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.47 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1compEex_type,type,
% 0.30/0.47 breln1compEex: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1compEex,definition,
% 0.30/0.47 ( breln1compEex
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.47 => ! [Xphi: $o] :
% 0.30/0.47 ( ! [Xz: $i] :
% 0.30/0.47 ( ( in @ Xz @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.30/0.47 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.30/0.47 => Xphi ) ) )
% 0.30/0.47 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionprop_type,type,
% 0.30/0.47 breln1unionprop: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionprop,definition,
% 0.30/0.47 ( breln1unionprop
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionIL_type,type,
% 0.30/0.47 breln1unionIL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionIL,definition,
% 0.30/0.47 ( breln1unionIL
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionIR_type,type,
% 0.30/0.47 breln1unionIR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionIR,definition,
% 0.30/0.47 ( breln1unionIR
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionI_type,type,
% 0.30/0.47 breln1unionI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionI,definition,
% 0.30/0.47 ( breln1unionI
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionE_type,type,
% 0.30/0.47 breln1unionE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionE,definition,
% 0.30/0.47 ( breln1unionE
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionEcases_type,type,
% 0.30/0.47 breln1unionEcases: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionEcases,definition,
% 0.30/0.47 ( breln1unionEcases
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.30/0.47 => ! [Xphi: $o] :
% 0.30/0.47 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionCommutes_type,type,
% 0.30/0.47 breln1unionCommutes: $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln1unionCommutes,definition,
% 0.30/0.47 ( breln1unionCommutes
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ( ( binunion @ R @ S )
% 0.30/0.47 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz2Ex_type,type,
% 0.30/0.47 woz2Ex: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz2Ex,definition,
% 0.30/0.47 ( woz2Ex
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ( R
% 0.30/0.47 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz2W_type,type,
% 0.30/0.47 woz2W: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz2W,definition,
% 0.30/0.47 ( woz2W
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.30/0.47 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz2A_type,type,
% 0.30/0.47 woz2A: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz2A,definition,
% 0.30/0.47 ( woz2A
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [T: $i] :
% 0.30/0.47 ( ( breln1 @ A @ T )
% 0.30/0.47 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.30/0.47 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz2B_type,type,
% 0.30/0.47 woz2B: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz2B,definition,
% 0.30/0.47 ( woz2B
% 0.30/0.47 = ( ! [A: $i,R: $i] :
% 0.30/0.47 ( ( breln1 @ A @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln1 @ A @ S )
% 0.30/0.47 => ! [T: $i] :
% 0.30/0.47 ( ( breln1 @ A @ T )
% 0.30/0.47 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.30/0.47 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1Ex_type,type,
% 0.30/0.47 image1Ex: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1Ex,definition,
% 0.30/0.47 ( image1Ex
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i] :
% 0.30/0.47 ? [B: $i] :
% 0.30/0.47 ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 <=> ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1Ex1_type,type,
% 0.30/0.47 image1Ex1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1Ex1,definition,
% 0.30/0.47 ( image1Ex1
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i] :
% 0.30/0.47 ( exu
% 0.30/0.47 @ ^ [B: $i] :
% 0.30/0.47 ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ B )
% 0.30/0.47 <=> ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1_type,type,
% 0.30/0.47 image1: $i > ( $i > $i ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(image1Equiv_type,type,
% 0.30/0.47 image1Equiv: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1Equiv,definition,
% 0.30/0.47 ( image1Equiv
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx
% 0.30/0.47 @ ( image1 @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.30/0.47 <=> ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1E_type,type,
% 0.30/0.47 image1E: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1E,definition,
% 0.30/0.47 ( image1E
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx
% 0.30/0.47 @ ( image1 @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.30/0.47 => ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(image1I_type,type,
% 0.30/0.47 image1I: $o ).
% 0.30/0.47
% 0.30/0.47 thf(image1I,definition,
% 0.30/0.47 ( image1I
% 0.30/0.47 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.30/0.47 ( ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 & ( Xx
% 0.30/0.47 = ( Xf @ Xy ) ) )
% 0.30/0.47 => ( in @ Xx
% 0.30/0.47 @ ( image1 @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(injective_type,type,
% 0.30/0.47 injective: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncSet_type,type,
% 0.30/0.47 injFuncSet: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncInInjFuncSet_type,type,
% 0.30/0.47 injFuncInInjFuncSet: $o ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncInInjFuncSet,definition,
% 0.30/0.47 ( injFuncInInjFuncSet
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( ( injective @ A @ B @ Xf )
% 0.30/0.47 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncSetFuncIn_type,type,
% 0.30/0.47 injFuncSetFuncIn: $o ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncSetFuncIn,definition,
% 0.30/0.47 ( injFuncSetFuncIn
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.30/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncSetFuncInj_type,type,
% 0.30/0.47 injFuncSetFuncInj: $o ).
% 0.30/0.47
% 0.30/0.47 thf(injFuncSetFuncInj,definition,
% 0.30/0.47 ( injFuncSetFuncInj
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.30/0.47 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(surjective_type,type,
% 0.30/0.47 surjective: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(surjFuncSet_type,type,
% 0.30/0.47 surjFuncSet: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(surjFuncSetFuncIn_type,type,
% 0.30/0.47 surjFuncSetFuncIn: $o ).
% 0.30/0.47
% 0.30/0.47 thf(surjFuncSetFuncIn,definition,
% 0.30/0.47 ( surjFuncSetFuncIn
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.30/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(surjFuncSetFuncSurj_type,type,
% 0.30/0.47 surjFuncSetFuncSurj: $o ).
% 0.30/0.47
% 0.30/0.47 thf(surjFuncSetFuncSurj,definition,
% 0.30/0.47 ( surjFuncSetFuncSurj
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 0.30/0.47 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(leftInvIsSurj_type,type,
% 0.30/0.47 leftInvIsSurj: $o ).
% 0.30/0.47
% 0.30/0.47 thf(leftInvIsSurj,definition,
% 0.30/0.47 ( leftInvIsSurj
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 0.30/0.47 = Xx ) )
% 0.30/0.47 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(surjCantorThm_type,type,
% 0.30/0.47 surjCantorThm: $o ).
% 0.30/0.47
% 0.30/0.47 thf(surjCantorThm,definition,
% 0.30/0.47 ( surjCantorThm
% 0.30/0.47 = ( ! [A: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 0.30/0.47 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(foundation2_type,type,
% 0.30/0.47 foundation2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(foundation2,definition,
% 0.30/0.47 ( foundation2
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ( ( nonempty @ A )
% 0.30/0.47 => ? [X: $i] :
% 0.30/0.47 ( ( in @ X @ A )
% 0.30/0.47 & ( ( binintersect @ X @ A )
% 0.30/0.47 = emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notinself_type,type,
% 0.30/0.47 notinself: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notinself,definition,
% 0.30/0.47 ( notinself
% 0.30/0.47 = ( ! [A: $i] :
% 0.30/0.47 ~ ( in @ A @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(notinself2_type,type,
% 0.30/0.47 notinself2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(notinself2,definition,
% 0.30/0.47 ( notinself2
% 0.30/0.47 = ( ! [A: $i,B: $i] :
% 0.30/0.47 ( ( in @ A @ B )
% 0.30/0.47 => ~ ( in @ B @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(omegaS_type,type,
% 0.30/0.47 omegaS: $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(omegaSp_type,type,
% 0.30/0.47 omegaSp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(omegaSp,definition,
% 0.30/0.47 ( omegaSp
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ omega )
% 0.30/0.47 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(omegaSclos_type,type,
% 0.30/0.47 omegaSclos: $o ).
% 0.30/0.47
% 0.30/0.47 thf(omegaSclos,definition,
% 0.30/0.47 ( omegaSclos
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ omega )
% 0.30/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(peano0notS_type,type,
% 0.30/0.47 peano0notS: $o ).
% 0.30/0.47
% 0.30/0.47 thf(peano0notS,definition,
% 0.30/0.47 ( peano0notS
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ omega )
% 0.30/0.47 => ( ( omegaS @ Xx )
% 0.30/0.47 != emptyset ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(peano3_type,type,
% 0.30/0.47 peano3: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(peano4_type,type,
% 0.30/0.47 peano4: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(peano5_type,type,
% 0.30/0.47 peano5: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(peanoSinj_type,type,
% 0.30/0.47 peanoSinj: $o ).
% 0.30/0.47
% 0.30/0.47 thf(peanoSinj,definition,
% 0.30/0.47 ( peanoSinj
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ omega )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ omega )
% 0.30/0.47 => ( ( ( omegaS @ Xx )
% 0.30/0.47 = ( omegaS @ Xy ) )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(peano_type,type,
% 0.30/0.47 peano: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(transitiveset_type,type,
% 0.30/0.47 transitiveset: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(transitiveset,definition,
% 0.30/0.47 ( transitiveset
% 0.30/0.47 = ( ^ [A: $i] :
% 0.30/0.47 ! [X: $i] :
% 0.30/0.47 ( ( in @ X @ A )
% 0.30/0.47 => ( subset @ X @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(transitivesetOp1,conjecture,
% 0.30/0.47 ( setextAx
% 0.30/0.47 => ( emptysetAx
% 0.30/0.47 => ( setadjoinAx
% 0.30/0.47 => ( powersetAx
% 0.30/0.47 => ( setunionAx
% 0.30/0.47 => ( omega0Ax
% 0.30/0.47 => ( omegaSAx
% 0.30/0.47 => ( omegaIndAx
% 0.30/0.47 => ( replAx
% 0.30/0.47 => ( foundationAx
% 0.30/0.47 => ( wellorderingAx
% 0.30/0.47 => ( descrp
% 0.30/0.47 => ( dsetconstrI
% 0.30/0.47 => ( dsetconstrEL
% 0.30/0.47 => ( dsetconstrER
% 0.30/0.47 => ( exuE1
% 0.30/0.47 => ( prop2setE
% 0.30/0.47 => ( emptysetE
% 0.30/0.47 => ( emptysetimpfalse
% 0.30/0.47 => ( notinemptyset
% 0.30/0.47 => ( exuE3e
% 0.30/0.47 => ( setext
% 0.30/0.47 => ( emptyI
% 0.30/0.47 => ( noeltsimpempty
% 0.30/0.47 => ( setbeta
% 0.30/0.47 => ( nonemptyE1
% 0.30/0.47 => ( nonemptyI
% 0.30/0.47 => ( nonemptyI1
% 0.30/0.47 => ( setadjoinIL
% 0.30/0.47 => ( emptyinunitempty
% 0.30/0.47 => ( setadjoinIR
% 0.30/0.47 => ( setadjoinE
% 0.30/0.47 => ( setadjoinOr
% 0.30/0.47 => ( setoftrueEq
% 0.30/0.47 => ( powersetI
% 0.30/0.47 => ( emptyinPowerset
% 0.30/0.47 => ( emptyInPowerset
% 0.30/0.47 => ( powersetE
% 0.30/0.47 => ( setunionI
% 0.30/0.47 => ( setunionE
% 0.30/0.47 => ( subPowSU
% 0.30/0.47 => ( exuE2
% 0.30/0.47 => ( nonemptyImpWitness
% 0.30/0.47 => ( uniqinunit
% 0.30/0.47 => ( notinsingleton
% 0.30/0.47 => ( eqinunit
% 0.30/0.47 => ( singletonsswitch
% 0.30/0.47 => ( upairsetE
% 0.30/0.47 => ( upairsetIL
% 0.30/0.47 => ( upairsetIR
% 0.30/0.47 => ( emptyE1
% 0.30/0.47 => ( vacuousDall
% 0.30/0.47 => ( quantDeMorgan1
% 0.30/0.47 => ( quantDeMorgan2
% 0.30/0.47 => ( quantDeMorgan3
% 0.30/0.47 => ( quantDeMorgan4
% 0.30/0.47 => ( prop2setI
% 0.30/0.47 => ( prop2set2propI
% 0.30/0.47 => ( notdexE
% 0.30/0.47 => ( notdallE
% 0.30/0.47 => ( exuI1
% 0.30/0.47 => ( exuI3
% 0.30/0.47 => ( exuI2
% 0.30/0.47 => ( inCongP
% 0.30/0.47 => ( in__Cong
% 0.30/0.47 => ( exuE3u
% 0.30/0.47 => ( exu__Cong
% 0.30/0.47 => ( emptyset__Cong
% 0.30/0.47 => ( setadjoin__Cong
% 0.30/0.47 => ( powerset__Cong
% 0.30/0.47 => ( setunion__Cong
% 0.30/0.47 => ( omega__Cong
% 0.30/0.47 => ( exuEu
% 0.30/0.47 => ( descr__Cong
% 0.30/0.47 => ( dsetconstr__Cong
% 0.30/0.47 => ( subsetI1
% 0.30/0.47 => ( eqimpsubset2
% 0.30/0.47 => ( eqimpsubset1
% 0.30/0.47 => ( subsetI2
% 0.30/0.47 => ( emptysetsubset
% 0.30/0.47 => ( subsetE
% 0.30/0.47 => ( subsetE2
% 0.30/0.47 => ( notsubsetI
% 0.30/0.47 => ( notequalI1
% 0.30/0.47 => ( notequalI2
% 0.30/0.47 => ( subsetRefl
% 0.30/0.47 => ( subsetTrans
% 0.30/0.47 => ( setadjoinSub
% 0.30/0.47 => ( setadjoinSub2
% 0.30/0.47 => ( subset2powerset
% 0.30/0.47 => ( setextsub
% 0.30/0.47 => ( subsetemptysetimpeq
% 0.30/0.47 => ( powersetI1
% 0.30/0.47 => ( powersetE1
% 0.30/0.47 => ( inPowerset
% 0.30/0.47 => ( powersetsubset
% 0.30/0.47 => ( sepInPowerset
% 0.30/0.47 => ( sepSubset
% 0.30/0.47 => ( binunionIL
% 0.30/0.47 => ( upairset2IR
% 0.30/0.47 => ( binunionIR
% 0.30/0.47 => ( binunionEcases
% 0.30/0.47 => ( binunionE
% 0.30/0.47 => ( binunionLsub
% 0.30/0.47 => ( binunionRsub
% 0.30/0.47 => ( binintersectI
% 0.30/0.47 => ( binintersectSubset5
% 0.30/0.47 => ( binintersectEL
% 0.30/0.47 => ( binintersectLsub
% 0.30/0.47 => ( binintersectSubset2
% 0.30/0.47 => ( binintersectSubset3
% 0.30/0.47 => ( binintersectER
% 0.30/0.47 => ( disjointsetsI1
% 0.30/0.47 => ( binintersectRsub
% 0.30/0.47 => ( binintersectSubset4
% 0.30/0.47 => ( binintersectSubset1
% 0.30/0.47 => ( bs114d
% 0.30/0.47 => ( setminusI
% 0.30/0.47 => ( setminusEL
% 0.30/0.47 => ( setminusER
% 0.30/0.47 => ( setminusSubset2
% 0.30/0.47 => ( setminusERneg
% 0.30/0.47 => ( setminusELneg
% 0.30/0.47 => ( setminusILneg
% 0.30/0.47 => ( setminusIRneg
% 0.30/0.47 => ( setminusLsub
% 0.30/0.47 => ( setminusSubset1
% 0.30/0.47 => ( symdiffE
% 0.30/0.47 => ( symdiffI1
% 0.30/0.47 => ( symdiffI2
% 0.30/0.47 => ( symdiffIneg1
% 0.30/0.47 => ( symdiffIneg2
% 0.30/0.47 => ( secondinupair
% 0.30/0.47 => ( setukpairIL
% 0.30/0.47 => ( setukpairIR
% 0.30/0.47 => ( kpairiskpair
% 0.30/0.47 => ( kpairp
% 0.30/0.47 => ( singletonsubset
% 0.30/0.47 => ( singletoninpowerset
% 0.30/0.47 => ( singletoninpowunion
% 0.30/0.47 => ( upairset2E
% 0.30/0.47 => ( upairsubunion
% 0.30/0.47 => ( upairinpowunion
% 0.30/0.47 => ( ubforcartprodlem1
% 0.30/0.47 => ( ubforcartprodlem2
% 0.30/0.47 => ( ubforcartprodlem3
% 0.30/0.47 => ( cartprodpairin
% 0.30/0.47 => ( cartprodmempair1
% 0.30/0.47 => ( cartprodmempair
% 0.30/0.47 => ( setunionE2
% 0.30/0.47 => ( setunionsingleton1
% 0.30/0.47 => ( setunionsingleton2
% 0.30/0.47 => ( setunionsingleton
% 0.30/0.47 => ( singletonprop
% 0.30/0.47 => ( ex1E1
% 0.30/0.47 => ( ex1I
% 0.30/0.47 => ( ex1I2
% 0.30/0.47 => ( singletonsuniq
% 0.30/0.47 => ( setukpairinjL1
% 0.30/0.47 => ( kfstsingleton
% 0.30/0.47 => ( theprop
% 0.30/0.47 => ( kfstpairEq
% 0.30/0.47 => ( cartprodfstin
% 0.30/0.47 => ( setukpairinjL2
% 0.30/0.47 => ( setukpairinjL
% 0.30/0.47 => ( setukpairinjR11
% 0.30/0.47 => ( setukpairinjR12
% 0.30/0.47 => ( setukpairinjR1
% 0.30/0.47 => ( upairequniteq
% 0.30/0.47 => ( setukpairinjR2
% 0.30/0.47 => ( setukpairinjR
% 0.30/0.47 => ( ksndsingleton
% 0.30/0.47 => ( ksndpairEq
% 0.30/0.47 => ( kpairsurjEq
% 0.30/0.47 => ( cartprodsndin
% 0.30/0.47 => ( cartprodpairmemEL
% 0.30/0.47 => ( cartprodpairmemER
% 0.30/0.47 => ( cartprodmempaircEq
% 0.30/0.47 => ( cartprodfstpairEq
% 0.30/0.47 => ( cartprodsndpairEq
% 0.30/0.47 => ( cartprodpairsurjEq
% 0.30/0.47 => ( dpsetconstrI
% 0.30/0.47 => ( dpsetconstrSub
% 0.30/0.47 => ( setOfPairsIsBReln
% 0.30/0.47 => ( dpsetconstrERa
% 0.30/0.47 => ( dpsetconstrEL1
% 0.30/0.47 => ( dpsetconstrEL2
% 0.30/0.47 => ( dpsetconstrER
% 0.30/0.47 => ( funcImageSingleton
% 0.30/0.47 => ( apProp
% 0.30/0.47 => ( app
% 0.30/0.47 => ( infuncsetfunc
% 0.30/0.47 => ( ap2p
% 0.30/0.47 => ( funcinfuncset
% 0.30/0.47 => ( lamProp
% 0.30/0.47 => ( lamp
% 0.30/0.47 => ( lam2p
% 0.30/0.47 => ( brelnall1
% 0.30/0.47 => ( brelnall2
% 0.30/0.47 => ( ex1E2
% 0.30/0.47 => ( funcGraphProp1
% 0.30/0.47 => ( funcGraphProp3
% 0.30/0.47 => ( funcGraphProp2
% 0.30/0.47 => ( funcextLem
% 0.30/0.47 => ( funcGraphProp4
% 0.30/0.47 => ( subbreln
% 0.30/0.47 => ( eqbreln
% 0.30/0.47 => ( funcext
% 0.30/0.47 => ( funcext2
% 0.30/0.47 => ( ap2apEq1
% 0.30/0.47 => ( ap2apEq2
% 0.30/0.47 => ( beta1
% 0.30/0.47 => ( eta1
% 0.30/0.47 => ( lam2lamEq
% 0.30/0.47 => ( beta2
% 0.30/0.47 => ( eta2
% 0.30/0.47 => ( iffalseProp1
% 0.30/0.47 => ( iffalseProp2
% 0.30/0.47 => ( iftrueProp1
% 0.30/0.47 => ( iftrueProp2
% 0.30/0.47 => ( ifSingleton
% 0.30/0.47 => ( ifp
% 0.30/0.47 => ( theeq
% 0.30/0.47 => ( iftrue
% 0.30/0.47 => ( iffalse
% 0.30/0.47 => ( iftrueorfalse
% 0.30/0.47 => ( binintersectT_lem
% 0.30/0.47 => ( binunionT_lem
% 0.30/0.47 => ( powersetT_lem
% 0.30/0.47 => ( setminusT_lem
% 0.30/0.47 => ( complementT_lem
% 0.30/0.47 => ( setextT
% 0.30/0.47 => ( subsetTI
% 0.30/0.47 => ( powersetTI1
% 0.30/0.47 => ( powersetTE1
% 0.30/0.47 => ( complementTI1
% 0.30/0.47 => ( complementTE1
% 0.30/0.47 => ( binintersectTELcontra
% 0.30/0.47 => ( binintersectTERcontra
% 0.30/0.47 => ( contrasubsetT
% 0.30/0.47 => ( contrasubsetT1
% 0.30/0.47 => ( contrasubsetT2
% 0.30/0.47 => ( contrasubsetT3
% 0.30/0.47 => ( doubleComplementI1
% 0.30/0.47 => ( doubleComplementE1
% 0.30/0.47 => ( doubleComplementSub1
% 0.30/0.47 => ( doubleComplementSub2
% 0.30/0.47 => ( doubleComplementEq
% 0.30/0.47 => ( complementTnotintersectT
% 0.30/0.47 => ( complementImpComplementIntersect
% 0.30/0.47 => ( complementSubsetComplementIntersect
% 0.30/0.47 => ( complementInPowersetComplementIntersect
% 0.30/0.47 => ( contraSubsetComplement
% 0.30/0.47 => ( complementTcontraSubset
% 0.30/0.47 => ( binunionTILcontra
% 0.30/0.47 => ( binunionTIRcontra
% 0.30/0.47 => ( inIntersectImpInUnion
% 0.30/0.47 => ( inIntersectImpInUnion2
% 0.30/0.47 => ( inIntersectImpInIntersectUnions
% 0.30/0.47 => ( intersectInPowersetIntersectUnions
% 0.30/0.47 => ( inComplementUnionImpNotIn1
% 0.30/0.47 => ( inComplementUnionImpInComplement1
% 0.30/0.47 => ( binunionTE
% 0.30/0.47 => ( binunionTEcontra
% 0.30/0.47 => ( demorgan2a1
% 0.30/0.47 => ( complementUnionInPowersetComplement
% 0.30/0.47 => ( demorgan2a2
% 0.30/0.47 => ( demorgan1a
% 0.30/0.47 => ( demorgan1b
% 0.30/0.47 => ( demorgan1
% 0.30/0.47 => ( demorgan2a
% 0.30/0.47 => ( demorgan2b2
% 0.30/0.47 => ( demorgan2b
% 0.30/0.47 => ( demorgan2
% 0.30/0.47 => ( woz13rule0
% 0.30/0.47 => ( woz13rule1
% 0.30/0.47 => ( woz13rule2
% 0.30/0.47 => ( woz13rule3
% 0.30/0.47 => ( woz13rule4
% 0.30/0.47 => ( woz1_1
% 0.30/0.47 => ( woz1_2
% 0.30/0.47 => ( woz1_3
% 0.30/0.47 => ( woz1_4
% 0.30/0.47 => ( woz1_5
% 0.30/0.47 => ( breln1all2
% 0.30/0.47 => ( breln1SetBreln1
% 0.30/0.47 => ( choice2fnsingleton
% 0.30/0.47 => ( setOfPairsIsBReln1
% 0.30/0.47 => ( breln1all1
% 0.30/0.47 => ( subbreln1
% 0.30/0.47 => ( eqbreln1
% 0.30/0.47 => ( breln1invprop
% 0.30/0.47 => ( breln1invI
% 0.30/0.47 => ( breln1invE
% 0.30/0.47 => ( breln1compprop
% 0.30/0.47 => ( breln1compI
% 0.30/0.47 => ( breln1compE
% 0.30/0.47 => ( breln1compEex
% 0.30/0.47 => ( breln1unionprop
% 0.30/0.47 => ( breln1unionIL
% 0.30/0.47 => ( breln1unionIR
% 0.30/0.47 => ( breln1unionI
% 0.30/0.47 => ( breln1unionE
% 0.30/0.47 => ( breln1unionEcases
% 0.30/0.47 => ( breln1unionCommutes
% 0.30/0.47 => ( woz2Ex
% 0.30/0.47 => ( woz2W
% 0.30/0.47 => ( woz2A
% 0.30/0.47 => ( woz2B
% 0.30/0.47 => ( image1Ex
% 0.30/0.47 => ( image1Ex1
% 0.30/0.47 => ( image1Equiv
% 0.30/0.47 => ( image1E
% 0.30/0.47 => ( image1I
% 0.30/0.47 => ( injFuncInInjFuncSet
% 0.30/0.47 => ( injFuncSetFuncIn
% 0.30/0.47 => ( injFuncSetFuncInj
% 0.30/0.47 => ( surjFuncSetFuncIn
% 0.30/0.47 => ( surjFuncSetFuncSurj
% 0.30/0.47 => ( leftInvIsSurj
% 0.30/0.47 => ( surjCantorThm
% 0.30/0.47 => ( foundation2
% 0.30/0.47 => ( notinself
% 0.30/0.47 => ( notinself2
% 0.30/0.47 => ( omegaSp
% 0.30/0.47 => ( omegaSclos
% 0.30/0.47 => ( peano0notS
% 0.30/0.47 => ( peanoSinj
% 0.30/0.47 => ! [X: $i] :
% 0.30/0.47 ( ( transitiveset @ X )
% 0.30/0.47 => ! [A: $i] :
% 0.30/0.47 ( ( in @ A @ X )
% 0.30/0.47 => ( subset @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.92/1.57 ---- Embedded in THF ---
% 2.92/1.57 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.92/1.57 %%% Generated on Tue Feb 24 12:34:39 EST 2026
% 2.92/1.57 %%% using '$$dhol' embedding, version 1.3.0.
% 2.92/1.57 %%% Logic specification used:
% 2.92/1.57 %%% thf(spec, logic, $$dhol).
% 2.92/1.57
% 2.92/1.57 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.4NXBcYPEpZ/DTF2DTF16671.p
% See solution above
% 2.92/1.57 ------------------------
% 3.13/1.65 ---- Cleaned THF ---
% 3.13/1.65 thf(in_type,type,
% 3.13/1.65 in: $i > $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(exu_type,type,
% 3.13/1.65 exu: ( $i > $o ) > $o ).
% 3.13/1.65
% 3.13/1.65 thf(exu,definition,
% 3.13/1.65 ( exu
% 3.13/1.65 = ( ^ [Xphi: $i > $o] :
% 3.13/1.65 ? [Xx: $i] :
% 3.13/1.65 ( ( Xphi @ Xx )
% 3.13/1.65 & ! [Xy: $i] :
% 3.13/1.65 ( ( Xphi @ Xy )
% 3.13/1.65 => ( Xx = Xy ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setextAx_type,type,
% 3.13/1.65 setextAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setextAx,definition,
% 3.13/1.65 ( setextAx
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 <=> ( in @ Xx @ B ) )
% 3.13/1.65 => ( A = B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptyset_type,type,
% 3.13/1.65 emptyset: $i ).
% 3.13/1.65
% 3.13/1.65 thf(emptysetAx_type,type,
% 3.13/1.65 emptysetAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptysetAx,definition,
% 3.13/1.65 ( emptysetAx
% 3.13/1.65 = ( ! [Xx: $i] :
% 3.13/1.65 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoin_type,type,
% 3.13/1.65 setadjoin: $i > $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinAx_type,type,
% 3.13/1.65 setadjoinAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinAx,definition,
% 3.13/1.65 ( setadjoinAx
% 3.13/1.65 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.13/1.65 <=> ( ( Xy = Xx )
% 3.13/1.65 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(powerset_type,type,
% 3.13/1.65 powerset: $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(powersetAx_type,type,
% 3.13/1.65 powersetAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(powersetAx,definition,
% 3.13/1.65 ( powersetAx
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( in @ B @ ( powerset @ A ) )
% 3.13/1.65 <=> ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setunion_type,type,
% 3.13/1.65 setunion: $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(setunionAx_type,type,
% 3.13/1.65 setunionAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setunionAx,definition,
% 3.13/1.65 ( setunionAx
% 3.13/1.65 = ( ! [A: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( setunion @ A ) )
% 3.13/1.65 <=> ? [B: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 & ( in @ B @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(omega_type,type,
% 3.13/1.65 omega: $i ).
% 3.13/1.65
% 3.13/1.65 thf(omega0Ax_type,type,
% 3.13/1.65 omega0Ax: $o ).
% 3.13/1.65
% 3.13/1.65 thf(omega0Ax,definition,
% 3.13/1.65 ( omega0Ax
% 3.13/1.65 = ( in @ emptyset @ omega ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(omegaSAx_type,type,
% 3.13/1.65 omegaSAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(omegaSAx,definition,
% 3.13/1.65 ( omegaSAx
% 3.13/1.65 = ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ omega )
% 3.13/1.65 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(omegaIndAx_type,type,
% 3.13/1.65 omegaIndAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(omegaIndAx,definition,
% 3.13/1.65 ( omegaIndAx
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ( ( ( in @ emptyset @ A )
% 3.13/1.65 & ! [Xx: $i] :
% 3.13/1.65 ( ( ( in @ Xx @ omega )
% 3.13/1.65 & ( in @ Xx @ A ) )
% 3.13/1.65 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.13/1.65 => ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ omega )
% 3.13/1.65 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(replAx_type,type,
% 3.13/1.65 replAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(replAx,definition,
% 3.13/1.65 ( replAx
% 3.13/1.65 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( exu
% 3.13/1.65 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.13/1.65 => ? [B: $i] :
% 3.13/1.65 ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 <=> ? [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ A )
% 3.13/1.65 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(foundationAx_type,type,
% 3.13/1.65 foundationAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(foundationAx,definition,
% 3.13/1.65 ( foundationAx
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.13/1.65 => ? [B: $i] :
% 3.13/1.65 ( ( in @ B @ A )
% 3.13/1.65 & ~ ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(wellorderingAx_type,type,
% 3.13/1.65 wellorderingAx: $o ).
% 3.13/1.65
% 3.13/1.65 thf(wellorderingAx,definition,
% 3.13/1.65 ( wellorderingAx
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ? [B: $i] :
% 3.13/1.65 ( ! [C: $i] :
% 3.13/1.65 ( ( in @ C @ B )
% 3.13/1.65 => ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ C )
% 3.13/1.65 => ( in @ Xx @ A ) ) )
% 3.13/1.65 & ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( ( in @ Xx @ A )
% 3.13/1.65 & ( in @ Xy @ A ) )
% 3.13/1.65 => ( ! [C: $i] :
% 3.13/1.65 ( ( in @ C @ B )
% 3.13/1.65 => ( ( in @ Xx @ C )
% 3.13/1.65 <=> ( in @ Xy @ C ) ) )
% 3.13/1.65 => ( Xx = Xy ) ) )
% 3.13/1.65 & ! [C: $i,D: $i] :
% 3.13/1.65 ( ( ( in @ C @ B )
% 3.13/1.65 & ( in @ D @ B ) )
% 3.13/1.65 => ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ C )
% 3.13/1.65 => ( in @ Xx @ D ) )
% 3.13/1.65 | ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ D )
% 3.13/1.65 => ( in @ Xx @ C ) ) ) )
% 3.13/1.65 & ! [C: $i] :
% 3.13/1.65 ( ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ C )
% 3.13/1.65 => ( in @ Xx @ A ) )
% 3.13/1.65 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.13/1.65 => ? [D: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ D @ B )
% 3.13/1.65 & ( in @ Xx @ C )
% 3.13/1.65 & ~ ? [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ D )
% 3.13/1.65 & ( in @ Xy @ C ) )
% 3.13/1.65 & ! [E: $i] :
% 3.13/1.65 ( ( in @ E @ B )
% 3.13/1.65 => ( ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ E )
% 3.13/1.65 => ( in @ Xy @ D ) )
% 3.13/1.65 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(descr_type,type,
% 3.13/1.65 descr: ( $i > $o ) > $i ).
% 3.13/1.65
% 3.13/1.65 thf(descrp_type,type,
% 3.13/1.65 descrp: $o ).
% 3.13/1.65
% 3.13/1.65 thf(descrp,definition,
% 3.13/1.65 ( descrp
% 3.13/1.65 = ( ! [Xphi: $i > $o] :
% 3.13/1.65 ( ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 => ( Xphi
% 3.13/1.65 @ ( descr
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(dsetconstr_type,type,
% 3.13/1.65 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.13/1.65
% 3.13/1.65 thf(dsetconstrI_type,type,
% 3.13/1.65 dsetconstrI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(dsetconstrI,definition,
% 3.13/1.65 ( dsetconstrI
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ( Xphi @ Xx )
% 3.13/1.65 => ( in @ Xx
% 3.13/1.65 @ ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(dsetconstrEL_type,type,
% 3.13/1.65 dsetconstrEL: $o ).
% 3.13/1.65
% 3.13/1.65 thf(dsetconstrEL,definition,
% 3.13/1.65 ( dsetconstrEL
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx
% 3.13/1.65 @ ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.13/1.65 => ( in @ Xx @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(dsetconstrER_type,type,
% 3.13/1.65 dsetconstrER: $o ).
% 3.13/1.65
% 3.13/1.65 thf(dsetconstrER,definition,
% 3.13/1.65 ( dsetconstrER
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx
% 3.13/1.65 @ ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.13/1.65 => ( Xphi @ Xx ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(exuE1_type,type,
% 3.13/1.65 exuE1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(exuE1,definition,
% 3.13/1.65 ( exuE1
% 3.13/1.65 = ( ! [Xphi: $i > $o] :
% 3.13/1.65 ( ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 => ? [Xx: $i] :
% 3.13/1.65 ( ( Xphi @ Xx )
% 3.13/1.65 & ! [Xy: $i] :
% 3.13/1.65 ( ( Xphi @ Xy )
% 3.13/1.65 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(prop2set_type,type,
% 3.13/1.65 prop2set: $o > $i ).
% 3.13/1.65
% 3.13/1.65 thf(prop2setE_type,type,
% 3.13/1.65 prop2setE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(prop2setE,definition,
% 3.13/1.65 ( prop2setE
% 3.13/1.65 = ( ! [Xphi: $o,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.13/1.65 => Xphi ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptysetE_type,type,
% 3.13/1.65 emptysetE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptysetE,definition,
% 3.13/1.65 ( emptysetE
% 3.13/1.65 = ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ emptyset )
% 3.13/1.65 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptysetimpfalse_type,type,
% 3.13/1.65 emptysetimpfalse: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptysetimpfalse,definition,
% 3.13/1.65 ( emptysetimpfalse
% 3.13/1.65 = ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ emptyset )
% 3.13/1.65 => $false ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(notinemptyset_type,type,
% 3.13/1.65 notinemptyset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(notinemptyset,definition,
% 3.13/1.65 ( notinemptyset
% 3.13/1.65 = ( ! [Xx: $i] :
% 3.13/1.65 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(exuE3e_type,type,
% 3.13/1.65 exuE3e: $o ).
% 3.13/1.65
% 3.13/1.65 thf(exuE3e,definition,
% 3.13/1.65 ( exuE3e
% 3.13/1.65 = ( ! [Xphi: $i > $o] :
% 3.13/1.65 ( ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setext_type,type,
% 3.13/1.65 setext: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setext,definition,
% 3.13/1.65 ( setext
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ Xx @ B ) )
% 3.13/1.65 => ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 => ( in @ Xx @ A ) )
% 3.13/1.65 => ( A = B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptyI_type,type,
% 3.13/1.65 emptyI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptyI,definition,
% 3.13/1.65 ( emptyI
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ~ ( in @ Xx @ A )
% 3.13/1.65 => ( A = emptyset ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(noeltsimpempty_type,type,
% 3.13/1.65 noeltsimpempty: $o ).
% 3.13/1.65
% 3.13/1.65 thf(noeltsimpempty,definition,
% 3.13/1.65 ( noeltsimpempty
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ~ ( in @ Xx @ A )
% 3.13/1.65 => ( A = emptyset ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setbeta_type,type,
% 3.13/1.65 setbeta: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setbeta,definition,
% 3.13/1.65 ( setbeta
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ( in @ Xx
% 3.13/1.65 @ ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.13/1.65 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(nonempty_type,type,
% 3.13/1.65 nonempty: $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(nonempty,definition,
% 3.13/1.65 ( nonempty
% 3.13/1.65 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(nonemptyE1_type,type,
% 3.13/1.65 nonemptyE1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(nonemptyE1,definition,
% 3.13/1.65 ( nonemptyE1
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ( ( nonempty @ A )
% 3.13/1.65 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(nonemptyI_type,type,
% 3.13/1.65 nonemptyI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(nonemptyI,definition,
% 3.13/1.65 ( nonemptyI
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ( Xphi @ Xx )
% 3.13/1.65 => ( nonempty
% 3.13/1.65 @ ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(nonemptyI1_type,type,
% 3.13/1.65 nonemptyI1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(nonemptyI1,definition,
% 3.13/1.65 ( nonemptyI1
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.13/1.65 => ( nonempty @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinIL_type,type,
% 3.13/1.65 setadjoinIL: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinIL,definition,
% 3.13/1.65 ( setadjoinIL
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptyinunitempty_type,type,
% 3.13/1.65 emptyinunitempty: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptyinunitempty,definition,
% 3.13/1.65 ( emptyinunitempty
% 3.13/1.65 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinIR_type,type,
% 3.13/1.65 setadjoinIR: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinIR,definition,
% 3.13/1.65 ( setadjoinIR
% 3.13/1.65 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ A )
% 3.13/1.65 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinE_type,type,
% 3.13/1.65 setadjoinE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinE,definition,
% 3.13/1.65 ( setadjoinE
% 3.13/1.65 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.13/1.65 => ! [Xphi: $o] :
% 3.13/1.65 ( ( ( Xy = Xx )
% 3.13/1.65 => Xphi )
% 3.13/1.65 => ( ( ( in @ Xy @ A )
% 3.13/1.65 => Xphi )
% 3.13/1.65 => Xphi ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinOr_type,type,
% 3.13/1.65 setadjoinOr: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinOr,definition,
% 3.13/1.65 ( setadjoinOr
% 3.13/1.65 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.13/1.65 => ( ( Xy = Xx )
% 3.13/1.65 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setoftrueEq_type,type,
% 3.13/1.65 setoftrueEq: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setoftrueEq,definition,
% 3.13/1.65 ( setoftrueEq
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ( ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xx: $i] : $true )
% 3.13/1.65 = A ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(powersetI_type,type,
% 3.13/1.65 powersetI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(powersetI,definition,
% 3.13/1.65 ( powersetI
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 => ( in @ Xx @ A ) )
% 3.13/1.65 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptyinPowerset_type,type,
% 3.13/1.65 emptyinPowerset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptyinPowerset,definition,
% 3.13/1.65 ( emptyinPowerset
% 3.13/1.65 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptyInPowerset_type,type,
% 3.13/1.65 emptyInPowerset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptyInPowerset,definition,
% 3.13/1.65 ( emptyInPowerset
% 3.13/1.65 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(powersetE_type,type,
% 3.13/1.65 powersetE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(powersetE,definition,
% 3.13/1.65 ( powersetE
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ B @ ( powerset @ A ) )
% 3.13/1.65 => ( ( in @ Xx @ B )
% 3.13/1.65 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setunionI_type,type,
% 3.13/1.65 setunionI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setunionI,definition,
% 3.13/1.65 ( setunionI
% 3.13/1.65 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 => ( ( in @ B @ A )
% 3.13/1.65 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setunionE_type,type,
% 3.13/1.65 setunionE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setunionE,definition,
% 3.13/1.65 ( setunionE
% 3.13/1.65 = ( ! [A: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( setunion @ A ) )
% 3.13/1.65 => ! [Xphi: $o] :
% 3.13/1.65 ( ! [B: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 => ( ( in @ B @ A )
% 3.13/1.65 => Xphi ) )
% 3.13/1.65 => Xphi ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(subPowSU_type,type,
% 3.13/1.65 subPowSU: $o ).
% 3.13/1.65
% 3.13/1.65 thf(subPowSU,definition,
% 3.13/1.65 ( subPowSU
% 3.13/1.65 = ( ! [A: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(exuE2_type,type,
% 3.13/1.65 exuE2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(exuE2,definition,
% 3.13/1.65 ( exuE2
% 3.13/1.65 = ( ! [Xphi: $i > $o] :
% 3.13/1.65 ( ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 => ? [Xx: $i] :
% 3.13/1.65 ! [Xy: $i] :
% 3.13/1.65 ( ( Xphi @ Xy )
% 3.13/1.65 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(nonemptyImpWitness_type,type,
% 3.13/1.65 nonemptyImpWitness: $o ).
% 3.13/1.65
% 3.13/1.65 thf(nonemptyImpWitness,definition,
% 3.13/1.65 ( nonemptyImpWitness
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ( ( nonempty @ A )
% 3.13/1.65 => ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & $true ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(uniqinunit_type,type,
% 3.13/1.65 uniqinunit: $o ).
% 3.13/1.65
% 3.13/1.65 thf(uniqinunit,definition,
% 3.13/1.65 ( uniqinunit
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.13/1.65 => ( Xx = Xy ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(notinsingleton_type,type,
% 3.13/1.65 notinsingleton: $o ).
% 3.13/1.65
% 3.13/1.65 thf(notinsingleton,definition,
% 3.13/1.65 ( notinsingleton
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xx != Xy )
% 3.13/1.65 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(eqinunit_type,type,
% 3.13/1.65 eqinunit: $o ).
% 3.13/1.65
% 3.13/1.65 thf(eqinunit,definition,
% 3.13/1.65 ( eqinunit
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xx = Xy )
% 3.13/1.65 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(singletonsswitch_type,type,
% 3.13/1.65 singletonsswitch: $o ).
% 3.13/1.65
% 3.13/1.65 thf(singletonsswitch,definition,
% 3.13/1.65 ( singletonsswitch
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.13/1.65 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(upairsetE_type,type,
% 3.13/1.65 upairsetE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(upairsetE,definition,
% 3.13/1.65 ( upairsetE
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.13/1.65 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.13/1.65 => ( ( Xz = Xx )
% 3.13/1.65 | ( Xz = Xy ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(upairsetIL_type,type,
% 3.13/1.65 upairsetIL: $o ).
% 3.13/1.65
% 3.13/1.65 thf(upairsetIL,definition,
% 3.13/1.65 ( upairsetIL
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(upairsetIR_type,type,
% 3.13/1.65 upairsetIR: $o ).
% 3.13/1.65
% 3.13/1.65 thf(upairsetIR,definition,
% 3.13/1.65 ( upairsetIR
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptyE1_type,type,
% 3.13/1.65 emptyE1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptyE1,definition,
% 3.13/1.65 ( emptyE1
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ( Xphi @ Xx ) )
% 3.13/1.65 => ( ( ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 = emptyset )
% 3.13/1.65 => $false ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(vacuousDall_type,type,
% 3.13/1.65 vacuousDall: $o ).
% 3.13/1.65
% 3.13/1.65 thf(vacuousDall,definition,
% 3.13/1.65 ( vacuousDall
% 3.13/1.65 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ emptyset )
% 3.13/1.65 => ( Xphi @ Xx ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(quantDeMorgan1_type,type,
% 3.13/1.65 quantDeMorgan1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(quantDeMorgan1,definition,
% 3.13/1.65 ( quantDeMorgan1
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ~ ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( Xphi @ Xx ) )
% 3.13/1.65 => ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(quantDeMorgan2_type,type,
% 3.13/1.65 quantDeMorgan2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(quantDeMorgan2,definition,
% 3.13/1.65 ( quantDeMorgan2
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ~ ( Xphi @ Xx ) )
% 3.13/1.65 => ~ ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(quantDeMorgan3_type,type,
% 3.13/1.65 quantDeMorgan3: $o ).
% 3.13/1.65
% 3.13/1.65 thf(quantDeMorgan3,definition,
% 3.13/1.65 ( quantDeMorgan3
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ~ ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ( Xphi @ Xx ) )
% 3.13/1.65 => ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(quantDeMorgan4_type,type,
% 3.13/1.65 quantDeMorgan4: $o ).
% 3.13/1.65
% 3.13/1.65 thf(quantDeMorgan4,definition,
% 3.13/1.65 ( quantDeMorgan4
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ~ ( Xphi @ Xx ) )
% 3.13/1.65 => ~ ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(prop2setI_type,type,
% 3.13/1.65 prop2setI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(prop2setI,definition,
% 3.13/1.65 ( prop2setI
% 3.13/1.65 = ( ! [Xphi: $o] :
% 3.13/1.65 ( Xphi
% 3.13/1.65 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(set2prop_type,type,
% 3.13/1.65 set2prop: $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(prop2set2propI_type,type,
% 3.13/1.65 prop2set2propI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(prop2set2propI,definition,
% 3.13/1.65 ( prop2set2propI
% 3.13/1.65 = ( ! [Xphi: $o] :
% 3.13/1.65 ( Xphi
% 3.13/1.65 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(notdexE_type,type,
% 3.13/1.65 notdexE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(notdexE,definition,
% 3.13/1.65 ( notdexE
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ~ ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ( Xphi @ Xx ) )
% 3.13/1.65 => ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(notdallE_type,type,
% 3.13/1.65 notdallE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(notdallE,definition,
% 3.13/1.65 ( notdallE
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ~ ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( Xphi @ Xx ) )
% 3.13/1.65 => ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(exuI1_type,type,
% 3.13/1.65 exuI1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(exuI1,definition,
% 3.13/1.65 ( exuI1
% 3.13/1.65 = ( ! [Xphi: $i > $o] :
% 3.13/1.65 ( ? [Xx: $i] :
% 3.13/1.65 ( ( Xphi @ Xx )
% 3.13/1.65 & ! [Xy: $i] :
% 3.13/1.65 ( ( Xphi @ Xy )
% 3.13/1.65 => ( Xx = Xy ) ) )
% 3.13/1.65 => ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(exuI3_type,type,
% 3.13/1.65 exuI3: $o ).
% 3.13/1.65
% 3.13/1.65 thf(exuI3,definition,
% 3.13/1.65 ( exuI3
% 3.13/1.65 = ( ! [Xphi: $i > $o] :
% 3.13/1.65 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.13/1.65 => ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xphi @ Xx )
% 3.13/1.65 => ( ( Xphi @ Xy )
% 3.13/1.65 => ( Xx = Xy ) ) )
% 3.13/1.65 => ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(exuI2_type,type,
% 3.13/1.65 exuI2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(exuI2,definition,
% 3.13/1.65 ( exuI2
% 3.13/1.65 = ( ! [Xphi: $i > $o] :
% 3.13/1.65 ( ? [Xx: $i] :
% 3.13/1.65 ! [Xy: $i] :
% 3.13/1.65 ( ( Xphi @ Xy )
% 3.13/1.65 <=> ( Xy = Xx ) )
% 3.13/1.65 => ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(inCongP_type,type,
% 3.13/1.65 inCongP: $o ).
% 3.13/1.65
% 3.13/1.65 thf(inCongP,definition,
% 3.13/1.65 ( inCongP
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( A = B )
% 3.13/1.65 => ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xx = Xy )
% 3.13/1.65 => ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(in__Cong_type,type,
% 3.13/1.65 in__Cong: $o ).
% 3.13/1.65
% 3.13/1.65 thf(in__Cong,definition,
% 3.13/1.65 ( in__Cong
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( A = B )
% 3.13/1.65 => ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xx = Xy )
% 3.13/1.65 => ( ( in @ Xx @ A )
% 3.13/1.65 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(exuE3u_type,type,
% 3.13/1.65 exuE3u: $o ).
% 3.13/1.65
% 3.13/1.65 thf(exuE3u,definition,
% 3.13/1.65 ( exuE3u
% 3.13/1.65 = ( ! [Xphi: $i > $o] :
% 3.13/1.65 ( ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 => ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xphi @ Xx )
% 3.13/1.65 => ( ( Xphi @ Xy )
% 3.13/1.65 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(exu__Cong_type,type,
% 3.13/1.65 exu__Cong: $o ).
% 3.13/1.65
% 3.13/1.65 thf(exu__Cong,definition,
% 3.13/1.65 ( exu__Cong
% 3.13/1.65 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.13/1.65 ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xx = Xy )
% 3.13/1.65 => ( ( Xphi @ Xx )
% 3.13/1.65 <=> ( Xpsi @ Xy ) ) )
% 3.13/1.65 => ( ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 <=> ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptyset__Cong_type,type,
% 3.13/1.65 emptyset__Cong: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptyset__Cong,definition,
% 3.13/1.65 ( emptyset__Cong
% 3.13/1.65 = ( emptyset = emptyset ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoin__Cong_type,type,
% 3.13/1.65 setadjoin__Cong: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoin__Cong,definition,
% 3.13/1.65 ( setadjoin__Cong
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xx = Xy )
% 3.13/1.65 => ! [Xz: $i,Xu: $i] :
% 3.13/1.65 ( ( Xz = Xu )
% 3.13/1.65 => ( ( setadjoin @ Xx @ Xz )
% 3.13/1.65 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(powerset__Cong_type,type,
% 3.13/1.65 powerset__Cong: $o ).
% 3.13/1.65
% 3.13/1.65 thf(powerset__Cong,definition,
% 3.13/1.65 ( powerset__Cong
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( A = B )
% 3.13/1.65 => ( ( powerset @ A )
% 3.13/1.65 = ( powerset @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setunion__Cong_type,type,
% 3.13/1.65 setunion__Cong: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setunion__Cong,definition,
% 3.13/1.65 ( setunion__Cong
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( A = B )
% 3.13/1.65 => ( ( setunion @ A )
% 3.13/1.65 = ( setunion @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(omega__Cong_type,type,
% 3.13/1.65 omega__Cong: $o ).
% 3.13/1.65
% 3.13/1.65 thf(omega__Cong,definition,
% 3.13/1.65 ( omega__Cong
% 3.13/1.65 = ( omega = omega ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(exuEu_type,type,
% 3.13/1.65 exuEu: $o ).
% 3.13/1.65
% 3.13/1.65 thf(exuEu,definition,
% 3.13/1.65 ( exuEu
% 3.13/1.65 = ( ! [Xphi: $i > $o] :
% 3.13/1.65 ( ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 => ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xphi @ Xx )
% 3.13/1.65 => ( ( Xphi @ Xy )
% 3.13/1.65 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(descr__Cong_type,type,
% 3.13/1.65 descr__Cong: $o ).
% 3.13/1.65
% 3.13/1.65 thf(descr__Cong,definition,
% 3.13/1.65 ( descr__Cong
% 3.13/1.65 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.13/1.65 ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xx = Xy )
% 3.13/1.65 => ( ( Xphi @ Xx )
% 3.13/1.65 <=> ( Xpsi @ Xy ) ) )
% 3.13/1.65 => ( ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 => ( ( exu
% 3.13/1.65 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.13/1.65 => ( ( descr
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 = ( descr
% 3.13/1.65 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(dsetconstr__Cong_type,type,
% 3.13/1.65 dsetconstr__Cong: $o ).
% 3.13/1.65
% 3.13/1.65 thf(dsetconstr__Cong,definition,
% 3.13/1.65 ( dsetconstr__Cong
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( A = B )
% 3.13/1.65 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ B )
% 3.13/1.65 => ( ( Xx = Xy )
% 3.13/1.65 => ( ( Xphi @ Xx )
% 3.13/1.65 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.13/1.65 => ( ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 = ( dsetconstr @ B
% 3.13/1.65 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(subset_type,type,
% 3.13/1.65 subset: $i > $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(disjoint_type,type,
% 3.13/1.65 disjoint: $i > $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(setsmeet_type,type,
% 3.13/1.65 setsmeet: $i > $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(subsetI1_type,type,
% 3.13/1.65 subsetI1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(subsetI1,definition,
% 3.13/1.65 ( subsetI1
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ Xx @ B ) )
% 3.13/1.65 => ( subset @ A @ B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(eqimpsubset2_type,type,
% 3.13/1.65 eqimpsubset2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(eqimpsubset2,definition,
% 3.13/1.65 ( eqimpsubset2
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( A = B )
% 3.13/1.65 => ( subset @ B @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(eqimpsubset1_type,type,
% 3.13/1.65 eqimpsubset1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(eqimpsubset1,definition,
% 3.13/1.65 ( eqimpsubset1
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( A = B )
% 3.13/1.65 => ( subset @ A @ B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(subsetI2_type,type,
% 3.13/1.65 subsetI2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(subsetI2,definition,
% 3.13/1.65 ( subsetI2
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ Xx @ B ) )
% 3.13/1.65 => ( subset @ A @ B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(emptysetsubset_type,type,
% 3.13/1.65 emptysetsubset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(emptysetsubset,definition,
% 3.13/1.65 ( emptysetsubset
% 3.13/1.65 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(subsetE_type,type,
% 3.13/1.65 subsetE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(subsetE,definition,
% 3.13/1.65 ( subsetE
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( subset @ A @ B )
% 3.13/1.65 => ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(subsetE2_type,type,
% 3.13/1.65 subsetE2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(subsetE2,definition,
% 3.13/1.65 ( subsetE2
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( subset @ A @ B )
% 3.13/1.65 => ( ~ ( in @ Xx @ B )
% 3.13/1.65 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(notsubsetI_type,type,
% 3.13/1.65 notsubsetI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(notsubsetI,definition,
% 3.13/1.65 ( notsubsetI
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ~ ( in @ Xx @ B )
% 3.13/1.65 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(notequalI1_type,type,
% 3.13/1.65 notequalI1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(notequalI1,definition,
% 3.13/1.65 ( notequalI1
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ~ ( subset @ A @ B )
% 3.13/1.65 => ( A != B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(notequalI2_type,type,
% 3.13/1.65 notequalI2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(notequalI2,definition,
% 3.13/1.65 ( notequalI2
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ~ ( in @ Xx @ B )
% 3.13/1.65 => ( A != B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(subsetRefl_type,type,
% 3.13/1.65 subsetRefl: $o ).
% 3.13/1.65
% 3.13/1.65 thf(subsetRefl,definition,
% 3.13/1.65 ( subsetRefl
% 3.13/1.65 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(subsetTrans_type,type,
% 3.13/1.65 subsetTrans: $o ).
% 3.13/1.65
% 3.13/1.65 thf(subsetTrans,definition,
% 3.13/1.65 ( subsetTrans
% 3.13/1.65 = ( ! [A: $i,B: $i,C: $i] :
% 3.13/1.65 ( ( subset @ A @ B )
% 3.13/1.65 => ( ( subset @ B @ C )
% 3.13/1.65 => ( subset @ A @ C ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinSub_type,type,
% 3.13/1.65 setadjoinSub: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinSub,definition,
% 3.13/1.65 ( setadjoinSub
% 3.13/1.65 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinSub2_type,type,
% 3.13/1.65 setadjoinSub2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setadjoinSub2,definition,
% 3.13/1.65 ( setadjoinSub2
% 3.13/1.65 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.13/1.65 ( ( subset @ A @ B )
% 3.13/1.65 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(subset2powerset_type,type,
% 3.13/1.65 subset2powerset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(subset2powerset,definition,
% 3.13/1.65 ( subset2powerset
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( subset @ A @ B )
% 3.13/1.65 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setextsub_type,type,
% 3.13/1.65 setextsub: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setextsub,definition,
% 3.13/1.65 ( setextsub
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( subset @ A @ B )
% 3.13/1.65 => ( ( subset @ B @ A )
% 3.13/1.65 => ( A = B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(subsetemptysetimpeq_type,type,
% 3.13/1.65 subsetemptysetimpeq: $o ).
% 3.13/1.65
% 3.13/1.65 thf(subsetemptysetimpeq,definition,
% 3.13/1.65 ( subsetemptysetimpeq
% 3.13/1.65 = ( ! [A: $i] :
% 3.13/1.65 ( ( subset @ A @ emptyset )
% 3.13/1.65 => ( A = emptyset ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(powersetI1_type,type,
% 3.13/1.65 powersetI1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(powersetI1,definition,
% 3.13/1.65 ( powersetI1
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( subset @ B @ A )
% 3.13/1.65 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(powersetE1_type,type,
% 3.13/1.65 powersetE1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(powersetE1,definition,
% 3.13/1.65 ( powersetE1
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( in @ B @ ( powerset @ A ) )
% 3.13/1.65 => ( subset @ B @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(inPowerset_type,type,
% 3.13/1.65 inPowerset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(inPowerset,definition,
% 3.13/1.65 ( inPowerset
% 3.13/1.65 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(powersetsubset_type,type,
% 3.13/1.65 powersetsubset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(powersetsubset,definition,
% 3.13/1.65 ( powersetsubset
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( subset @ A @ B )
% 3.13/1.65 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(sepInPowerset_type,type,
% 3.13/1.65 sepInPowerset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(sepInPowerset,definition,
% 3.13/1.65 ( sepInPowerset
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( in
% 3.13/1.65 @ ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 @ ( powerset @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(sepSubset_type,type,
% 3.13/1.65 sepSubset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(sepSubset,definition,
% 3.13/1.65 ( sepSubset
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( subset
% 3.13/1.65 @ ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 @ A ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binunion_type,type,
% 3.13/1.65 binunion: $i > $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(binunionIL_type,type,
% 3.13/1.65 binunionIL: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binunionIL,definition,
% 3.13/1.65 ( binunionIL
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(upairset2IR_type,type,
% 3.13/1.65 upairset2IR: $o ).
% 3.13/1.65
% 3.13/1.65 thf(upairset2IR,definition,
% 3.13/1.65 ( upairset2IR
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binunionIR_type,type,
% 3.13/1.65 binunionIR: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binunionIR,definition,
% 3.13/1.65 ( binunionIR
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binunionEcases_type,type,
% 3.13/1.65 binunionEcases: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binunionEcases,definition,
% 3.13/1.65 ( binunionEcases
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.13/1.65 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.13/1.65 => ( ( ( in @ Xx @ A )
% 3.13/1.65 => Xphi )
% 3.13/1.65 => ( ( ( in @ Xx @ B )
% 3.13/1.65 => Xphi )
% 3.13/1.65 => Xphi ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binunionE_type,type,
% 3.13/1.65 binunionE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binunionE,definition,
% 3.13/1.65 ( binunionE
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.13/1.65 => ( ( in @ Xx @ A )
% 3.13/1.65 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binunionLsub_type,type,
% 3.13/1.65 binunionLsub: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binunionLsub,definition,
% 3.13/1.65 ( binunionLsub
% 3.13/1.65 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binunionRsub_type,type,
% 3.13/1.65 binunionRsub: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binunionRsub,definition,
% 3.13/1.65 ( binunionRsub
% 3.13/1.65 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersect_type,type,
% 3.13/1.65 binintersect: $i > $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectI_type,type,
% 3.13/1.65 binintersectI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectI,definition,
% 3.13/1.65 ( binintersectI
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ( in @ Xx @ B )
% 3.13/1.65 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset5_type,type,
% 3.13/1.65 binintersectSubset5: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset5,definition,
% 3.13/1.65 ( binintersectSubset5
% 3.13/1.65 = ( ! [A: $i,B: $i,C: $i] :
% 3.13/1.65 ( ( subset @ C @ A )
% 3.13/1.65 => ( ( subset @ C @ B )
% 3.13/1.65 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectEL_type,type,
% 3.13/1.65 binintersectEL: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectEL,definition,
% 3.13/1.65 ( binintersectEL
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.13/1.65 => ( in @ Xx @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectLsub_type,type,
% 3.13/1.65 binintersectLsub: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectLsub,definition,
% 3.13/1.65 ( binintersectLsub
% 3.13/1.65 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset2_type,type,
% 3.13/1.65 binintersectSubset2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset2,definition,
% 3.13/1.65 ( binintersectSubset2
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( subset @ A @ B )
% 3.13/1.65 => ( ( binintersect @ A @ B )
% 3.13/1.65 = A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset3_type,type,
% 3.13/1.65 binintersectSubset3: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset3,definition,
% 3.13/1.65 ( binintersectSubset3
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( ( binintersect @ A @ B )
% 3.13/1.65 = B )
% 3.13/1.65 => ( subset @ B @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectER_type,type,
% 3.13/1.65 binintersectER: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectER,definition,
% 3.13/1.65 ( binintersectER
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.13/1.65 => ( in @ Xx @ B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(disjointsetsI1_type,type,
% 3.13/1.65 disjointsetsI1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(disjointsetsI1,definition,
% 3.13/1.65 ( disjointsetsI1
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ~ ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ( in @ Xx @ B ) )
% 3.13/1.65 => ( ( binintersect @ A @ B )
% 3.13/1.65 = emptyset ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectRsub_type,type,
% 3.13/1.65 binintersectRsub: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectRsub,definition,
% 3.13/1.65 ( binintersectRsub
% 3.13/1.65 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset4_type,type,
% 3.13/1.65 binintersectSubset4: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset4,definition,
% 3.13/1.65 ( binintersectSubset4
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( subset @ B @ A )
% 3.13/1.65 => ( ( binintersect @ A @ B )
% 3.13/1.65 = B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset1_type,type,
% 3.13/1.65 binintersectSubset1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(binintersectSubset1,definition,
% 3.13/1.65 ( binintersectSubset1
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( ( binintersect @ A @ B )
% 3.13/1.65 = A )
% 3.13/1.65 => ( subset @ A @ B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(bs114d_type,type,
% 3.13/1.65 bs114d: $o ).
% 3.13/1.65
% 3.13/1.65 thf(bs114d,definition,
% 3.13/1.65 ( bs114d
% 3.13/1.65 = ( ! [A: $i,B: $i,C: $i] :
% 3.13/1.65 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.13/1.65 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(regular_type,type,
% 3.13/1.65 regular: $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminus_type,type,
% 3.13/1.65 setminus: $i > $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(setminusI_type,type,
% 3.13/1.65 setminusI: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusI,definition,
% 3.13/1.65 ( setminusI
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ~ ( in @ Xx @ B )
% 3.13/1.65 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setminusEL_type,type,
% 3.13/1.65 setminusEL: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusEL,definition,
% 3.13/1.65 ( setminusEL
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.13/1.65 => ( in @ Xx @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setminusER_type,type,
% 3.13/1.65 setminusER: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusER,definition,
% 3.13/1.65 ( setminusER
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.13/1.65 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setminusSubset2_type,type,
% 3.13/1.65 setminusSubset2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusSubset2,definition,
% 3.13/1.65 ( setminusSubset2
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( subset @ A @ B )
% 3.13/1.65 => ( ( setminus @ A @ B )
% 3.13/1.65 = emptyset ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setminusERneg_type,type,
% 3.13/1.65 setminusERneg: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusERneg,definition,
% 3.13/1.65 ( setminusERneg
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.13/1.65 => ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setminusELneg_type,type,
% 3.13/1.65 setminusELneg: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusELneg,definition,
% 3.13/1.65 ( setminusELneg
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.13/1.65 => ( ~ ( in @ Xx @ B )
% 3.13/1.65 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setminusILneg_type,type,
% 3.13/1.65 setminusILneg: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusILneg,definition,
% 3.13/1.65 ( setminusILneg
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ~ ( in @ Xx @ A )
% 3.13/1.65 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setminusIRneg_type,type,
% 3.13/1.65 setminusIRneg: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusIRneg,definition,
% 3.13/1.65 ( setminusIRneg
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ B )
% 3.13/1.65 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setminusLsub_type,type,
% 3.13/1.65 setminusLsub: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusLsub,definition,
% 3.13/1.65 ( setminusLsub
% 3.13/1.65 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setminusSubset1_type,type,
% 3.13/1.65 setminusSubset1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setminusSubset1,definition,
% 3.13/1.65 ( setminusSubset1
% 3.13/1.65 = ( ! [A: $i,B: $i] :
% 3.13/1.65 ( ( ( setminus @ A @ B )
% 3.13/1.65 = emptyset )
% 3.13/1.65 => ( subset @ A @ B ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(symdiff_type,type,
% 3.13/1.65 symdiff: $i > $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffE_type,type,
% 3.13/1.65 symdiffE: $o ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffE,definition,
% 3.13/1.65 ( symdiffE
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.13/1.65 => ! [Xphi: $o] :
% 3.13/1.65 ( ( ( in @ Xx @ A )
% 3.13/1.65 => ( ~ ( in @ Xx @ B )
% 3.13/1.65 => Xphi ) )
% 3.13/1.65 => ( ( ~ ( in @ Xx @ A )
% 3.13/1.65 => ( ( in @ Xx @ B )
% 3.13/1.65 => Xphi ) )
% 3.13/1.65 => Xphi ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffI1_type,type,
% 3.13/1.65 symdiffI1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffI1,definition,
% 3.13/1.65 ( symdiffI1
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ~ ( in @ Xx @ B )
% 3.13/1.65 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffI2_type,type,
% 3.13/1.65 symdiffI2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffI2,definition,
% 3.13/1.65 ( symdiffI2
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ~ ( in @ Xx @ A )
% 3.13/1.65 => ( ( in @ Xx @ B )
% 3.13/1.65 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffIneg1_type,type,
% 3.13/1.65 symdiffIneg1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffIneg1,definition,
% 3.13/1.65 ( symdiffIneg1
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ( in @ Xx @ B )
% 3.13/1.65 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffIneg2_type,type,
% 3.13/1.65 symdiffIneg2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(symdiffIneg2,definition,
% 3.13/1.65 ( symdiffIneg2
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ~ ( in @ Xx @ A )
% 3.13/1.65 => ( ~ ( in @ Xx @ B )
% 3.13/1.65 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(iskpair_type,type,
% 3.13/1.65 iskpair: $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(secondinupair_type,type,
% 3.13/1.65 secondinupair: $o ).
% 3.13/1.65
% 3.13/1.65 thf(secondinupair,definition,
% 3.13/1.65 ( secondinupair
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairIL_type,type,
% 3.13/1.65 setukpairIL: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairIL,definition,
% 3.13/1.65 ( setukpairIL
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairIR_type,type,
% 3.13/1.65 setukpairIR: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairIR,definition,
% 3.13/1.65 ( setukpairIR
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(kpairiskpair_type,type,
% 3.13/1.65 kpairiskpair: $o ).
% 3.13/1.65
% 3.13/1.65 thf(kpairiskpair,definition,
% 3.13/1.65 ( kpairiskpair
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(kpair_type,type,
% 3.13/1.65 kpair: $i > $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(kpairp_type,type,
% 3.13/1.65 kpairp: $o ).
% 3.13/1.65
% 3.13/1.65 thf(kpairp,definition,
% 3.13/1.65 ( kpairp
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(cartprod_type,type,
% 3.13/1.65 cartprod: $i > $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(singletonsubset_type,type,
% 3.13/1.65 singletonsubset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(singletonsubset,definition,
% 3.13/1.65 ( singletonsubset
% 3.13/1.65 = ( ! [A: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(singletoninpowerset_type,type,
% 3.13/1.65 singletoninpowerset: $o ).
% 3.13/1.65
% 3.13/1.65 thf(singletoninpowerset,definition,
% 3.13/1.65 ( singletoninpowerset
% 3.13/1.65 = ( ! [A: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(singletoninpowunion_type,type,
% 3.13/1.65 singletoninpowunion: $o ).
% 3.13/1.65
% 3.13/1.65 thf(singletoninpowunion,definition,
% 3.13/1.65 ( singletoninpowunion
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(upairset2E_type,type,
% 3.13/1.65 upairset2E: $o ).
% 3.13/1.65
% 3.13/1.65 thf(upairset2E,definition,
% 3.13/1.65 ( upairset2E
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.13/1.65 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.13/1.65 => ( ( Xz = Xx )
% 3.13/1.65 | ( Xz = Xy ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(upairsubunion_type,type,
% 3.13/1.65 upairsubunion: $o ).
% 3.13/1.65
% 3.13/1.65 thf(upairsubunion,definition,
% 3.13/1.65 ( upairsubunion
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ B )
% 3.13/1.65 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(upairinpowunion_type,type,
% 3.13/1.65 upairinpowunion: $o ).
% 3.13/1.65
% 3.13/1.65 thf(upairinpowunion,definition,
% 3.13/1.65 ( upairinpowunion
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ B )
% 3.13/1.65 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(ubforcartprodlem1_type,type,
% 3.13/1.65 ubforcartprodlem1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(ubforcartprodlem1,definition,
% 3.13/1.65 ( ubforcartprodlem1
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ B )
% 3.13/1.65 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(ubforcartprodlem2_type,type,
% 3.13/1.65 ubforcartprodlem2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(ubforcartprodlem2,definition,
% 3.13/1.65 ( ubforcartprodlem2
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ B )
% 3.13/1.65 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(ubforcartprodlem3_type,type,
% 3.13/1.65 ubforcartprodlem3: $o ).
% 3.13/1.65
% 3.13/1.65 thf(ubforcartprodlem3,definition,
% 3.13/1.65 ( ubforcartprodlem3
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ B )
% 3.13/1.65 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(cartprodpairin_type,type,
% 3.13/1.65 cartprodpairin: $o ).
% 3.13/1.65
% 3.13/1.65 thf(cartprodpairin,definition,
% 3.13/1.65 ( cartprodpairin
% 3.13/1.65 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ B )
% 3.13/1.65 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(cartprodmempair1_type,type,
% 3.13/1.65 cartprodmempair1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(cartprodmempair1,definition,
% 3.13/1.65 ( cartprodmempair1
% 3.13/1.65 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.13/1.65 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.13/1.65 => ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ? [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ B )
% 3.13/1.65 & ( Xu
% 3.13/1.65 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(cartprodmempair_type,type,
% 3.13/1.65 cartprodmempair: $o ).
% 3.13/1.65
% 3.13/1.65 thf(cartprodmempair,definition,
% 3.13/1.65 ( cartprodmempair
% 3.13/1.65 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.13/1.65 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.13/1.65 => ( iskpair @ Xu ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setunionE2_type,type,
% 3.13/1.65 setunionE2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setunionE2,definition,
% 3.13/1.65 ( setunionE2
% 3.13/1.65 = ( ! [A: $i,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ ( setunion @ A ) )
% 3.13/1.65 => ? [X: $i] :
% 3.13/1.65 ( ( in @ X @ A )
% 3.13/1.65 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setunionsingleton1_type,type,
% 3.13/1.65 setunionsingleton1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setunionsingleton1,definition,
% 3.13/1.65 ( setunionsingleton1
% 3.13/1.65 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setunionsingleton2_type,type,
% 3.13/1.65 setunionsingleton2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setunionsingleton2,definition,
% 3.13/1.65 ( setunionsingleton2
% 3.13/1.65 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setunionsingleton_type,type,
% 3.13/1.65 setunionsingleton: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setunionsingleton,definition,
% 3.13/1.65 ( setunionsingleton
% 3.13/1.65 = ( ! [Xx: $i] :
% 3.13/1.65 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.13/1.65 = Xx ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(singleton_type,type,
% 3.13/1.65 singleton: $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(singletonprop_type,type,
% 3.13/1.65 singletonprop: $o ).
% 3.13/1.65
% 3.13/1.65 thf(singletonprop,definition,
% 3.13/1.65 ( singletonprop
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ A )
% 3.13/1.65 => ( ( Xphi @ Xx )
% 3.13/1.65 => ( ( Xphi @ Xy )
% 3.13/1.65 => ( Xx = Xy ) ) ) ) )
% 3.13/1.65 => ( ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ( Xphi @ Xx ) )
% 3.13/1.65 => ( singleton
% 3.13/1.65 @ ( dsetconstr @ A
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(ex1_type,type,
% 3.13/1.65 ex1: $i > ( $i > $o ) > $o ).
% 3.13/1.65
% 3.13/1.65 thf(ex1E1_type,type,
% 3.13/1.65 ex1E1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(ex1E1,definition,
% 3.13/1.65 ( ex1E1
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ( ex1 @ A
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.65 => ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(ex1I_type,type,
% 3.13/1.65 ex1I: $o ).
% 3.13/1.65
% 3.13/1.65 thf(ex1I,definition,
% 3.13/1.65 ( ex1I
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ( ( Xphi @ Xx )
% 3.13/1.65 => ( ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ A )
% 3.13/1.65 => ( ( Xphi @ Xy )
% 3.13/1.65 => ( Xy = Xx ) ) )
% 3.13/1.65 => ( ex1 @ A
% 3.13/1.65 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(ex1I2_type,type,
% 3.13/1.65 ex1I2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(ex1I2,definition,
% 3.13/1.65 ( ex1I2
% 3.13/1.65 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.65 ( ! [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 => ! [Xy: $i] :
% 3.13/1.65 ( ( in @ Xy @ A )
% 3.13/1.65 => ( ( Xphi @ Xx )
% 3.13/1.65 => ( ( Xphi @ Xy )
% 3.13/1.65 => ( Xx = Xy ) ) ) ) )
% 3.13/1.65 => ( ? [Xx: $i] :
% 3.13/1.65 ( ( in @ Xx @ A )
% 3.13/1.65 & ( Xphi @ Xx ) )
% 3.13/1.65 => ( ex1 @ A
% 3.13/1.65 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(singletonsuniq_type,type,
% 3.13/1.65 singletonsuniq: $o ).
% 3.13/1.65
% 3.13/1.65 thf(singletonsuniq,definition,
% 3.13/1.65 ( singletonsuniq
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( ( setadjoin @ Xx @ emptyset )
% 3.13/1.65 = ( setadjoin @ Xy @ emptyset ) )
% 3.13/1.65 => ( Xx = Xy ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(atmost1p_type,type,
% 3.13/1.65 atmost1p: $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(atleast2p_type,type,
% 3.13/1.65 atleast2p: $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(atmost2p_type,type,
% 3.13/1.65 atmost2p: $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(upairsetp_type,type,
% 3.13/1.65 upairsetp: $i > $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjL1_type,type,
% 3.13/1.65 setukpairinjL1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjL1,definition,
% 3.13/1.65 ( setukpairinjL1
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.13/1.65 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.13/1.65 => ( Xx = Xz ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(kfstsingleton_type,type,
% 3.13/1.65 kfstsingleton: $o ).
% 3.13/1.65
% 3.13/1.65 thf(kfstsingleton,definition,
% 3.13/1.65 ( kfstsingleton
% 3.13/1.65 = ( ! [Xu: $i] :
% 3.13/1.65 ( ( iskpair @ Xu )
% 3.13/1.65 => ( singleton
% 3.13/1.65 @ ( dsetconstr @ ( setunion @ Xu )
% 3.13/1.65 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(theprop_type,type,
% 3.13/1.65 theprop: $o ).
% 3.13/1.65
% 3.13/1.65 thf(theprop,definition,
% 3.13/1.65 ( theprop
% 3.13/1.65 = ( ! [X: $i] :
% 3.13/1.65 ( ( singleton @ X )
% 3.13/1.65 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(kfst_type,type,
% 3.13/1.65 kfst: $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(kfstpairEq_type,type,
% 3.13/1.65 kfstpairEq: $o ).
% 3.13/1.65
% 3.13/1.65 thf(kfstpairEq,definition,
% 3.13/1.65 ( kfstpairEq
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.13/1.65 = Xx ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(cartprodfstin_type,type,
% 3.13/1.65 cartprodfstin: $o ).
% 3.13/1.65
% 3.13/1.65 thf(cartprodfstin,definition,
% 3.13/1.65 ( cartprodfstin
% 3.13/1.65 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.13/1.65 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.13/1.65 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjL2_type,type,
% 3.13/1.65 setukpairinjL2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjL2,definition,
% 3.13/1.65 ( setukpairinjL2
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.13/1.65 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.13/1.65 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.13/1.65 => ( Xx = Xz ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjL_type,type,
% 3.13/1.65 setukpairinjL: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjL,definition,
% 3.13/1.65 ( setukpairinjL
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.13/1.65 ( ( ( kpair @ Xx @ Xy )
% 3.13/1.65 = ( kpair @ Xz @ Xu ) )
% 3.13/1.65 => ( Xx = Xz ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR11_type,type,
% 3.13/1.65 setukpairinjR11: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR11,definition,
% 3.13/1.65 ( setukpairinjR11
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xx = Xy )
% 3.13/1.65 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.13/1.65 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR12_type,type,
% 3.13/1.65 setukpairinjR12: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR12,definition,
% 3.13/1.65 ( setukpairinjR12
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( Xx = Xy )
% 3.13/1.65 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.13/1.65 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR1_type,type,
% 3.13/1.65 setukpairinjR1: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR1,definition,
% 3.13/1.65 ( setukpairinjR1
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.13/1.65 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.13/1.65 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.13/1.65 => ( ( Xz = Xu )
% 3.13/1.65 => ( Xy = Xu ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(upairequniteq_type,type,
% 3.13/1.65 upairequniteq: $o ).
% 3.13/1.65
% 3.13/1.65 thf(upairequniteq,definition,
% 3.13/1.65 ( upairequniteq
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.13/1.65 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.13/1.65 = ( setadjoin @ Xz @ emptyset ) )
% 3.13/1.65 => ( Xx = Xy ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR2_type,type,
% 3.13/1.65 setukpairinjR2: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR2,definition,
% 3.13/1.65 ( setukpairinjR2
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.13/1.65 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.13/1.65 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.13/1.65 => ( Xy = Xu ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR_type,type,
% 3.13/1.65 setukpairinjR: $o ).
% 3.13/1.65
% 3.13/1.65 thf(setukpairinjR,definition,
% 3.13/1.65 ( setukpairinjR
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.13/1.65 ( ( ( kpair @ Xx @ Xy )
% 3.13/1.65 = ( kpair @ Xz @ Xu ) )
% 3.13/1.65 => ( Xy = Xu ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(ksndsingleton_type,type,
% 3.13/1.65 ksndsingleton: $o ).
% 3.13/1.65
% 3.13/1.65 thf(ksndsingleton,definition,
% 3.13/1.65 ( ksndsingleton
% 3.13/1.65 = ( ! [Xu: $i] :
% 3.13/1.65 ( ( iskpair @ Xu )
% 3.13/1.65 => ( singleton
% 3.13/1.65 @ ( dsetconstr @ ( setunion @ Xu )
% 3.13/1.65 @ ^ [Xx: $i] :
% 3.13/1.65 ( Xu
% 3.13/1.65 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(ksnd_type,type,
% 3.13/1.65 ksnd: $i > $i ).
% 3.13/1.65
% 3.13/1.65 thf(ksndpairEq_type,type,
% 3.13/1.65 ksndpairEq: $o ).
% 3.13/1.65
% 3.13/1.65 thf(ksndpairEq,definition,
% 3.13/1.65 ( ksndpairEq
% 3.13/1.65 = ( ! [Xx: $i,Xy: $i] :
% 3.13/1.65 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.13/1.65 = Xy ) ) ) ).
% 3.13/1.65
% 3.13/1.65 thf(kpairsurjEq_type,type,
% 3.13/1.65 kpairsurjEq: $o ).
% 3.13/1.65
% 3.13/1.65 thf(kpairsurjEq,definition,
% 3.13/1.66 ( kpairsurjEq
% 3.13/1.66 = ( ! [Xu: $i] :
% 3.13/1.66 ( ( iskpair @ Xu )
% 3.13/1.66 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.13/1.66 = Xu ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodsndin_type,type,
% 3.13/1.66 cartprodsndin: $o ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodsndin,definition,
% 3.13/1.66 ( cartprodsndin
% 3.13/1.66 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.13/1.66 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.13/1.66 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodpairmemEL_type,type,
% 3.13/1.66 cartprodpairmemEL: $o ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodpairmemEL,definition,
% 3.13/1.66 ( cartprodpairmemEL
% 3.13/1.66 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.13/1.66 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.13/1.66 => ( in @ Xx @ A ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodpairmemER_type,type,
% 3.13/1.66 cartprodpairmemER: $o ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodpairmemER,definition,
% 3.13/1.66 ( cartprodpairmemER
% 3.13/1.66 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.13/1.66 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.13/1.66 => ( in @ Xy @ B ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodmempaircEq_type,type,
% 3.13/1.66 cartprodmempaircEq: $o ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodmempaircEq,definition,
% 3.13/1.66 ( cartprodmempaircEq
% 3.13/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( kpair @ Xx @ Xy )
% 3.13/1.66 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodfstpairEq_type,type,
% 3.13/1.66 cartprodfstpairEq: $o ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodfstpairEq,definition,
% 3.13/1.66 ( cartprodfstpairEq
% 3.13/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.13/1.66 = Xx ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodsndpairEq_type,type,
% 3.13/1.66 cartprodsndpairEq: $o ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodsndpairEq,definition,
% 3.13/1.66 ( cartprodsndpairEq
% 3.13/1.66 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.13/1.66 = Xy ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodpairsurjEq_type,type,
% 3.13/1.66 cartprodpairsurjEq: $o ).
% 3.13/1.66
% 3.13/1.66 thf(cartprodpairsurjEq,definition,
% 3.13/1.66 ( cartprodpairsurjEq
% 3.13/1.66 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.13/1.66 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.13/1.66 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.13/1.66 = Xu ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln_type,type,
% 3.13/1.66 breln: $i > $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstr_type,type,
% 3.13/1.66 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrI_type,type,
% 3.13/1.66 dpsetconstrI: $o ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrI,definition,
% 3.13/1.66 ( dpsetconstrI
% 3.13/1.66 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( Xphi @ Xx @ Xy )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy )
% 3.13/1.66 @ ( dpsetconstr @ A @ B
% 3.13/1.66 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrSub_type,type,
% 3.13/1.66 dpsetconstrSub: $o ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrSub,definition,
% 3.13/1.66 ( dpsetconstrSub
% 3.13/1.66 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.13/1.66 ( subset
% 3.13/1.66 @ ( dpsetconstr @ A @ B
% 3.13/1.66 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.13/1.66 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(setOfPairsIsBReln_type,type,
% 3.13/1.66 setOfPairsIsBReln: $o ).
% 3.13/1.66
% 3.13/1.66 thf(setOfPairsIsBReln,definition,
% 3.13/1.66 ( setOfPairsIsBReln
% 3.13/1.66 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.13/1.66 ( breln @ A @ B
% 3.13/1.66 @ ( dpsetconstr @ A @ B
% 3.13/1.66 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrERa_type,type,
% 3.13/1.66 dpsetconstrERa: $o ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrERa,definition,
% 3.13/1.66 ( dpsetconstrERa
% 3.13/1.66 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.13/1.66 @ ( dpsetconstr @ A @ B
% 3.13/1.66 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.13/1.66 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrEL1_type,type,
% 3.13/1.66 dpsetconstrEL1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrEL1,definition,
% 3.13/1.66 ( dpsetconstrEL1
% 3.13/1.66 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.13/1.66 ( ( in @ ( kpair @ Xx @ Xy )
% 3.13/1.66 @ ( dpsetconstr @ A @ B
% 3.13/1.66 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.13/1.66 => ( in @ Xx @ A ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrEL2_type,type,
% 3.13/1.66 dpsetconstrEL2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrEL2,definition,
% 3.13/1.66 ( dpsetconstrEL2
% 3.13/1.66 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.13/1.66 ( ( in @ ( kpair @ Xx @ Xy )
% 3.13/1.66 @ ( dpsetconstr @ A @ B
% 3.13/1.66 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.13/1.66 => ( in @ Xy @ B ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrER_type,type,
% 3.13/1.66 dpsetconstrER: $o ).
% 3.13/1.66
% 3.13/1.66 thf(dpsetconstrER,definition,
% 3.13/1.66 ( dpsetconstrER
% 3.13/1.66 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.13/1.66 ( ( in @ ( kpair @ Xx @ Xy )
% 3.13/1.66 @ ( dpsetconstr @ A @ B
% 3.13/1.66 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.13/1.66 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(func_type,type,
% 3.13/1.66 func: $i > $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcSet_type,type,
% 3.13/1.66 funcSet: $i > $i > $i ).
% 3.13/1.66
% 3.13/1.66 thf(funcImageSingleton_type,type,
% 3.13/1.66 funcImageSingleton: $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcImageSingleton,definition,
% 3.13/1.66 ( funcImageSingleton
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( singleton
% 3.13/1.66 @ ( dsetconstr @ B
% 3.13/1.66 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(apProp_type,type,
% 3.13/1.66 apProp: $o ).
% 3.13/1.66
% 3.13/1.66 thf(apProp,definition,
% 3.13/1.66 ( apProp
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in
% 3.13/1.66 @ ( setunion
% 3.13/1.66 @ ( dsetconstr @ B
% 3.13/1.66 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.13/1.66 @ B ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(ap_type,type,
% 3.13/1.66 ap: $i > $i > $i > $i > $i ).
% 3.13/1.66
% 3.13/1.66 thf(app_type,type,
% 3.13/1.66 app: $o ).
% 3.13/1.66
% 3.13/1.66 thf(app,definition,
% 3.13/1.66 ( app
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(infuncsetfunc_type,type,
% 3.13/1.66 infuncsetfunc: $o ).
% 3.13/1.66
% 3.13/1.66 thf(infuncsetfunc,definition,
% 3.13/1.66 ( infuncsetfunc
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.13/1.66 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(ap2p_type,type,
% 3.13/1.66 ap2p: $o ).
% 3.13/1.66
% 3.13/1.66 thf(ap2p,definition,
% 3.13/1.66 ( ap2p
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(funcinfuncset_type,type,
% 3.13/1.66 funcinfuncset: $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcinfuncset,definition,
% 3.13/1.66 ( funcinfuncset
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(lamProp_type,type,
% 3.13/1.66 lamProp: $o ).
% 3.13/1.66
% 3.13/1.66 thf(lamProp,definition,
% 3.13/1.66 ( lamProp
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.13/1.66 => ( func @ A @ B
% 3.13/1.66 @ ( dpsetconstr @ A @ B
% 3.13/1.66 @ ^ [Xx: $i,Xy: $i] :
% 3.13/1.66 ( ( Xf @ Xx )
% 3.13/1.66 = Xy ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(lam_type,type,
% 3.13/1.66 lam: $i > $i > ( $i > $i ) > $i ).
% 3.13/1.66
% 3.13/1.66 thf(lamp_type,type,
% 3.13/1.66 lamp: $o ).
% 3.13/1.66
% 3.13/1.66 thf(lamp,definition,
% 3.13/1.66 ( lamp
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.13/1.66 => ( func @ A @ B
% 3.13/1.66 @ ( lam @ A @ B
% 3.13/1.66 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(lam2p_type,type,
% 3.13/1.66 lam2p: $o ).
% 3.13/1.66
% 3.13/1.66 thf(lam2p,definition,
% 3.13/1.66 ( lam2p
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.13/1.66 => ( in
% 3.13/1.66 @ ( lam @ A @ B
% 3.13/1.66 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.13/1.66 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(brelnall1_type,type,
% 3.13/1.66 brelnall1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(brelnall1,definition,
% 3.13/1.66 ( brelnall1
% 3.13/1.66 = ( ! [A: $i,B: $i,R: $i] :
% 3.13/1.66 ( ( breln @ A @ B @ R )
% 3.13/1.66 => ! [Xphi: $i > $o] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ R )
% 3.13/1.66 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(brelnall2_type,type,
% 3.13/1.66 brelnall2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(brelnall2,definition,
% 3.13/1.66 ( brelnall2
% 3.13/1.66 = ( ! [A: $i,B: $i,R: $i] :
% 3.13/1.66 ( ( breln @ A @ B @ R )
% 3.13/1.66 => ! [Xphi: $i > $o] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ R )
% 3.13/1.66 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(ex1E2_type,type,
% 3.13/1.66 ex1E2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(ex1E2,definition,
% 3.13/1.66 ( ex1E2
% 3.13/1.66 = ( ! [A: $i,Xphi: $i > $o] :
% 3.13/1.66 ( ( ex1 @ A
% 3.13/1.66 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( Xphi @ Xx )
% 3.13/1.66 => ( ( Xphi @ Xy )
% 3.13/1.66 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(funcGraphProp1_type,type,
% 3.13/1.66 funcGraphProp1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcGraphProp1,definition,
% 3.13/1.66 ( funcGraphProp1
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(funcGraphProp3_type,type,
% 3.13/1.66 funcGraphProp3: $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcGraphProp3,definition,
% 3.13/1.66 ( funcGraphProp3
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(funcGraphProp2_type,type,
% 3.13/1.66 funcGraphProp2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcGraphProp2,definition,
% 3.13/1.66 ( funcGraphProp2
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.13/1.66 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.13/1.66 = Xy ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(funcextLem_type,type,
% 3.13/1.66 funcextLem: $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcextLem,definition,
% 3.13/1.66 ( funcextLem
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ! [Xg: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xg )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.13/1.66 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(funcGraphProp4_type,type,
% 3.13/1.66 funcGraphProp4: $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcGraphProp4,definition,
% 3.13/1.66 ( funcGraphProp4
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.13/1.66 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.13/1.66 = Xy ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(subbreln_type,type,
% 3.13/1.66 subbreln: $o ).
% 3.13/1.66
% 3.13/1.66 thf(subbreln,definition,
% 3.13/1.66 ( subbreln
% 3.13/1.66 = ( ! [A: $i,B: $i,R: $i] :
% 3.13/1.66 ( ( breln @ A @ B @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln @ A @ B @ S )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.13/1.66 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(eqbreln_type,type,
% 3.13/1.66 eqbreln: $o ).
% 3.13/1.66
% 3.13/1.66 thf(eqbreln,definition,
% 3.13/1.66 ( eqbreln
% 3.13/1.66 = ( ! [A: $i,B: $i,R: $i] :
% 3.13/1.66 ( ( breln @ A @ B @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln @ A @ B @ S )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.13/1.66 => ( R = S ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(funcext_type,type,
% 3.13/1.66 funcext: $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcext,definition,
% 3.13/1.66 ( funcext
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ! [Xg: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xg )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.13/1.66 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.13/1.66 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(funcext2_type,type,
% 3.13/1.66 funcext2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(funcext2,definition,
% 3.13/1.66 ( funcext2
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.13/1.66 => ! [Xg: $i] :
% 3.13/1.66 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.13/1.66 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.13/1.66 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(ap2apEq1_type,type,
% 3.13/1.66 ap2apEq1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(ap2apEq1,definition,
% 3.13/1.66 ( ap2apEq1
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.13/1.66 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(ap2apEq2_type,type,
% 3.13/1.66 ap2apEq2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(ap2apEq2,definition,
% 3.13/1.66 ( ap2apEq2
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.13/1.66 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(beta1_type,type,
% 3.13/1.66 beta1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(beta1,definition,
% 3.13/1.66 ( beta1
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( ap @ A @ B
% 3.13/1.66 @ ( lam @ A @ B
% 3.13/1.66 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.13/1.66 @ Xx )
% 3.13/1.66 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(eta1_type,type,
% 3.13/1.66 eta1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(eta1,definition,
% 3.13/1.66 ( eta1
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( func @ A @ B @ Xf )
% 3.13/1.66 => ( ( lam @ A @ B
% 3.13/1.66 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.13/1.66 = Xf ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(lam2lamEq_type,type,
% 3.13/1.66 lam2lamEq: $o ).
% 3.13/1.66
% 3.13/1.66 thf(lam2lamEq,definition,
% 3.13/1.66 ( lam2lamEq
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.13/1.66 => ( ( lam @ A @ B
% 3.13/1.66 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.13/1.66 = ( lam @ A @ B
% 3.13/1.66 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(beta2_type,type,
% 3.13/1.66 beta2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(beta2,definition,
% 3.13/1.66 ( beta2
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( ap @ A @ B
% 3.13/1.66 @ ( lam @ A @ B
% 3.13/1.66 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.13/1.66 @ Xx )
% 3.13/1.66 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(eta2_type,type,
% 3.13/1.66 eta2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(eta2,definition,
% 3.13/1.66 ( eta2
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.13/1.66 => ( ( lam @ A @ B
% 3.13/1.66 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.13/1.66 = Xf ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(iffalseProp1_type,type,
% 3.13/1.66 iffalseProp1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(iffalseProp1,definition,
% 3.13/1.66 ( iffalseProp1
% 3.13/1.66 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ~ Xphi
% 3.13/1.66 => ( in @ Xy
% 3.13/1.66 @ ( dsetconstr @ A
% 3.13/1.66 @ ^ [Xz: $i] :
% 3.13/1.66 ( ( Xphi
% 3.13/1.66 & ( Xz = Xx ) )
% 3.13/1.66 | ( ~ Xphi
% 3.13/1.66 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(iffalseProp2_type,type,
% 3.13/1.66 iffalseProp2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(iffalseProp2,definition,
% 3.13/1.66 ( iffalseProp2
% 3.13/1.66 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ~ Xphi
% 3.13/1.66 => ( ( dsetconstr @ A
% 3.13/1.66 @ ^ [Xz: $i] :
% 3.13/1.66 ( ( Xphi
% 3.13/1.66 & ( Xz = Xx ) )
% 3.13/1.66 | ( ~ Xphi
% 3.13/1.66 & ( Xz = Xy ) ) ) )
% 3.13/1.66 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(iftrueProp1_type,type,
% 3.13/1.66 iftrueProp1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(iftrueProp1,definition,
% 3.13/1.66 ( iftrueProp1
% 3.13/1.66 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( Xphi
% 3.13/1.66 => ( in @ Xx
% 3.13/1.66 @ ( dsetconstr @ A
% 3.13/1.66 @ ^ [Xz: $i] :
% 3.13/1.66 ( ( Xphi
% 3.13/1.66 & ( Xz = Xx ) )
% 3.13/1.66 | ( ~ Xphi
% 3.13/1.66 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(iftrueProp2_type,type,
% 3.13/1.66 iftrueProp2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(iftrueProp2,definition,
% 3.13/1.66 ( iftrueProp2
% 3.13/1.66 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( Xphi
% 3.13/1.66 => ( ( dsetconstr @ A
% 3.13/1.66 @ ^ [Xz: $i] :
% 3.13/1.66 ( ( Xphi
% 3.13/1.66 & ( Xz = Xx ) )
% 3.13/1.66 | ( ~ Xphi
% 3.13/1.66 & ( Xz = Xy ) ) ) )
% 3.13/1.66 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(ifSingleton_type,type,
% 3.13/1.66 ifSingleton: $o ).
% 3.13/1.66
% 3.13/1.66 thf(ifSingleton,definition,
% 3.13/1.66 ( ifSingleton
% 3.13/1.66 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( singleton
% 3.13/1.66 @ ( dsetconstr @ A
% 3.13/1.66 @ ^ [Xz: $i] :
% 3.13/1.66 ( ( Xphi
% 3.13/1.66 & ( Xz = Xx ) )
% 3.13/1.66 | ( ~ Xphi
% 3.13/1.66 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(if_type,type,
% 3.13/1.66 if: $i > $o > $i > $i > $i ).
% 3.13/1.66
% 3.13/1.66 thf(ifp_type,type,
% 3.13/1.66 ifp: $o ).
% 3.13/1.66
% 3.13/1.66 thf(ifp,definition,
% 3.13/1.66 ( ifp
% 3.13/1.66 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(theeq_type,type,
% 3.13/1.66 theeq: $o ).
% 3.13/1.66
% 3.13/1.66 thf(theeq,definition,
% 3.13/1.66 ( theeq
% 3.13/1.66 = ( ! [X: $i] :
% 3.13/1.66 ( ( singleton @ X )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ X )
% 3.13/1.66 => ( ( setunion @ X )
% 3.13/1.66 = Xx ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(iftrue_type,type,
% 3.13/1.66 iftrue: $o ).
% 3.13/1.66
% 3.13/1.66 thf(iftrue,definition,
% 3.13/1.66 ( iftrue
% 3.13/1.66 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( Xphi
% 3.13/1.66 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.13/1.66 = Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(iffalse_type,type,
% 3.13/1.66 iffalse: $o ).
% 3.13/1.66
% 3.13/1.66 thf(iffalse,definition,
% 3.13/1.66 ( iffalse
% 3.13/1.66 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ~ Xphi
% 3.13/1.66 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.13/1.66 = Xy ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(iftrueorfalse_type,type,
% 3.13/1.66 iftrueorfalse: $o ).
% 3.13/1.66
% 3.13/1.66 thf(iftrueorfalse,definition,
% 3.13/1.66 ( iftrueorfalse
% 3.13/1.66 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(binintersectT_lem_type,type,
% 3.13/1.66 binintersectT_lem: $o ).
% 3.13/1.66
% 3.13/1.66 thf(binintersectT_lem,definition,
% 3.13/1.66 ( binintersectT_lem
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(binunionT_lem_type,type,
% 3.13/1.66 binunionT_lem: $o ).
% 3.13/1.66
% 3.13/1.66 thf(binunionT_lem,definition,
% 3.13/1.66 ( binunionT_lem
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(powersetT_lem_type,type,
% 3.13/1.66 powersetT_lem: $o ).
% 3.13/1.66
% 3.13/1.66 thf(powersetT_lem,definition,
% 3.13/1.66 ( powersetT_lem
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(setminusT_lem_type,type,
% 3.13/1.66 setminusT_lem: $o ).
% 3.13/1.66
% 3.13/1.66 thf(setminusT_lem,definition,
% 3.13/1.66 ( setminusT_lem
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(complementT_lem_type,type,
% 3.13/1.66 complementT_lem: $o ).
% 3.13/1.66
% 3.13/1.66 thf(complementT_lem,definition,
% 3.13/1.66 ( complementT_lem
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(setextT_type,type,
% 3.13/1.66 setextT: $o ).
% 3.13/1.66
% 3.13/1.66 thf(setextT,definition,
% 3.13/1.66 ( setextT
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ X )
% 3.13/1.66 => ( in @ Xx @ Y ) ) )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ Y )
% 3.13/1.66 => ( in @ Xx @ X ) ) )
% 3.13/1.66 => ( X = Y ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(subsetTI_type,type,
% 3.13/1.66 subsetTI: $o ).
% 3.13/1.66
% 3.13/1.66 thf(subsetTI,definition,
% 3.13/1.66 ( subsetTI
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ X )
% 3.13/1.66 => ( in @ Xx @ Y ) ) )
% 3.13/1.66 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(powersetTI1_type,type,
% 3.13/1.66 powersetTI1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(powersetTI1,definition,
% 3.13/1.66 ( powersetTI1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ X )
% 3.13/1.66 => ( in @ Xx @ Y ) ) )
% 3.13/1.66 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(powersetTE1_type,type,
% 3.13/1.66 powersetTE1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(powersetTE1,definition,
% 3.13/1.66 ( powersetTE1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ X @ ( powerset @ Y ) )
% 3.13/1.66 => ( ( in @ Xx @ X )
% 3.13/1.66 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(complementTI1_type,type,
% 3.13/1.66 complementTI1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(complementTI1,definition,
% 3.13/1.66 ( complementTI1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ X )
% 3.13/1.66 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(complementTE1_type,type,
% 3.13/1.66 complementTE1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(complementTE1,definition,
% 3.13/1.66 ( complementTE1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.13/1.66 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(binintersectTELcontra_type,type,
% 3.13/1.66 binintersectTELcontra: $o ).
% 3.13/1.66
% 3.13/1.66 thf(binintersectTELcontra,definition,
% 3.13/1.66 ( binintersectTELcontra
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ~ ( in @ Xx @ X )
% 3.13/1.66 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(binintersectTERcontra_type,type,
% 3.13/1.66 binintersectTERcontra: $o ).
% 3.13/1.66
% 3.13/1.66 thf(binintersectTERcontra,definition,
% 3.13/1.66 ( binintersectTERcontra
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ~ ( in @ Xx @ Y )
% 3.13/1.66 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(contrasubsetT_type,type,
% 3.13/1.66 contrasubsetT: $o ).
% 3.13/1.66
% 3.13/1.66 thf(contrasubsetT,definition,
% 3.13/1.66 ( contrasubsetT
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.13/1.66 => ( ( in @ Xx @ Y )
% 3.13/1.66 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(contrasubsetT1_type,type,
% 3.13/1.66 contrasubsetT1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(contrasubsetT1,definition,
% 3.13/1.66 ( contrasubsetT1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( subset @ X @ Y )
% 3.13/1.66 => ( ~ ( in @ Xx @ Y )
% 3.13/1.66 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(contrasubsetT2_type,type,
% 3.13/1.66 contrasubsetT2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(contrasubsetT2,definition,
% 3.13/1.66 ( contrasubsetT2
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ( subset @ X @ Y )
% 3.13/1.66 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(contrasubsetT3_type,type,
% 3.13/1.66 contrasubsetT3: $o ).
% 3.13/1.66
% 3.13/1.66 thf(contrasubsetT3,definition,
% 3.13/1.66 ( contrasubsetT3
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.13/1.66 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementI1_type,type,
% 3.13/1.66 doubleComplementI1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementI1,definition,
% 3.13/1.66 ( doubleComplementI1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ X )
% 3.13/1.66 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementE1_type,type,
% 3.13/1.66 doubleComplementE1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementE1,definition,
% 3.13/1.66 ( doubleComplementE1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.13/1.66 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementSub1_type,type,
% 3.13/1.66 doubleComplementSub1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementSub1,definition,
% 3.13/1.66 ( doubleComplementSub1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementSub2_type,type,
% 3.13/1.66 doubleComplementSub2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementSub2,definition,
% 3.13/1.66 ( doubleComplementSub2
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementEq_type,type,
% 3.13/1.66 doubleComplementEq: $o ).
% 3.13/1.66
% 3.13/1.66 thf(doubleComplementEq,definition,
% 3.13/1.66 ( doubleComplementEq
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ( X
% 3.13/1.66 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(complementTnotintersectT_type,type,
% 3.13/1.66 complementTnotintersectT: $o ).
% 3.13/1.66
% 3.13/1.66 thf(complementTnotintersectT,definition,
% 3.13/1.66 ( complementTnotintersectT
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.13/1.66 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(complementImpComplementIntersect_type,type,
% 3.13/1.66 complementImpComplementIntersect: $o ).
% 3.13/1.66
% 3.13/1.66 thf(complementImpComplementIntersect,definition,
% 3.13/1.66 ( complementImpComplementIntersect
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.13/1.66 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(complementSubsetComplementIntersect_type,type,
% 3.13/1.66 complementSubsetComplementIntersect: $o ).
% 3.13/1.66
% 3.13/1.66 thf(complementSubsetComplementIntersect,definition,
% 3.13/1.66 ( complementSubsetComplementIntersect
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(complementInPowersetComplementIntersect_type,type,
% 3.13/1.66 complementInPowersetComplementIntersect: $o ).
% 3.13/1.66
% 3.13/1.66 thf(complementInPowersetComplementIntersect,definition,
% 3.13/1.66 ( complementInPowersetComplementIntersect
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(contraSubsetComplement_type,type,
% 3.13/1.66 contraSubsetComplement: $o ).
% 3.13/1.66
% 3.13/1.66 thf(contraSubsetComplement,definition,
% 3.13/1.66 ( contraSubsetComplement
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ Y )
% 3.13/1.66 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(complementTcontraSubset_type,type,
% 3.13/1.66 complementTcontraSubset: $o ).
% 3.13/1.66
% 3.13/1.66 thf(complementTcontraSubset,definition,
% 3.13/1.66 ( complementTcontraSubset
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.13/1.66 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(binunionTILcontra_type,type,
% 3.13/1.66 binunionTILcontra: $o ).
% 3.13/1.66
% 3.13/1.66 thf(binunionTILcontra,definition,
% 3.13/1.66 ( binunionTILcontra
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.13/1.66 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(binunionTIRcontra_type,type,
% 3.13/1.66 binunionTIRcontra: $o ).
% 3.13/1.66
% 3.13/1.66 thf(binunionTIRcontra,definition,
% 3.13/1.66 ( binunionTIRcontra
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.13/1.66 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(inIntersectImpInUnion_type,type,
% 3.13/1.66 inIntersectImpInUnion: $o ).
% 3.13/1.66
% 3.13/1.66 thf(inIntersectImpInUnion,definition,
% 3.13/1.66 ( inIntersectImpInUnion
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.13/1.66 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(inIntersectImpInUnion2_type,type,
% 3.13/1.66 inIntersectImpInUnion2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(inIntersectImpInUnion2,definition,
% 3.13/1.66 ( inIntersectImpInUnion2
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.13/1.66 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(inIntersectImpInIntersectUnions_type,type,
% 3.13/1.66 inIntersectImpInIntersectUnions: $o ).
% 3.13/1.66
% 3.13/1.66 thf(inIntersectImpInIntersectUnions,definition,
% 3.13/1.66 ( inIntersectImpInIntersectUnions
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.13/1.66 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(intersectInPowersetIntersectUnions_type,type,
% 3.13/1.66 intersectInPowersetIntersectUnions: $o ).
% 3.13/1.66
% 3.13/1.66 thf(intersectInPowersetIntersectUnions,definition,
% 3.13/1.66 ( intersectInPowersetIntersectUnions
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(inComplementUnionImpNotIn1_type,type,
% 3.13/1.66 inComplementUnionImpNotIn1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(inComplementUnionImpNotIn1,definition,
% 3.13/1.66 ( inComplementUnionImpNotIn1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.13/1.66 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(inComplementUnionImpInComplement1_type,type,
% 3.13/1.66 inComplementUnionImpInComplement1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(inComplementUnionImpInComplement1,definition,
% 3.13/1.66 ( inComplementUnionImpInComplement1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.13/1.66 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(binunionTE_type,type,
% 3.13/1.66 binunionTE: $o ).
% 3.13/1.66
% 3.13/1.66 thf(binunionTE,definition,
% 3.13/1.66 ( binunionTE
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xphi: $o,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.13/1.66 => ( ( ( in @ Xx @ X )
% 3.13/1.66 => Xphi )
% 3.13/1.66 => ( ( ( in @ Xx @ Y )
% 3.13/1.66 => Xphi )
% 3.13/1.66 => Xphi ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(binunionTEcontra_type,type,
% 3.13/1.66 binunionTEcontra: $o ).
% 3.13/1.66
% 3.13/1.66 thf(binunionTEcontra,definition,
% 3.13/1.66 ( binunionTEcontra
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ~ ( in @ Xx @ X )
% 3.13/1.66 => ( ~ ( in @ Xx @ Y )
% 3.13/1.66 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2a1_type,type,
% 3.13/1.66 demorgan2a1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2a1,definition,
% 3.13/1.66 ( demorgan2a1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.13/1.66 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(complementUnionInPowersetComplement_type,type,
% 3.13/1.66 complementUnionInPowersetComplement: $o ).
% 3.13/1.66
% 3.13/1.66 thf(complementUnionInPowersetComplement,definition,
% 3.13/1.66 ( complementUnionInPowersetComplement
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2a2_type,type,
% 3.13/1.66 demorgan2a2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2a2,definition,
% 3.13/1.66 ( demorgan2a2
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.13/1.66 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan1a_type,type,
% 3.13/1.66 demorgan1a: $o ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan1a,definition,
% 3.13/1.66 ( demorgan1a
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.13/1.66 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan1b_type,type,
% 3.13/1.66 demorgan1b: $o ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan1b,definition,
% 3.13/1.66 ( demorgan1b
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.13/1.66 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan1_type,type,
% 3.13/1.66 demorgan1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan1,definition,
% 3.13/1.66 ( demorgan1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.13/1.66 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2a_type,type,
% 3.13/1.66 demorgan2a: $o ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2a,definition,
% 3.13/1.66 ( demorgan2a
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.13/1.66 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2b2_type,type,
% 3.13/1.66 demorgan2b2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2b2,definition,
% 3.13/1.66 ( demorgan2b2
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.13/1.66 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.13/1.66 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2b_type,type,
% 3.13/1.66 demorgan2b: $o ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2b,definition,
% 3.13/1.66 ( demorgan2b
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.13/1.66 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2_type,type,
% 3.13/1.66 demorgan2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(demorgan2,definition,
% 3.13/1.66 ( demorgan2
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.13/1.66 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule0_type,type,
% 3.13/1.66 woz13rule0: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule0,definition,
% 3.13/1.66 ( woz13rule0
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.13/1.66 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule1_type,type,
% 3.13/1.66 woz13rule1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule1,definition,
% 3.13/1.66 ( woz13rule1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ( ( subset @ X @ Z )
% 3.13/1.66 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule2_type,type,
% 3.13/1.66 woz13rule2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule2,definition,
% 3.13/1.66 ( woz13rule2
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ( ( subset @ Y @ Z )
% 3.13/1.66 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule3_type,type,
% 3.13/1.66 woz13rule3: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule3,definition,
% 3.13/1.66 ( woz13rule3
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ( ( subset @ X @ Y )
% 3.13/1.66 => ( ( subset @ X @ Z )
% 3.13/1.66 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule4_type,type,
% 3.13/1.66 woz13rule4: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz13rule4,definition,
% 3.13/1.66 ( woz13rule4
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ! [W: $i] :
% 3.13/1.66 ( ( in @ W @ ( powerset @ A ) )
% 3.13/1.66 => ( ( subset @ X @ Z )
% 3.13/1.66 => ( ( subset @ Y @ W )
% 3.13/1.66 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_1_type,type,
% 3.13/1.66 woz1_1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_1,definition,
% 3.13/1.66 ( woz1_1
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_2_type,type,
% 3.13/1.66 woz1_2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_2,definition,
% 3.13/1.66 ( woz1_2
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ! [W: $i] :
% 3.13/1.66 ( ( in @ W @ ( powerset @ A ) )
% 3.13/1.66 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.13/1.66 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_3_type,type,
% 3.13/1.66 woz1_3: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_3,definition,
% 3.13/1.66 ( woz1_3
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ! [Z: $i] :
% 3.13/1.66 ( ( in @ Z @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_4_type,type,
% 3.13/1.66 woz1_4: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_4,definition,
% 3.13/1.66 ( woz1_4
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.13/1.66 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_5_type,type,
% 3.13/1.66 woz1_5: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz1_5,definition,
% 3.13/1.66 ( woz1_5
% 3.13/1.66 = ( ! [A: $i,X: $i] :
% 3.13/1.66 ( ( in @ X @ ( powerset @ A ) )
% 3.13/1.66 => ! [Y: $i] :
% 3.13/1.66 ( ( in @ Y @ ( powerset @ A ) )
% 3.13/1.66 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1_type,type,
% 3.13/1.66 breln1: $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1all2_type,type,
% 3.13/1.66 breln1all2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1all2,definition,
% 3.13/1.66 ( breln1all2
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [Xphi: $i > $o] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ R )
% 3.13/1.66 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1Set_type,type,
% 3.13/1.66 breln1Set: $i > $i ).
% 3.13/1.66
% 3.13/1.66 thf(breln1SetBreln1_type,type,
% 3.13/1.66 breln1SetBreln1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1SetBreln1,definition,
% 3.13/1.66 ( breln1SetBreln1
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( in @ R @ ( breln1Set @ A ) )
% 3.13/1.66 => ( breln1 @ A @ R ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(transitive_type,type,
% 3.13/1.66 transitive: $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(antisymmetric_type,type,
% 3.13/1.66 antisymmetric: $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(reflexive_type,type,
% 3.13/1.66 reflexive: $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(refltransitive_type,type,
% 3.13/1.66 refltransitive: $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(refllinearorder_type,type,
% 3.13/1.66 refllinearorder: $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(reflwellordering_type,type,
% 3.13/1.66 reflwellordering: $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(choice2fnsingleton_type,type,
% 3.13/1.66 choice2fnsingleton: $o ).
% 3.13/1.66
% 3.13/1.66 thf(choice2fnsingleton,definition,
% 3.13/1.66 ( choice2fnsingleton
% 3.13/1.66 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ? [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ B )
% 3.13/1.66 & ( Xphi @ Xx @ Xy ) ) )
% 3.13/1.66 => ! [R: $i] :
% 3.13/1.66 ( ( in @ R @ ( breln1Set @ B ) )
% 3.13/1.66 => ( ( reflwellordering @ B @ R )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( singleton
% 3.13/1.66 @ ( dsetconstr @ B
% 3.13/1.66 @ ^ [Xy: $i] :
% 3.13/1.66 ( ( Xphi @ Xx @ Xy )
% 3.13/1.66 & ! [Xz: $i] :
% 3.13/1.66 ( ( in @ Xz @ B )
% 3.13/1.66 => ( ( Xphi @ Xx @ Xz )
% 3.13/1.66 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(setOfPairsIsBReln1_type,type,
% 3.13/1.66 setOfPairsIsBReln1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(setOfPairsIsBReln1,definition,
% 3.13/1.66 ( setOfPairsIsBReln1
% 3.13/1.66 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.13/1.66 ( breln1 @ A
% 3.13/1.66 @ ( dpsetconstr @ A @ A
% 3.13/1.66 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1all1_type,type,
% 3.13/1.66 breln1all1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1all1,definition,
% 3.13/1.66 ( breln1all1
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [Xphi: $i > $o] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ R )
% 3.13/1.66 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(subbreln1_type,type,
% 3.13/1.66 subbreln1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(subbreln1,definition,
% 3.13/1.66 ( subbreln1
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.13/1.66 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(eqbreln1_type,type,
% 3.13/1.66 eqbreln1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(eqbreln1,definition,
% 3.13/1.66 ( eqbreln1
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.13/1.66 => ( R = S ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1invset_type,type,
% 3.13/1.66 breln1invset: $i > $i > $i ).
% 3.13/1.66
% 3.13/1.66 thf(breln1invprop_type,type,
% 3.13/1.66 breln1invprop: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1invprop,definition,
% 3.13/1.66 ( breln1invprop
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1invI_type,type,
% 3.13/1.66 breln1invI: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1invI,definition,
% 3.13/1.66 ( breln1invI
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1invE_type,type,
% 3.13/1.66 breln1invE: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1invE,definition,
% 3.13/1.66 ( breln1invE
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1compset_type,type,
% 3.13/1.66 breln1compset: $i > $i > $i > $i ).
% 3.13/1.66
% 3.13/1.66 thf(breln1compprop_type,type,
% 3.13/1.66 breln1compprop: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1compprop,definition,
% 3.13/1.66 ( breln1compprop
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1compI_type,type,
% 3.13/1.66 breln1compI: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1compI,definition,
% 3.13/1.66 ( breln1compI
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ! [Xz: $i] :
% 3.13/1.66 ( ( in @ Xz @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.13/1.66 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1compE_type,type,
% 3.13/1.66 breln1compE: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1compE,definition,
% 3.13/1.66 ( breln1compE
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.13/1.66 => ? [Xz: $i] :
% 3.13/1.66 ( ( in @ Xz @ A )
% 3.13/1.66 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.13/1.66 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1compEex_type,type,
% 3.13/1.66 breln1compEex: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1compEex,definition,
% 3.13/1.66 ( breln1compEex
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.13/1.66 => ! [Xphi: $o] :
% 3.13/1.66 ( ! [Xz: $i] :
% 3.13/1.66 ( ( in @ Xz @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.13/1.66 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.13/1.66 => Xphi ) ) )
% 3.13/1.66 => Xphi ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionprop_type,type,
% 3.13/1.66 breln1unionprop: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionprop,definition,
% 3.13/1.66 ( breln1unionprop
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionIL_type,type,
% 3.13/1.66 breln1unionIL: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionIL,definition,
% 3.13/1.66 ( breln1unionIL
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionIR_type,type,
% 3.13/1.66 breln1unionIR: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionIR,definition,
% 3.13/1.66 ( breln1unionIR
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionI_type,type,
% 3.13/1.66 breln1unionI: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionI,definition,
% 3.13/1.66 ( breln1unionI
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.13/1.66 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionE_type,type,
% 3.13/1.66 breln1unionE: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionE,definition,
% 3.13/1.66 ( breln1unionE
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionEcases_type,type,
% 3.13/1.66 breln1unionEcases: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionEcases,definition,
% 3.13/1.66 ( breln1unionEcases
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.13/1.66 => ! [Xphi: $o] :
% 3.13/1.66 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.13/1.66 => Xphi )
% 3.13/1.66 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.13/1.66 => Xphi )
% 3.13/1.66 => Xphi ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionCommutes_type,type,
% 3.13/1.66 breln1unionCommutes: $o ).
% 3.13/1.66
% 3.13/1.66 thf(breln1unionCommutes,definition,
% 3.13/1.66 ( breln1unionCommutes
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ( ( binunion @ R @ S )
% 3.13/1.66 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz2Ex_type,type,
% 3.13/1.66 woz2Ex: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz2Ex,definition,
% 3.13/1.66 ( woz2Ex
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ( R
% 3.13/1.66 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz2W_type,type,
% 3.13/1.66 woz2W: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz2W,definition,
% 3.13/1.66 ( woz2W
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 3.13/1.66 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz2A_type,type,
% 3.13/1.66 woz2A: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz2A,definition,
% 3.13/1.66 ( woz2A
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [T: $i] :
% 3.13/1.66 ( ( breln1 @ A @ T )
% 3.13/1.66 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.13/1.66 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(woz2B_type,type,
% 3.13/1.66 woz2B: $o ).
% 3.13/1.66
% 3.13/1.66 thf(woz2B,definition,
% 3.13/1.66 ( woz2B
% 3.13/1.66 = ( ! [A: $i,R: $i] :
% 3.13/1.66 ( ( breln1 @ A @ R )
% 3.13/1.66 => ! [S: $i] :
% 3.13/1.66 ( ( breln1 @ A @ S )
% 3.13/1.66 => ! [T: $i] :
% 3.13/1.66 ( ( breln1 @ A @ T )
% 3.13/1.66 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.13/1.66 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(image1Ex_type,type,
% 3.13/1.66 image1Ex: $o ).
% 3.13/1.66
% 3.13/1.66 thf(image1Ex,definition,
% 3.13/1.66 ( image1Ex
% 3.13/1.66 = ( ! [A: $i,Xf: $i > $i] :
% 3.13/1.66 ? [B: $i] :
% 3.13/1.66 ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ B )
% 3.13/1.66 <=> ? [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 & ( Xx
% 3.13/1.66 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(image1Ex1_type,type,
% 3.13/1.66 image1Ex1: $o ).
% 3.13/1.66
% 3.13/1.66 thf(image1Ex1,definition,
% 3.13/1.66 ( image1Ex1
% 3.13/1.66 = ( ! [A: $i,Xf: $i > $i] :
% 3.13/1.66 ( exu
% 3.13/1.66 @ ^ [B: $i] :
% 3.13/1.66 ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ B )
% 3.13/1.66 <=> ? [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 & ( Xx
% 3.13/1.66 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(image1_type,type,
% 3.13/1.66 image1: $i > ( $i > $i ) > $i ).
% 3.13/1.66
% 3.13/1.66 thf(image1Equiv_type,type,
% 3.13/1.66 image1Equiv: $o ).
% 3.13/1.66
% 3.13/1.66 thf(image1Equiv,definition,
% 3.13/1.66 ( image1Equiv
% 3.13/1.66 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx
% 3.13/1.66 @ ( image1 @ A
% 3.13/1.66 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.13/1.66 <=> ? [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 & ( Xx
% 3.13/1.66 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(image1E_type,type,
% 3.13/1.66 image1E: $o ).
% 3.13/1.66
% 3.13/1.66 thf(image1E,definition,
% 3.13/1.66 ( image1E
% 3.13/1.66 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.13/1.66 ( ( in @ Xx
% 3.13/1.66 @ ( image1 @ A
% 3.13/1.66 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.13/1.66 => ? [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 & ( Xx
% 3.13/1.66 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(image1I_type,type,
% 3.13/1.66 image1I: $o ).
% 3.13/1.66
% 3.13/1.66 thf(image1I,definition,
% 3.13/1.66 ( image1I
% 3.13/1.66 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.13/1.66 ( ? [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ A )
% 3.13/1.66 & ( Xx
% 3.13/1.66 = ( Xf @ Xy ) ) )
% 3.13/1.66 => ( in @ Xx
% 3.13/1.66 @ ( image1 @ A
% 3.13/1.66 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(injective_type,type,
% 3.13/1.66 injective: $i > $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(injFuncSet_type,type,
% 3.13/1.66 injFuncSet: $i > $i > $i ).
% 3.13/1.66
% 3.13/1.66 thf(injFuncInInjFuncSet_type,type,
% 3.13/1.66 injFuncInInjFuncSet: $o ).
% 3.13/1.66
% 3.13/1.66 thf(injFuncInInjFuncSet,definition,
% 3.13/1.66 ( injFuncInInjFuncSet
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.13/1.66 => ( ( injective @ A @ B @ Xf )
% 3.13/1.66 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(injFuncSetFuncIn_type,type,
% 3.13/1.66 injFuncSetFuncIn: $o ).
% 3.13/1.66
% 3.13/1.66 thf(injFuncSetFuncIn,definition,
% 3.13/1.66 ( injFuncSetFuncIn
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 3.13/1.66 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(injFuncSetFuncInj_type,type,
% 3.13/1.66 injFuncSetFuncInj: $o ).
% 3.13/1.66
% 3.13/1.66 thf(injFuncSetFuncInj,definition,
% 3.13/1.66 ( injFuncSetFuncInj
% 3.13/1.66 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 3.13/1.66 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(surjective_type,type,
% 3.13/1.66 surjective: $i > $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(surjFuncSet_type,type,
% 3.13/1.66 surjFuncSet: $i > $i > $i ).
% 3.13/1.66
% 3.13/1.66 thf(surjFuncSetFuncIn_type,type,
% 3.13/1.66 surjFuncSetFuncIn: $o ).
% 3.13/1.66
% 3.13/1.66 thf(surjFuncSetFuncIn,definition,
% 3.13/1.66 ( surjFuncSetFuncIn
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 3.13/1.66 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(surjFuncSetFuncSurj_type,type,
% 3.13/1.66 surjFuncSetFuncSurj: $o ).
% 3.13/1.66
% 3.13/1.66 thf(surjFuncSetFuncSurj,definition,
% 3.13/1.66 ( surjFuncSetFuncSurj
% 3.13/1.66 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 3.13/1.66 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(leftInvIsSurj_type,type,
% 3.13/1.66 leftInvIsSurj: $o ).
% 3.13/1.66
% 3.13/1.66 thf(leftInvIsSurj,definition,
% 3.13/1.66 ( leftInvIsSurj
% 3.13/1.66 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.13/1.66 ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.13/1.66 => ! [Xg: $i] :
% 3.13/1.66 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 3.13/1.66 => ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ A )
% 3.13/1.66 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 3.13/1.66 = Xx ) )
% 3.13/1.66 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(surjCantorThm_type,type,
% 3.13/1.66 surjCantorThm: $o ).
% 3.13/1.66
% 3.13/1.66 thf(surjCantorThm,definition,
% 3.13/1.66 ( surjCantorThm
% 3.13/1.66 = ( ! [A: $i,Xf: $i] :
% 3.13/1.66 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 3.13/1.66 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(foundation2_type,type,
% 3.13/1.66 foundation2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(foundation2,definition,
% 3.13/1.66 ( foundation2
% 3.13/1.66 = ( ! [A: $i] :
% 3.13/1.66 ( ( nonempty @ A )
% 3.13/1.66 => ? [X: $i] :
% 3.13/1.66 ( ( in @ X @ A )
% 3.13/1.66 & ( ( binintersect @ X @ A )
% 3.13/1.66 = emptyset ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(notinself_type,type,
% 3.13/1.66 notinself: $o ).
% 3.13/1.66
% 3.13/1.66 thf(notinself,definition,
% 3.13/1.66 ( notinself
% 3.13/1.66 = ( ! [A: $i] :
% 3.13/1.66 ~ ( in @ A @ A ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(notinself2_type,type,
% 3.13/1.66 notinself2: $o ).
% 3.13/1.66
% 3.13/1.66 thf(notinself2,definition,
% 3.13/1.66 ( notinself2
% 3.13/1.66 = ( ! [A: $i,B: $i] :
% 3.13/1.66 ( ( in @ A @ B )
% 3.13/1.66 => ~ ( in @ B @ A ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(omegaS_type,type,
% 3.13/1.66 omegaS: $i > $i ).
% 3.13/1.66
% 3.13/1.66 thf(omegaSp_type,type,
% 3.13/1.66 omegaSp: $o ).
% 3.13/1.66
% 3.13/1.66 thf(omegaSp,definition,
% 3.13/1.66 ( omegaSp
% 3.13/1.66 = ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ omega )
% 3.13/1.66 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(omegaSclos_type,type,
% 3.13/1.66 omegaSclos: $o ).
% 3.13/1.66
% 3.13/1.66 thf(omegaSclos,definition,
% 3.13/1.66 ( omegaSclos
% 3.13/1.66 = ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ omega )
% 3.13/1.66 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(peano0notS_type,type,
% 3.13/1.66 peano0notS: $o ).
% 3.13/1.66
% 3.13/1.66 thf(peano0notS,definition,
% 3.13/1.66 ( peano0notS
% 3.13/1.66 = ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ omega )
% 3.13/1.66 => ( ( omegaS @ Xx )
% 3.13/1.66 != emptyset ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(peano3_type,type,
% 3.13/1.66 peano3: $i > $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(peano4_type,type,
% 3.13/1.66 peano4: $i > $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(peano5_type,type,
% 3.13/1.66 peano5: $i > $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(peanoSinj_type,type,
% 3.13/1.66 peanoSinj: $o ).
% 3.13/1.66
% 3.13/1.66 thf(peanoSinj,definition,
% 3.13/1.66 ( peanoSinj
% 3.13/1.66 = ( ! [Xx: $i] :
% 3.13/1.66 ( ( in @ Xx @ omega )
% 3.13/1.66 => ! [Xy: $i] :
% 3.13/1.66 ( ( in @ Xy @ omega )
% 3.13/1.66 => ( ( ( omegaS @ Xx )
% 3.13/1.66 = ( omegaS @ Xy ) )
% 3.13/1.66 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(peano_type,type,
% 3.13/1.66 peano: $i > $i > $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(transitiveset_type,type,
% 3.13/1.66 transitiveset: $i > $o ).
% 3.13/1.66
% 3.13/1.66 thf(transitiveset,definition,
% 3.13/1.66 ( transitiveset
% 3.13/1.66 = ( ^ [A: $i] :
% 3.13/1.66 ! [X: $i] :
% 3.13/1.66 ( ( in @ X @ A )
% 3.13/1.66 => ( subset @ X @ A ) ) ) ) ).
% 3.13/1.66
% 3.13/1.66 thf(transitivesetOp1,conjecture,
% 3.13/1.66 ( setextAx
% 3.13/1.66 => ( emptysetAx
% 3.13/1.66 => ( setadjoinAx
% 3.13/1.66 => ( powersetAx
% 3.13/1.66 => ( setunionAx
% 3.13/1.66 => ( omega0Ax
% 3.13/1.66 => ( omegaSAx
% 3.13/1.66 => ( omegaIndAx
% 3.13/1.66 => ( replAx
% 3.13/1.66 => ( foundationAx
% 3.13/1.66 => ( wellorderingAx
% 3.13/1.66 => ( descrp
% 3.13/1.66 => ( dsetconstrI
% 3.13/1.66 => ( dsetconstrEL
% 3.13/1.66 => ( dsetconstrER
% 3.13/1.66 => ( exuE1
% 3.13/1.66 => ( prop2setE
% 3.13/1.66 => ( emptysetE
% 3.13/1.66 => ( emptysetimpfalse
% 3.13/1.66 => ( notinemptyset
% 3.13/1.66 => ( exuE3e
% 3.13/1.66 => ( setext
% 3.13/1.66 => ( emptyI
% 3.13/1.66 => ( noeltsimpempty
% 3.13/1.66 => ( setbeta
% 3.13/1.66 => ( nonemptyE1
% 3.13/1.66 => ( nonemptyI
% 3.13/1.66 => ( nonemptyI1
% 3.13/1.66 => ( setadjoinIL
% 3.13/1.66 => ( emptyinunitempty
% 3.13/1.66 => ( setadjoinIR
% 3.13/1.66 => ( setadjoinE
% 3.13/1.66 => ( setadjoinOr
% 3.13/1.66 => ( setoftrueEq
% 3.13/1.66 => ( powersetI
% 3.13/1.66 => ( emptyinPowerset
% 3.13/1.66 => ( emptyInPowerset
% 3.13/1.66 => ( powersetE
% 3.13/1.66 => ( setunionI
% 3.13/1.66 => ( setunionE
% 3.13/1.66 => ( subPowSU
% 3.13/1.66 => ( exuE2
% 3.13/1.66 => ( nonemptyImpWitness
% 3.13/1.66 => ( uniqinunit
% 3.13/1.66 => ( notinsingleton
% 3.13/1.66 => ( eqinunit
% 3.13/1.66 => ( singletonsswitch
% 3.13/1.66 => ( upairsetE
% 3.13/1.66 => ( upairsetIL
% 3.13/1.66 => ( upairsetIR
% 3.13/1.66 => ( emptyE1
% 3.13/1.66 => ( vacuousDall
% 3.13/1.66 => ( quantDeMorgan1
% 3.13/1.66 => ( quantDeMorgan2
% 3.13/1.66 => ( quantDeMorgan3
% 3.13/1.66 => ( quantDeMorgan4
% 3.13/1.66 => ( prop2setI
% 3.13/1.66 => ( prop2set2propI
% 3.13/1.66 => ( notdexE
% 3.13/1.66 => ( notdallE
% 3.13/1.66 => ( exuI1
% 3.13/1.66 => ( exuI3
% 3.13/1.66 => ( exuI2
% 3.13/1.66 => ( inCongP
% 3.13/1.66 => ( in__Cong
% 3.13/1.66 => ( exuE3u
% 3.13/1.66 => ( exu__Cong
% 3.13/1.66 => ( emptyset__Cong
% 3.13/1.66 => ( setadjoin__Cong
% 3.13/1.66 => ( powerset__Cong
% 3.13/1.66 => ( setunion__Cong
% 3.13/1.66 => ( omega__Cong
% 3.13/1.66 => ( exuEu
% 3.13/1.66 => ( descr__Cong
% 3.13/1.66 => ( dsetconstr__Cong
% 3.13/1.66 => ( subsetI1
% 3.13/1.66 => ( eqimpsubset2
% 3.13/1.66 => ( eqimpsubset1
% 3.13/1.66 => ( subsetI2
% 3.13/1.66 => ( emptysetsubset
% 3.13/1.66 => ( subsetE
% 3.13/1.66 => ( subsetE2
% 3.13/1.66 => ( notsubsetI
% 3.13/1.66 => ( notequalI1
% 3.13/1.66 => ( notequalI2
% 3.13/1.66 => ( subsetRefl
% 3.13/1.66 => ( subsetTrans
% 3.13/1.66 => ( setadjoinSub
% 3.13/1.66 => ( setadjoinSub2
% 3.13/1.66 => ( subset2powerset
% 3.13/1.66 => ( setextsub
% 3.13/1.66 => ( subsetemptysetimpeq
% 3.13/1.66 => ( powersetI1
% 3.13/1.66 => ( powersetE1
% 3.13/1.66 => ( inPowerset
% 3.13/1.66 => ( powersetsubset
% 3.13/1.66 => ( sepInPowerset
% 3.13/1.66 => ( sepSubset
% 3.13/1.66 => ( binunionIL
% 3.13/1.66 => ( upairset2IR
% 3.13/1.66 => ( binunionIR
% 3.13/1.66 => ( binunionEcases
% 3.13/1.66 => ( binunionE
% 3.13/1.66 => ( binunionLsub
% 3.13/1.66 => ( binunionRsub
% 3.13/1.66 => ( binintersectI
% 3.13/1.66 => ( binintersectSubset5
% 3.13/1.66 => ( binintersectEL
% 3.13/1.66 => ( binintersectLsub
% 3.13/1.66 => ( binintersectSubset2
% 3.13/1.66 => ( binintersectSubset3
% 3.13/1.66 => ( binintersectER
% 3.13/1.66 => ( disjointsetsI1
% 3.13/1.66 => ( binintersectRsub
% 3.13/1.66 => ( binintersectSubset4
% 3.13/1.66 => ( binintersectSubset1
% 3.13/1.66 => ( bs114d
% 3.13/1.66 => ( setminusI
% 3.13/1.66 => ( setminusEL
% 3.13/1.66 => ( setminusER
% 3.13/1.66 => ( setminusSubset2
% 3.13/1.66 => ( setminusERneg
% 3.13/1.66 => ( setminusELneg
% 3.13/1.66 => ( setminusILneg
% 3.13/1.66 => ( setminusIRneg
% 3.13/1.66 => ( setminusLsub
% 3.13/1.66 => ( setminusSubset1
% 3.13/1.66 => ( symdiffE
% 3.13/1.66 => ( symdiffI1
% 3.13/1.66 => ( symdiffI2
% 3.13/1.66 => ( symdiffIneg1
% 3.13/1.66 => ( symdiffIneg2
% 3.13/1.66 => ( secondinupair
% 3.13/1.66 => ( setukpairIL
% 3.13/1.66 => ( setukpairIR
% 3.13/1.66 => ( kpairiskpair
% 3.13/1.66 => ( kpairp
% 3.13/1.66 => ( singletonsubset
% 3.13/1.66 => ( singletoninpowerset
% 3.13/1.66 => ( singletoninpowunion
% 3.13/1.66 => ( upairset2E
% 3.13/1.66 => ( upairsubunion
% 3.13/1.66 => ( upairinpowunion
% 3.13/1.66 => ( ubforcartprodlem1
% 3.13/1.66 => ( ubforcartprodlem2
% 3.13/1.66 => ( ubforcartprodlem3
% 3.13/1.66 => ( cartprodpairin
% 3.13/1.66 => ( cartprodmempair1
% 3.13/1.66 => ( cartprodmempair
% 3.13/1.66 => ( setunionE2
% 3.13/1.66 => ( setunionsingleton1
% 3.13/1.66 => ( setunionsingleton2
% 3.13/1.66 => ( setunionsingleton
% 3.13/1.66 => ( singletonprop
% 3.13/1.66 => ( ex1E1
% 3.13/1.66 => ( ex1I
% 3.13/1.66 => ( ex1I2
% 3.13/1.66 => ( singletonsuniq
% 3.13/1.66 => ( setukpairinjL1
% 3.13/1.66 => ( kfstsingleton
% 3.13/1.66 => ( theprop
% 3.13/1.66 => ( kfstpairEq
% 3.13/1.66 => ( cartprodfstin
% 3.13/1.66 => ( setukpairinjL2
% 3.13/1.66 => ( setukpairinjL
% 3.13/1.66 => ( setukpairinjR11
% 3.13/1.66 => ( setukpairinjR12
% 3.13/1.66 => ( setukpairinjR1
% 3.13/1.66 => ( upairequniteq
% 3.13/1.66 => ( setukpairinjR2
% 3.13/1.66 => ( setukpairinjR
% 3.13/1.66 => ( ksndsingleton
% 3.13/1.66 => ( ksndpairEq
% 3.13/1.66 => ( kpairsurjEq
% 3.13/1.66 => ( cartprodsndin
% 3.13/1.66 => ( cartprodpairmemEL
% 3.13/1.66 => ( cartprodpairmemER
% 3.13/1.66 => ( cartprodmempaircEq
% 3.13/1.66 => ( cartprodfstpairEq
% 3.13/1.66 => ( cartprodsndpairEq
% 3.13/1.66 => ( cartprodpairsurjEq
% 3.13/1.66 => ( dpsetconstrI
% 3.13/1.66 => ( dpsetconstrSub
% 3.13/1.66 => ( setOfPairsIsBReln
% 3.13/1.66 => ( dpsetconstrERa
% 3.13/1.66 => ( dpsetconstrEL1
% 3.13/1.66 => ( dpsetconstrEL2
% 3.13/1.66 => ( dpsetconstrER
% 3.13/1.66 => ( funcImageSingleton
% 3.13/1.66 => ( apProp
% 3.13/1.66 => ( app
% 3.13/1.66 => ( infuncsetfunc
% 3.13/1.66 => ( ap2p
% 3.13/1.66 => ( funcinfuncset
% 3.13/1.66 => ( lamProp
% 3.13/1.66 => ( lamp
% 3.13/1.66 => ( lam2p
% 3.13/1.66 => ( brelnall1
% 3.13/1.66 => ( brelnall2
% 3.13/1.66 => ( ex1E2
% 3.13/1.66 => ( funcGraphProp1
% 3.13/1.66 => ( funcGraphProp3
% 3.13/1.66 => ( funcGraphProp2
% 3.13/1.66 => ( funcextLem
% 3.13/1.66 => ( funcGraphProp4
% 3.13/1.66 => ( subbreln
% 3.13/1.66 => ( eqbreln
% 3.13/1.66 => ( funcext
% 3.13/1.66 => ( funcext2
% 3.13/1.66 => ( ap2apEq1
% 3.13/1.66 => ( ap2apEq2
% 3.13/1.66 => ( beta1
% 3.13/1.66 => ( eta1
% 3.13/1.66 => ( lam2lamEq
% 3.13/1.66 => ( beta2
% 3.13/1.66 => ( eta2
% 3.13/1.66 => ( iffalseProp1
% 3.13/1.66 => ( iffalseProp2
% 3.13/1.66 => ( iftrueProp1
% 3.13/1.66 => ( iftrueProp2
% 3.13/1.66 => ( ifSingleton
% 3.13/1.66 => ( ifp
% 3.13/1.66 => ( theeq
% 3.13/1.66 => ( iftrue
% 3.13/1.66 => ( iffalse
% 3.13/1.66 => ( iftrueorfalse
% 3.13/1.66 => ( binintersectT_lem
% 3.13/1.66 => ( binunionT_lem
% 3.13/1.66 => ( powersetT_lem
% 3.13/1.66 => ( setminusT_lem
% 3.13/1.66 => ( complementT_lem
% 3.13/1.66 => ( setextT
% 3.13/1.66 => ( subsetTI
% 3.13/1.66 => ( powersetTI1
% 3.13/1.66 => ( powersetTE1
% 3.13/1.66 => ( complementTI1
% 3.13/1.66 => ( complementTE1
% 3.13/1.66 => ( binintersectTELcontra
% 3.13/1.66 => ( binintersectTERcontra
% 3.13/1.66 => ( contrasubsetT
% 3.13/1.66 => ( contrasubsetT1
% 3.13/1.66 => ( contrasubsetT2
% 3.13/1.66 => ( contrasubsetT3
% 3.13/1.66 => ( doubleComplementI1
% 3.13/1.66 => ( doubleComplementE1
% 3.13/1.66 => ( doubleComplementSub1
% 3.13/1.66 => ( doubleComplementSub2
% 3.13/1.66 => ( doubleComplementEq
% 3.13/1.66 => ( complementTnotintersectT
% 3.13/1.66 => ( complementImpComplementIntersect
% 3.13/1.66 => ( complementSubsetComplementIntersect
% 3.13/1.66 => ( complementInPowersetComplementIntersect
% 3.13/1.66 => ( contraSubsetComplement
% 3.13/1.66 => ( complementTcontraSubset
% 3.13/1.66 => ( binunionTILcontra
% 3.13/1.66 => ( binunionTIRcontra
% 3.13/1.66 => ( inIntersectImpInUnion
% 3.13/1.66 => ( inIntersectImpInUnion2
% 3.13/1.66 => ( inIntersectImpInIntersectUnions
% 3.13/1.66 => ( intersectInPowersetIntersectUnions
% 3.13/1.66 => ( inComplementUnionImpNotIn1
% 3.13/1.66 => ( inComplementUnionImpInComplement1
% 3.13/1.66 => ( binunionTE
% 3.13/1.66 => ( binunionTEcontra
% 3.13/1.66 => ( demorgan2a1
% 3.13/1.66 => ( complementUnionInPowersetComplement
% 3.13/1.66 => ( demorgan2a2
% 3.13/1.66 => ( demorgan1a
% 3.13/1.66 => ( demorgan1b
% 3.13/1.66 => ( demorgan1
% 3.13/1.66 => ( demorgan2a
% 3.13/1.66 => ( demorgan2b2
% 3.13/1.66 => ( demorgan2b
% 3.13/1.66 => ( demorgan2
% 3.13/1.66 => ( woz13rule0
% 3.13/1.66 => ( woz13rule1
% 3.13/1.66 => ( woz13rule2
% 3.13/1.66 => ( woz13rule3
% 3.13/1.66 => ( woz13rule4
% 3.13/1.66 => ( woz1_1
% 3.13/1.66 => ( woz1_2
% 3.13/1.66 => ( woz1_3
% 3.13/1.66 => ( woz1_4
% 3.13/1.66 => ( woz1_5
% 3.13/1.66 => ( breln1all2
% 3.13/1.66 => ( breln1SetBreln1
% 3.13/1.66 => ( choice2fnsingleton
% 3.13/1.66 => ( setOfPairsIsBReln1
% 3.13/1.66 => ( breln1all1
% 3.13/1.66 => ( subbreln1
% 3.13/1.66 => ( eqbreln1
% 3.13/1.66 => ( breln1invprop
% 3.13/1.66 => ( breln1invI
% 3.13/1.66 => ( breln1invE
% 3.13/1.66 => ( breln1compprop
% 3.13/1.66 => ( breln1compI
% 3.13/1.66 => ( breln1compE
% 3.13/1.66 => ( breln1compEex
% 3.13/1.66 => ( breln1unionprop
% 3.13/1.66 => ( breln1unionIL
% 3.13/1.66 => ( breln1unionIR
% 3.13/1.66 => ( breln1unionI
% 3.13/1.66 => ( breln1unionE
% 3.13/1.66 => ( breln1unionEcases
% 3.13/1.66 => ( breln1unionCommutes
% 3.13/1.66 => ( woz2Ex
% 3.13/1.66 => ( woz2W
% 3.13/1.66 => ( woz2A
% 3.13/1.66 => ( woz2B
% 3.13/1.66 => ( image1Ex
% 3.13/1.66 => ( image1Ex1
% 3.13/1.66 => ( image1Equiv
% 3.13/1.66 => ( image1E
% 3.13/1.66 => ( image1I
% 3.13/1.66 => ( injFuncInInjFuncSet
% 3.13/1.66 => ( injFuncSetFuncIn
% 3.13/1.66 => ( injFuncSetFuncInj
% 3.13/1.66 => ( surjFuncSetFuncIn
% 3.13/1.66 => ( surjFuncSetFuncSurj
% 3.13/1.66 => ( leftInvIsSurj
% 3.13/1.66 => ( surjCantorThm
% 3.13/1.66 => ( foundation2
% 3.13/1.66 => ( notinself
% 3.13/1.66 => ( notinself2
% 3.13/1.66 => ( omegaSp
% 3.13/1.66 => ( omegaSclos
% 3.13/1.66 => ( peano0notS
% 3.13/1.66 => ( peanoSinj
% 3.13/1.66 => ! [X: $i] :
% 3.13/1.66 ( ( transitiveset @ X )
% 3.13/1.66 => ! [A: $i] :
% 3.13/1.66 ( ( in @ A @ X )
% 3.13/1.66 => ( subset @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.13/1.66 ------------------------
% 3.20/1.70 % (16886)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_16671 for (2999ds/2Mi)
% 3.20/1.70 % (16886)Instruction limit reached!
% 3.20/1.70 % (16886)------------------------------
% 3.20/1.70 % (16886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.70 % (16886)Termination reason: Unknown
% 3.20/1.70 % (16886)Termination phase: shuffling
% 3.20/1.70
% 3.20/1.70 % (16886)Memory used [KB]: 1663
% 3.20/1.70 % (16886)Time elapsed: 0.003 s
% 3.20/1.70 % (16886)Instructions burned: 3 (million)
% 3.20/1.70 % (16886)------------------------------
% 3.20/1.70 % (16886)------------------------------
% 3.20/1.70 % (16882)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/183Mi)
% 3.20/1.70 % (16883)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_16671 for (2999ds/4Mi)
% 3.20/1.70 % (16884)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_16671 for (2999ds/27Mi)
% 3.20/1.70 % (16887)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_16671 for (2999ds/275Mi)
% 3.20/1.70 % (16888)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/18Mi)
% 3.20/1.70 % (16885)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/2Mi)
% 3.20/1.70 % (16885)Instruction limit reached!
% 3.20/1.70 % (16885)------------------------------
% 3.20/1.70 % (16885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.70 % (16885)Termination reason: Unknown
% 3.20/1.70 % (16885)Termination phase: shuffling
% 3.20/1.70
% 3.20/1.70 % (16885)Memory used [KB]: 1663
% 3.20/1.70 % (16885)Time elapsed: 0.003 s
% 3.20/1.70 % (16885)Instructions burned: 2 (million)
% 3.20/1.70 % (16885)------------------------------
% 3.20/1.70 % (16885)------------------------------
% 3.20/1.70 % (16883)Instruction limit reached!
% 3.20/1.70 % (16883)------------------------------
% 3.20/1.70 % (16883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.70 % (16883)Termination reason: Unknown
% 3.20/1.70 % (16883)Termination phase: shuffling
% 3.20/1.70
% 3.20/1.70 % (16883)Memory used [KB]: 1663
% 3.20/1.70 % (16883)Time elapsed: 0.005 s
% 3.20/1.70 % (16883)Instructions burned: 5 (million)
% 3.20/1.70 % (16883)------------------------------
% 3.20/1.70 % (16883)------------------------------
% 3.20/1.71 % (16889)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/3Mi)
% 3.20/1.71 % (16888)Instruction limit reached!
% 3.20/1.71 % (16888)------------------------------
% 3.20/1.71 % (16888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.71 % (16888)Termination reason: Unknown
% 3.20/1.71 % (16888)Termination phase: shuffling
% 3.20/1.71
% 3.20/1.71 % (16888)Memory used [KB]: 1918
% 3.20/1.71 % (16888)Time elapsed: 0.014 s
% 3.20/1.71 % (16888)Instructions burned: 19 (million)
% 3.20/1.71 % (16888)------------------------------
% 3.20/1.71 % (16888)------------------------------
% 3.20/1.71 % (16889)Instruction limit reached!
% 3.20/1.71 % (16889)------------------------------
% 3.20/1.71 % (16889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.71 % (16889)Termination reason: Unknown
% 3.20/1.71 % (16889)Termination phase: shuffling
% 3.20/1.71
% 3.20/1.71 % (16889)Memory used [KB]: 1663
% 3.20/1.71 % (16889)Time elapsed: 0.003 s
% 3.20/1.71 % (16889)Instructions burned: 5 (million)
% 3.20/1.71 % (16889)------------------------------
% 3.20/1.71 % (16889)------------------------------
% 3.20/1.71 % (16884)Instruction limit reached!
% 3.20/1.71 % (16884)------------------------------
% 3.20/1.71 % (16884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.71 % (16884)Termination reason: Unknown
% 3.20/1.71 % (16884)Termination phase: shuffling
% 3.20/1.71
% 3.20/1.71 % (16884)Memory used [KB]: 2046
% 3.20/1.71 % (16884)Time elapsed: 0.015 s
% 3.20/1.71 % (16884)Instructions burned: 27 (million)
% 3.20/1.71 % (16884)------------------------------
% 3.20/1.71 % (16884)------------------------------
% 3.20/1.72 % (16890)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_16671 for (2999ds/37Mi)
% 3.20/1.72 % (16891)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_16671 for (2999ds/15Mi)
% 3.20/1.72 % (16893)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_16671 for (2999ds/1041Mi)
% 3.20/1.73 % (16894)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/7Mi)
% 3.20/1.73 % (16892)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_16671 for (2999ds/3Mi)
% 3.20/1.73 % (16891)Instruction limit reached!
% 3.20/1.73 % (16891)------------------------------
% 3.20/1.73 % (16891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.73 % (16891)Termination reason: Unknown
% 3.20/1.73 % (16891)Termination phase: shuffling
% 3.20/1.73
% 3.20/1.73 % (16891)Memory used [KB]: 1918
% 3.20/1.73 % (16891)Time elapsed: 0.012 s
% 3.20/1.73 % (16891)Instructions burned: 16 (million)
% 3.20/1.73 % (16891)------------------------------
% 3.20/1.73 % (16891)------------------------------
% 3.20/1.73 % (16892)Instruction limit reached!
% 3.20/1.73 % (16892)------------------------------
% 3.20/1.73 % (16892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.73 % (16892)Termination reason: Unknown
% 3.20/1.73 % (16892)Termination phase: shuffling
% 3.20/1.73
% 3.20/1.73 % (16892)Memory used [KB]: 1663
% 3.20/1.73 % (16892)Time elapsed: 0.004 s
% 3.20/1.73 % (16892)Instructions burned: 3 (million)
% 3.20/1.73 % (16892)------------------------------
% 3.20/1.73 % (16892)------------------------------
% 3.20/1.73 % (16894)Instruction limit reached!
% 3.20/1.73 % (16894)------------------------------
% 3.20/1.73 % (16894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.73 % (16894)Termination reason: Unknown
% 3.20/1.73 % (16894)Termination phase: shuffling
% 3.20/1.73
% 3.20/1.73 % (16894)Memory used [KB]: 1791
% 3.20/1.73 % (16894)Time elapsed: 0.004 s
% 3.20/1.73 % (16894)Instructions burned: 10 (million)
% 3.20/1.73 % (16894)------------------------------
% 3.20/1.73 % (16894)------------------------------
% 3.20/1.74 % (16890)Instruction limit reached!
% 3.20/1.74 % (16890)------------------------------
% 3.20/1.74 % (16890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.74 % (16890)Termination reason: Unknown
% 3.20/1.74 % (16890)Termination phase: shuffling
% 3.20/1.74
% 3.20/1.74 % (16890)Memory used [KB]: 2430
% 3.20/1.74 % (16890)Time elapsed: 0.021 s
% 3.20/1.74 % (16890)Instructions burned: 37 (million)
% 3.20/1.74 % (16890)------------------------------
% 3.20/1.74 % (16890)------------------------------
% 3.20/1.74 % (16897)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_16671 for (2999ds/3Mi)
% 3.20/1.74 % (16897)Instruction limit reached!
% 3.20/1.74 % (16897)------------------------------
% 3.20/1.74 % (16897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.74 % (16897)Termination reason: Unknown
% 3.20/1.74 % (16897)Termination phase: shuffling
% 3.20/1.74
% 3.20/1.74 % (16897)Memory used [KB]: 1663
% 3.20/1.74 % (16897)Time elapsed: 0.003 s
% 3.20/1.74 % (16897)Instructions burned: 6 (million)
% 3.20/1.74 % (16897)------------------------------
% 3.20/1.74 % (16897)------------------------------
% 3.20/1.75 % (16895)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/16Mi)
% 3.20/1.75 % (16896)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_16671 for (2999ds/3Mi)
% 3.20/1.75 % (16896)Instruction limit reached!
% 3.20/1.75 % (16896)------------------------------
% 3.20/1.75 % (16896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.75 % (16896)Termination reason: Unknown
% 3.20/1.75 % (16896)Termination phase: shuffling
% 3.20/1.75
% 3.20/1.75 % (16896)Memory used [KB]: 1663
% 3.20/1.75 % (16896)Time elapsed: 0.004 s
% 3.20/1.75 % (16896)Instructions burned: 5 (million)
% 3.20/1.75 % (16896)------------------------------
% 3.20/1.75 % (16896)------------------------------
% 3.20/1.75 % (16895)Instruction limit reached!
% 3.20/1.75 % (16895)------------------------------
% 3.20/1.75 % (16895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.75 % (16895)Termination reason: Unknown
% 3.20/1.75 % (16895)Termination phase: shuffling
% 3.20/1.75
% 3.20/1.75 % (16895)Memory used [KB]: 1918
% 3.20/1.75 % (16895)Time elapsed: 0.008 s
% 3.20/1.75 % (16895)Instructions burned: 18 (million)
% 3.20/1.75 % (16895)------------------------------
% 3.20/1.75 % (16895)------------------------------
% 3.20/1.75 % (16899)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/3Mi)
% 3.20/1.76 % (16899)Instruction limit reached!
% 3.20/1.76 % (16899)------------------------------
% 3.20/1.76 % (16899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.76 % (16899)Termination reason: Unknown
% 3.20/1.76 % (16899)Termination phase: shuffling
% 3.20/1.76
% 3.20/1.76 % (16899)Memory used [KB]: 1663
% 3.20/1.76 % (16899)Time elapsed: 0.002 s
% 3.20/1.76 % (16899)Instructions burned: 4 (million)
% 3.20/1.76 % (16899)------------------------------
% 3.20/1.76 % (16899)------------------------------
% 3.20/1.76 % (16898)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/7Mi)
% 3.20/1.76 % (16882)Instruction limit reached!
% 3.20/1.76 % (16882)------------------------------
% 3.20/1.76 % (16882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.76 % (16882)Termination reason: Unknown
% 3.20/1.76 % (16882)Termination phase: Property scanning
% 3.20/1.76
% 3.20/1.76 % (16882)Memory used [KB]: 3070
% 3.20/1.76 % (16882)Time elapsed: 0.059 s
% 3.20/1.76 % (16882)Instructions burned: 184 (million)
% 3.20/1.76 % (16882)------------------------------
% 3.20/1.76 % (16882)------------------------------
% 3.20/1.76 % (16898)Instruction limit reached!
% 3.20/1.76 % (16898)------------------------------
% 3.20/1.76 % (16898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.76 % (16898)Termination reason: Unknown
% 3.20/1.76 % (16898)Termination phase: shuffling
% 3.20/1.76
% 3.20/1.76 % (16898)Memory used [KB]: 1663
% 3.20/1.76 % (16898)Time elapsed: 0.006 s
% 3.20/1.76 % (16898)Instructions burned: 7 (million)
% 3.20/1.76 % (16898)------------------------------
% 3.20/1.76 % (16898)------------------------------
% 3.20/1.76 % (16901)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/18Mi)
% 3.20/1.76 % (16900)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/4Mi)
% 3.20/1.76 % (16900)Instruction limit reached!
% 3.20/1.76 % (16900)------------------------------
% 3.20/1.76 % (16900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.76 % (16900)Termination reason: Unknown
% 3.20/1.76 % (16900)Termination phase: shuffling
% 3.20/1.76
% 3.20/1.76 % (16900)Memory used [KB]: 1663
% 3.20/1.76 % (16900)Time elapsed: 0.002 s
% 3.20/1.76 % (16900)Instructions burned: 4 (million)
% 3.20/1.76 % (16900)------------------------------
% 3.20/1.76 % (16900)------------------------------
% 3.20/1.76 % (16902)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/710Mi)
% 3.20/1.77 % (16903)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/6Mi)
% 3.20/1.77 % (16901)Instruction limit reached!
% 3.20/1.77 % (16901)------------------------------
% 3.20/1.77 % (16901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.77 % (16901)Termination reason: Unknown
% 3.20/1.77 % (16901)Termination phase: shuffling
% 3.20/1.77
% 3.20/1.77 % (16901)Memory used [KB]: 1918
% 3.20/1.77 % (16901)Time elapsed: 0.007 s
% 3.20/1.77 % (16901)Instructions burned: 21 (million)
% 3.20/1.77 % (16901)------------------------------
% 3.20/1.77 % (16901)------------------------------
% 3.20/1.77 % (16903)Instruction limit reached!
% 3.20/1.77 % (16903)------------------------------
% 3.20/1.77 % (16903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.77 % (16903)Termination reason: Unknown
% 3.20/1.77 % (16903)Termination phase: shuffling
% 3.20/1.77
% 3.20/1.77 % (16903)Memory used [KB]: 1663
% 3.20/1.77 % (16903)Time elapsed: 0.003 s
% 3.20/1.77 % (16903)Instructions burned: 7 (million)
% 3.20/1.77 % (16903)------------------------------
% 3.20/1.77 % (16903)------------------------------
% 3.20/1.77 % (16905)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_16671 for (2999ds/21Mi)
% 3.20/1.78 % (16904)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_16671 for (2999ds/902Mi)
% 3.20/1.78 % (16906)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_16671 for (2999ds/5Mi)
% 3.20/1.78 % (16905)Instruction limit reached!
% 3.20/1.78 % (16905)------------------------------
% 3.20/1.78 % (16905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.78 % (16905)Termination reason: Unknown
% 3.20/1.78 % (16905)Termination phase: shuffling
% 3.20/1.78
% 3.20/1.78 % (16905)Memory used [KB]: 1918
% 3.20/1.78 % (16905)Time elapsed: 0.007 s
% 3.20/1.78 % (16905)Instructions burned: 21 (million)
% 3.20/1.78 % (16905)------------------------------
% 3.20/1.78 % (16905)------------------------------
% 3.20/1.78 % (16907)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_16671 for (2999ds/6Mi)
% 3.20/1.78 % (16906)Instruction limit reached!
% 3.20/1.78 % (16906)------------------------------
% 3.20/1.78 % (16906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.78 % (16906)Termination reason: Unknown
% 3.20/1.78 % (16906)Termination phase: shuffling
% 3.20/1.78
% 3.20/1.78 % (16906)Memory used [KB]: 1663
% 3.20/1.78 % (16906)Time elapsed: 0.003 s
% 3.20/1.78 % (16906)Instructions burned: 7 (million)
% 3.20/1.78 % (16906)------------------------------
% 3.20/1.78 % (16906)------------------------------
% 3.20/1.78 % (16907)Instruction limit reached!
% 3.20/1.78 % (16907)------------------------------
% 3.20/1.78 % (16907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.78 % (16907)Termination reason: Unknown
% 3.20/1.78 % (16907)Termination phase: shuffling
% 3.20/1.78
% 3.20/1.78 % (16907)Memory used [KB]: 1663
% 3.20/1.78 % (16907)Time elapsed: 0.003 s
% 3.20/1.78 % (16907)Instructions burned: 7 (million)
% 3.20/1.78 % (16907)------------------------------
% 3.20/1.78 % (16907)------------------------------
% 3.20/1.79 % (16908)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_16671 for (2998ds/377Mi)
% 3.20/1.79 % (16909)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_16671 for (2998ds/779Mi)
% 3.20/1.79 % (16910)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_16671 for (2998ds/19Mi)
% 3.20/1.80 % (16910)Instruction limit reached!
% 3.20/1.80 % (16910)------------------------------
% 3.20/1.80 % (16910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.80 % (16910)Termination reason: Unknown
% 3.20/1.80 % (16910)Termination phase: shuffling
% 3.20/1.80
% 3.20/1.80 % (16910)Memory used [KB]: 1918
% 3.20/1.80 % (16910)Time elapsed: 0.007 s
% 3.20/1.80 % (16910)Instructions burned: 21 (million)
% 3.20/1.80 % (16910)------------------------------
% 3.20/1.80 % (16910)------------------------------
% 3.20/1.80 % (16887)Instruction limit reached!
% 3.20/1.80 % (16887)------------------------------
% 3.20/1.80 % (16887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.80 % (16887)Termination reason: Unknown
% 3.20/1.80 % (16887)Termination phase: Property scanning
% 3.20/1.80
% 3.20/1.80 % (16887)Memory used [KB]: 6012
% 3.20/1.80 % (16887)Time elapsed: 0.103 s
% 3.20/1.80 % (16887)Instructions burned: 277 (million)
% 3.20/1.80 % (16887)------------------------------
% 3.20/1.80 % (16887)------------------------------
% 3.20/1.81 % (16911)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_16671 for (2998ds/879Mi)
% 3.20/1.81 % (16912)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_16671 for (2998ds/17Mi)
% 3.20/1.82 % (16912)Instruction limit reached!
% 3.20/1.82 % (16912)------------------------------
% 3.20/1.82 % (16912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.82 % (16912)Termination reason: Unknown
% 3.20/1.82 % (16912)Termination phase: shuffling
% 3.20/1.82
% 3.20/1.82 % (16912)Memory used [KB]: 1918
% 3.20/1.82 % (16912)Time elapsed: 0.006 s
% 3.20/1.82 % (16912)Instructions burned: 17 (million)
% 3.20/1.82 % (16912)------------------------------
% 3.20/1.82 % (16912)------------------------------
% 3.20/1.83 % (16913)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_16671 for (2998ds/3Mi)
% 3.20/1.83 % (16913)Instruction limit reached!
% 3.20/1.83 % (16913)------------------------------
% 3.20/1.83 % (16913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.83 % (16913)Termination reason: Unknown
% 3.20/1.83 % (16913)Termination phase: shuffling
% 3.20/1.83
% 3.20/1.83 % (16913)Memory used [KB]: 1663
% 3.20/1.83 % (16913)Time elapsed: 0.002 s
% 3.20/1.83 % (16913)Instructions burned: 3 (million)
% 3.20/1.83 % (16913)------------------------------
% 3.20/1.83 % (16913)------------------------------
% 3.20/1.84 % (16914)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_16671 for (2998ds/30Mi)
% 3.20/1.85 % (16914)Instruction limit reached!
% 3.20/1.85 % (16914)------------------------------
% 3.20/1.85 % (16914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.85 % (16914)Termination reason: Unknown
% 3.20/1.85 % (16914)Termination phase: shuffling
% 3.20/1.85
% 3.20/1.85 % (16914)Memory used [KB]: 2174
% 3.20/1.85 % (16914)Time elapsed: 0.010 s
% 3.20/1.85 % (16914)Instructions burned: 30 (million)
% 3.20/1.85 % (16914)------------------------------
% 3.20/1.85 % (16914)------------------------------
% 3.20/1.86 % (16915)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_16671 for (2998ds/127Mi)
% 3.20/1.89 % (16908)Instruction limit reached!
% 3.20/1.89 % (16908)------------------------------
% 3.20/1.89 % (16908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.89 % (16908)Termination reason: Unknown
% 3.20/1.89 % (16908)Termination phase: Saturation
% 3.20/1.89
% 3.20/1.89 % (16908)Memory used [KB]: 9466
% 3.20/1.89 % (16908)Time elapsed: 0.101 s
% 3.20/1.89 % (16908)Instructions burned: 380 (million)
% 3.20/1.89 % (16908)------------------------------
% 3.20/1.89 % (16908)------------------------------
% 3.20/1.90 % (16915)Instruction limit reached!
% 3.20/1.90 % (16915)------------------------------
% 3.20/1.90 % (16915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.90 % (16915)Termination reason: Unknown
% 3.20/1.90 % (16915)Termination phase: Property scanning
% 3.20/1.90
% 3.20/1.90 % (16915)Memory used [KB]: 2942
% 3.20/1.90 % (16915)Time elapsed: 0.037 s
% 3.20/1.90 % (16915)Instructions burned: 130 (million)
% 3.20/1.90 % (16915)------------------------------
% 3.20/1.90 % (16915)------------------------------
% 3.20/1.90 % (16916)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_16671 for (2997ds/100Mi)
% 3.20/1.91 % (16917)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_16671 for (2997ds/3Mi)
% 3.20/1.91 % (16917)Instruction limit reached!
% 3.20/1.91 % (16917)------------------------------
% 3.20/1.91 % (16917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.91 % (16917)Termination reason: Unknown
% 3.20/1.91 % (16917)Termination phase: shuffling
% 3.20/1.91
% 3.20/1.91 % (16917)Memory used [KB]: 1663
% 3.20/1.91 % (16917)Time elapsed: 0.002 s
% 3.20/1.91 % (16917)Instructions burned: 3 (million)
% 3.20/1.91 % (16917)------------------------------
% 3.20/1.91 % (16917)------------------------------
% 3.20/1.92 % (16918)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_16671 for (2997ds/20Mi)
% 3.20/1.92 % (16918)Instruction limit reached!
% 3.20/1.92 % (16918)------------------------------
% 3.20/1.92 % (16918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.92 % (16918)Termination reason: Unknown
% 3.20/1.92 % (16918)Termination phase: shuffling
% 3.20/1.92
% 3.20/1.92 % (16918)Memory used [KB]: 1918
% 3.20/1.92 % (16918)Time elapsed: 0.009 s
% 3.20/1.92 % (16918)Instructions burned: 20 (million)
% 3.20/1.92 % (16918)------------------------------
% 3.20/1.92 % (16918)------------------------------
% 3.20/1.93 % (16916)Instruction limit reached!
% 3.20/1.93 % (16916)------------------------------
% 3.20/1.93 % (16916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.93 % (16916)Termination reason: Unknown
% 3.20/1.93 % (16916)Termination phase: Property scanning
% 3.20/1.93
% 3.20/1.93 % (16916)Memory used [KB]: 2814
% 3.20/1.93 % (16916)Time elapsed: 0.027 s
% 3.20/1.93 % (16916)Instructions burned: 100 (million)
% 3.20/1.93 % (16916)------------------------------
% 3.20/1.93 % (16916)------------------------------
% 3.20/1.93 % (16919)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_16671 for (2997ds/86Mi)
% 3.20/1.94 % (16920)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_16671 for (2997ds/107Mi)
% 3.20/1.96 % (16919)Instruction limit reached!
% 3.20/1.96 % (16919)------------------------------
% 3.20/1.96 % (16919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.96 % (16919)Termination reason: Unknown
% 3.20/1.96 % (16919)Termination phase: Property scanning
% 3.20/1.96
% 3.20/1.96 % (16919)Memory used [KB]: 2814
% 3.20/1.96 % (16919)Time elapsed: 0.026 s
% 3.20/1.96 % (16919)Instructions burned: 87 (million)
% 3.20/1.96 % (16919)------------------------------
% 3.20/1.96 % (16919)------------------------------
% 3.20/1.96 % (16920)Instruction limit reached!
% 3.20/1.96 % (16920)------------------------------
% 3.20/1.96 % (16920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.96 % (16920)Termination reason: Unknown
% 3.20/1.96 % (16920)Termination phase: SInE selection
% 3.20/1.96
% 3.20/1.96 % (16920)Memory used [KB]: 2814
% 3.20/1.96 % (16920)Time elapsed: 0.030 s
% 3.20/1.96 % (16920)Instructions burned: 107 (million)
% 3.20/1.96 % (16920)------------------------------
% 3.20/1.96 % (16920)------------------------------
% 3.20/1.97 % (16921)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_16671 for (2997ds/39Mi)
% 3.20/1.97 % (16922)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_16671 for (2997ds/448Mi)
% 3.20/1.98 % (16902)Instruction limit reached!
% 3.20/1.98 % (16902)------------------------------
% 3.20/1.98 % (16902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.98 % (16902)Termination reason: Unknown
% 3.20/1.98 % (16902)Termination phase: Saturation
% 3.20/1.98
% 3.20/1.98 % (16902)Memory used [KB]: 11769
% 3.20/1.98 % (16902)Time elapsed: 0.217 s
% 3.20/1.98 % (16902)Instructions burned: 710 (million)
% 3.20/1.98 % (16902)------------------------------
% 3.20/1.98 % (16902)------------------------------
% 3.20/1.98 % (16921)Instruction limit reached!
% 3.20/1.98 % (16921)------------------------------
% 3.20/1.98 % (16921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.98 % (16921)Termination reason: Unknown
% 3.20/1.98 % (16921)Termination phase: shuffling
% 3.20/1.98
% 3.20/1.98 % (16921)Memory used [KB]: 2430
% 3.20/1.98 % (16921)Time elapsed: 0.013 s
% 3.20/1.98 % (16921)Instructions burned: 41 (million)
% 3.20/1.98 % (16921)------------------------------
% 3.20/1.98 % (16921)------------------------------
% 3.20/1.99 % (16893)Instruction limit reached!
% 3.20/1.99 % (16893)------------------------------
% 3.20/1.99 % (16893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/1.99 % (16893)Termination reason: Unknown
% 3.20/1.99 % (16893)Termination phase: Saturation
% 3.20/1.99
% 3.20/1.99 % (16893)Memory used [KB]: 18933
% 3.20/1.99 % (16893)Time elapsed: 0.266 s
% 3.20/1.99 % (16893)Instructions burned: 1042 (million)
% 3.20/1.99 % (16893)------------------------------
% 3.20/1.99 % (16893)------------------------------
% 3.20/1.99 % (16923)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_16671 for (2997ds/46Mi)
% 3.20/1.99 % (16924)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_16671 for (2996ds/98Mi)
% 3.20/2.00 % (16925)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_16671 for (2996ds/507Mi)
% 3.20/2.00 % (16909)Instruction limit reached!
% 3.20/2.00 % (16909)------------------------------
% 3.20/2.00 % (16909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/2.00 % (16909)Termination reason: Unknown
% 3.20/2.00 % (16909)Termination phase: Saturation
% 3.20/2.00
% 3.20/2.00 % (16909)Memory used [KB]: 7931
% 3.20/2.00 % (16909)Time elapsed: 0.211 s
% 3.20/2.00 % (16909)Instructions burned: 781 (million)
% 3.20/2.00 % (16909)------------------------------
% 3.20/2.00 % (16909)------------------------------
% 3.20/2.00 % (16923)Instruction limit reached!
% 3.20/2.00 % (16923)------------------------------
% 3.20/2.00 % (16923)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.20/2.00 % (16923)Termination reason: Unknown
% 3.20/2.00 % (16923)Termination phase: shuffling
% 3.20/2.00
% 3.20/2.00 % (16923)Memory used [KB]: 2558
% 3.20/2.00 % (16923)Time elapsed: 0.014 s
% 3.20/2.00 % (16923)Instructions burned: 47 (million)
% 3.20/2.00 % (16923)------------------------------
% 3.20/2.00 % (16923)------------------------------
% 5.44/2.01 % (16926)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_16671 for (2996ds/149Mi)
% 5.44/2.01 % (16927)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_16671 for (2996ds/39Mi)
% 5.44/2.02 % (16924)Instruction limit reached!
% 5.44/2.02 % (16924)------------------------------
% 5.44/2.02 % (16924)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.44/2.02 % (16924)Termination reason: Unknown
% 5.44/2.02 % (16924)Termination phase: Property scanning
% 5.44/2.02
% 5.44/2.02 % (16924)Memory used [KB]: 2814
% 5.44/2.02 % (16924)Time elapsed: 0.026 s
% 5.44/2.02 % (16924)Instructions burned: 99 (million)
% 5.44/2.02 % (16924)------------------------------
% 5.44/2.02 % (16924)------------------------------
% 5.44/2.02 % (16927)Instruction limit reached!
% 5.44/2.02 % (16927)------------------------------
% 5.44/2.02 % (16927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.44/2.02 % (16927)Termination reason: Unknown
% 5.44/2.02 % (16927)Termination phase: shuffling
% 5.44/2.02
% 5.44/2.02 % (16927)Memory used [KB]: 2430
% 5.44/2.02 % (16927)Time elapsed: 0.012 s
% 5.44/2.02 % (16927)Instructions burned: 40 (million)
% 5.44/2.02 % (16927)------------------------------
% 5.44/2.02 % (16927)------------------------------
% 5.44/2.03 % (16928)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_16671 for (2996ds/6Mi)
% 5.44/2.03 % (16928)Instruction limit reached!
% 5.44/2.03 % (16928)------------------------------
% 5.44/2.03 % (16928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.44/2.03 % (16928)Termination reason: Unknown
% 5.44/2.03 % (16928)Termination phase: shuffling
% 5.44/2.03
% 5.44/2.03 % (16928)Memory used [KB]: 1663
% 5.44/2.03 % (16928)Time elapsed: 0.003 s
% 5.44/2.03 % (16928)Instructions burned: 7 (million)
% 5.44/2.03 % (16928)------------------------------
% 5.44/2.03 % (16928)------------------------------
% 5.44/2.03 % (16929)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_16671 for (2996ds/1087Mi)
% 5.44/2.04 % (16930)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_16671 for (2996ds/47Mi)
% 5.44/2.05 % (16930)Instruction limit reached!
% 5.44/2.05 % (16930)------------------------------
% 5.44/2.05 % (16930)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.44/2.05 % (16930)Termination reason: Unknown
% 5.44/2.05 % (16930)Termination phase: shuffling
% 5.44/2.05
% 5.44/2.05 % (16930)Memory used [KB]: 2558
% 5.44/2.05 % (16930)Time elapsed: 0.014 s
% 5.44/2.05 % (16930)Instructions burned: 48 (million)
% 5.44/2.05 % (16930)------------------------------
% 5.44/2.05 % (16930)------------------------------
% 5.44/2.05 % (16926)Instruction limit reached!
% 5.44/2.05 % (16926)------------------------------
% 5.44/2.05 % (16926)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.44/2.05 % (16926)Termination reason: Unknown
% 5.44/2.05 % (16926)Termination phase: Function definition elimination
% 5.44/2.05
% 5.44/2.05 % (16926)Memory used [KB]: 3070
% 5.44/2.05 % (16926)Time elapsed: 0.043 s
% 5.44/2.05 % (16926)Instructions burned: 151 (million)
% 5.44/2.05 % (16926)------------------------------
% 5.44/2.05 % (16926)------------------------------
% 5.44/2.06 % (16931)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_16671 for (2996ds/44Mi)
% 5.44/2.07 % (16932)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_16671 for (2996ds/432Mi)
% 5.44/2.07 % (16931)Instruction limit reached!
% 5.44/2.07 % (16931)------------------------------
% 5.44/2.07 % (16931)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.44/2.07 % (16931)Termination reason: Unknown
% 5.44/2.07 % (16931)Termination phase: shuffling
% 5.44/2.07
% 5.44/2.07 % (16931)Memory used [KB]: 2430
% 5.44/2.07 % (16931)Time elapsed: 0.013 s
% 5.44/2.07 % (16931)Instructions burned: 45 (million)
% 5.44/2.07 % (16931)------------------------------
% 5.44/2.07 % (16931)------------------------------
% 5.44/2.08 % (16922)Instruction limit reached!
% 5.44/2.08 % (16922)------------------------------
% 5.44/2.08 % (16922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.44/2.08 % (16922)Termination reason: Unknown
% 5.44/2.08 % (16922)Termination phase: Saturation
% 5.44/2.08
% 5.44/2.08 % (16922)Memory used [KB]: 7803
% 5.44/2.08 % (16922)Time elapsed: 0.110 s
% 5.44/2.08 % (16922)Instructions burned: 448 (million)
% 5.44/2.08 % (16922)------------------------------
% 5.44/2.08 % (16922)------------------------------
% 5.44/2.08 % (16933)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_16671 for (2995ds/62Mi)
% 5.44/2.08 % (16911)Instruction limit reached!
% 5.44/2.08 % (16911)------------------------------
% 5.44/2.08 % (16911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.44/2.08 % (16911)Termination reason: Unknown
% 5.44/2.08 % (16911)Termination phase: Saturation
% 5.44/2.08
% 5.44/2.08 % (16911)Memory used [KB]: 10746
% 5.44/2.08 % (16911)Time elapsed: 0.274 s
% 5.44/2.08 % (16911)Instructions burned: 880 (million)
% 5.44/2.08 % (16911)------------------------------
% 5.44/2.08 % (16911)------------------------------
% 5.44/2.08 % (16925)First to succeed.
% 5.44/2.09 % (16925)Refutation found. Thanks to Tanya!
% 5.44/2.09 % SZS status Theorem for DTF2THF_16671
% 5.44/2.09 % SZS output start Proof for DTF2THF_16671
% 5.44/2.09 thf(func_def_0, type, in: $i > $i > $o).
% 5.44/2.09 thf(func_def_1, type, exu: ($i > $o) > $o).
% 5.44/2.09 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 5.44/2.09 thf(func_def_8, type, powerset: $i > $i).
% 5.44/2.09 thf(func_def_10, type, setunion: $i > $i).
% 5.44/2.09 thf(func_def_19, type, descr: ($i > $o) > $i).
% 5.44/2.09 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 5.44/2.09 thf(func_def_26, type, prop2set: $o > $i).
% 5.44/2.09 thf(func_def_36, type, nonempty: $i > $o).
% 5.44/2.09 thf(func_def_69, type, set2prop: $i > $o).
% 5.44/2.09 thf(func_def_88, type, subset: $i > $i > $o).
% 5.44/2.09 thf(func_def_89, type, disjoint: $i > $i > $o).
% 5.44/2.09 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 5.44/2.09 thf(func_def_114, type, binunion: $i > $i > $i).
% 5.44/2.09 thf(func_def_122, type, binintersect: $i > $i > $i).
% 5.44/2.09 thf(func_def_135, type, regular: $i > $o).
% 5.44/2.09 thf(func_def_136, type, setminus: $i > $i > $i).
% 5.44/2.09 thf(func_def_147, type, symdiff: $i > $i > $i).
% 5.44/2.09 thf(func_def_153, type, iskpair: $i > $o).
% 5.44/2.09 thf(func_def_158, type, kpair: $i > $i > $i).
% 5.44/2.09 thf(func_def_160, type, cartprod: $i > $i > $i).
% 5.44/2.09 thf(func_def_177, type, singleton: $i > $o).
% 5.44/2.09 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 5.44/2.09 thf(func_def_184, type, atmost1p: $i > $o).
% 5.44/2.09 thf(func_def_185, type, atleast2p: $i > $o).
% 5.44/2.09 thf(func_def_186, type, atmost2p: $i > $o).
% 5.44/2.09 thf(func_def_187, type, upairsetp: $i > $o).
% 5.44/2.09 thf(func_def_191, type, kfst: $i > $i).
% 5.44/2.09 thf(func_def_203, type, ksnd: $i > $i).
% 5.44/2.09 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 5.44/2.09 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 5.44/2.09 thf(func_def_222, type, func: $i > $i > $i > $o).
% 5.44/2.09 thf(func_def_223, type, funcSet: $i > $i > $i).
% 5.44/2.09 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 5.44/2.09 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 5.44/2.09 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 5.44/2.09 thf(func_def_323, type, breln1: $i > $i > $o).
% 5.44/2.09 thf(func_def_325, type, breln1Set: $i > $i).
% 5.44/2.09 thf(func_def_327, type, transitive: $i > $i > $o).
% 5.44/2.09 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 5.44/2.09 thf(func_def_329, type, reflexive: $i > $i > $o).
% 5.44/2.09 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 5.44/2.09 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 5.44/2.09 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 5.44/2.09 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 5.44/2.09 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 5.44/2.09 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 5.44/2.09 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 5.44/2.09 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 5.44/2.09 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 5.44/2.09 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 5.44/2.09 thf(func_def_378, type, omegaS: $i > $i).
% 5.44/2.09 thf(func_def_382, type, peano3: $i > $i > $i > $o).
% 5.44/2.09 thf(func_def_383, type, peano4: $i > $i > $i > $o).
% 5.44/2.09 thf(func_def_384, type, peano5: $i > $i > $i > $o).
% 5.44/2.09 thf(func_def_386, type, peano: $i > $i > $i > $o).
% 5.44/2.09 thf(func_def_387, type, transitiveset: $i > $o).
% 5.44/2.09 thf(func_def_406, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.44/2.09 thf(func_def_407, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 5.44/2.09 thf(func_def_411, type, ph5: !>[X0: $tType]:(X0)).
% 5.44/2.09 thf(f2422,plain,(
% 5.44/2.09 $false),
% 5.44/2.09 inference(subsumption_resolution,[status(thm)],[f2421,f1483])).
% 5.44/2.09 thf(f1483,plain,(
% 5.44/2.09 ((subset @ sK3 @ sK2) != $true)),
% 5.44/2.09 inference(cnf_transformation,[status(thm)],[f1016])).
% 5.44/2.09 thf(f1016,plain,(
% 5.44/2.09 (foundation2 = $true) & (notinself2 = $true) & (setextT = $true) & (cartprodfstin = $true) & (emptyset__Cong = $true) & (woz1_1 = $true) & (breln1compprop = $true) & (setadjoinE = $true) & (breln1unionIL = $true) & (notinsingleton = $true) & (lam2p = $true) & (binunionTIRcontra = $true) & (infuncsetfunc = $true) & (demorgan2a2 = $true) & (binunionT_lem = $true) & (setOfPairsIsBReln = $true) & (setminusSubset2 = $true) & (setext = $true) & (symdiffI2 = $true) & (omegaIndAx = $true) & (setukpairIR = $true) & (eta2 = $true) & (injFuncSetFuncInj = $true) & (binintersectSubset5 = $true) & (setukpairinjR2 = $true) & (setminusELneg = $true) & (beta2 = $true) & (emptysetimpfalse = $true) & (eta1 = $true) & (contraSubsetComplement = $true) & (nonemptyE1 = $true) & (singletonsubset = $true) & (exuI1 = $true) & (((transitiveset @ sK2) = $true) & (((in @ sK3 @ sK2) = $true) & ((subset @ sK3 @ sK2) != $true))) & (breln1unionE = $true) & (powersetTE1 = $true) & (woz1_3 = $true) & (cartprodpairsurjEq = $true) & (setadjoinSub2 = $true) & (funcGraphProp1 = $true) & (sepInPowerset = $true) & (cartprodmempair1 = $true) & (brelnall2 = $true) & (ap2p = $true) & (powerset__Cong = $true) & (ex1E2 = $true) & (in__Cong = $true) & (binunionTE = $true) & (setunionsingleton2 = $true) & (app = $true) & (setukpairinjR = $true) & (emptyI = $true) & (ex1I = $true) & (setminusSubset1 = $true) & (demorgan1 = $true) & (woz2A = $true) & (setukpairinjR1 = $true) & (setukpairinjL1 = $true) & (binintersectI = $true) & (powersetT_lem = $true) & (ap2apEq1 = $true) & (demorgan2a = $true) & (binintersectTERcontra = $true) & (eqinunit = $true) & (subsetI2 = $true) & (lamProp = $true) & (setadjoinIR = $true) & (omegaSp = $true) & (setadjoinOr = $true) & (sepSubset = $true) & (dsetconstrER = $true) & (setunionAx = $true) & (upairequniteq = $true) & (powersetAx = $true) & (apProp = $true) & (ex1I2 = $true) & (contrasubsetT1 = $true) & (wellorderingAx = $true) & (setukpairinjR12 = $true) & (powersetE1 = $true) & (dpsetconstrSub = $true) & (iftrueProp2 = $true) & (exu__Cong = $true) & (symdiffIneg1 = $true) & (setunionsingleton1 = $true) & (powersetsubset = $true) & (singletonsuniq = $true) & (woz13rule3 = $true) & (setextAx = $true) & (uniqinunit = $true) & (powersetE = $true) & (breln1unionIR = $true) & (exuE2 = $true) & (singletonsswitch = $true) & (upairsubunion = $true) & (image1E = $true) & (demorgan1a = $true) & (dsetconstrI = $true) & (breln1unionI = $true) & (complementTnotintersectT = $true) & (subsetE2 = $true) & (binunionIR = $true) & (upairset2E = $true) & (iffalseProp1 = $true) & (doubleComplementE1 = $true) & (quantDeMorgan3 = $true) & (setminusILneg = $true) & (breln1all2 = $true) & (iftrueorfalse = $true) & (omegaSAx = $true) & (binintersectSubset2 = $true) & (emptyinunitempty = $true) & (doubleComplementI1 = $true) & (setminusIRneg = $true) & (complementT_lem = $true) & (funcGraphProp2 = $true) & (dsetconstr__Cong = $true) & (singletoninpowunion = $true) & (singletonprop = $true) & (setminusT_lem = $true) & (inIntersectImpInIntersectUnions = $true) & (complementUnionInPowersetComplement = $true) & (doubleComplementEq = $true) & (replAx = $true) & (exuE1 = $true) & (descr__Cong = $true) & (woz13rule1 = $true) & (ifp = $true) & (vacuousDall = $true) & (binintersectLsub = $true) & (breln1invI = $true) & (setminusI = $true) & (inComplementUnionImpInComplement1 = $true) & (cartprodfstpairEq = $true) & (eqbreln = $true) & (setunionI = $true) & (binintersectTELcontra = $true) & (contrasubsetT = $true) & (inIntersectImpInUnion2 = $true) & (complementSubsetComplementIntersect = $true) & (powersetI = $true) & (singletoninpowerset = $true) & (setextsub = $true) & (iftrue = $true) & (cartprodsndin = $true) & (brelnall1 = $true) & (setunionE2 = $true) & (binintersectSubset3 = $true) & (binunionTILcontra = $true) & (subsetTI = $true) & (peano0notS = $true) & (lam2lamEq = $true) & (funcGraphProp4 = $true) & (woz2B = $true) & (inPowerset = $true) & (inCongP = $true) & (funcextLem = $true) & (nonemptyI = $true) & (binunionEcases = $true) & (complementTcontraSubset = $true) & (binunionLsub = $true) & (upairsetIR = $true) & (exuE3e = $true) & (surjCantorThm = $true) & (setukpairinjL2 = $true) & (notinself = $true) & (notdexE = $true) & (intersectInPowersetIntersectUnions = $true) & (setadjoin__Cong = $true) & (ubforcartprodlem1 = $true) & (ap2apEq2 = $true) & (funcext2 = $true) & (woz2W = $true) & (choice2fnsingleton = $true) & (dpsetconstrI = $true) & (setminusEL = $true) & (subPowSU = $true) & (binunionIL = $true) & (eqbreln1 = $true) & (image1Ex1 = $true) & (surjFuncSetFuncSurj = $true) & (image1Ex = $true) & (subsetI1 = $true) & (cartprodmempair = $true) & (breln1SetBreln1 = $true) & (exuI3 = $true) & (breln1invE = $true) & (notequalI1 = $true) & (kfstpairEq = $true) & (subsetRefl = $true) & (ksndsingleton = $true) & (emptyinPowerset = $true) & (injFuncSetFuncIn = $true) & (kpairp = $true) & (dsetconstrEL = $true) & (quantDeMorgan1 = $true) & (binintersectSubset1 = $true) & (binintersectER = $true) & (funcImageSingleton = $true) & (woz13rule2 = $true) & (breln1invprop = $true) & (demorgan2b2 = $true) & (setukpairinjL = $true) & (foundationAx = $true) & (iftrueProp1 = $true) & (complementTI1 = $true) & (inIntersectImpInUnion = $true) & (image1I = $true) & (kfstsingleton = $true) & (setadjoinIL = $true) & (inComplementUnionImpNotIn1 = $true) & (omegaSclos = $true) & (upairinpowunion = $true) & (woz2Ex = $true) & (demorgan2a1 = $true) & (doubleComplementSub1 = $true) & (prop2set2propI = $true) & (contrasubsetT3 = $true) & (nonemptyI1 = $true) & (upairsetE = $true) & (breln1unionprop = $true) & (demorgan2b = $true) & (binintersectRsub = $true) & (contrasubsetT2 = $true) & (kpairiskpair = $true) & (woz1_5 = $true) & (cartprodpairmemER = $true) & (setunionE = $true) & (notsubsetI = $true) & (setunion__Cong = $true) & (dpsetconstrERa = $true) & (breln1all1 = $true) & (breln1compEex = $true) & (nonemptyImpWitness = $true) & (powersetI1 = $true) & (ifSingleton = $true) & (complementTE1 = $true) & (binintersectSubset4 = $true) & (surjFuncSetFuncIn = $true) & (setminusLsub = $true) & (breln1compE = $true) & (upairset2IR = $true) & (omega0Ax = $true) & (funcGraphProp3 = $true) & (quantDeMorgan2 = $true) & (subsetE = $true) & (noeltsimpempty = $true) & (ubforcartprodlem3 = $true) & (leftInvIsSurj = $true) & (demorgan1b = $true) & (lamp = $true) & (theeq = $true) & (setoftrueEq = $true) & (dpsetconstrEL1 = $true) & (upairsetIL = $true) & (prop2setE = $true) & (complementImpComplementIntersect = $true) & (emptysetAx = $true) & (dpsetconstrEL2 = $true) & (binintersectT_lem = $true) & (demorgan2 = $true) & (dpsetconstrER = $true) & (woz13rule4 = $true) & (funcinfuncset = $true) & (notdallE = $true) & (image1Equiv = $true) & (emptysetsubset = $true) & (subsetemptysetimpeq = $true) & (woz1_4 = $true) & (binunionTEcontra = $true) & (complementInPowersetComplementIntersect = $true) & (iffalse = $true) & (descrp = $true) & (emptyE1 = $true) & (subset2powerset = $true) & (cartprodpairmemEL = $true) & (subbreln1 = $true) & (binunionRsub = $true) & (theprop = $true) & (setminusER = $true) & (notinemptyset = $true) & (exuE3u = $true) & (cartprodsndpairEq = $true) & (binintersectEL = $true) & (symdiffI1 = $true) & (breln1unionEcases = $true) & (injFuncInInjFuncSet = $true) & (setminusERneg = $true) & (ex1E1 = $true) & (cartprodmempaircEq = $true) & (setukpairinjR11 = $true) & (woz1_2 = $true) & (subsetTrans = $true) & (setadjoinSub = $true) & (cartprodpairin = $true) & (woz13rule0 = $true) & (setOfPairsIsBReln1 = $true) & (breln1unionCommutes = $true) & (prop2setI = $true) & (breln1compI = $true) & (exuI2 = $true) & (emptyInPowerset = $true) & (funcext = $true) & (setukpairIL = $true) & (secondinupair = $true) & (symdiffIneg2 = $true) & (ubforcartprodlem2 = $true) & (quantDeMorgan4 = $true) & (bs114d = $true) & (powersetTI1 = $true) & (beta1 = $true) & (binunionE = $true) & (setunionsingleton = $true) & (iffalseProp2 = $true) & (disjointsetsI1 = $true) & (eqimpsubset2 = $true) & (eqimpsubset1 = $true) & (symdiffE = $true) & (setadjoinAx = $true) & (omega__Cong = $true) & (kpairsurjEq = $true) & (emptysetE = $true) & (ksndpairEq = $true) & (subbreln = $true) & (setbeta = $true) & (notequalI2 = $true) & (exuEu = $true) & (doubleComplementSub2 = $true) & (peanoSinj = $true)),
% 5.44/2.09 inference(skolemisation,[status(esa),new_symbols(skolem,[sK2,sK3])],[f1013,f1015,f1014])).
% 5.44/2.09 thf(f1014,plain,(
% 5.44/2.09 ? [X0 : $i] : (($true = (transitiveset @ X0)) & ? [X1 : $i] : (($true = (in @ X1 @ X0)) & ((subset @ X1 @ X0) != $true))) => (((transitiveset @ sK2) = $true) & ? [X1 : $i] : (((in @ X1 @ sK2) = $true) & ((subset @ X1 @ sK2) != $true)))),
% 5.44/2.09 introduced(definition,[],[choice_axiom])).
% 5.44/2.09 thf(f1015,plain,(
% 5.44/2.09 ? [X1 : $i] : (((in @ X1 @ sK2) = $true) & ((subset @ X1 @ sK2) != $true)) => (((in @ sK3 @ sK2) = $true) & ((subset @ sK3 @ sK2) != $true))),
% 5.44/2.09 introduced(definition,[],[choice_axiom])).
% 5.44/2.09 thf(f1013,plain,(
% 5.44/2.09 (foundation2 = $true) & (notinself2 = $true) & (setextT = $true) & (cartprodfstin = $true) & (emptyset__Cong = $true) & (woz1_1 = $true) & (breln1compprop = $true) & (setadjoinE = $true) & (breln1unionIL = $true) & (notinsingleton = $true) & (lam2p = $true) & (binunionTIRcontra = $true) & (infuncsetfunc = $true) & (demorgan2a2 = $true) & (binunionT_lem = $true) & (setOfPairsIsBReln = $true) & (setminusSubset2 = $true) & (setext = $true) & (symdiffI2 = $true) & (omegaIndAx = $true) & (setukpairIR = $true) & (eta2 = $true) & (injFuncSetFuncInj = $true) & (binintersectSubset5 = $true) & (setukpairinjR2 = $true) & (setminusELneg = $true) & (beta2 = $true) & (emptysetimpfalse = $true) & (eta1 = $true) & (contraSubsetComplement = $true) & (nonemptyE1 = $true) & (singletonsubset = $true) & (exuI1 = $true) & ? [X0 : $i] : (($true = (transitiveset @ X0)) & ? [X1 : $i] : (($true = (in @ X1 @ X0)) & ((subset @ X1 @ X0) != $true))) & (breln1unionE = $true) & (powersetTE1 = $true) & (woz1_3 = $true) & (cartprodpairsurjEq = $true) & (setadjoinSub2 = $true) & (funcGraphProp1 = $true) & (sepInPowerset = $true) & (cartprodmempair1 = $true) & (brelnall2 = $true) & (ap2p = $true) & (powerset__Cong = $true) & (ex1E2 = $true) & (in__Cong = $true) & (binunionTE = $true) & (setunionsingleton2 = $true) & (app = $true) & (setukpairinjR = $true) & (emptyI = $true) & (ex1I = $true) & (setminusSubset1 = $true) & (demorgan1 = $true) & (woz2A = $true) & (setukpairinjR1 = $true) & (setukpairinjL1 = $true) & (binintersectI = $true) & (powersetT_lem = $true) & (ap2apEq1 = $true) & (demorgan2a = $true) & (binintersectTERcontra = $true) & (eqinunit = $true) & (subsetI2 = $true) & (lamProp = $true) & (setadjoinIR = $true) & (omegaSp = $true) & (setadjoinOr = $true) & (sepSubset = $true) & (dsetconstrER = $true) & (setunionAx = $true) & (upairequniteq = $true) & (powersetAx = $true) & (apProp = $true) & (ex1I2 = $true) & (contrasubsetT1 = $true) & (wellorderingAx = $true) & (setukpairinjR12 = $true) & (powersetE1 = $true) & (dpsetconstrSub = $true) & (iftrueProp2 = $true) & (exu__Cong = $true) & (symdiffIneg1 = $true) & (setunionsingleton1 = $true) & (powersetsubset = $true) & (singletonsuniq = $true) & (woz13rule3 = $true) & (setextAx = $true) & (uniqinunit = $true) & (powersetE = $true) & (breln1unionIR = $true) & (exuE2 = $true) & (singletonsswitch = $true) & (upairsubunion = $true) & (image1E = $true) & (demorgan1a = $true) & (dsetconstrI = $true) & (breln1unionI = $true) & (complementTnotintersectT = $true) & (subsetE2 = $true) & (binunionIR = $true) & (upairset2E = $true) & (iffalseProp1 = $true) & (doubleComplementE1 = $true) & (quantDeMorgan3 = $true) & (setminusILneg = $true) & (breln1all2 = $true) & (iftrueorfalse = $true) & (omegaSAx = $true) & (binintersectSubset2 = $true) & (emptyinunitempty = $true) & (doubleComplementI1 = $true) & (setminusIRneg = $true) & (complementT_lem = $true) & (funcGraphProp2 = $true) & (dsetconstr__Cong = $true) & (singletoninpowunion = $true) & (singletonprop = $true) & (setminusT_lem = $true) & (inIntersectImpInIntersectUnions = $true) & (complementUnionInPowersetComplement = $true) & (doubleComplementEq = $true) & (replAx = $true) & (exuE1 = $true) & (descr__Cong = $true) & (woz13rule1 = $true) & (ifp = $true) & (vacuousDall = $true) & (binintersectLsub = $true) & (breln1invI = $true) & (setminusI = $true) & (inComplementUnionImpInComplement1 = $true) & (cartprodfstpairEq = $true) & (eqbreln = $true) & (setunionI = $true) & (binintersectTELcontra = $true) & (contrasubsetT = $true) & (inIntersectImpInUnion2 = $true) & (complementSubsetComplementIntersect = $true) & (powersetI = $true) & (singletoninpowerset = $true) & (setextsub = $true) & (iftrue = $true) & (cartprodsndin = $true) & (brelnall1 = $true) & (setunionE2 = $true) & (binintersectSubset3 = $true) & (binunionTILcontra = $true) & (subsetTI = $true) & (peano0notS = $true) & (lam2lamEq = $true) & (funcGraphProp4 = $true) & (woz2B = $true) & (inPowerset = $true) & (inCongP = $true) & (funcextLem = $true) & (nonemptyI = $true) & (binunionEcases = $true) & (complementTcontraSubset = $true) & (binunionLsub = $true) & (upairsetIR = $true) & (exuE3e = $true) & (surjCantorThm = $true) & (setukpairinjL2 = $true) & (notinself = $true) & (notdexE = $true) & (intersectInPowersetIntersectUnions = $true) & (setadjoin__Cong = $true) & (ubforcartprodlem1 = $true) & (ap2apEq2 = $true) & (funcext2 = $true) & (woz2W = $true) & (choice2fnsingleton = $true) & (dpsetconstrI = $true) & (setminusEL = $true) & (subPowSU = $true) & (binunionIL = $true) & (eqbreln1 = $true) & (image1Ex1 = $true) & (surjFuncSetFuncSurj = $true) & (image1Ex = $true) & (subsetI1 = $true) & (cartprodmempair = $true) & (breln1SetBreln1 = $true) & (exuI3 = $true) & (breln1invE = $true) & (notequalI1 = $true) & (kfstpairEq = $true) & (subsetRefl = $true) & (ksndsingleton = $true) & (emptyinPowerset = $true) & (injFuncSetFuncIn = $true) & (kpairp = $true) & (dsetconstrEL = $true) & (quantDeMorgan1 = $true) & (binintersectSubset1 = $true) & (binintersectER = $true) & (funcImageSingleton = $true) & (woz13rule2 = $true) & (breln1invprop = $true) & (demorgan2b2 = $true) & (setukpairinjL = $true) & (foundationAx = $true) & (iftrueProp1 = $true) & (complementTI1 = $true) & (inIntersectImpInUnion = $true) & (image1I = $true) & (kfstsingleton = $true) & (setadjoinIL = $true) & (inComplementUnionImpNotIn1 = $true) & (omegaSclos = $true) & (upairinpowunion = $true) & (woz2Ex = $true) & (demorgan2a1 = $true) & (doubleComplementSub1 = $true) & (prop2set2propI = $true) & (contrasubsetT3 = $true) & (nonemptyI1 = $true) & (upairsetE = $true) & (breln1unionprop = $true) & (demorgan2b = $true) & (binintersectRsub = $true) & (contrasubsetT2 = $true) & (kpairiskpair = $true) & (woz1_5 = $true) & (cartprodpairmemER = $true) & (setunionE = $true) & (notsubsetI = $true) & (setunion__Cong = $true) & (dpsetconstrERa = $true) & (breln1all1 = $true) & (breln1compEex = $true) & (nonemptyImpWitness = $true) & (powersetI1 = $true) & (ifSingleton = $true) & (complementTE1 = $true) & (binintersectSubset4 = $true) & (surjFuncSetFuncIn = $true) & (setminusLsub = $true) & (breln1compE = $true) & (upairset2IR = $true) & (omega0Ax = $true) & (funcGraphProp3 = $true) & (quantDeMorgan2 = $true) & (subsetE = $true) & (noeltsimpempty = $true) & (ubforcartprodlem3 = $true) & (leftInvIsSurj = $true) & (demorgan1b = $true) & (lamp = $true) & (theeq = $true) & (setoftrueEq = $true) & (dpsetconstrEL1 = $true) & (upairsetIL = $true) & (prop2setE = $true) & (complementImpComplementIntersect = $true) & (emptysetAx = $true) & (dpsetconstrEL2 = $true) & (binintersectT_lem = $true) & (demorgan2 = $true) & (dpsetconstrER = $true) & (woz13rule4 = $true) & (funcinfuncset = $true) & (notdallE = $true) & (image1Equiv = $true) & (emptysetsubset = $true) & (subsetemptysetimpeq = $true) & (woz1_4 = $true) & (binunionTEcontra = $true) & (complementInPowersetComplementIntersect = $true) & (iffalse = $true) & (descrp = $true) & (emptyE1 = $true) & (subset2powerset = $true) & (cartprodpairmemEL = $true) & (subbreln1 = $true) & (binunionRsub = $true) & (theprop = $true) & (setminusER = $true) & (notinemptyset = $true) & (exuE3u = $true) & (cartprodsndpairEq = $true) & (binintersectEL = $true) & (symdiffI1 = $true) & (breln1unionEcases = $true) & (injFuncInInjFuncSet = $true) & (setminusERneg = $true) & (ex1E1 = $true) & (cartprodmempaircEq = $true) & (setukpairinjR11 = $true) & (woz1_2 = $true) & (subsetTrans = $true) & (setadjoinSub = $true) & (cartprodpairin = $true) & (woz13rule0 = $true) & (setOfPairsIsBReln1 = $true) & (breln1unionCommutes = $true) & (prop2setI = $true) & (breln1compI = $true) & (exuI2 = $true) & (emptyInPowerset = $true) & (funcext = $true) & (setukpairIL = $true) & (secondinupair = $true) & (symdiffIneg2 = $true) & (ubforcartprodlem2 = $true) & (quantDeMorgan4 = $true) & (bs114d = $true) & (powersetTI1 = $true) & (beta1 = $true) & (binunionE = $true) & (setunionsingleton = $true) & (iffalseProp2 = $true) & (disjointsetsI1 = $true) & (eqimpsubset2 = $true) & (eqimpsubset1 = $true) & (symdiffE = $true) & (setadjoinAx = $true) & (omega__Cong = $true) & (kpairsurjEq = $true) & (emptysetE = $true) & (ksndpairEq = $true) & (subbreln = $true) & (setbeta = $true) & (notequalI2 = $true) & (exuEu = $true) & (doubleComplementSub2 = $true) & (peanoSinj = $true)),
% 5.44/2.09 inference(flattening,[status(thm)],[f1012])).
% 5.44/2.09 thf(f1012,plain,(
% 5.44/2.09 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (($true = (transitiveset @ X0)) & ? [X1 : $i] : (($true = (in @ X1 @ X0)) & ((subset @ X1 @ X0) != $true))) & (peanoSinj = $true)) & (peano0notS = $true)) & (omegaSclos = $true)) & (omegaSp = $true)) & (notinself2 = $true)) & (notinself = $true)) & (foundation2 = $true)) & (surjCantorThm = $true)) & (leftInvIsSurj = $true)) & (surjFuncSetFuncSurj = $true)) & (surjFuncSetFuncIn = $true)) & (injFuncSetFuncInj = $true)) & (injFuncSetFuncIn = $true)) & (injFuncInInjFuncSet = $true)) & (image1I = $true)) & (image1E = $true)) & (image1Equiv = $true)) & (image1Ex1 = $true)) & (image1Ex = $true)) & (woz2B = $true)) & (woz2A = $true)) & (woz2W = $true)) & (woz2Ex = $true)) & (breln1unionCommutes = $true)) & (breln1unionEcases = $true)) & (breln1unionE = $true)) & (breln1unionI = $true)) & (breln1unionIR = $true)) & (breln1unionIL = $true)) & (breln1unionprop = $true)) & (breln1compEex = $true)) & (breln1compE = $true)) & (breln1compI = $true)) & (breln1compprop = $true)) & (breln1invE = $true)) & (breln1invI = $true)) & (breln1invprop = $true)) & (eqbreln1 = $true)) & (subbreln1 = $true)) & (breln1all1 = $true)) & (setOfPairsIsBReln1 = $true)) & (choice2fnsingleton = $true)) & (breln1SetBreln1 = $true)) & (breln1all2 = $true)) & (woz1_5 = $true)) & (woz1_4 = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 5.44/2.09 inference(ennf_transformation,[status(thm)],[f910])).
% 5.44/2.09 thf(f910,plain,(
% 5.44/2.09 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ((woz1_4 = $true) => ((woz1_5 = $true) => ((breln1all2 = $true) => ((breln1SetBreln1 = $true) => ((choice2fnsingleton = $true) => ((setOfPairsIsBReln1 = $true) => ((breln1all1 = $true) => ((subbreln1 = $true) => ((eqbreln1 = $true) => ((breln1invprop = $true) => ((breln1invI = $true) => ((breln1invE = $true) => ((breln1compprop = $true) => ((breln1compI = $true) => ((breln1compE = $true) => ((breln1compEex = $true) => ((breln1unionprop = $true) => ((breln1unionIL = $true) => ((breln1unionIR = $true) => ((breln1unionI = $true) => ((breln1unionE = $true) => ((breln1unionEcases = $true) => ((breln1unionCommutes = $true) => ((woz2Ex = $true) => ((woz2W = $true) => ((woz2A = $true) => ((woz2B = $true) => ((image1Ex = $true) => ((image1Ex1 = $true) => ((image1Equiv = $true) => ((image1E = $true) => ((image1I = $true) => ((injFuncInInjFuncSet = $true) => ((injFuncSetFuncIn = $true) => ((injFuncSetFuncInj = $true) => ((surjFuncSetFuncIn = $true) => ((surjFuncSetFuncSurj = $true) => ((leftInvIsSurj = $true) => ((surjCantorThm = $true) => ((foundation2 = $true) => ((notinself = $true) => ((notinself2 = $true) => ((omegaSp = $true) => ((omegaSclos = $true) => ((peano0notS = $true) => ((peanoSinj = $true) => ! [X0 : $i] : (($true = (transitiveset @ X0)) => ! [X1 : $i] : (($true = (in @ X1 @ X0)) => ((subset @ X1 @ X0) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.44/2.09 inference(fool_elimination,[status(thm)],[f909])).
% 5.44/2.09 thf(f909,plain,(
% 5.44/2.09 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => ! [X0 : $i] : ((transitiveset @ X0) => ! [X1 : $i] : ((in @ X1 @ X0) => (subset @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.44/2.09 inference(rectify,[status(thm)],[f333])).
% 5.44/2.09 thf(f333,negated_conjecture,(
% 5.44/2.09 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => ! [X11 : $i] : ((transitiveset @ X11) => ! [X3 : $i] : ((in @ X3 @ X11) => (subset @ X3 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.44/2.09 inference(negated_conjecture,[status(cth)],[f332])).
% 5.44/2.09 thf(f332,conjecture,(
% 5.44/2.09 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => (woz1_4 => (woz1_5 => (breln1all2 => (breln1SetBreln1 => (choice2fnsingleton => (setOfPairsIsBReln1 => (breln1all1 => (subbreln1 => (eqbreln1 => (breln1invprop => (breln1invI => (breln1invE => (breln1compprop => (breln1compI => (breln1compE => (breln1compEex => (breln1unionprop => (breln1unionIL => (breln1unionIR => (breln1unionI => (breln1unionE => (breln1unionEcases => (breln1unionCommutes => (woz2Ex => (woz2W => (woz2A => (woz2B => (image1Ex => (image1Ex1 => (image1Equiv => (image1E => (image1I => (injFuncInInjFuncSet => (injFuncSetFuncIn => (injFuncSetFuncInj => (surjFuncSetFuncIn => (surjFuncSetFuncSurj => (leftInvIsSurj => (surjCantorThm => (foundation2 => (notinself => (notinself2 => (omegaSp => (omegaSclos => (peano0notS => (peanoSinj => ! [X11 : $i] : ((transitiveset @ X11) => ! [X3 : $i] : ((in @ X3 @ X11) => (subset @ X3 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 5.44/2.09 file('/export/starexec/sandbox2/tmp/tmp.4NXBcYPEpZ/DTF2THF_16671.p',transitivesetOp1)).
% 5.44/2.09 thf(f2421,plain,(
% 5.44/2.09 ((subset @ sK3 @ sK2) = $true)),
% 5.44/2.09 inference(trivial_inequality_removal,[status(thm)],[f2420])).
% 5.44/2.09 thf(f2420,plain,(
% 5.44/2.09 ((subset @ sK3 @ sK2) = $true) | ($false = $true)),
% 5.44/2.09 inference(duplicate_literal_removal,[status(thm)],[f2417])).
% 5.44/2.09 thf(f2417,plain,(
% 5.44/2.09 ($false = $true) | ($false = $true) | ((subset @ sK3 @ sK2) = $true)),
% 5.44/2.09 inference(superposition,[status(thm)],[f1005,f2378])).
% 5.44/2.09 thf(f2378,plain,(
% 5.44/2.09 (($true => (subset @ sK3 @ sK2)) = $true)),
% 5.44/2.09 inference(superposition,[status(thm)],[f2206,f1484])).
% 5.44/2.09 thf(f1484,plain,(
% 5.44/2.09 ((in @ sK3 @ sK2) = $true)),
% 5.44/2.09 inference(cnf_transformation,[status(thm)],[f1016])).
% 5.44/2.09 thf(f2206,plain,(
% 5.44/2.09 ( ! [X0 : $i] : (($true = ((in @ X0 @ sK2) => (subset @ X0 @ sK2)))) )),
% 5.44/2.09 inference(beta_eta_normalization,[status(thm)],[f2205])).
% 5.44/2.09 thf(f2205,plain,(
% 5.44/2.09 ( ! [X0 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ sK2) => (subset @ Y0 @ sK2))) @ X0))) )),
% 5.44/2.09 inference(trivial_inequality_removal,[status(thm)],[f2202])).
% 5.44/2.09 thf(f2202,plain,(
% 5.44/2.09 ( ! [X0 : $i] : (($true = ((^[Y0 : $i]: ((in @ Y0 @ sK2) => (subset @ Y0 @ sK2))) @ X0)) | ($false = $true)) )),
% 5.44/2.09 inference(superposition,[status(thm)],[f1000,f2061])).
% 5.44/2.09 thf(f2061,plain,(
% 5.44/2.09 ((!! @ $i @ (^[Y0 : $i]: ((in @ Y0 @ sK2) => (subset @ Y0 @ sK2)))) = $true)),
% 5.44/2.09 inference(beta_eta_normalization,[status(thm)],[f1707])).
% 5.44/2.09 thf(f1707,plain,(
% 5.44/2.09 (((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (subset @ Y1 @ Y0))))) @ sK2) = $true)),
% 5.44/2.09 inference(definition_unfolding,[status(thm)],[f1485,f1677])).
% 5.44/2.09 thf(f1677,plain,(
% 5.44/2.09 (transitiveset = (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (subset @ Y1 @ Y0))))))),
% 5.44/2.09 inference(cnf_transformation,[status(thm)],[f522])).
% 5.44/2.09 thf(f522,plain,(
% 5.44/2.09 (transitiveset = (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: ((in @ Y1 @ Y0) => (subset @ Y1 @ Y0))))))),
% 5.44/2.09 inference(fool_elimination,[status(thm)],[f521])).
% 5.44/2.09 thf(f521,plain,(
% 5.44/2.09 ((^[X0 : $i] : (! [X1 : $i] : ((in @ X1 @ X0) => (subset @ X1 @ X0)))) = transitiveset)),
% 5.44/2.09 inference(rectify,[status(thm)],[f331])).
% 5.44/2.09 thf(f331,axiom,(
% 5.44/2.09 ((^[X3 : $i] : (! [X11 : $i] : ((in @ X11 @ X3) => (subset @ X11 @ X3)))) = transitiveset)),
% 5.44/2.09 file('/export/starexec/sandbox2/tmp/tmp.4NXBcYPEpZ/DTF2THF_16671.p',transitiveset)).
% 5.44/2.09 thf(f1485,plain,(
% 5.44/2.09 ((transitiveset @ sK2) = $true)),
% 5.44/2.09 inference(cnf_transformation,[status(thm)],[f1016])).
% 5.44/2.09 thf(f1000,plain,(
% 5.44/2.09 ( ! [X0 : $tType,X2 : X0,X1 : X0 > $o] : (($false = (!! @ X0 @ X1)) | ((X1 @ X2) = $true)) )),
% 5.44/2.09 introduced(definition,[],[pi_proxy_axiom])).
% 5.44/2.09 thf(f1005,plain,(
% 5.44/2.09 ( ! [X2 : $o,X1 : $o] : (($false = (X1 => X2)) | ($true = X2) | ($false = X1)) )),
% 5.44/2.09 introduced(definition,[],[implies_proxy_axiom])).
% 5.44/2.09 % SZS output end Proof for DTF2THF_16671
% 5.44/2.09 % (16925)------------------------------
% 5.44/2.09 % (16925)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.44/2.09 % (16925)Termination reason: Refutation
% 5.44/2.09
% 5.44/2.09 % (16925)Memory used [KB]: 8315
% 5.44/2.09 % (16925)Time elapsed: 0.088 s
% 5.44/2.09 % (16925)Instructions burned: 311 (million)
% 5.44/2.09 % (16925)------------------------------
% 5.44/2.09 % (16925)------------------------------
% 5.44/2.09 % (16881)Success in time 0.415 s
%------------------------------------------------------------------------------