%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU810^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n029.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 09:37:23 AM UTC 2026
% Result : Theorem 6.90s 2.56s
% Output : Refutation 6.90s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 7
% Syntax : Number of formulae : 42 ( 28 unt; 0 typ; 0 def)
% Number of atoms : 5079 (1358 equ; 0 cnn)
% Maximal formula atoms : 329 ( 120 avg)
% Number of connectives : 2484 ( 17 ~; 6 |; 983 &; 155 @)
% ( 0 <=>;1309 =>; 0 <=; 0 <~>)
% Maximal formula depth : 332 ( 56 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 1 ( 1 >; 0 *; 0 +; 0 <<)
% Number of symbols : 388 ( 384 usr; 332 con; 0-4 aty)
% ( 14 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 56 ( 25 ^; 28 !; 3 ?; 56 :)
% 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_399,type,
sK0:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_400,type,
sK1:
!>[X0: $tType] : ( ( X0 > $o ) > X0 ) ).
thf(func_def_403,type,
ph4:
!>[X0: $tType] : X0 ).
thf(f2411,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2409]) ).
thf(f2409,plain,
$true = $false,
inference(superposition,[status(thm)],[f2205,f2400]) ).
thf(f2400,plain,
! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2399]) ).
thf(f2399,plain,
! [X0: $i,X1: $i] :
( $true
= ( ^ [Y0: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ X0 ) )
@ X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2396]) ).
thf(f2396,plain,
! [X0: $i,X1: $i] :
( ( $true = $false )
| ( $true
= ( ^ [Y0: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ X0 ) )
@ X1 ) ) ),
inference(superposition,[status(thm)],[f994,f2219]) ).
thf(f2219,plain,
! [X0: $i] :
( ( !! @ $i
@ ^ [Y0: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ X0 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f2218]) ).
thf(f2218,plain,
! [X0: $i] :
( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y1 @ Y0 ) ) )
@ X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2214]) ).
thf(f2214,plain,
! [X0: $i] :
( ( $true
= ( ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y1 @ Y0 ) ) )
@ X0 ) )
| ( $true = $false ) ),
inference(superposition,[status(thm)],[f1868,f994]) ).
thf(f1868,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y1 @ Y0 ) ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1521,f1338]) ).
thf(f1338,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f1009]) ).
thf(f1009,plain,
( ( subsetRefl = $true )
& ( symdiffIneg2 = $true )
& ( funcGraphProp4 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( subsetemptysetimpeq = $true )
& ( singletonsuniq = $true )
& ( dpsetconstrERa = $true )
& ( breln1all1 = $true )
& ( $true
= ( in @ sK2 @ omega ) )
& ( emptyset
= ( omegaS @ sK2 ) )
& ( binunionLsub = $true )
& ( notsubsetI = $true )
& ( binintersectRsub = $true )
& ( setukpairinjR11 = $true )
& ( notinself = $true )
& ( ex1E2 = $true )
& ( subsetE2 = $true )
& ( powersetE = $true )
& ( binunionTILcontra = $true )
& ( binintersectT_lem = $true )
& ( singletonprop = $true )
& ( ubforcartprodlem1 = $true )
& ( kfstpairEq = $true )
& ( setextT = $true )
& ( quantDeMorgan1 = $true )
& ( symdiffIneg1 = $true )
& ( contraSubsetComplement = $true )
& ( upairinpowunion = $true )
& ( symdiffE = $true )
& ( kpairsurjEq = $true )
& ( notequalI2 = $true )
& ( lam2lamEq = $true )
& ( binunionTEcontra = $true )
& ( exuE3u = $true )
& ( complementImpComplementIntersect = $true )
& ( woz2B = $true )
& ( setunion__Cong = $true )
& ( prop2set2propI = $true )
& ( notequalI1 = $true )
& ( omegaSAx = $true )
& ( eta1 = $true )
& ( breln1compprop = $true )
& ( app = $true )
& ( ap2p = $true )
& ( powersetTE1 = $true )
& ( dsetconstrI = $true )
& ( in__Cong = $true )
& ( image1Ex1 = $true )
& ( quantDeMorgan2 = $true )
& ( leftInvIsSurj = $true )
& ( singletonsubset = $true )
& ( setminusIRneg = $true )
& ( notinsingleton = $true )
& ( dpsetconstrSub = $true )
& ( symdiffI2 = $true )
& ( woz2W = $true )
& ( emptyInPowerset = $true )
& ( setunionAx = $true )
& ( cartprodpairmemER = $true )
& ( image1Equiv = $true )
& ( ksndpairEq = $true )
& ( contrasubsetT1 = $true )
& ( nonemptyE1 = $true )
& ( ap2apEq2 = $true )
& ( eqbreln = $true )
& ( setminusEL = $true )
& ( symdiffI1 = $true )
& ( singletoninpowerset = $true )
& ( nonemptyI1 = $true )
& ( omegaSclos = $true )
& ( binintersectSubset5 = $true )
& ( inCongP = $true )
& ( sepSubset = $true )
& ( setadjoinIL = $true )
& ( brelnall1 = $true )
& ( setukpairinjR2 = $true )
& ( subbreln1 = $true )
& ( contrasubsetT = $true )
& ( funcGraphProp2 = $true )
& ( powersetI = $true )
& ( ubforcartprodlem3 = $true )
& ( dsetconstrEL = $true )
& ( setukpairinjL1 = $true )
& ( woz2Ex = $true )
& ( woz2A = $true )
& ( quantDeMorgan3 = $true )
& ( beta1 = $true )
& ( setukpairinjR12 = $true )
& ( kfstsingleton = $true )
& ( ex1I = $true )
& ( secondinupair = $true )
& ( inIntersectImpInUnion2 = $true )
& ( kpairiskpair = $true )
& ( emptyinunitempty = $true )
& ( inIntersectImpInUnion = $true )
& ( breln1invE = $true )
& ( bs114d = $true )
& ( setminusSubset2 = $true )
& ( iffalse = $true )
& ( setunionsingleton1 = $true )
& ( omega0Ax = $true )
& ( setextsub = $true )
& ( lam2p = $true )
& ( complementTcontraSubset = $true )
& ( setunionsingleton2 = $true )
& ( emptysetAx = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binunionIR = $true )
& ( cartprodpairmemEL = $true )
& ( dsetconstr__Cong = $true )
& ( doubleComplementI1 = $true )
& ( woz13rule1 = $true )
& ( breln1unionprop = $true )
& ( vacuousDall = $true )
& ( binintersectI = $true )
& ( funcinfuncset = $true )
& ( ifp = $true )
& ( infuncsetfunc = $true )
& ( exuE1 = $true )
& ( doubleComplementSub2 = $true )
& ( funcImageSingleton = $true )
& ( binunionTIRcontra = $true )
& ( binunionRsub = $true )
& ( demorgan2 = $true )
& ( funcext = $true )
& ( binunionEcases = $true )
& ( subsetI2 = $true )
& ( demorgan2b2 = $true )
& ( setminusERneg = $true )
& ( ksndsingleton = $true )
& ( woz1_4 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( powersetT_lem = $true )
& ( breln1unionIL = $true )
& ( omegaIndAx = $true )
& ( setukpairinjR1 = $true )
& ( setadjoinE = $true )
& ( cartprodpairin = $true )
& ( complementUnionInPowersetComplement = $true )
& ( subsetTI = $true )
& ( setadjoinIR = $true )
& ( image1E = $true )
& ( setukpairIL = $true )
& ( demorgan2a = $true )
& ( subbreln = $true )
& ( binunionTE = $true )
& ( eqinunit = $true )
& ( injFuncSetFuncInj = $true )
& ( wellorderingAx = $true )
& ( setminusER = $true )
& ( prop2setE = $true )
& ( complementTI1 = $true )
& ( complementTE1 = $true )
& ( setadjoinSub = $true )
& ( cartprodfstpairEq = $true )
& ( surjCantorThm = $true )
& ( setunionE = $true )
& ( subPowSU = $true )
& ( funcext2 = $true )
& ( cartprodfstin = $true )
& ( surjFuncSetFuncSurj = $true )
& ( woz1_2 = $true )
& ( subset2powerset = $true )
& ( ex1E1 = $true )
& ( subsetE = $true )
& ( lamProp = $true )
& ( woz1_3 = $true )
& ( powersetE1 = $true )
& ( breln1all2 = $true )
& ( binintersectSubset1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setukpairIR = $true )
& ( setminusI = $true )
& ( cartprodpairsurjEq = $true )
& ( notdexE = $true )
& ( singletoninpowunion = $true )
& ( upairsetIL = $true )
& ( exu__Cong = $true )
& ( funcGraphProp1 = $true )
& ( binintersectSubset4 = $true )
& ( exuE2 = $true )
& ( exuI3 = $true )
& ( demorgan1a = $true )
& ( disjointsetsI1 = $true )
& ( complementTnotintersectT = $true )
& ( binintersectSubset2 = $true )
& ( breln1invI = $true )
& ( powerset__Cong = $true )
& ( binintersectEL = $true )
& ( woz1_1 = $true )
& ( descr__Cong = $true )
& ( setukpairinjL = $true )
& ( image1Ex = $true )
& ( breln1unionIR = $true )
& ( emptyI = $true )
& ( cartprodmempaircEq = $true )
& ( singletonsswitch = $true )
& ( notinself2 = $true )
& ( ex1I2 = $true )
& ( powersetI1 = $true )
& ( demorgan1b = $true )
& ( setoftrueEq = $true )
& ( powersetsubset = $true )
& ( subsetTrans = $true )
& ( woz13rule4 = $true )
& ( woz13rule0 = $true )
& ( ubforcartprodlem2 = $true )
& ( eqbreln1 = $true )
& ( binunionT_lem = $true )
& ( omegaSp = $true )
& ( binunionIL = $true )
& ( exuE3e = $true )
& ( dpsetconstrI = $true )
& ( noeltsimpempty = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( eta2 = $true )
& ( binintersectTERcontra = $true )
& ( beta2 = $true )
& ( setOfPairsIsBReln = $true )
& ( iftrue = $true )
& ( demorgan2a1 = $true )
& ( dpsetconstrER = $true )
& ( binintersectLsub = $true )
& ( lamp = $true )
& ( complementT_lem = $true )
& ( iftrueorfalse = $true )
& ( demorgan2b = $true )
& ( cartprodsndin = $true )
& ( image1I = $true )
& ( iffalseProp1 = $true )
& ( upairsetIR = $true )
& ( omega__Cong = $true )
& ( replAx = $true )
& ( sepInPowerset = $true )
& ( nonemptyImpWitness = $true )
& ( setextAx = $true )
& ( iftrueProp2 = $true )
& ( binintersectTELcontra = $true )
& ( upairset2E = $true )
& ( doubleComplementEq = $true )
& ( surjFuncSetFuncIn = $true )
& ( eqimpsubset1 = $true )
& ( emptyset__Cong = $true )
& ( setunionI = $true )
& ( emptysetsubset = $true )
& ( cartprodmempair = $true )
& ( exuI1 = $true )
& ( woz1_5 = $true )
& ( prop2setI = $true )
& ( emptyE1 = $true )
& ( breln1compE = $true )
& ( breln1unionCommutes = $true )
& ( demorgan1 = $true )
& ( injFuncInInjFuncSet = $true )
& ( funcextLem = $true )
& ( setminusT_lem = $true )
& ( emptysetE = $true )
& ( theprop = $true )
& ( doubleComplementSub1 = $true )
& ( binintersectSubset3 = $true )
& ( cartprodsndpairEq = $true )
& ( contrasubsetT2 = $true )
& ( cartprodmempair1 = $true )
& ( descrp = $true )
& ( breln1compI = $true )
& ( powersetAx = $true )
& ( injFuncSetFuncIn = $true )
& ( funcGraphProp3 = $true )
& ( upairsubunion = $true )
& ( uniqinunit = $true )
& ( iftrueProp1 = $true )
& ( woz13rule3 = $true )
& ( dsetconstrER = $true )
& ( setbeta = $true )
& ( setext = $true )
& ( setminusELneg = $true )
& ( setukpairinjL2 = $true )
& ( upairsetE = $true )
& ( contrasubsetT3 = $true )
& ( foundationAx = $true )
& ( notinemptyset = $true )
& ( setadjoinAx = $true )
& ( upairequniteq = $true )
& ( setadjoinSub2 = $true )
& ( breln1unionE = $true )
& ( choice2fnsingleton = $true )
& ( upairset2IR = $true )
& ( eqimpsubset2 = $true )
& ( exuI2 = $true )
& ( quantDeMorgan4 = $true )
& ( brelnall2 = $true )
& ( powersetTI1 = $true )
& ( binunionE = $true )
& ( binintersectER = $true )
& ( setunionE2 = $true )
& ( breln1SetBreln1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( dpsetconstrEL1 = $true )
& ( ifSingleton = $true )
& ( apProp = $true )
& ( breln1unionI = $true )
& ( emptyinPowerset = $true )
& ( setminusLsub = $true )
& ( setukpairinjR = $true )
& ( inPowerset = $true )
& ( dpsetconstrEL2 = $true )
& ( iffalseProp2 = $true )
& ( notdallE = $true )
& ( setminusSubset1 = $true )
& ( breln1compEex = $true )
& ( breln1unionEcases = $true )
& ( subsetI1 = $true )
& ( kpairp = $true )
& ( doubleComplementE1 = $true )
& ( ap2apEq1 = $true )
& ( setminusILneg = $true )
& ( setunionsingleton = $true )
& ( demorgan2a2 = $true )
& ( breln1invprop = $true )
& ( nonemptyI = $true )
& ( woz13rule2 = $true )
& ( emptysetimpfalse = $true )
& ( setadjoin__Cong = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( theeq = $true )
& ( foundation2 = $true )
& ( setadjoinOr = $true )
& ( exuEu = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK2])],[f1007,f1008]) ).
thf(f1008,plain,
( ? [X0: $i] :
( ( $true
= ( in @ X0 @ omega ) )
& ( emptyset
= ( omegaS @ X0 ) ) )
=> ( ( $true
= ( in @ sK2 @ omega ) )
& ( emptyset
= ( omegaS @ sK2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1007,plain,
( ( subsetRefl = $true )
& ( symdiffIneg2 = $true )
& ( funcGraphProp4 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( subsetemptysetimpeq = $true )
& ( singletonsuniq = $true )
& ( dpsetconstrERa = $true )
& ( breln1all1 = $true )
& ? [X0: $i] :
( ( $true
= ( in @ X0 @ omega ) )
& ( emptyset
= ( omegaS @ X0 ) ) )
& ( binunionLsub = $true )
& ( notsubsetI = $true )
& ( binintersectRsub = $true )
& ( setukpairinjR11 = $true )
& ( notinself = $true )
& ( ex1E2 = $true )
& ( subsetE2 = $true )
& ( powersetE = $true )
& ( binunionTILcontra = $true )
& ( binintersectT_lem = $true )
& ( singletonprop = $true )
& ( ubforcartprodlem1 = $true )
& ( kfstpairEq = $true )
& ( setextT = $true )
& ( quantDeMorgan1 = $true )
& ( symdiffIneg1 = $true )
& ( contraSubsetComplement = $true )
& ( upairinpowunion = $true )
& ( symdiffE = $true )
& ( kpairsurjEq = $true )
& ( notequalI2 = $true )
& ( lam2lamEq = $true )
& ( binunionTEcontra = $true )
& ( exuE3u = $true )
& ( complementImpComplementIntersect = $true )
& ( woz2B = $true )
& ( setunion__Cong = $true )
& ( prop2set2propI = $true )
& ( notequalI1 = $true )
& ( omegaSAx = $true )
& ( eta1 = $true )
& ( breln1compprop = $true )
& ( app = $true )
& ( ap2p = $true )
& ( powersetTE1 = $true )
& ( dsetconstrI = $true )
& ( in__Cong = $true )
& ( image1Ex1 = $true )
& ( quantDeMorgan2 = $true )
& ( leftInvIsSurj = $true )
& ( singletonsubset = $true )
& ( setminusIRneg = $true )
& ( notinsingleton = $true )
& ( dpsetconstrSub = $true )
& ( symdiffI2 = $true )
& ( woz2W = $true )
& ( emptyInPowerset = $true )
& ( setunionAx = $true )
& ( cartprodpairmemER = $true )
& ( image1Equiv = $true )
& ( ksndpairEq = $true )
& ( contrasubsetT1 = $true )
& ( nonemptyE1 = $true )
& ( ap2apEq2 = $true )
& ( eqbreln = $true )
& ( setminusEL = $true )
& ( symdiffI1 = $true )
& ( singletoninpowerset = $true )
& ( nonemptyI1 = $true )
& ( omegaSclos = $true )
& ( binintersectSubset5 = $true )
& ( inCongP = $true )
& ( sepSubset = $true )
& ( setadjoinIL = $true )
& ( brelnall1 = $true )
& ( setukpairinjR2 = $true )
& ( subbreln1 = $true )
& ( contrasubsetT = $true )
& ( funcGraphProp2 = $true )
& ( powersetI = $true )
& ( ubforcartprodlem3 = $true )
& ( dsetconstrEL = $true )
& ( setukpairinjL1 = $true )
& ( woz2Ex = $true )
& ( woz2A = $true )
& ( quantDeMorgan3 = $true )
& ( beta1 = $true )
& ( setukpairinjR12 = $true )
& ( kfstsingleton = $true )
& ( ex1I = $true )
& ( secondinupair = $true )
& ( inIntersectImpInUnion2 = $true )
& ( kpairiskpair = $true )
& ( emptyinunitempty = $true )
& ( inIntersectImpInUnion = $true )
& ( breln1invE = $true )
& ( bs114d = $true )
& ( setminusSubset2 = $true )
& ( iffalse = $true )
& ( setunionsingleton1 = $true )
& ( omega0Ax = $true )
& ( setextsub = $true )
& ( lam2p = $true )
& ( complementTcontraSubset = $true )
& ( setunionsingleton2 = $true )
& ( emptysetAx = $true )
& ( complementSubsetComplementIntersect = $true )
& ( binunionIR = $true )
& ( cartprodpairmemEL = $true )
& ( dsetconstr__Cong = $true )
& ( doubleComplementI1 = $true )
& ( woz13rule1 = $true )
& ( breln1unionprop = $true )
& ( vacuousDall = $true )
& ( binintersectI = $true )
& ( funcinfuncset = $true )
& ( ifp = $true )
& ( infuncsetfunc = $true )
& ( exuE1 = $true )
& ( doubleComplementSub2 = $true )
& ( funcImageSingleton = $true )
& ( binunionTIRcontra = $true )
& ( binunionRsub = $true )
& ( demorgan2 = $true )
& ( funcext = $true )
& ( binunionEcases = $true )
& ( subsetI2 = $true )
& ( demorgan2b2 = $true )
& ( setminusERneg = $true )
& ( ksndsingleton = $true )
& ( woz1_4 = $true )
& ( setOfPairsIsBReln1 = $true )
& ( powersetT_lem = $true )
& ( breln1unionIL = $true )
& ( omegaIndAx = $true )
& ( setukpairinjR1 = $true )
& ( setadjoinE = $true )
& ( cartprodpairin = $true )
& ( complementUnionInPowersetComplement = $true )
& ( subsetTI = $true )
& ( setadjoinIR = $true )
& ( image1E = $true )
& ( setukpairIL = $true )
& ( demorgan2a = $true )
& ( subbreln = $true )
& ( binunionTE = $true )
& ( eqinunit = $true )
& ( injFuncSetFuncInj = $true )
& ( wellorderingAx = $true )
& ( setminusER = $true )
& ( prop2setE = $true )
& ( complementTI1 = $true )
& ( complementTE1 = $true )
& ( setadjoinSub = $true )
& ( cartprodfstpairEq = $true )
& ( surjCantorThm = $true )
& ( setunionE = $true )
& ( subPowSU = $true )
& ( funcext2 = $true )
& ( cartprodfstin = $true )
& ( surjFuncSetFuncSurj = $true )
& ( woz1_2 = $true )
& ( subset2powerset = $true )
& ( ex1E1 = $true )
& ( subsetE = $true )
& ( lamProp = $true )
& ( woz1_3 = $true )
& ( powersetE1 = $true )
& ( breln1all2 = $true )
& ( binintersectSubset1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( setukpairIR = $true )
& ( setminusI = $true )
& ( cartprodpairsurjEq = $true )
& ( notdexE = $true )
& ( singletoninpowunion = $true )
& ( upairsetIL = $true )
& ( exu__Cong = $true )
& ( funcGraphProp1 = $true )
& ( binintersectSubset4 = $true )
& ( exuE2 = $true )
& ( exuI3 = $true )
& ( demorgan1a = $true )
& ( disjointsetsI1 = $true )
& ( complementTnotintersectT = $true )
& ( binintersectSubset2 = $true )
& ( breln1invI = $true )
& ( powerset__Cong = $true )
& ( binintersectEL = $true )
& ( woz1_1 = $true )
& ( descr__Cong = $true )
& ( setukpairinjL = $true )
& ( image1Ex = $true )
& ( breln1unionIR = $true )
& ( emptyI = $true )
& ( cartprodmempaircEq = $true )
& ( singletonsswitch = $true )
& ( notinself2 = $true )
& ( ex1I2 = $true )
& ( powersetI1 = $true )
& ( demorgan1b = $true )
& ( setoftrueEq = $true )
& ( powersetsubset = $true )
& ( subsetTrans = $true )
& ( woz13rule4 = $true )
& ( woz13rule0 = $true )
& ( ubforcartprodlem2 = $true )
& ( eqbreln1 = $true )
& ( binunionT_lem = $true )
& ( omegaSp = $true )
& ( binunionIL = $true )
& ( exuE3e = $true )
& ( dpsetconstrI = $true )
& ( noeltsimpempty = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( eta2 = $true )
& ( binintersectTERcontra = $true )
& ( beta2 = $true )
& ( setOfPairsIsBReln = $true )
& ( iftrue = $true )
& ( demorgan2a1 = $true )
& ( dpsetconstrER = $true )
& ( binintersectLsub = $true )
& ( lamp = $true )
& ( complementT_lem = $true )
& ( iftrueorfalse = $true )
& ( demorgan2b = $true )
& ( cartprodsndin = $true )
& ( image1I = $true )
& ( iffalseProp1 = $true )
& ( upairsetIR = $true )
& ( omega__Cong = $true )
& ( replAx = $true )
& ( sepInPowerset = $true )
& ( nonemptyImpWitness = $true )
& ( setextAx = $true )
& ( iftrueProp2 = $true )
& ( binintersectTELcontra = $true )
& ( upairset2E = $true )
& ( doubleComplementEq = $true )
& ( surjFuncSetFuncIn = $true )
& ( eqimpsubset1 = $true )
& ( emptyset__Cong = $true )
& ( setunionI = $true )
& ( emptysetsubset = $true )
& ( cartprodmempair = $true )
& ( exuI1 = $true )
& ( woz1_5 = $true )
& ( prop2setI = $true )
& ( emptyE1 = $true )
& ( breln1compE = $true )
& ( breln1unionCommutes = $true )
& ( demorgan1 = $true )
& ( injFuncInInjFuncSet = $true )
& ( funcextLem = $true )
& ( setminusT_lem = $true )
& ( emptysetE = $true )
& ( theprop = $true )
& ( doubleComplementSub1 = $true )
& ( binintersectSubset3 = $true )
& ( cartprodsndpairEq = $true )
& ( contrasubsetT2 = $true )
& ( cartprodmempair1 = $true )
& ( descrp = $true )
& ( breln1compI = $true )
& ( powersetAx = $true )
& ( injFuncSetFuncIn = $true )
& ( funcGraphProp3 = $true )
& ( upairsubunion = $true )
& ( uniqinunit = $true )
& ( iftrueProp1 = $true )
& ( woz13rule3 = $true )
& ( dsetconstrER = $true )
& ( setbeta = $true )
& ( setext = $true )
& ( setminusELneg = $true )
& ( setukpairinjL2 = $true )
& ( upairsetE = $true )
& ( contrasubsetT3 = $true )
& ( foundationAx = $true )
& ( notinemptyset = $true )
& ( setadjoinAx = $true )
& ( upairequniteq = $true )
& ( setadjoinSub2 = $true )
& ( breln1unionE = $true )
& ( choice2fnsingleton = $true )
& ( upairset2IR = $true )
& ( eqimpsubset2 = $true )
& ( exuI2 = $true )
& ( quantDeMorgan4 = $true )
& ( brelnall2 = $true )
& ( powersetTI1 = $true )
& ( binunionE = $true )
& ( binintersectER = $true )
& ( setunionE2 = $true )
& ( breln1SetBreln1 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( dpsetconstrEL1 = $true )
& ( ifSingleton = $true )
& ( apProp = $true )
& ( breln1unionI = $true )
& ( emptyinPowerset = $true )
& ( setminusLsub = $true )
& ( setukpairinjR = $true )
& ( inPowerset = $true )
& ( dpsetconstrEL2 = $true )
& ( iffalseProp2 = $true )
& ( notdallE = $true )
& ( setminusSubset1 = $true )
& ( breln1compEex = $true )
& ( breln1unionEcases = $true )
& ( subsetI1 = $true )
& ( kpairp = $true )
& ( doubleComplementE1 = $true )
& ( ap2apEq1 = $true )
& ( setminusILneg = $true )
& ( setunionsingleton = $true )
& ( demorgan2a2 = $true )
& ( breln1invprop = $true )
& ( nonemptyI = $true )
& ( woz13rule2 = $true )
& ( emptysetimpfalse = $true )
& ( setadjoin__Cong = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( theeq = $true )
& ( foundation2 = $true )
& ( setadjoinOr = $true )
& ( exuEu = $true ) ),
inference(flattening,[status(thm)],[f1006]) ).
thf(f1006,plain,
( ? [X0: $i] :
( ( $true
= ( in @ X0 @ omega ) )
& ( emptyset
= ( omegaS @ X0 ) ) )
& ( 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)],[f702]) ).
thf(f702,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 )
=> ! [X0: $i] :
( ( $true
= ( in @ X0 @ omega ) )
=> ( emptyset
!= ( omegaS @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f701]) ).
thf(f701,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
=> ! [X0: $i] :
( ( in @ X0 @ omega )
=> ( emptyset
!= ( omegaS @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f331]) ).
thf(f331,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
=> ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( emptyset
!= ( omegaS @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f330]) ).
thf(f330,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
=> ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( emptyset
!= ( omegaS @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.y5OtfSgMuG/DTF2THF_27360.p',peano0notS) ).
thf(f1521,plain,
( setadjoinIL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y1 @ Y0 ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f432]) ).
thf(f432,plain,
( setadjoinIL
= ( !! @ $i
@ ^ [Y0: $i] :
( !! @ $i
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y1 @ Y0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f431]) ).
thf(f431,plain,
( setadjoinIL
= ( ! [X0: $i,X1: $i] : ( in @ X0 @ ( setadjoin @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f31]) ).
thf(f31,axiom,
( setadjoinIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.y5OtfSgMuG/DTF2THF_27360.p',setadjoinIL) ).
thf(f994,plain,
! [X0: $tType,X2: X0,X1: X0 > $o] :
( ( $false
= ( !! @ X0 @ X1 ) )
| ( ( X1 @ X2 )
= $true ) ),
introduced(definition,[],[pi_proxy_axiom]) ).
thf(f2205,plain,
! [X0: $i] :
( ( in @ X0 @ ( setadjoin @ sK2 @ sK2 ) )
= $false ),
inference(trivial_inequality_removal,[status(thm)],[f2202]) ).
thf(f2202,plain,
! [X0: $i] :
( ( $true = $false )
| ( ( in @ X0 @ ( setadjoin @ sK2 @ sK2 ) )
= $false ) ),
inference(superposition,[status(thm)],[f2157,f992]) ).
thf(f992,plain,
! [X1: $o] :
( ( $false = ( ~ X1 ) )
| ( $false = X1 ) ),
introduced(definition,[],[logical_not_proxy_axiom]) ).
thf(f2157,plain,
! [X0: $i] :
( $true
= ( ~ ( in @ X0 @ ( setadjoin @ sK2 @ sK2 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2156]) ).
thf(f2156,plain,
! [X0: $i] :
( $true
= ( ^ [Y0: $i] :
~ ( in @ Y0 @ ( setadjoin @ sK2 @ sK2 ) )
@ X0 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2152]) ).
thf(f2152,plain,
! [X0: $i] :
( ( $true
= ( ^ [Y0: $i] :
~ ( in @ Y0 @ ( setadjoin @ sK2 @ sK2 ) )
@ X0 ) )
| ( $true = $false ) ),
inference(superposition,[status(thm)],[f2065,f994]) ).
thf(f2065,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ ( setadjoin @ sK2 @ sK2 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1773]) ).
thf(f1773,plain,
( ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0
@ ( ^ [Y1: $i] : ( setadjoin @ Y1 @ Y1 )
@ sK2 ) ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1425,f1306,f1667]) ).
thf(f1667,plain,
( emptyset
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 )
@ sK2 ) ),
inference(definition_unfolding,[status(thm)],[f1402,f1560]) ).
thf(f1560,plain,
( omegaS
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f764]) ).
thf(f764,plain,
( omegaS
= ( ^ [Y0: $i] : ( setadjoin @ Y0 @ Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f763]) ).
thf(f763,plain,
( omegaS
= ( ^ [X0: $i] : ( setadjoin @ X0 @ X0 ) ) ),
inference(rectify,[status(thm)],[f327]) ).
thf(f327,axiom,
( omegaS
= ( ^ [X1: $i] : ( setadjoin @ X1 @ X1 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.y5OtfSgMuG/DTF2THF_27360.p',omegaS) ).
thf(f1402,plain,
( emptyset
= ( omegaS @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f1009]) ).
thf(f1306,plain,
emptysetAx = $true,
inference(cnf_transformation,[status(thm)],[f1009]) ).
thf(f1425,plain,
( emptysetAx
= ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset ) ) ),
inference(cnf_transformation,[status(thm)],[f676]) ).
thf(f676,plain,
( emptysetAx
= ( !! @ $i
@ ^ [Y0: $i] :
~ ( in @ Y0 @ emptyset ) ) ),
inference(fool_elimination,[status(thm)],[f675]) ).
thf(f675,plain,
! [X0: $i] :
( ( ~ ( in @ X0 @ emptyset ) )
= emptysetAx ),
inference(rectify,[status(thm)],[f3]) ).
thf(f3,axiom,
! [X1: $i] :
( ( ~ ( in @ X1 @ emptyset ) )
= emptysetAx ),
file('/export/starexec/sandbox/tmp/tmp.y5OtfSgMuG/DTF2THF_27360.p',emptysetAx) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU810^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n029.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 12:34:44 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.48 ---- Original DTF file ---
% 0.19/0.48 thf(spec,logic,$$dhol).
% 0.19/0.48 %------------------------------------------------------------------------------
% 0.19/0.48 % File : SEU810^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.48 % Domain : Set Theory
% 0.19/0.48 % Problem : Omega and Peano
% 0.19/0.48 % Version : Especial.
% 0.19/0.48 % English : (! x:i.in x omega -> ~(omegaS x = emptyset))
% 0.19/0.48
% 0.19/0.48 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.48 % Source : [Bro08]
% 0.19/0.48 % Names : ZFC312g [Bro08]
% 0.19/0.48
% 0.19/0.48 % Status : Theorem
% 0.19/0.48 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v5.1.0, 1.00 v5.0.0, 0.80 v4.1.0, 0.67 v3.7.0
% 0.19/0.48 % Syntax : Number of formulae : 710 ( 329 unt; 380 typ; 329 def)
% 0.19/0.48 % Number of atoms : 2019 ( 472 equ; 0 cnn)
% 0.19/0.48 % Maximal formula atoms : 328 ( 6 avg)
% 0.19/0.48 % Number of connectives : 4417 ( 77 ~; 14 |; 54 &;3214 @)
% 0.19/0.48 % ( 17 <=>;1041 =>; 0 <=; 0 <~>)
% 0.19/0.48 % Maximal formula depth : 331 ( 2 avg)
% 0.19/0.48 % Number of types : 2 ( 0 usr)
% 0.19/0.48 % Number of type conns : 177 ( 177 >; 0 *; 0 +; 0 <<)
% 0.19/0.48 % Number of symbols : 383 ( 380 usr; 330 con; 0-4 aty)
% 0.19/0.48 % Number of variables : 1150 ( 76 ^;1030 !; 44 ?;1150 :)
% 0.19/0.48 % SPC : TH0_THM_EQU_NAR
% 0.19/0.48
% 0.19/0.48 % Comments : http://mathgate.info/detsetitem.php?id=443
% 0.19/0.48 %------------------------------------------------------------------------------
% 0.19/0.48 thf(in_type,type,
% 0.19/0.48 in: $i > $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(exu_type,type,
% 0.19/0.48 exu: ( $i > $o ) > $o ).
% 0.19/0.48
% 0.19/0.48 thf(exu,definition,
% 0.19/0.48 ( exu
% 0.19/0.48 = ( ^ [Xphi: $i > $o] :
% 0.19/0.48 ? [Xx: $i] :
% 0.19/0.48 ( ( Xphi @ Xx )
% 0.19/0.48 & ! [Xy: $i] :
% 0.19/0.48 ( ( Xphi @ Xy )
% 0.19/0.48 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setextAx_type,type,
% 0.19/0.48 setextAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setextAx,definition,
% 0.19/0.48 ( setextAx
% 0.19/0.48 = ( ! [A: $i,B: $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 <=> ( in @ Xx @ B ) )
% 0.19/0.48 => ( A = B ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(emptyset_type,type,
% 0.19/0.48 emptyset: $i ).
% 0.19/0.48
% 0.19/0.48 thf(emptysetAx_type,type,
% 0.19/0.48 emptysetAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(emptysetAx,definition,
% 0.19/0.48 ( emptysetAx
% 0.19/0.48 = ( ! [Xx: $i] :
% 0.19/0.48 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoin_type,type,
% 0.19/0.48 setadjoin: $i > $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoinAx_type,type,
% 0.19/0.48 setadjoinAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoinAx,definition,
% 0.19/0.48 ( setadjoinAx
% 0.19/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.48 <=> ( ( Xy = Xx )
% 0.19/0.48 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(powerset_type,type,
% 0.19/0.48 powerset: $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(powersetAx_type,type,
% 0.19/0.48 powersetAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(powersetAx,definition,
% 0.19/0.48 ( powersetAx
% 0.19/0.48 = ( ! [A: $i,B: $i] :
% 0.19/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.48 <=> ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ B )
% 0.19/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setunion_type,type,
% 0.19/0.48 setunion: $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(setunionAx_type,type,
% 0.19/0.48 setunionAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setunionAx,definition,
% 0.19/0.48 ( setunionAx
% 0.19/0.48 = ( ! [A: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.48 <=> ? [B: $i] :
% 0.19/0.48 ( ( in @ Xx @ B )
% 0.19/0.48 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(omega_type,type,
% 0.19/0.48 omega: $i ).
% 0.19/0.48
% 0.19/0.48 thf(omega0Ax_type,type,
% 0.19/0.48 omega0Ax: $o ).
% 0.19/0.48
% 0.19/0.48 thf(omega0Ax,definition,
% 0.19/0.48 ( omega0Ax
% 0.19/0.48 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(omegaSAx_type,type,
% 0.19/0.48 omegaSAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(omegaSAx,definition,
% 0.19/0.48 ( omegaSAx
% 0.19/0.48 = ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ omega )
% 0.19/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(omegaIndAx_type,type,
% 0.19/0.48 omegaIndAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(omegaIndAx,definition,
% 0.19/0.48 ( omegaIndAx
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ( ( ( in @ emptyset @ A )
% 0.19/0.48 & ! [Xx: $i] :
% 0.19/0.48 ( ( ( in @ Xx @ omega )
% 0.19/0.48 & ( in @ Xx @ A ) )
% 0.19/0.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ omega )
% 0.19/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(replAx_type,type,
% 0.19/0.48 replAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(replAx,definition,
% 0.19/0.48 ( replAx
% 0.19/0.48 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( exu
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.48 => ? [B: $i] :
% 0.19/0.48 ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ B )
% 0.19/0.48 <=> ? [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(foundationAx_type,type,
% 0.19/0.48 foundationAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(foundationAx,definition,
% 0.19/0.48 ( foundationAx
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.48 => ? [B: $i] :
% 0.19/0.48 ( ( in @ B @ A )
% 0.19/0.48 & ~ ? [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ B )
% 0.19/0.48 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(wellorderingAx_type,type,
% 0.19/0.48 wellorderingAx: $o ).
% 0.19/0.48
% 0.19/0.48 thf(wellorderingAx,definition,
% 0.19/0.48 ( wellorderingAx
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ? [B: $i] :
% 0.19/0.48 ( ! [C: $i] :
% 0.19/0.48 ( ( in @ C @ B )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ C )
% 0.19/0.48 => ( in @ Xx @ A ) ) )
% 0.19/0.48 & ! [Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( ( in @ Xx @ A )
% 0.19/0.48 & ( in @ Xy @ A ) )
% 0.19/0.48 => ( ! [C: $i] :
% 0.19/0.48 ( ( in @ C @ B )
% 0.19/0.48 => ( ( in @ Xx @ C )
% 0.19/0.48 <=> ( in @ Xy @ C ) ) )
% 0.19/0.48 => ( Xx = Xy ) ) )
% 0.19/0.48 & ! [C: $i,D: $i] :
% 0.19/0.48 ( ( ( in @ C @ B )
% 0.19/0.48 & ( in @ D @ B ) )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ C )
% 0.19/0.48 => ( in @ Xx @ D ) )
% 0.19/0.48 | ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ D )
% 0.19/0.48 => ( in @ Xx @ C ) ) ) )
% 0.19/0.48 & ! [C: $i] :
% 0.19/0.48 ( ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ C )
% 0.19/0.48 => ( in @ Xx @ A ) )
% 0.19/0.48 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.48 => ? [D: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ D @ B )
% 0.19/0.48 & ( in @ Xx @ C )
% 0.19/0.48 & ~ ? [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ D )
% 0.19/0.48 & ( in @ Xy @ C ) )
% 0.19/0.48 & ! [E: $i] :
% 0.19/0.48 ( ( in @ E @ B )
% 0.19/0.48 => ( ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ E )
% 0.19/0.48 => ( in @ Xy @ D ) )
% 0.19/0.48 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(descr_type,type,
% 0.19/0.48 descr: ( $i > $o ) > $i ).
% 0.19/0.48
% 0.19/0.48 thf(descrp_type,type,
% 0.19/0.48 descrp: $o ).
% 0.19/0.48
% 0.19/0.48 thf(descrp,definition,
% 0.19/0.48 ( descrp
% 0.19/0.48 = ( ! [Xphi: $i > $o] :
% 0.19/0.48 ( ( exu
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.48 => ( Xphi
% 0.19/0.48 @ ( descr
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstr_type,type,
% 0.19/0.48 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrI_type,type,
% 0.19/0.48 dsetconstrI: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrI,definition,
% 0.19/0.48 ( dsetconstrI
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( Xphi @ Xx )
% 0.19/0.48 => ( in @ Xx
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrEL_type,type,
% 0.19/0.48 dsetconstrEL: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrEL,definition,
% 0.19/0.48 ( dsetconstrEL
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrER_type,type,
% 0.19/0.48 dsetconstrER: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dsetconstrER,definition,
% 0.19/0.48 ( dsetconstrER
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.48 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(exuE1_type,type,
% 0.19/0.48 exuE1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(exuE1,definition,
% 0.19/0.48 ( exuE1
% 0.19/0.48 = ( ! [Xphi: $i > $o] :
% 0.19/0.48 ( ( exu
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.48 => ? [Xx: $i] :
% 0.19/0.48 ( ( Xphi @ Xx )
% 0.19/0.48 & ! [Xy: $i] :
% 0.19/0.48 ( ( Xphi @ Xy )
% 0.19/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(prop2set_type,type,
% 0.19/0.48 prop2set: $o > $i ).
% 0.19/0.48
% 0.19/0.48 thf(prop2setE_type,type,
% 0.19/0.48 prop2setE: $o ).
% 0.19/0.48
% 0.19/0.48 thf(prop2setE,definition,
% 0.19/0.48 ( prop2setE
% 0.19/0.48 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.48 => Xphi ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(emptysetE_type,type,
% 0.19/0.48 emptysetE: $o ).
% 0.19/0.48
% 0.19/0.48 thf(emptysetE,definition,
% 0.19/0.48 ( emptysetE
% 0.19/0.48 = ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ emptyset )
% 0.19/0.48 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(emptysetimpfalse_type,type,
% 0.19/0.48 emptysetimpfalse: $o ).
% 0.19/0.48
% 0.19/0.48 thf(emptysetimpfalse,definition,
% 0.19/0.48 ( emptysetimpfalse
% 0.19/0.48 = ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ emptyset )
% 0.19/0.48 => $false ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(notinemptyset_type,type,
% 0.19/0.48 notinemptyset: $o ).
% 0.19/0.48
% 0.19/0.48 thf(notinemptyset,definition,
% 0.19/0.48 ( notinemptyset
% 0.19/0.48 = ( ! [Xx: $i] :
% 0.19/0.48 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(exuE3e_type,type,
% 0.19/0.48 exuE3e: $o ).
% 0.19/0.48
% 0.19/0.48 thf(exuE3e,definition,
% 0.19/0.48 ( exuE3e
% 0.19/0.48 = ( ! [Xphi: $i > $o] :
% 0.19/0.48 ( ( exu
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.48 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setext_type,type,
% 0.19/0.48 setext: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setext,definition,
% 0.19/0.48 ( setext
% 0.19/0.48 = ( ! [A: $i,B: $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ Xx @ B ) )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ B )
% 0.19/0.48 => ( in @ Xx @ A ) )
% 0.19/0.48 => ( A = B ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(emptyI_type,type,
% 0.19/0.48 emptyI: $o ).
% 0.19/0.48
% 0.19/0.48 thf(emptyI,definition,
% 0.19/0.48 ( emptyI
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ~ ( in @ Xx @ A )
% 0.19/0.48 => ( A = emptyset ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(noeltsimpempty_type,type,
% 0.19/0.48 noeltsimpempty: $o ).
% 0.19/0.48
% 0.19/0.48 thf(noeltsimpempty,definition,
% 0.19/0.48 ( noeltsimpempty
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ~ ( in @ Xx @ A )
% 0.19/0.48 => ( A = emptyset ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setbeta_type,type,
% 0.19/0.48 setbeta: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setbeta,definition,
% 0.19/0.48 ( setbeta
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.48 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(nonempty_type,type,
% 0.19/0.48 nonempty: $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(nonempty,definition,
% 0.19/0.48 ( nonempty
% 0.19/0.48 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(nonemptyE1_type,type,
% 0.19/0.48 nonemptyE1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(nonemptyE1,definition,
% 0.19/0.48 ( nonemptyE1
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ( ( nonempty @ A )
% 0.19/0.48 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(nonemptyI_type,type,
% 0.19/0.48 nonemptyI: $o ).
% 0.19/0.48
% 0.19/0.48 thf(nonemptyI,definition,
% 0.19/0.48 ( nonemptyI
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( Xphi @ Xx )
% 0.19/0.48 => ( nonempty
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(nonemptyI1_type,type,
% 0.19/0.48 nonemptyI1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(nonemptyI1,definition,
% 0.19/0.48 ( nonemptyI1
% 0.19/0.48 = ( ! [A: $i] :
% 0.19/0.48 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.48 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoinIL_type,type,
% 0.19/0.48 setadjoinIL: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoinIL,definition,
% 0.19/0.48 ( setadjoinIL
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(emptyinunitempty_type,type,
% 0.19/0.48 emptyinunitempty: $o ).
% 0.19/0.48
% 0.19/0.48 thf(emptyinunitempty,definition,
% 0.19/0.48 ( emptyinunitempty
% 0.19/0.48 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoinIR_type,type,
% 0.19/0.48 setadjoinIR: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setadjoinIR,definition,
% 0.19/0.48 ( setadjoinIR
% 0.19/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ A )
% 0.32/0.48 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoinE_type,type,
% 0.32/0.48 setadjoinE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoinE,definition,
% 0.32/0.48 ( setadjoinE
% 0.32/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.48 => ! [Xphi: $o] :
% 0.32/0.48 ( ( ( Xy = Xx )
% 0.32/0.48 => Xphi )
% 0.32/0.48 => ( ( ( in @ Xy @ A )
% 0.32/0.48 => Xphi )
% 0.32/0.48 => Xphi ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoinOr_type,type,
% 0.32/0.48 setadjoinOr: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoinOr,definition,
% 0.32/0.48 ( setadjoinOr
% 0.32/0.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.48 => ( ( Xy = Xx )
% 0.32/0.48 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setoftrueEq_type,type,
% 0.32/0.48 setoftrueEq: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setoftrueEq,definition,
% 0.32/0.48 ( setoftrueEq
% 0.32/0.48 = ( ! [A: $i] :
% 0.32/0.48 ( ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xx: $i] : $true )
% 0.32/0.48 = A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(powersetI_type,type,
% 0.32/0.48 powersetI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(powersetI,definition,
% 0.32/0.48 ( powersetI
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ B )
% 0.32/0.48 => ( in @ Xx @ A ) )
% 0.32/0.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(emptyinPowerset_type,type,
% 0.32/0.48 emptyinPowerset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(emptyinPowerset,definition,
% 0.32/0.48 ( emptyinPowerset
% 0.32/0.48 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(emptyInPowerset_type,type,
% 0.32/0.48 emptyInPowerset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(emptyInPowerset,definition,
% 0.32/0.48 ( emptyInPowerset
% 0.32/0.48 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(powersetE_type,type,
% 0.32/0.48 powersetE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(powersetE,definition,
% 0.32/0.48 ( powersetE
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.48 => ( ( in @ Xx @ B )
% 0.32/0.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setunionI_type,type,
% 0.32/0.48 setunionI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setunionI,definition,
% 0.32/0.48 ( setunionI
% 0.32/0.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.48 ( ( in @ Xx @ B )
% 0.32/0.48 => ( ( in @ B @ A )
% 0.32/0.48 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setunionE_type,type,
% 0.32/0.48 setunionE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setunionE,definition,
% 0.32/0.48 ( setunionE
% 0.32/0.48 = ( ! [A: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.48 => ! [Xphi: $o] :
% 0.32/0.48 ( ! [B: $i] :
% 0.32/0.48 ( ( in @ Xx @ B )
% 0.32/0.48 => ( ( in @ B @ A )
% 0.32/0.48 => Xphi ) )
% 0.32/0.48 => Xphi ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subPowSU_type,type,
% 0.32/0.48 subPowSU: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subPowSU,definition,
% 0.32/0.48 ( subPowSU
% 0.32/0.48 = ( ! [A: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(exuE2_type,type,
% 0.32/0.48 exuE2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(exuE2,definition,
% 0.32/0.48 ( exuE2
% 0.32/0.48 = ( ! [Xphi: $i > $o] :
% 0.32/0.48 ( ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 => ? [Xx: $i] :
% 0.32/0.48 ! [Xy: $i] :
% 0.32/0.48 ( ( Xphi @ Xy )
% 0.32/0.48 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(nonemptyImpWitness_type,type,
% 0.32/0.48 nonemptyImpWitness: $o ).
% 0.32/0.48
% 0.32/0.48 thf(nonemptyImpWitness,definition,
% 0.32/0.48 ( nonemptyImpWitness
% 0.32/0.48 = ( ! [A: $i] :
% 0.32/0.48 ( ( nonempty @ A )
% 0.32/0.48 => ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & $true ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(uniqinunit_type,type,
% 0.32/0.48 uniqinunit: $o ).
% 0.32/0.48
% 0.32/0.48 thf(uniqinunit,definition,
% 0.32/0.48 ( uniqinunit
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(notinsingleton_type,type,
% 0.32/0.48 notinsingleton: $o ).
% 0.32/0.48
% 0.32/0.48 thf(notinsingleton,definition,
% 0.32/0.48 ( notinsingleton
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xx != Xy )
% 0.32/0.48 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(eqinunit_type,type,
% 0.32/0.48 eqinunit: $o ).
% 0.32/0.48
% 0.32/0.48 thf(eqinunit,definition,
% 0.32/0.48 ( eqinunit
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xx = Xy )
% 0.32/0.48 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(singletonsswitch_type,type,
% 0.32/0.48 singletonsswitch: $o ).
% 0.32/0.48
% 0.32/0.48 thf(singletonsswitch,definition,
% 0.32/0.48 ( singletonsswitch
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.48 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(upairsetE_type,type,
% 0.32/0.48 upairsetE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(upairsetE,definition,
% 0.32/0.48 ( upairsetE
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/0.48 => ( ( Xz = Xx )
% 0.32/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(upairsetIL_type,type,
% 0.32/0.48 upairsetIL: $o ).
% 0.32/0.48
% 0.32/0.48 thf(upairsetIL,definition,
% 0.32/0.48 ( upairsetIL
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(upairsetIR_type,type,
% 0.32/0.48 upairsetIR: $o ).
% 0.32/0.48
% 0.32/0.48 thf(upairsetIR,definition,
% 0.32/0.48 ( upairsetIR
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(emptyE1_type,type,
% 0.32/0.48 emptyE1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(emptyE1,definition,
% 0.32/0.48 ( emptyE1
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.48 ( ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & ( Xphi @ Xx ) )
% 0.32/0.48 => ( ( ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 = emptyset )
% 0.32/0.48 => $false ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(vacuousDall_type,type,
% 0.32/0.48 vacuousDall: $o ).
% 0.32/0.48
% 0.32/0.48 thf(vacuousDall,definition,
% 0.32/0.48 ( vacuousDall
% 0.32/0.48 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ emptyset )
% 0.32/0.48 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(quantDeMorgan1_type,type,
% 0.32/0.48 quantDeMorgan1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(quantDeMorgan1,definition,
% 0.32/0.48 ( quantDeMorgan1
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.48 ( ~ ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( Xphi @ Xx ) )
% 0.32/0.48 => ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(quantDeMorgan2_type,type,
% 0.32/0.48 quantDeMorgan2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(quantDeMorgan2,definition,
% 0.32/0.48 ( quantDeMorgan2
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.48 ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ~ ( Xphi @ Xx ) )
% 0.32/0.48 => ~ ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(quantDeMorgan3_type,type,
% 0.32/0.48 quantDeMorgan3: $o ).
% 0.32/0.48
% 0.32/0.48 thf(quantDeMorgan3,definition,
% 0.32/0.48 ( quantDeMorgan3
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.48 ( ~ ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & ( Xphi @ Xx ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(quantDeMorgan4_type,type,
% 0.32/0.48 quantDeMorgan4: $o ).
% 0.32/0.48
% 0.32/0.48 thf(quantDeMorgan4,definition,
% 0.32/0.48 ( quantDeMorgan4
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.48 ( ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & ~ ( Xphi @ Xx ) )
% 0.32/0.48 => ~ ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(prop2setI_type,type,
% 0.32/0.48 prop2setI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(prop2setI,definition,
% 0.32/0.48 ( prop2setI
% 0.32/0.48 = ( ! [Xphi: $o] :
% 0.32/0.48 ( Xphi
% 0.32/0.48 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(set2prop_type,type,
% 0.32/0.48 set2prop: $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(prop2set2propI_type,type,
% 0.32/0.48 prop2set2propI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(prop2set2propI,definition,
% 0.32/0.48 ( prop2set2propI
% 0.32/0.48 = ( ! [Xphi: $o] :
% 0.32/0.48 ( Xphi
% 0.32/0.48 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(notdexE_type,type,
% 0.32/0.48 notdexE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(notdexE,definition,
% 0.32/0.48 ( notdexE
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.48 ( ~ ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & ( Xphi @ Xx ) )
% 0.32/0.48 => ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(notdallE_type,type,
% 0.32/0.48 notdallE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(notdallE,definition,
% 0.32/0.48 ( notdallE
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.48 ( ~ ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( Xphi @ Xx ) )
% 0.32/0.48 => ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(exuI1_type,type,
% 0.32/0.48 exuI1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(exuI1,definition,
% 0.32/0.48 ( exuI1
% 0.32/0.48 = ( ! [Xphi: $i > $o] :
% 0.32/0.48 ( ? [Xx: $i] :
% 0.32/0.48 ( ( Xphi @ Xx )
% 0.32/0.48 & ! [Xy: $i] :
% 0.32/0.48 ( ( Xphi @ Xy )
% 0.32/0.48 => ( Xx = Xy ) ) )
% 0.32/0.48 => ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(exuI3_type,type,
% 0.32/0.48 exuI3: $o ).
% 0.32/0.48
% 0.32/0.48 thf(exuI3,definition,
% 0.32/0.48 ( exuI3
% 0.32/0.48 = ( ! [Xphi: $i > $o] :
% 0.32/0.48 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.32/0.48 => ( ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xphi @ Xx )
% 0.32/0.48 => ( ( Xphi @ Xy )
% 0.32/0.48 => ( Xx = Xy ) ) )
% 0.32/0.48 => ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(exuI2_type,type,
% 0.32/0.48 exuI2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(exuI2,definition,
% 0.32/0.48 ( exuI2
% 0.32/0.48 = ( ! [Xphi: $i > $o] :
% 0.32/0.48 ( ? [Xx: $i] :
% 0.32/0.48 ! [Xy: $i] :
% 0.32/0.48 ( ( Xphi @ Xy )
% 0.32/0.48 <=> ( Xy = Xx ) )
% 0.32/0.48 => ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(inCongP_type,type,
% 0.32/0.48 inCongP: $o ).
% 0.32/0.48
% 0.32/0.48 thf(inCongP,definition,
% 0.32/0.48 ( inCongP
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xx = Xy )
% 0.32/0.48 => ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(in__Cong_type,type,
% 0.32/0.48 in__Cong: $o ).
% 0.32/0.48
% 0.32/0.48 thf(in__Cong,definition,
% 0.32/0.48 ( in__Cong
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xx = Xy )
% 0.32/0.48 => ( ( in @ Xx @ A )
% 0.32/0.48 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(exuE3u_type,type,
% 0.32/0.48 exuE3u: $o ).
% 0.32/0.48
% 0.32/0.48 thf(exuE3u,definition,
% 0.32/0.48 ( exuE3u
% 0.32/0.48 = ( ! [Xphi: $i > $o] :
% 0.32/0.48 ( ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 => ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xphi @ Xx )
% 0.32/0.48 => ( ( Xphi @ Xy )
% 0.32/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(exu__Cong_type,type,
% 0.32/0.48 exu__Cong: $o ).
% 0.32/0.48
% 0.32/0.48 thf(exu__Cong,definition,
% 0.32/0.48 ( exu__Cong
% 0.32/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xx = Xy )
% 0.32/0.48 => ( ( Xphi @ Xx )
% 0.32/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.48 => ( ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 <=> ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(emptyset__Cong_type,type,
% 0.32/0.48 emptyset__Cong: $o ).
% 0.32/0.48
% 0.32/0.48 thf(emptyset__Cong,definition,
% 0.32/0.48 ( emptyset__Cong
% 0.32/0.48 = ( emptyset = emptyset ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoin__Cong_type,type,
% 0.32/0.48 setadjoin__Cong: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoin__Cong,definition,
% 0.32/0.48 ( setadjoin__Cong
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xx = Xy )
% 0.32/0.48 => ! [Xz: $i,Xu: $i] :
% 0.32/0.48 ( ( Xz = Xu )
% 0.32/0.48 => ( ( setadjoin @ Xx @ Xz )
% 0.32/0.48 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(powerset__Cong_type,type,
% 0.32/0.48 powerset__Cong: $o ).
% 0.32/0.48
% 0.32/0.48 thf(powerset__Cong,definition,
% 0.32/0.48 ( powerset__Cong
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ( ( powerset @ A )
% 0.32/0.48 = ( powerset @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setunion__Cong_type,type,
% 0.32/0.48 setunion__Cong: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setunion__Cong,definition,
% 0.32/0.48 ( setunion__Cong
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ( ( setunion @ A )
% 0.32/0.48 = ( setunion @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(omega__Cong_type,type,
% 0.32/0.48 omega__Cong: $o ).
% 0.32/0.48
% 0.32/0.48 thf(omega__Cong,definition,
% 0.32/0.48 ( omega__Cong
% 0.32/0.48 = ( omega = omega ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(exuEu_type,type,
% 0.32/0.48 exuEu: $o ).
% 0.32/0.48
% 0.32/0.48 thf(exuEu,definition,
% 0.32/0.48 ( exuEu
% 0.32/0.48 = ( ! [Xphi: $i > $o] :
% 0.32/0.48 ( ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 => ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xphi @ Xx )
% 0.32/0.48 => ( ( Xphi @ Xy )
% 0.32/0.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(descr__Cong_type,type,
% 0.32/0.48 descr__Cong: $o ).
% 0.32/0.48
% 0.32/0.48 thf(descr__Cong,definition,
% 0.32/0.48 ( descr__Cong
% 0.32/0.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.48 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.48 ( ( Xx = Xy )
% 0.32/0.48 => ( ( Xphi @ Xx )
% 0.32/0.48 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.48 => ( ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 => ( ( exu
% 0.32/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.32/0.48 => ( ( descr
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 = ( descr
% 0.32/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(dsetconstr__Cong_type,type,
% 0.32/0.48 dsetconstr__Cong: $o ).
% 0.32/0.48
% 0.32/0.48 thf(dsetconstr__Cong,definition,
% 0.32/0.48 ( dsetconstr__Cong
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.48 ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ B )
% 0.32/0.48 => ( ( Xx = Xy )
% 0.32/0.48 => ( ( Xphi @ Xx )
% 0.32/0.48 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.32/0.48 => ( ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 = ( dsetconstr @ B
% 0.32/0.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subset_type,type,
% 0.32/0.48 subset: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(disjoint_type,type,
% 0.32/0.48 disjoint: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(setsmeet_type,type,
% 0.32/0.48 setsmeet: $i > $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(subsetI1_type,type,
% 0.32/0.48 subsetI1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subsetI1,definition,
% 0.32/0.48 ( subsetI1
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ Xx @ B ) )
% 0.32/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(eqimpsubset2_type,type,
% 0.32/0.48 eqimpsubset2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(eqimpsubset2,definition,
% 0.32/0.48 ( eqimpsubset2
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(eqimpsubset1_type,type,
% 0.32/0.48 eqimpsubset1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(eqimpsubset1,definition,
% 0.32/0.48 ( eqimpsubset1
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( A = B )
% 0.32/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subsetI2_type,type,
% 0.32/0.48 subsetI2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subsetI2,definition,
% 0.32/0.48 ( subsetI2
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ! [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ Xx @ B ) )
% 0.32/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(emptysetsubset_type,type,
% 0.32/0.48 emptysetsubset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(emptysetsubset,definition,
% 0.32/0.48 ( emptysetsubset
% 0.32/0.48 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subsetE_type,type,
% 0.32/0.48 subsetE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subsetE,definition,
% 0.32/0.48 ( subsetE
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( subset @ A @ B )
% 0.32/0.48 => ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subsetE2_type,type,
% 0.32/0.48 subsetE2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subsetE2,definition,
% 0.32/0.48 ( subsetE2
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( subset @ A @ B )
% 0.32/0.48 => ( ~ ( in @ Xx @ B )
% 0.32/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(notsubsetI_type,type,
% 0.32/0.48 notsubsetI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(notsubsetI,definition,
% 0.32/0.48 ( notsubsetI
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ B )
% 0.32/0.48 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(notequalI1_type,type,
% 0.32/0.48 notequalI1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(notequalI1,definition,
% 0.32/0.48 ( notequalI1
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ~ ( subset @ A @ B )
% 0.32/0.48 => ( A != B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(notequalI2_type,type,
% 0.32/0.48 notequalI2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(notequalI2,definition,
% 0.32/0.48 ( notequalI2
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ B )
% 0.32/0.48 => ( A != B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subsetRefl_type,type,
% 0.32/0.48 subsetRefl: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subsetRefl,definition,
% 0.32/0.48 ( subsetRefl
% 0.32/0.48 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subsetTrans_type,type,
% 0.32/0.48 subsetTrans: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subsetTrans,definition,
% 0.32/0.48 ( subsetTrans
% 0.32/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.48 ( ( subset @ A @ B )
% 0.32/0.48 => ( ( subset @ B @ C )
% 0.32/0.48 => ( subset @ A @ C ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoinSub_type,type,
% 0.32/0.48 setadjoinSub: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoinSub,definition,
% 0.32/0.48 ( setadjoinSub
% 0.32/0.48 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoinSub2_type,type,
% 0.32/0.48 setadjoinSub2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setadjoinSub2,definition,
% 0.32/0.48 ( setadjoinSub2
% 0.32/0.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.48 ( ( subset @ A @ B )
% 0.32/0.48 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subset2powerset_type,type,
% 0.32/0.48 subset2powerset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subset2powerset,definition,
% 0.32/0.48 ( subset2powerset
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( subset @ A @ B )
% 0.32/0.48 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setextsub_type,type,
% 0.32/0.48 setextsub: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setextsub,definition,
% 0.32/0.48 ( setextsub
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( subset @ A @ B )
% 0.32/0.48 => ( ( subset @ B @ A )
% 0.32/0.48 => ( A = B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(subsetemptysetimpeq_type,type,
% 0.32/0.48 subsetemptysetimpeq: $o ).
% 0.32/0.48
% 0.32/0.48 thf(subsetemptysetimpeq,definition,
% 0.32/0.48 ( subsetemptysetimpeq
% 0.32/0.48 = ( ! [A: $i] :
% 0.32/0.48 ( ( subset @ A @ emptyset )
% 0.32/0.48 => ( A = emptyset ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(powersetI1_type,type,
% 0.32/0.48 powersetI1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(powersetI1,definition,
% 0.32/0.48 ( powersetI1
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( subset @ B @ A )
% 0.32/0.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(powersetE1_type,type,
% 0.32/0.48 powersetE1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(powersetE1,definition,
% 0.32/0.48 ( powersetE1
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(inPowerset_type,type,
% 0.32/0.48 inPowerset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(inPowerset,definition,
% 0.32/0.48 ( inPowerset
% 0.32/0.48 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(powersetsubset_type,type,
% 0.32/0.48 powersetsubset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(powersetsubset,definition,
% 0.32/0.48 ( powersetsubset
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( subset @ A @ B )
% 0.32/0.48 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(sepInPowerset_type,type,
% 0.32/0.48 sepInPowerset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(sepInPowerset,definition,
% 0.32/0.48 ( sepInPowerset
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.48 ( in
% 0.32/0.48 @ ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 @ ( powerset @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(sepSubset_type,type,
% 0.32/0.48 sepSubset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(sepSubset,definition,
% 0.32/0.48 ( sepSubset
% 0.32/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.48 ( subset
% 0.32/0.48 @ ( dsetconstr @ A
% 0.32/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.48 @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunion_type,type,
% 0.32/0.48 binunion: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(binunionIL_type,type,
% 0.32/0.48 binunionIL: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionIL,definition,
% 0.32/0.48 ( binunionIL
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(upairset2IR_type,type,
% 0.32/0.48 upairset2IR: $o ).
% 0.32/0.48
% 0.32/0.48 thf(upairset2IR,definition,
% 0.32/0.48 ( upairset2IR
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionIR_type,type,
% 0.32/0.48 binunionIR: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionIR,definition,
% 0.32/0.48 ( binunionIR
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ B )
% 0.32/0.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionEcases_type,type,
% 0.32/0.48 binunionEcases: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionEcases,definition,
% 0.32/0.48 ( binunionEcases
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.32/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/0.48 => ( ( ( in @ Xx @ A )
% 0.32/0.48 => Xphi )
% 0.32/0.48 => ( ( ( in @ Xx @ B )
% 0.32/0.48 => Xphi )
% 0.32/0.48 => Xphi ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionE_type,type,
% 0.32/0.48 binunionE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionE,definition,
% 0.32/0.48 ( binunionE
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/0.48 => ( ( in @ Xx @ A )
% 0.32/0.48 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionLsub_type,type,
% 0.32/0.48 binunionLsub: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionLsub,definition,
% 0.32/0.48 ( binunionLsub
% 0.32/0.48 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binunionRsub_type,type,
% 0.32/0.48 binunionRsub: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binunionRsub,definition,
% 0.32/0.48 ( binunionRsub
% 0.32/0.48 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersect_type,type,
% 0.32/0.48 binintersect: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectI_type,type,
% 0.32/0.48 binintersectI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectI,definition,
% 0.32/0.48 ( binintersectI
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ B )
% 0.32/0.48 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset5_type,type,
% 0.32/0.48 binintersectSubset5: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset5,definition,
% 0.32/0.48 ( binintersectSubset5
% 0.32/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.48 ( ( subset @ C @ A )
% 0.32/0.48 => ( ( subset @ C @ B )
% 0.32/0.48 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectEL_type,type,
% 0.32/0.48 binintersectEL: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectEL,definition,
% 0.32/0.48 ( binintersectEL
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectLsub_type,type,
% 0.32/0.48 binintersectLsub: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectLsub,definition,
% 0.32/0.48 ( binintersectLsub
% 0.32/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset2_type,type,
% 0.32/0.48 binintersectSubset2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset2,definition,
% 0.32/0.48 ( binintersectSubset2
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( subset @ A @ B )
% 0.32/0.48 => ( ( binintersect @ A @ B )
% 0.32/0.48 = A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset3_type,type,
% 0.32/0.48 binintersectSubset3: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset3,definition,
% 0.32/0.48 ( binintersectSubset3
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( ( binintersect @ A @ B )
% 0.32/0.48 = B )
% 0.32/0.48 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectER_type,type,
% 0.32/0.48 binintersectER: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectER,definition,
% 0.32/0.48 ( binintersectER
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/0.48 => ( in @ Xx @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(disjointsetsI1_type,type,
% 0.32/0.48 disjointsetsI1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(disjointsetsI1,definition,
% 0.32/0.48 ( disjointsetsI1
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ~ ? [Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 & ( in @ Xx @ B ) )
% 0.32/0.48 => ( ( binintersect @ A @ B )
% 0.32/0.48 = emptyset ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectRsub_type,type,
% 0.32/0.48 binintersectRsub: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectRsub,definition,
% 0.32/0.48 ( binintersectRsub
% 0.32/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset4_type,type,
% 0.32/0.48 binintersectSubset4: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset4,definition,
% 0.32/0.48 ( binintersectSubset4
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( subset @ B @ A )
% 0.32/0.48 => ( ( binintersect @ A @ B )
% 0.32/0.48 = B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset1_type,type,
% 0.32/0.48 binintersectSubset1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(binintersectSubset1,definition,
% 0.32/0.48 ( binintersectSubset1
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( ( binintersect @ A @ B )
% 0.32/0.48 = A )
% 0.32/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(bs114d_type,type,
% 0.32/0.48 bs114d: $o ).
% 0.32/0.48
% 0.32/0.48 thf(bs114d,definition,
% 0.32/0.48 ( bs114d
% 0.32/0.48 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.48 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.32/0.48 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(regular_type,type,
% 0.32/0.48 regular: $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminus_type,type,
% 0.32/0.48 setminus: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(setminusI_type,type,
% 0.32/0.48 setminusI: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusI,definition,
% 0.32/0.48 ( setminusI
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ B )
% 0.32/0.48 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusEL_type,type,
% 0.32/0.48 setminusEL: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusEL,definition,
% 0.32/0.48 ( setminusEL
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusER_type,type,
% 0.32/0.48 setminusER: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusER,definition,
% 0.32/0.48 ( setminusER
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.48 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusSubset2_type,type,
% 0.32/0.48 setminusSubset2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusSubset2,definition,
% 0.32/0.48 ( setminusSubset2
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( subset @ A @ B )
% 0.32/0.48 => ( ( setminus @ A @ B )
% 0.32/0.48 = emptyset ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusERneg_type,type,
% 0.32/0.48 setminusERneg: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusERneg,definition,
% 0.32/0.48 ( setminusERneg
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.48 => ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusELneg_type,type,
% 0.32/0.48 setminusELneg: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusELneg,definition,
% 0.32/0.48 ( setminusELneg
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.48 => ( ~ ( in @ Xx @ B )
% 0.32/0.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusILneg_type,type,
% 0.32/0.48 setminusILneg: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusILneg,definition,
% 0.32/0.48 ( setminusILneg
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ~ ( in @ Xx @ A )
% 0.32/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusIRneg_type,type,
% 0.32/0.48 setminusIRneg: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusIRneg,definition,
% 0.32/0.48 ( setminusIRneg
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ B )
% 0.32/0.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusLsub_type,type,
% 0.32/0.48 setminusLsub: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusLsub,definition,
% 0.32/0.48 ( setminusLsub
% 0.32/0.48 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setminusSubset1_type,type,
% 0.32/0.48 setminusSubset1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setminusSubset1,definition,
% 0.32/0.48 ( setminusSubset1
% 0.32/0.48 = ( ! [A: $i,B: $i] :
% 0.32/0.48 ( ( ( setminus @ A @ B )
% 0.32/0.48 = emptyset )
% 0.32/0.48 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(symdiff_type,type,
% 0.32/0.48 symdiff: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffE_type,type,
% 0.32/0.48 symdiffE: $o ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffE,definition,
% 0.32/0.48 ( symdiffE
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.32/0.48 => ! [Xphi: $o] :
% 0.32/0.48 ( ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ B )
% 0.32/0.48 => Xphi ) )
% 0.32/0.48 => ( ( ~ ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ B )
% 0.32/0.48 => Xphi ) )
% 0.32/0.48 => Xphi ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffI1_type,type,
% 0.32/0.48 symdiffI1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffI1,definition,
% 0.32/0.48 ( symdiffI1
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ B )
% 0.32/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffI2_type,type,
% 0.32/0.48 symdiffI2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffI2,definition,
% 0.32/0.48 ( symdiffI2
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ~ ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ B )
% 0.32/0.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffIneg1_type,type,
% 0.32/0.48 symdiffIneg1: $o ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffIneg1,definition,
% 0.32/0.48 ( symdiffIneg1
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( ( in @ Xx @ B )
% 0.32/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffIneg2_type,type,
% 0.32/0.48 symdiffIneg2: $o ).
% 0.32/0.48
% 0.32/0.48 thf(symdiffIneg2,definition,
% 0.32/0.48 ( symdiffIneg2
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ~ ( in @ Xx @ A )
% 0.32/0.48 => ( ~ ( in @ Xx @ B )
% 0.32/0.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(iskpair_type,type,
% 0.32/0.48 iskpair: $i > $o ).
% 0.32/0.48
% 0.32/0.48 thf(secondinupair_type,type,
% 0.32/0.48 secondinupair: $o ).
% 0.32/0.48
% 0.32/0.48 thf(secondinupair,definition,
% 0.32/0.48 ( secondinupair
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setukpairIL_type,type,
% 0.32/0.48 setukpairIL: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setukpairIL,definition,
% 0.32/0.48 ( setukpairIL
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(setukpairIR_type,type,
% 0.32/0.48 setukpairIR: $o ).
% 0.32/0.48
% 0.32/0.48 thf(setukpairIR,definition,
% 0.32/0.48 ( setukpairIR
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(kpairiskpair_type,type,
% 0.32/0.48 kpairiskpair: $o ).
% 0.32/0.48
% 0.32/0.48 thf(kpairiskpair,definition,
% 0.32/0.48 ( kpairiskpair
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(kpair_type,type,
% 0.32/0.48 kpair: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(kpairp_type,type,
% 0.32/0.48 kpairp: $o ).
% 0.32/0.48
% 0.32/0.48 thf(kpairp,definition,
% 0.32/0.48 ( kpairp
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(cartprod_type,type,
% 0.32/0.48 cartprod: $i > $i > $i ).
% 0.32/0.48
% 0.32/0.48 thf(singletonsubset_type,type,
% 0.32/0.48 singletonsubset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(singletonsubset,definition,
% 0.32/0.48 ( singletonsubset
% 0.32/0.48 = ( ! [A: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(singletoninpowerset_type,type,
% 0.32/0.48 singletoninpowerset: $o ).
% 0.32/0.48
% 0.32/0.48 thf(singletoninpowerset,definition,
% 0.32/0.48 ( singletoninpowerset
% 0.32/0.48 = ( ! [A: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(singletoninpowunion_type,type,
% 0.32/0.48 singletoninpowunion: $o ).
% 0.32/0.48
% 0.32/0.48 thf(singletoninpowunion,definition,
% 0.32/0.48 ( singletoninpowunion
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(upairset2E_type,type,
% 0.32/0.48 upairset2E: $o ).
% 0.32/0.48
% 0.32/0.48 thf(upairset2E,definition,
% 0.32/0.48 ( upairset2E
% 0.32/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/0.48 => ( ( Xz = Xx )
% 0.32/0.48 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/0.48
% 0.32/0.48 thf(upairsubunion_type,type,
% 0.32/0.48 upairsubunion: $o ).
% 0.32/0.48
% 0.32/0.48 thf(upairsubunion,definition,
% 0.32/0.48 ( upairsubunion
% 0.32/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.48 ( ( in @ Xx @ A )
% 0.32/0.48 => ! [Xy: $i] :
% 0.32/0.48 ( ( in @ Xy @ B )
% 0.32/0.48 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(upairinpowunion_type,type,
% 0.32/0.49 upairinpowunion: $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairinpowunion,definition,
% 0.32/0.49 ( upairinpowunion
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem1_type,type,
% 0.32/0.49 ubforcartprodlem1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem1,definition,
% 0.32/0.49 ( ubforcartprodlem1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem2_type,type,
% 0.32/0.49 ubforcartprodlem2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem2,definition,
% 0.32/0.49 ( ubforcartprodlem2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem3_type,type,
% 0.32/0.49 ubforcartprodlem3: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem3,definition,
% 0.32/0.49 ( ubforcartprodlem3
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairin_type,type,
% 0.32/0.49 cartprodpairin: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairin,definition,
% 0.32/0.49 ( cartprodpairin
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempair1_type,type,
% 0.32/0.49 cartprodmempair1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempair1,definition,
% 0.32/0.49 ( cartprodmempair1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.49 => ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 & ( Xu
% 0.32/0.49 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempair_type,type,
% 0.32/0.49 cartprodmempair: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempair,definition,
% 0.32/0.49 ( cartprodmempair
% 0.32/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.49 => ( iskpair @ Xu ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionE2_type,type,
% 0.32/0.49 setunionE2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionE2,definition,
% 0.32/0.49 ( setunionE2
% 0.32/0.49 = ( ! [A: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.49 => ? [X: $i] :
% 0.32/0.49 ( ( in @ X @ A )
% 0.32/0.49 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton1_type,type,
% 0.32/0.49 setunionsingleton1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton1,definition,
% 0.32/0.49 ( setunionsingleton1
% 0.32/0.49 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton2_type,type,
% 0.32/0.49 setunionsingleton2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton2,definition,
% 0.32/0.49 ( setunionsingleton2
% 0.32/0.49 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton_type,type,
% 0.32/0.49 setunionsingleton: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton,definition,
% 0.32/0.49 ( setunionsingleton
% 0.32/0.49 = ( ! [Xx: $i] :
% 0.32/0.49 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.32/0.49 = Xx ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(singleton_type,type,
% 0.32/0.49 singleton: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(singletonprop_type,type,
% 0.32/0.49 singletonprop: $o ).
% 0.32/0.49
% 0.32/0.49 thf(singletonprop,definition,
% 0.32/0.49 ( singletonprop
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 => ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) )
% 0.32/0.49 => ( ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ( Xphi @ Xx ) )
% 0.32/0.49 => ( singleton
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ex1_type,type,
% 0.32/0.49 ex1: $i > ( $i > $o ) > $o ).
% 0.32/0.49
% 0.32/0.49 thf(ex1E1_type,type,
% 0.32/0.49 ex1E1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ex1E1,definition,
% 0.32/0.49 ( ex1E1
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ( ex1 @ A
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 => ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ex1I_type,type,
% 0.32/0.49 ex1I: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ex1I,definition,
% 0.32/0.49 ( ex1I
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 => ( ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xy = Xx ) ) )
% 0.32/0.49 => ( ex1 @ A
% 0.32/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ex1I2_type,type,
% 0.32/0.49 ex1I2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ex1I2,definition,
% 0.32/0.49 ( ex1I2
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 => ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) )
% 0.32/0.49 => ( ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ( Xphi @ Xx ) )
% 0.32/0.49 => ( ex1 @ A
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(singletonsuniq_type,type,
% 0.32/0.49 singletonsuniq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(singletonsuniq,definition,
% 0.32/0.49 ( singletonsuniq
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( ( setadjoin @ Xx @ emptyset )
% 0.32/0.49 = ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(atmost1p_type,type,
% 0.32/0.49 atmost1p: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(atleast2p_type,type,
% 0.32/0.49 atleast2p: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(atmost2p_type,type,
% 0.32/0.49 atmost2p: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairsetp_type,type,
% 0.32/0.49 upairsetp: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjL1_type,type,
% 0.32/0.49 setukpairinjL1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjL1,definition,
% 0.32/0.49 ( setukpairinjL1
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.49 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(kfstsingleton_type,type,
% 0.32/0.49 kfstsingleton: $o ).
% 0.32/0.49
% 0.32/0.49 thf(kfstsingleton,definition,
% 0.32/0.49 ( kfstsingleton
% 0.32/0.49 = ( ! [Xu: $i] :
% 0.32/0.49 ( ( iskpair @ Xu )
% 0.32/0.49 => ( singleton
% 0.32/0.49 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/0.49 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(theprop_type,type,
% 0.32/0.49 theprop: $o ).
% 0.32/0.49
% 0.32/0.49 thf(theprop,definition,
% 0.32/0.49 ( theprop
% 0.32/0.49 = ( ! [X: $i] :
% 0.32/0.49 ( ( singleton @ X )
% 0.32/0.49 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(kfst_type,type,
% 0.32/0.49 kfst: $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(kfstpairEq_type,type,
% 0.32/0.49 kfstpairEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(kfstpairEq,definition,
% 0.32/0.49 ( kfstpairEq
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.32/0.49 = Xx ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodfstin_type,type,
% 0.32/0.49 cartprodfstin: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodfstin,definition,
% 0.32/0.49 ( cartprodfstin
% 0.32/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.49 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjL2_type,type,
% 0.32/0.49 setukpairinjL2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjL2,definition,
% 0.32/0.49 ( setukpairinjL2
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjL_type,type,
% 0.32/0.49 setukpairinjL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjL,definition,
% 0.32/0.49 ( setukpairinjL
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.49 ( ( ( kpair @ Xx @ Xy )
% 0.32/0.49 = ( kpair @ Xz @ Xu ) )
% 0.32/0.49 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR11_type,type,
% 0.32/0.49 setukpairinjR11: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR11,definition,
% 0.32/0.49 ( setukpairinjR11
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xx = Xy )
% 0.32/0.49 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.49 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR12_type,type,
% 0.32/0.49 setukpairinjR12: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR12,definition,
% 0.32/0.49 ( setukpairinjR12
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xx = Xy )
% 0.32/0.49 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.49 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR1_type,type,
% 0.32/0.49 setukpairinjR1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR1,definition,
% 0.32/0.49 ( setukpairinjR1
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.49 => ( ( Xz = Xu )
% 0.32/0.49 => ( Xy = Xu ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(upairequniteq_type,type,
% 0.32/0.49 upairequniteq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairequniteq,definition,
% 0.32/0.49 ( upairequniteq
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.49 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.49 = ( setadjoin @ Xz @ emptyset ) )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR2_type,type,
% 0.32/0.49 setukpairinjR2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR2,definition,
% 0.32/0.49 ( setukpairinjR2
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.49 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.49 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.49 => ( Xy = Xu ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR_type,type,
% 0.32/0.49 setukpairinjR: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairinjR,definition,
% 0.32/0.49 ( setukpairinjR
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.49 ( ( ( kpair @ Xx @ Xy )
% 0.32/0.49 = ( kpair @ Xz @ Xu ) )
% 0.32/0.49 => ( Xy = Xu ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ksndsingleton_type,type,
% 0.32/0.49 ksndsingleton: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ksndsingleton,definition,
% 0.32/0.49 ( ksndsingleton
% 0.32/0.49 = ( ! [Xu: $i] :
% 0.32/0.49 ( ( iskpair @ Xu )
% 0.32/0.49 => ( singleton
% 0.32/0.49 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/0.49 @ ^ [Xx: $i] :
% 0.32/0.49 ( Xu
% 0.32/0.49 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ksnd_type,type,
% 0.32/0.49 ksnd: $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(ksndpairEq_type,type,
% 0.32/0.49 ksndpairEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ksndpairEq,definition,
% 0.32/0.49 ( ksndpairEq
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.32/0.49 = Xy ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(kpairsurjEq_type,type,
% 0.32/0.49 kpairsurjEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(kpairsurjEq,definition,
% 0.32/0.49 ( kpairsurjEq
% 0.32/0.49 = ( ! [Xu: $i] :
% 0.32/0.49 ( ( iskpair @ Xu )
% 0.32/0.49 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.32/0.49 = Xu ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodsndin_type,type,
% 0.32/0.49 cartprodsndin: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodsndin,definition,
% 0.32/0.49 ( cartprodsndin
% 0.32/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.49 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairmemEL_type,type,
% 0.32/0.49 cartprodpairmemEL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairmemEL,definition,
% 0.32/0.49 ( cartprodpairmemEL
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairmemER_type,type,
% 0.32/0.49 cartprodpairmemER: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairmemER,definition,
% 0.32/0.49 ( cartprodpairmemER
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/0.49 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempaircEq_type,type,
% 0.32/0.49 cartprodmempaircEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempaircEq,definition,
% 0.32/0.49 ( cartprodmempaircEq
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( kpair @ Xx @ Xy )
% 0.32/0.49 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodfstpairEq_type,type,
% 0.32/0.49 cartprodfstpairEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodfstpairEq,definition,
% 0.32/0.49 ( cartprodfstpairEq
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.32/0.49 = Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodsndpairEq_type,type,
% 0.32/0.49 cartprodsndpairEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodsndpairEq,definition,
% 0.32/0.49 ( cartprodsndpairEq
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.32/0.49 = Xy ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairsurjEq_type,type,
% 0.32/0.49 cartprodpairsurjEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairsurjEq,definition,
% 0.32/0.49 ( cartprodpairsurjEq
% 0.32/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.49 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.32/0.49 = Xu ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln_type,type,
% 0.32/0.49 breln: $i > $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstr_type,type,
% 0.32/0.49 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrI_type,type,
% 0.32/0.49 dpsetconstrI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrI,definition,
% 0.32/0.49 ( dpsetconstrI
% 0.32/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( Xphi @ Xx @ Xy )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.49 @ ( dpsetconstr @ A @ B
% 0.32/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrSub_type,type,
% 0.32/0.49 dpsetconstrSub: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrSub,definition,
% 0.32/0.49 ( dpsetconstrSub
% 0.32/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.49 ( subset
% 0.32/0.49 @ ( dpsetconstr @ A @ B
% 0.32/0.49 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.32/0.49 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setOfPairsIsBReln_type,type,
% 0.32/0.49 setOfPairsIsBReln: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setOfPairsIsBReln,definition,
% 0.32/0.49 ( setOfPairsIsBReln
% 0.32/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.49 ( breln @ A @ B
% 0.32/0.49 @ ( dpsetconstr @ A @ B
% 0.32/0.49 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrERa_type,type,
% 0.32/0.49 dpsetconstrERa: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrERa,definition,
% 0.32/0.49 ( dpsetconstrERa
% 0.32/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.49 @ ( dpsetconstr @ A @ B
% 0.32/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.49 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrEL1_type,type,
% 0.32/0.49 dpsetconstrEL1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrEL1,definition,
% 0.32/0.49 ( dpsetconstrEL1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.49 @ ( dpsetconstr @ A @ B
% 0.32/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrEL2_type,type,
% 0.32/0.49 dpsetconstrEL2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrEL2,definition,
% 0.32/0.49 ( dpsetconstrEL2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.49 @ ( dpsetconstr @ A @ B
% 0.32/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.49 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrER_type,type,
% 0.32/0.49 dpsetconstrER: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dpsetconstrER,definition,
% 0.32/0.49 ( dpsetconstrER
% 0.32/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.49 @ ( dpsetconstr @ A @ B
% 0.32/0.49 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.49 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(func_type,type,
% 0.32/0.49 func: $i > $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcSet_type,type,
% 0.32/0.49 funcSet: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(funcImageSingleton_type,type,
% 0.32/0.49 funcImageSingleton: $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcImageSingleton,definition,
% 0.32/0.49 ( funcImageSingleton
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( singleton
% 0.32/0.49 @ ( dsetconstr @ B
% 0.32/0.49 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(apProp_type,type,
% 0.32/0.49 apProp: $o ).
% 0.32/0.49
% 0.32/0.49 thf(apProp,definition,
% 0.32/0.49 ( apProp
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in
% 0.32/0.49 @ ( setunion
% 0.32/0.49 @ ( dsetconstr @ B
% 0.32/0.49 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.32/0.49 @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ap_type,type,
% 0.32/0.49 ap: $i > $i > $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(app_type,type,
% 0.32/0.49 app: $o ).
% 0.32/0.49
% 0.32/0.49 thf(app,definition,
% 0.32/0.49 ( app
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(infuncsetfunc_type,type,
% 0.32/0.49 infuncsetfunc: $o ).
% 0.32/0.49
% 0.32/0.49 thf(infuncsetfunc,definition,
% 0.32/0.49 ( infuncsetfunc
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.49 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ap2p_type,type,
% 0.32/0.49 ap2p: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ap2p,definition,
% 0.32/0.49 ( ap2p
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(funcinfuncset_type,type,
% 0.32/0.49 funcinfuncset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcinfuncset,definition,
% 0.32/0.49 ( funcinfuncset
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(lamProp_type,type,
% 0.32/0.49 lamProp: $o ).
% 0.32/0.49
% 0.32/0.49 thf(lamProp,definition,
% 0.32/0.49 ( lamProp
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.49 => ( func @ A @ B
% 0.32/0.49 @ ( dpsetconstr @ A @ B
% 0.32/0.49 @ ^ [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xf @ Xx )
% 0.32/0.49 = Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(lam_type,type,
% 0.32/0.49 lam: $i > $i > ( $i > $i ) > $i ).
% 0.32/0.49
% 0.32/0.49 thf(lamp_type,type,
% 0.32/0.49 lamp: $o ).
% 0.32/0.49
% 0.32/0.49 thf(lamp,definition,
% 0.32/0.49 ( lamp
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.49 => ( func @ A @ B
% 0.32/0.49 @ ( lam @ A @ B
% 0.32/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(lam2p_type,type,
% 0.32/0.49 lam2p: $o ).
% 0.32/0.49
% 0.32/0.49 thf(lam2p,definition,
% 0.32/0.49 ( lam2p
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.49 => ( in
% 0.32/0.49 @ ( lam @ A @ B
% 0.32/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.32/0.49 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(brelnall1_type,type,
% 0.32/0.49 brelnall1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(brelnall1,definition,
% 0.32/0.49 ( brelnall1
% 0.32/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.49 ( ( breln @ A @ B @ R )
% 0.32/0.49 => ! [Xphi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ R )
% 0.32/0.49 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(brelnall2_type,type,
% 0.32/0.49 brelnall2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(brelnall2,definition,
% 0.32/0.49 ( brelnall2
% 0.32/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.49 ( ( breln @ A @ B @ R )
% 0.32/0.49 => ! [Xphi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ R )
% 0.32/0.49 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ex1E2_type,type,
% 0.32/0.49 ex1E2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ex1E2,definition,
% 0.32/0.49 ( ex1E2
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ( ex1 @ A
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 => ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(funcGraphProp1_type,type,
% 0.32/0.49 funcGraphProp1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcGraphProp1,definition,
% 0.32/0.49 ( funcGraphProp1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(funcGraphProp3_type,type,
% 0.32/0.49 funcGraphProp3: $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcGraphProp3,definition,
% 0.32/0.49 ( funcGraphProp3
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(funcGraphProp2_type,type,
% 0.32/0.49 funcGraphProp2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcGraphProp2,definition,
% 0.32/0.49 ( funcGraphProp2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.32/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.49 = Xy ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(funcextLem_type,type,
% 0.32/0.49 funcextLem: $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcextLem,definition,
% 0.32/0.49 ( funcextLem
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ! [Xg: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xg )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.49 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(funcGraphProp4_type,type,
% 0.32/0.49 funcGraphProp4: $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcGraphProp4,definition,
% 0.32/0.49 ( funcGraphProp4
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.32/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.49 = Xy ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subbreln_type,type,
% 0.32/0.49 subbreln: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subbreln,definition,
% 0.32/0.49 ( subbreln
% 0.32/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.49 ( ( breln @ A @ B @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln @ A @ B @ S )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.49 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(eqbreln_type,type,
% 0.32/0.49 eqbreln: $o ).
% 0.32/0.49
% 0.32/0.49 thf(eqbreln,definition,
% 0.32/0.49 ( eqbreln
% 0.32/0.49 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.49 ( ( breln @ A @ B @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln @ A @ B @ S )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.32/0.49 => ( R = S ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(funcext_type,type,
% 0.32/0.49 funcext: $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcext,definition,
% 0.32/0.49 ( funcext
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ! [Xg: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xg )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.49 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.32/0.49 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(funcext2_type,type,
% 0.32/0.49 funcext2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(funcext2,definition,
% 0.32/0.49 ( funcext2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.49 => ! [Xg: $i] :
% 0.32/0.49 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.49 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.32/0.49 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ap2apEq1_type,type,
% 0.32/0.49 ap2apEq1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ap2apEq1,definition,
% 0.32/0.49 ( ap2apEq1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.49 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ap2apEq2_type,type,
% 0.32/0.49 ap2apEq2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ap2apEq2,definition,
% 0.32/0.49 ( ap2apEq2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.49 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(beta1_type,type,
% 0.32/0.49 beta1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(beta1,definition,
% 0.32/0.49 ( beta1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( ap @ A @ B
% 0.32/0.49 @ ( lam @ A @ B
% 0.32/0.49 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.32/0.49 @ Xx )
% 0.32/0.49 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(eta1_type,type,
% 0.32/0.49 eta1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(eta1,definition,
% 0.32/0.49 ( eta1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( func @ A @ B @ Xf )
% 0.32/0.49 => ( ( lam @ A @ B
% 0.32/0.49 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.32/0.49 = Xf ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(lam2lamEq_type,type,
% 0.32/0.49 lam2lamEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(lam2lamEq,definition,
% 0.32/0.49 ( lam2lamEq
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.49 => ( ( lam @ A @ B
% 0.32/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.32/0.49 = ( lam @ A @ B
% 0.32/0.49 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(beta2_type,type,
% 0.32/0.49 beta2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(beta2,definition,
% 0.32/0.49 ( beta2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( ap @ A @ B
% 0.32/0.49 @ ( lam @ A @ B
% 0.32/0.49 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.32/0.49 @ Xx )
% 0.32/0.49 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(eta2_type,type,
% 0.32/0.49 eta2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(eta2,definition,
% 0.32/0.49 ( eta2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.49 => ( ( lam @ A @ B
% 0.32/0.49 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.32/0.49 = Xf ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(iffalseProp1_type,type,
% 0.32/0.49 iffalseProp1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(iffalseProp1,definition,
% 0.32/0.49 ( iffalseProp1
% 0.32/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ~ Xphi
% 0.32/0.49 => ( in @ Xy
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xz: $i] :
% 0.32/0.49 ( ( Xphi
% 0.32/0.49 & ( Xz = Xx ) )
% 0.32/0.49 | ( ~ Xphi
% 0.32/0.49 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(iffalseProp2_type,type,
% 0.32/0.49 iffalseProp2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(iffalseProp2,definition,
% 0.32/0.49 ( iffalseProp2
% 0.32/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ~ Xphi
% 0.32/0.49 => ( ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xz: $i] :
% 0.32/0.49 ( ( Xphi
% 0.32/0.49 & ( Xz = Xx ) )
% 0.32/0.49 | ( ~ Xphi
% 0.32/0.49 & ( Xz = Xy ) ) ) )
% 0.32/0.49 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(iftrueProp1_type,type,
% 0.32/0.49 iftrueProp1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(iftrueProp1,definition,
% 0.32/0.49 ( iftrueProp1
% 0.32/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( Xphi
% 0.32/0.49 => ( in @ Xx
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xz: $i] :
% 0.32/0.49 ( ( Xphi
% 0.32/0.49 & ( Xz = Xx ) )
% 0.32/0.49 | ( ~ Xphi
% 0.32/0.49 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(iftrueProp2_type,type,
% 0.32/0.49 iftrueProp2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(iftrueProp2,definition,
% 0.32/0.49 ( iftrueProp2
% 0.32/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( Xphi
% 0.32/0.49 => ( ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xz: $i] :
% 0.32/0.49 ( ( Xphi
% 0.32/0.49 & ( Xz = Xx ) )
% 0.32/0.49 | ( ~ Xphi
% 0.32/0.49 & ( Xz = Xy ) ) ) )
% 0.32/0.49 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ifSingleton_type,type,
% 0.32/0.49 ifSingleton: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ifSingleton,definition,
% 0.32/0.49 ( ifSingleton
% 0.32/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( singleton
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xz: $i] :
% 0.32/0.49 ( ( Xphi
% 0.32/0.49 & ( Xz = Xx ) )
% 0.32/0.49 | ( ~ Xphi
% 0.32/0.49 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(if_type,type,
% 0.32/0.49 if: $i > $o > $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(ifp_type,type,
% 0.32/0.49 ifp: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ifp,definition,
% 0.32/0.49 ( ifp
% 0.32/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(theeq_type,type,
% 0.32/0.49 theeq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(theeq,definition,
% 0.32/0.49 ( theeq
% 0.32/0.49 = ( ! [X: $i] :
% 0.32/0.49 ( ( singleton @ X )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ X )
% 0.32/0.49 => ( ( setunion @ X )
% 0.32/0.49 = Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(iftrue_type,type,
% 0.32/0.49 iftrue: $o ).
% 0.32/0.49
% 0.32/0.49 thf(iftrue,definition,
% 0.32/0.49 ( iftrue
% 0.32/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( Xphi
% 0.32/0.49 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.32/0.49 = Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(iffalse_type,type,
% 0.32/0.49 iffalse: $o ).
% 0.32/0.49
% 0.32/0.49 thf(iffalse,definition,
% 0.32/0.49 ( iffalse
% 0.32/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ~ Xphi
% 0.32/0.49 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.32/0.49 = Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(iftrueorfalse_type,type,
% 0.32/0.49 iftrueorfalse: $o ).
% 0.32/0.49
% 0.32/0.49 thf(iftrueorfalse,definition,
% 0.32/0.49 ( iftrueorfalse
% 0.32/0.49 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectT_lem_type,type,
% 0.32/0.49 binintersectT_lem: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectT_lem,definition,
% 0.32/0.49 ( binintersectT_lem
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionT_lem_type,type,
% 0.32/0.49 binunionT_lem: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionT_lem,definition,
% 0.32/0.49 ( binunionT_lem
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powersetT_lem_type,type,
% 0.32/0.49 powersetT_lem: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powersetT_lem,definition,
% 0.32/0.49 ( powersetT_lem
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusT_lem_type,type,
% 0.32/0.49 setminusT_lem: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusT_lem,definition,
% 0.32/0.49 ( setminusT_lem
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(complementT_lem_type,type,
% 0.32/0.49 complementT_lem: $o ).
% 0.32/0.49
% 0.32/0.49 thf(complementT_lem,definition,
% 0.32/0.49 ( complementT_lem
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setextT_type,type,
% 0.32/0.49 setextT: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setextT,definition,
% 0.32/0.49 ( setextT
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ X )
% 0.32/0.49 => ( in @ Xx @ Y ) ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ Y )
% 0.32/0.49 => ( in @ Xx @ X ) ) )
% 0.32/0.49 => ( X = Y ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subsetTI_type,type,
% 0.32/0.49 subsetTI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subsetTI,definition,
% 0.32/0.49 ( subsetTI
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ X )
% 0.32/0.49 => ( in @ Xx @ Y ) ) )
% 0.32/0.49 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powersetTI1_type,type,
% 0.32/0.49 powersetTI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powersetTI1,definition,
% 0.32/0.49 ( powersetTI1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ X )
% 0.32/0.49 => ( in @ Xx @ Y ) ) )
% 0.32/0.49 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powersetTE1_type,type,
% 0.32/0.49 powersetTE1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powersetTE1,definition,
% 0.32/0.49 ( powersetTE1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ X @ ( powerset @ Y ) )
% 0.32/0.49 => ( ( in @ Xx @ X )
% 0.32/0.49 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(complementTI1_type,type,
% 0.32/0.49 complementTI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(complementTI1,definition,
% 0.32/0.49 ( complementTI1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ X )
% 0.32/0.49 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(complementTE1_type,type,
% 0.32/0.49 complementTE1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(complementTE1,definition,
% 0.32/0.49 ( complementTE1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.49 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectTELcontra_type,type,
% 0.32/0.49 binintersectTELcontra: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectTELcontra,definition,
% 0.32/0.49 ( binintersectTELcontra
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ X )
% 0.32/0.49 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectTERcontra_type,type,
% 0.32/0.49 binintersectTERcontra: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectTERcontra,definition,
% 0.32/0.49 ( binintersectTERcontra
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ Y )
% 0.32/0.49 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(contrasubsetT_type,type,
% 0.32/0.49 contrasubsetT: $o ).
% 0.32/0.49
% 0.32/0.49 thf(contrasubsetT,definition,
% 0.32/0.49 ( contrasubsetT
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.49 => ( ( in @ Xx @ Y )
% 0.32/0.49 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(contrasubsetT1_type,type,
% 0.32/0.49 contrasubsetT1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(contrasubsetT1,definition,
% 0.32/0.49 ( contrasubsetT1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( subset @ X @ Y )
% 0.32/0.49 => ( ~ ( in @ Xx @ Y )
% 0.32/0.49 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(contrasubsetT2_type,type,
% 0.32/0.49 contrasubsetT2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(contrasubsetT2,definition,
% 0.32/0.49 ( contrasubsetT2
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ( subset @ X @ Y )
% 0.32/0.49 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(contrasubsetT3_type,type,
% 0.32/0.49 contrasubsetT3: $o ).
% 0.32/0.49
% 0.32/0.49 thf(contrasubsetT3,definition,
% 0.32/0.49 ( contrasubsetT3
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.32/0.49 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementI1_type,type,
% 0.32/0.49 doubleComplementI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementI1,definition,
% 0.32/0.49 ( doubleComplementI1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ X )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementE1_type,type,
% 0.32/0.49 doubleComplementE1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementE1,definition,
% 0.32/0.49 ( doubleComplementE1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.32/0.49 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementSub1_type,type,
% 0.32/0.49 doubleComplementSub1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementSub1,definition,
% 0.32/0.49 ( doubleComplementSub1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementSub2_type,type,
% 0.32/0.49 doubleComplementSub2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementSub2,definition,
% 0.32/0.49 ( doubleComplementSub2
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementEq_type,type,
% 0.32/0.49 doubleComplementEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(doubleComplementEq,definition,
% 0.32/0.49 ( doubleComplementEq
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ( X
% 0.32/0.49 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(complementTnotintersectT_type,type,
% 0.32/0.49 complementTnotintersectT: $o ).
% 0.32/0.49
% 0.32/0.49 thf(complementTnotintersectT,definition,
% 0.32/0.49 ( complementTnotintersectT
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.49 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(complementImpComplementIntersect_type,type,
% 0.32/0.49 complementImpComplementIntersect: $o ).
% 0.32/0.49
% 0.32/0.49 thf(complementImpComplementIntersect,definition,
% 0.32/0.49 ( complementImpComplementIntersect
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(complementSubsetComplementIntersect_type,type,
% 0.32/0.49 complementSubsetComplementIntersect: $o ).
% 0.32/0.49
% 0.32/0.49 thf(complementSubsetComplementIntersect,definition,
% 0.32/0.49 ( complementSubsetComplementIntersect
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(complementInPowersetComplementIntersect_type,type,
% 0.32/0.49 complementInPowersetComplementIntersect: $o ).
% 0.32/0.49
% 0.32/0.49 thf(complementInPowersetComplementIntersect,definition,
% 0.32/0.49 ( complementInPowersetComplementIntersect
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(contraSubsetComplement_type,type,
% 0.32/0.49 contraSubsetComplement: $o ).
% 0.32/0.49
% 0.32/0.49 thf(contraSubsetComplement,definition,
% 0.32/0.49 ( contraSubsetComplement
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ Y )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(complementTcontraSubset_type,type,
% 0.32/0.49 complementTcontraSubset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(complementTcontraSubset,definition,
% 0.32/0.49 ( complementTcontraSubset
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.49 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionTILcontra_type,type,
% 0.32/0.49 binunionTILcontra: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionTILcontra,definition,
% 0.32/0.49 ( binunionTILcontra
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.32/0.49 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionTIRcontra_type,type,
% 0.32/0.49 binunionTIRcontra: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionTIRcontra,definition,
% 0.32/0.49 ( binunionTIRcontra
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.32/0.49 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(inIntersectImpInUnion_type,type,
% 0.32/0.49 inIntersectImpInUnion: $o ).
% 0.32/0.49
% 0.32/0.49 thf(inIntersectImpInUnion,definition,
% 0.32/0.49 ( inIntersectImpInUnion
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.49 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(inIntersectImpInUnion2_type,type,
% 0.32/0.49 inIntersectImpInUnion2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(inIntersectImpInUnion2,definition,
% 0.32/0.49 ( inIntersectImpInUnion2
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.49 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(inIntersectImpInIntersectUnions_type,type,
% 0.32/0.49 inIntersectImpInIntersectUnions: $o ).
% 0.32/0.49
% 0.32/0.49 thf(inIntersectImpInIntersectUnions,definition,
% 0.32/0.49 ( inIntersectImpInIntersectUnions
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.49 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(intersectInPowersetIntersectUnions_type,type,
% 0.32/0.49 intersectInPowersetIntersectUnions: $o ).
% 0.32/0.49
% 0.32/0.49 thf(intersectInPowersetIntersectUnions,definition,
% 0.32/0.49 ( intersectInPowersetIntersectUnions
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(inComplementUnionImpNotIn1_type,type,
% 0.32/0.49 inComplementUnionImpNotIn1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(inComplementUnionImpNotIn1,definition,
% 0.32/0.49 ( inComplementUnionImpNotIn1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.49 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(inComplementUnionImpInComplement1_type,type,
% 0.32/0.49 inComplementUnionImpInComplement1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(inComplementUnionImpInComplement1,definition,
% 0.32/0.49 ( inComplementUnionImpInComplement1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionTE_type,type,
% 0.32/0.49 binunionTE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionTE,definition,
% 0.32/0.49 ( binunionTE
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.32/0.49 => ( ( ( in @ Xx @ X )
% 0.32/0.49 => Xphi )
% 0.32/0.49 => ( ( ( in @ Xx @ Y )
% 0.32/0.49 => Xphi )
% 0.32/0.49 => Xphi ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionTEcontra_type,type,
% 0.32/0.49 binunionTEcontra: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionTEcontra,definition,
% 0.32/0.49 ( binunionTEcontra
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ X )
% 0.32/0.49 => ( ~ ( in @ Xx @ Y )
% 0.32/0.49 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2a1_type,type,
% 0.32/0.49 demorgan2a1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2a1,definition,
% 0.32/0.49 ( demorgan2a1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(complementUnionInPowersetComplement_type,type,
% 0.32/0.49 complementUnionInPowersetComplement: $o ).
% 0.32/0.49
% 0.32/0.49 thf(complementUnionInPowersetComplement,definition,
% 0.32/0.49 ( complementUnionInPowersetComplement
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2a2_type,type,
% 0.32/0.49 demorgan2a2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2a2,definition,
% 0.32/0.49 ( demorgan2a2
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan1a_type,type,
% 0.32/0.49 demorgan1a: $o ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan1a,definition,
% 0.32/0.49 ( demorgan1a
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.32/0.49 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan1b_type,type,
% 0.32/0.49 demorgan1b: $o ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan1b,definition,
% 0.32/0.49 ( demorgan1b
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan1_type,type,
% 0.32/0.49 demorgan1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan1,definition,
% 0.32/0.49 ( demorgan1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.32/0.49 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2a_type,type,
% 0.32/0.49 demorgan2a: $o ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2a,definition,
% 0.32/0.49 ( demorgan2a
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.32/0.49 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2b2_type,type,
% 0.32/0.49 demorgan2b2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2b2,definition,
% 0.32/0.49 ( demorgan2b2
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.49 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2b_type,type,
% 0.32/0.49 demorgan2b: $o ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2b,definition,
% 0.32/0.49 ( demorgan2b
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2_type,type,
% 0.32/0.49 demorgan2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(demorgan2,definition,
% 0.32/0.49 ( demorgan2
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.32/0.49 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule0_type,type,
% 0.32/0.49 woz13rule0: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule0,definition,
% 0.32/0.49 ( woz13rule0
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.49 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule1_type,type,
% 0.32/0.49 woz13rule1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule1,definition,
% 0.32/0.49 ( woz13rule1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ( ( subset @ X @ Z )
% 0.32/0.49 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule2_type,type,
% 0.32/0.49 woz13rule2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule2,definition,
% 0.32/0.49 ( woz13rule2
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ( ( subset @ Y @ Z )
% 0.32/0.49 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule3_type,type,
% 0.32/0.49 woz13rule3: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule3,definition,
% 0.32/0.49 ( woz13rule3
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ( ( subset @ X @ Y )
% 0.32/0.49 => ( ( subset @ X @ Z )
% 0.32/0.49 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule4_type,type,
% 0.32/0.49 woz13rule4: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz13rule4,definition,
% 0.32/0.49 ( woz13rule4
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ! [W: $i] :
% 0.32/0.49 ( ( in @ W @ ( powerset @ A ) )
% 0.32/0.49 => ( ( subset @ X @ Z )
% 0.32/0.49 => ( ( subset @ Y @ W )
% 0.32/0.49 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_1_type,type,
% 0.32/0.49 woz1_1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_1,definition,
% 0.32/0.49 ( woz1_1
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_2_type,type,
% 0.32/0.49 woz1_2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_2,definition,
% 0.32/0.49 ( woz1_2
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ! [W: $i] :
% 0.32/0.49 ( ( in @ W @ ( powerset @ A ) )
% 0.32/0.49 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.32/0.49 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_3_type,type,
% 0.32/0.49 woz1_3: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_3,definition,
% 0.32/0.49 ( woz1_3
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ! [Z: $i] :
% 0.32/0.49 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_4_type,type,
% 0.32/0.49 woz1_4: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_4,definition,
% 0.32/0.49 ( woz1_4
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.49 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_5_type,type,
% 0.32/0.49 woz1_5: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz1_5,definition,
% 0.32/0.49 ( woz1_5
% 0.32/0.49 = ( ! [A: $i,X: $i] :
% 0.32/0.49 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.49 => ! [Y: $i] :
% 0.32/0.49 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.49 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1_type,type,
% 0.32/0.49 breln1: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1all2_type,type,
% 0.32/0.49 breln1all2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1all2,definition,
% 0.32/0.49 ( breln1all2
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [Xphi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ R )
% 0.32/0.49 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1Set_type,type,
% 0.32/0.49 breln1Set: $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(breln1SetBreln1_type,type,
% 0.32/0.49 breln1SetBreln1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1SetBreln1,definition,
% 0.32/0.49 ( breln1SetBreln1
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( in @ R @ ( breln1Set @ A ) )
% 0.32/0.49 => ( breln1 @ A @ R ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(transitive_type,type,
% 0.32/0.49 transitive: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(antisymmetric_type,type,
% 0.32/0.49 antisymmetric: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(reflexive_type,type,
% 0.32/0.49 reflexive: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(refltransitive_type,type,
% 0.32/0.49 refltransitive: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(refllinearorder_type,type,
% 0.32/0.49 refllinearorder: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(reflwellordering_type,type,
% 0.32/0.49 reflwellordering: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(choice2fnsingleton_type,type,
% 0.32/0.49 choice2fnsingleton: $o ).
% 0.32/0.49
% 0.32/0.49 thf(choice2fnsingleton,definition,
% 0.32/0.49 ( choice2fnsingleton
% 0.32/0.49 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 & ( Xphi @ Xx @ Xy ) ) )
% 0.32/0.49 => ! [R: $i] :
% 0.32/0.49 ( ( in @ R @ ( breln1Set @ B ) )
% 0.32/0.49 => ( ( reflwellordering @ B @ R )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( singleton
% 0.32/0.49 @ ( dsetconstr @ B
% 0.32/0.49 @ ^ [Xy: $i] :
% 0.32/0.49 ( ( Xphi @ Xx @ Xy )
% 0.32/0.49 & ! [Xz: $i] :
% 0.32/0.49 ( ( in @ Xz @ B )
% 0.32/0.49 => ( ( Xphi @ Xx @ Xz )
% 0.32/0.49 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setOfPairsIsBReln1_type,type,
% 0.32/0.49 setOfPairsIsBReln1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setOfPairsIsBReln1,definition,
% 0.32/0.49 ( setOfPairsIsBReln1
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 0.32/0.49 ( breln1 @ A
% 0.32/0.49 @ ( dpsetconstr @ A @ A
% 0.32/0.49 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1all1_type,type,
% 0.32/0.49 breln1all1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1all1,definition,
% 0.32/0.49 ( breln1all1
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [Xphi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ R )
% 0.32/0.49 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subbreln1_type,type,
% 0.32/0.49 subbreln1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subbreln1,definition,
% 0.32/0.49 ( subbreln1
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.49 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(eqbreln1_type,type,
% 0.32/0.49 eqbreln1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(eqbreln1,definition,
% 0.32/0.49 ( eqbreln1
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.32/0.49 => ( R = S ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1invset_type,type,
% 0.32/0.49 breln1invset: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(breln1invprop_type,type,
% 0.32/0.49 breln1invprop: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1invprop,definition,
% 0.32/0.49 ( breln1invprop
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1invI_type,type,
% 0.32/0.49 breln1invI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1invI,definition,
% 0.32/0.49 ( breln1invI
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1invE_type,type,
% 0.32/0.49 breln1invE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1invE,definition,
% 0.32/0.49 ( breln1invE
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1compset_type,type,
% 0.32/0.49 breln1compset: $i > $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(breln1compprop_type,type,
% 0.32/0.49 breln1compprop: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1compprop,definition,
% 0.32/0.49 ( breln1compprop
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1compI_type,type,
% 0.32/0.49 breln1compI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1compI,definition,
% 0.32/0.49 ( breln1compI
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ! [Xz: $i] :
% 0.32/0.49 ( ( in @ Xz @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.32/0.49 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1compE_type,type,
% 0.32/0.49 breln1compE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1compE,definition,
% 0.32/0.49 ( breln1compE
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.32/0.49 => ? [Xz: $i] :
% 0.32/0.49 ( ( in @ Xz @ A )
% 0.32/0.49 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.32/0.49 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1compEex_type,type,
% 0.32/0.49 breln1compEex: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1compEex,definition,
% 0.32/0.49 ( breln1compEex
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 0.32/0.49 => ! [Xphi: $o] :
% 0.32/0.49 ( ! [Xz: $i] :
% 0.32/0.49 ( ( in @ Xz @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 0.32/0.49 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 0.32/0.49 => Xphi ) ) )
% 0.32/0.49 => Xphi ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionprop_type,type,
% 0.32/0.49 breln1unionprop: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionprop,definition,
% 0.32/0.49 ( breln1unionprop
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionIL_type,type,
% 0.32/0.49 breln1unionIL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionIL,definition,
% 0.32/0.49 ( breln1unionIL
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionIR_type,type,
% 0.32/0.49 breln1unionIR: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionIR,definition,
% 0.32/0.49 ( breln1unionIR
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionI_type,type,
% 0.32/0.49 breln1unionI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionI,definition,
% 0.32/0.49 ( breln1unionI
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionE_type,type,
% 0.32/0.49 breln1unionE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionE,definition,
% 0.32/0.49 ( breln1unionE
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionEcases_type,type,
% 0.32/0.49 breln1unionEcases: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionEcases,definition,
% 0.32/0.49 ( breln1unionEcases
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 0.32/0.49 => ! [Xphi: $o] :
% 0.32/0.49 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.49 => Xphi )
% 0.32/0.49 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.32/0.49 => Xphi )
% 0.32/0.49 => Xphi ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionCommutes_type,type,
% 0.32/0.49 breln1unionCommutes: $o ).
% 0.32/0.49
% 0.32/0.49 thf(breln1unionCommutes,definition,
% 0.32/0.49 ( breln1unionCommutes
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ( ( binunion @ R @ S )
% 0.32/0.49 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz2Ex_type,type,
% 0.32/0.49 woz2Ex: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz2Ex,definition,
% 0.32/0.49 ( woz2Ex
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ( R
% 0.32/0.49 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz2W_type,type,
% 0.32/0.49 woz2W: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz2W,definition,
% 0.32/0.49 ( woz2W
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 0.32/0.49 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz2A_type,type,
% 0.32/0.49 woz2A: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz2A,definition,
% 0.32/0.49 ( woz2A
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [T: $i] :
% 0.32/0.49 ( ( breln1 @ A @ T )
% 0.32/0.49 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.32/0.49 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(woz2B_type,type,
% 0.32/0.49 woz2B: $o ).
% 0.32/0.49
% 0.32/0.49 thf(woz2B,definition,
% 0.32/0.49 ( woz2B
% 0.32/0.49 = ( ! [A: $i,R: $i] :
% 0.32/0.49 ( ( breln1 @ A @ R )
% 0.32/0.49 => ! [S: $i] :
% 0.32/0.49 ( ( breln1 @ A @ S )
% 0.32/0.49 => ! [T: $i] :
% 0.32/0.49 ( ( breln1 @ A @ T )
% 0.32/0.49 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 0.32/0.49 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(image1Ex_type,type,
% 0.32/0.49 image1Ex: $o ).
% 0.32/0.49
% 0.32/0.49 thf(image1Ex,definition,
% 0.32/0.49 ( image1Ex
% 0.32/0.49 = ( ! [A: $i,Xf: $i > $i] :
% 0.32/0.49 ? [B: $i] :
% 0.32/0.49 ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 <=> ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 & ( Xx
% 0.32/0.49 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(image1Ex1_type,type,
% 0.32/0.49 image1Ex1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(image1Ex1,definition,
% 0.32/0.49 ( image1Ex1
% 0.32/0.49 = ( ! [A: $i,Xf: $i > $i] :
% 0.32/0.49 ( exu
% 0.32/0.49 @ ^ [B: $i] :
% 0.32/0.49 ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 <=> ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 & ( Xx
% 0.32/0.49 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(image1_type,type,
% 0.32/0.49 image1: $i > ( $i > $i ) > $i ).
% 0.32/0.49
% 0.32/0.49 thf(image1Equiv_type,type,
% 0.32/0.49 image1Equiv: $o ).
% 0.32/0.49
% 0.32/0.49 thf(image1Equiv,definition,
% 0.32/0.49 ( image1Equiv
% 0.32/0.49 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx
% 0.32/0.49 @ ( image1 @ A
% 0.32/0.49 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.32/0.49 <=> ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 & ( Xx
% 0.32/0.49 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(image1E_type,type,
% 0.32/0.49 image1E: $o ).
% 0.32/0.49
% 0.32/0.49 thf(image1E,definition,
% 0.32/0.49 ( image1E
% 0.32/0.49 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx
% 0.32/0.49 @ ( image1 @ A
% 0.32/0.49 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 0.32/0.49 => ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 & ( Xx
% 0.32/0.49 = ( Xf @ Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(image1I_type,type,
% 0.32/0.49 image1I: $o ).
% 0.32/0.49
% 0.32/0.49 thf(image1I,definition,
% 0.32/0.49 ( image1I
% 0.32/0.49 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 0.32/0.49 ( ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 & ( Xx
% 0.32/0.49 = ( Xf @ Xy ) ) )
% 0.32/0.49 => ( in @ Xx
% 0.32/0.49 @ ( image1 @ A
% 0.32/0.49 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(injective_type,type,
% 0.32/0.49 injective: $i > $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(injFuncSet_type,type,
% 0.32/0.49 injFuncSet: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(injFuncInInjFuncSet_type,type,
% 0.32/0.49 injFuncInInjFuncSet: $o ).
% 0.32/0.49
% 0.32/0.49 thf(injFuncInInjFuncSet,definition,
% 0.32/0.49 ( injFuncInInjFuncSet
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.49 => ( ( injective @ A @ B @ Xf )
% 0.32/0.49 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(injFuncSetFuncIn_type,type,
% 0.32/0.49 injFuncSetFuncIn: $o ).
% 0.32/0.49
% 0.32/0.49 thf(injFuncSetFuncIn,definition,
% 0.32/0.49 ( injFuncSetFuncIn
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 0.32/0.49 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(injFuncSetFuncInj_type,type,
% 0.32/0.49 injFuncSetFuncInj: $o ).
% 0.32/0.49
% 0.32/0.49 thf(injFuncSetFuncInj,definition,
% 0.32/0.49 ( injFuncSetFuncInj
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 0.32/0.49 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(surjective_type,type,
% 0.32/0.49 surjective: $i > $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(surjFuncSet_type,type,
% 0.32/0.49 surjFuncSet: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(surjFuncSetFuncIn_type,type,
% 0.32/0.49 surjFuncSetFuncIn: $o ).
% 0.32/0.49
% 0.32/0.49 thf(surjFuncSetFuncIn,definition,
% 0.32/0.49 ( surjFuncSetFuncIn
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 0.32/0.49 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(surjFuncSetFuncSurj_type,type,
% 0.32/0.49 surjFuncSetFuncSurj: $o ).
% 0.32/0.49
% 0.32/0.49 thf(surjFuncSetFuncSurj,definition,
% 0.32/0.49 ( surjFuncSetFuncSurj
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 0.32/0.49 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(leftInvIsSurj_type,type,
% 0.32/0.49 leftInvIsSurj: $o ).
% 0.32/0.49
% 0.32/0.49 thf(leftInvIsSurj,definition,
% 0.32/0.49 ( leftInvIsSurj
% 0.32/0.49 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.49 => ! [Xg: $i] :
% 0.32/0.49 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 0.32/0.49 = Xx ) )
% 0.32/0.49 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(surjCantorThm_type,type,
% 0.32/0.49 surjCantorThm: $o ).
% 0.32/0.49
% 0.32/0.49 thf(surjCantorThm,definition,
% 0.32/0.49 ( surjCantorThm
% 0.32/0.49 = ( ! [A: $i,Xf: $i] :
% 0.32/0.49 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 0.32/0.49 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(foundation2_type,type,
% 0.32/0.49 foundation2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(foundation2,definition,
% 0.32/0.49 ( foundation2
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ( nonempty @ A )
% 0.32/0.49 => ? [X: $i] :
% 0.32/0.49 ( ( in @ X @ A )
% 0.32/0.49 & ( ( binintersect @ X @ A )
% 0.32/0.49 = emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(notinself_type,type,
% 0.32/0.49 notinself: $o ).
% 0.32/0.49
% 0.32/0.49 thf(notinself,definition,
% 0.32/0.49 ( notinself
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ~ ( in @ A @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(notinself2_type,type,
% 0.32/0.49 notinself2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(notinself2,definition,
% 0.32/0.49 ( notinself2
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( in @ A @ B )
% 0.32/0.49 => ~ ( in @ B @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(omegaS_type,type,
% 0.32/0.49 omegaS: $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(omegaS,definition,
% 0.32/0.49 ( omegaS
% 0.32/0.49 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(omegaSp_type,type,
% 0.32/0.49 omegaSp: $o ).
% 0.32/0.49
% 0.32/0.49 thf(omegaSp,definition,
% 0.32/0.49 ( omegaSp
% 0.32/0.49 = ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ omega )
% 0.32/0.49 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(omegaSclos_type,type,
% 0.32/0.49 omegaSclos: $o ).
% 0.32/0.49
% 0.32/0.49 thf(omegaSclos,definition,
% 0.32/0.49 ( omegaSclos
% 0.32/0.49 = ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ omega )
% 0.32/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(peano0notS,conjecture,
% 0.32/0.49 ( setextAx
% 0.32/0.49 => ( emptysetAx
% 0.32/0.49 => ( setadjoinAx
% 0.32/0.49 => ( powersetAx
% 0.32/0.49 => ( setunionAx
% 0.32/0.49 => ( omega0Ax
% 0.32/0.49 => ( omegaSAx
% 0.32/0.49 => ( omegaIndAx
% 0.32/0.49 => ( replAx
% 0.32/0.49 => ( foundationAx
% 0.32/0.49 => ( wellorderingAx
% 0.32/0.49 => ( descrp
% 0.32/0.49 => ( dsetconstrI
% 0.32/0.49 => ( dsetconstrEL
% 0.32/0.49 => ( dsetconstrER
% 0.32/0.49 => ( exuE1
% 0.32/0.49 => ( prop2setE
% 0.32/0.49 => ( emptysetE
% 0.32/0.49 => ( emptysetimpfalse
% 0.32/0.49 => ( notinemptyset
% 0.32/0.49 => ( exuE3e
% 0.32/0.49 => ( setext
% 0.32/0.49 => ( emptyI
% 0.32/0.49 => ( noeltsimpempty
% 0.32/0.49 => ( setbeta
% 0.32/0.49 => ( nonemptyE1
% 0.32/0.49 => ( nonemptyI
% 0.32/0.49 => ( nonemptyI1
% 0.32/0.49 => ( setadjoinIL
% 0.32/0.49 => ( emptyinunitempty
% 0.32/0.49 => ( setadjoinIR
% 0.32/0.49 => ( setadjoinE
% 0.32/0.49 => ( setadjoinOr
% 0.32/0.49 => ( setoftrueEq
% 0.32/0.49 => ( powersetI
% 0.32/0.49 => ( emptyinPowerset
% 0.32/0.49 => ( emptyInPowerset
% 0.32/0.49 => ( powersetE
% 0.32/0.49 => ( setunionI
% 0.32/0.49 => ( setunionE
% 0.32/0.49 => ( subPowSU
% 0.32/0.49 => ( exuE2
% 0.32/0.49 => ( nonemptyImpWitness
% 0.32/0.49 => ( uniqinunit
% 0.32/0.49 => ( notinsingleton
% 0.32/0.49 => ( eqinunit
% 0.32/0.49 => ( singletonsswitch
% 0.32/0.49 => ( upairsetE
% 0.32/0.49 => ( upairsetIL
% 0.32/0.49 => ( upairsetIR
% 0.32/0.49 => ( emptyE1
% 0.32/0.49 => ( vacuousDall
% 0.32/0.49 => ( quantDeMorgan1
% 0.32/0.49 => ( quantDeMorgan2
% 0.32/0.49 => ( quantDeMorgan3
% 0.32/0.49 => ( quantDeMorgan4
% 0.32/0.49 => ( prop2setI
% 0.32/0.49 => ( prop2set2propI
% 0.32/0.49 => ( notdexE
% 0.32/0.49 => ( notdallE
% 0.32/0.49 => ( exuI1
% 0.32/0.49 => ( exuI3
% 0.32/0.49 => ( exuI2
% 0.32/0.49 => ( inCongP
% 0.32/0.49 => ( in__Cong
% 0.32/0.49 => ( exuE3u
% 0.32/0.49 => ( exu__Cong
% 0.32/0.49 => ( emptyset__Cong
% 0.32/0.49 => ( setadjoin__Cong
% 0.32/0.49 => ( powerset__Cong
% 0.32/0.49 => ( setunion__Cong
% 0.32/0.49 => ( omega__Cong
% 0.32/0.49 => ( exuEu
% 0.32/0.49 => ( descr__Cong
% 0.32/0.49 => ( dsetconstr__Cong
% 0.32/0.49 => ( subsetI1
% 0.32/0.49 => ( eqimpsubset2
% 0.32/0.49 => ( eqimpsubset1
% 0.32/0.49 => ( subsetI2
% 0.32/0.49 => ( emptysetsubset
% 0.32/0.49 => ( subsetE
% 0.32/0.49 => ( subsetE2
% 0.32/0.49 => ( notsubsetI
% 0.32/0.49 => ( notequalI1
% 0.32/0.49 => ( notequalI2
% 0.32/0.49 => ( subsetRefl
% 0.32/0.49 => ( subsetTrans
% 0.32/0.49 => ( setadjoinSub
% 0.32/0.49 => ( setadjoinSub2
% 0.32/0.49 => ( subset2powerset
% 0.32/0.49 => ( setextsub
% 0.32/0.49 => ( subsetemptysetimpeq
% 0.32/0.49 => ( powersetI1
% 0.32/0.49 => ( powersetE1
% 0.32/0.49 => ( inPowerset
% 0.32/0.49 => ( powersetsubset
% 0.32/0.49 => ( sepInPowerset
% 0.32/0.49 => ( sepSubset
% 0.32/0.49 => ( binunionIL
% 0.32/0.49 => ( upairset2IR
% 0.32/0.49 => ( binunionIR
% 0.32/0.49 => ( binunionEcases
% 0.32/0.49 => ( binunionE
% 0.32/0.49 => ( binunionLsub
% 0.32/0.49 => ( binunionRsub
% 0.32/0.49 => ( binintersectI
% 0.32/0.49 => ( binintersectSubset5
% 0.32/0.49 => ( binintersectEL
% 0.32/0.49 => ( binintersectLsub
% 0.32/0.49 => ( binintersectSubset2
% 0.32/0.49 => ( binintersectSubset3
% 0.32/0.49 => ( binintersectER
% 0.32/0.49 => ( disjointsetsI1
% 0.32/0.49 => ( binintersectRsub
% 0.32/0.49 => ( binintersectSubset4
% 0.32/0.49 => ( binintersectSubset1
% 0.32/0.49 => ( bs114d
% 0.32/0.49 => ( setminusI
% 0.32/0.49 => ( setminusEL
% 0.32/0.49 => ( setminusER
% 0.32/0.49 => ( setminusSubset2
% 0.32/0.49 => ( setminusERneg
% 0.32/0.49 => ( setminusELneg
% 0.32/0.49 => ( setminusILneg
% 0.32/0.49 => ( setminusIRneg
% 0.32/0.49 => ( setminusLsub
% 0.32/0.49 => ( setminusSubset1
% 0.32/0.49 => ( symdiffE
% 0.32/0.49 => ( symdiffI1
% 0.32/0.49 => ( symdiffI2
% 0.32/0.49 => ( symdiffIneg1
% 0.32/0.49 => ( symdiffIneg2
% 0.32/0.49 => ( secondinupair
% 0.32/0.49 => ( setukpairIL
% 0.32/0.49 => ( setukpairIR
% 0.32/0.49 => ( kpairiskpair
% 0.32/0.49 => ( kpairp
% 0.32/0.49 => ( singletonsubset
% 0.32/0.49 => ( singletoninpowerset
% 0.32/0.49 => ( singletoninpowunion
% 0.32/0.49 => ( upairset2E
% 0.32/0.49 => ( upairsubunion
% 0.32/0.49 => ( upairinpowunion
% 0.32/0.49 => ( ubforcartprodlem1
% 0.32/0.49 => ( ubforcartprodlem2
% 0.32/0.49 => ( ubforcartprodlem3
% 0.32/0.49 => ( cartprodpairin
% 0.32/0.49 => ( cartprodmempair1
% 0.32/0.49 => ( cartprodmempair
% 0.32/0.49 => ( setunionE2
% 0.32/0.49 => ( setunionsingleton1
% 0.32/0.49 => ( setunionsingleton2
% 0.32/0.49 => ( setunionsingleton
% 0.32/0.49 => ( singletonprop
% 0.32/0.49 => ( ex1E1
% 0.32/0.49 => ( ex1I
% 0.32/0.49 => ( ex1I2
% 0.32/0.49 => ( singletonsuniq
% 0.32/0.49 => ( setukpairinjL1
% 0.32/0.49 => ( kfstsingleton
% 0.32/0.49 => ( theprop
% 0.32/0.49 => ( kfstpairEq
% 0.32/0.49 => ( cartprodfstin
% 0.32/0.49 => ( setukpairinjL2
% 0.32/0.49 => ( setukpairinjL
% 0.32/0.49 => ( setukpairinjR11
% 0.32/0.49 => ( setukpairinjR12
% 0.32/0.49 => ( setukpairinjR1
% 0.32/0.49 => ( upairequniteq
% 0.32/0.49 => ( setukpairinjR2
% 0.32/0.49 => ( setukpairinjR
% 0.32/0.49 => ( ksndsingleton
% 0.32/0.49 => ( ksndpairEq
% 0.32/0.49 => ( kpairsurjEq
% 0.32/0.49 => ( cartprodsndin
% 0.32/0.49 => ( cartprodpairmemEL
% 0.32/0.49 => ( cartprodpairmemER
% 0.32/0.49 => ( cartprodmempaircEq
% 0.32/0.49 => ( cartprodfstpairEq
% 0.32/0.49 => ( cartprodsndpairEq
% 0.32/0.49 => ( cartprodpairsurjEq
% 0.32/0.49 => ( dpsetconstrI
% 0.32/0.49 => ( dpsetconstrSub
% 0.32/0.49 => ( setOfPairsIsBReln
% 0.32/0.49 => ( dpsetconstrERa
% 0.32/0.49 => ( dpsetconstrEL1
% 0.32/0.49 => ( dpsetconstrEL2
% 0.32/0.49 => ( dpsetconstrER
% 0.32/0.49 => ( funcImageSingleton
% 0.32/0.49 => ( apProp
% 0.32/0.49 => ( app
% 0.32/0.49 => ( infuncsetfunc
% 0.32/0.49 => ( ap2p
% 0.32/0.49 => ( funcinfuncset
% 0.32/0.49 => ( lamProp
% 0.32/0.49 => ( lamp
% 0.32/0.49 => ( lam2p
% 0.32/0.49 => ( brelnall1
% 0.32/0.49 => ( brelnall2
% 0.32/0.49 => ( ex1E2
% 0.32/0.49 => ( funcGraphProp1
% 0.32/0.49 => ( funcGraphProp3
% 0.32/0.49 => ( funcGraphProp2
% 0.32/0.49 => ( funcextLem
% 0.32/0.49 => ( funcGraphProp4
% 0.32/0.49 => ( subbreln
% 0.32/0.49 => ( eqbreln
% 0.32/0.49 => ( funcext
% 0.32/0.49 => ( funcext2
% 0.32/0.49 => ( ap2apEq1
% 0.32/0.49 => ( ap2apEq2
% 0.32/0.49 => ( beta1
% 0.32/0.49 => ( eta1
% 0.32/0.49 => ( lam2lamEq
% 0.32/0.49 => ( beta2
% 0.32/0.49 => ( eta2
% 0.32/0.49 => ( iffalseProp1
% 0.32/0.49 => ( iffalseProp2
% 0.32/0.49 => ( iftrueProp1
% 0.32/0.49 => ( iftrueProp2
% 0.32/0.49 => ( ifSingleton
% 0.32/0.49 => ( ifp
% 0.32/0.49 => ( theeq
% 0.32/0.49 => ( iftrue
% 0.32/0.49 => ( iffalse
% 0.32/0.49 => ( iftrueorfalse
% 0.32/0.49 => ( binintersectT_lem
% 0.32/0.49 => ( binunionT_lem
% 0.32/0.49 => ( powersetT_lem
% 0.32/0.49 => ( setminusT_lem
% 0.32/0.49 => ( complementT_lem
% 0.32/0.49 => ( setextT
% 0.32/0.49 => ( subsetTI
% 0.32/0.49 => ( powersetTI1
% 0.32/0.49 => ( powersetTE1
% 0.32/0.49 => ( complementTI1
% 0.32/0.49 => ( complementTE1
% 0.32/0.49 => ( binintersectTELcontra
% 0.32/0.49 => ( binintersectTERcontra
% 0.32/0.49 => ( contrasubsetT
% 0.32/0.49 => ( contrasubsetT1
% 0.32/0.49 => ( contrasubsetT2
% 0.32/0.49 => ( contrasubsetT3
% 0.32/0.49 => ( doubleComplementI1
% 0.32/0.49 => ( doubleComplementE1
% 0.32/0.49 => ( doubleComplementSub1
% 0.32/0.49 => ( doubleComplementSub2
% 0.32/0.49 => ( doubleComplementEq
% 0.32/0.49 => ( complementTnotintersectT
% 0.32/0.49 => ( complementImpComplementIntersect
% 0.32/0.49 => ( complementSubsetComplementIntersect
% 0.32/0.49 => ( complementInPowersetComplementIntersect
% 0.32/0.49 => ( contraSubsetComplement
% 0.32/0.49 => ( complementTcontraSubset
% 0.32/0.49 => ( binunionTILcontra
% 0.32/0.49 => ( binunionTIRcontra
% 0.32/0.49 => ( inIntersectImpInUnion
% 0.32/0.49 => ( inIntersectImpInUnion2
% 0.32/0.49 => ( inIntersectImpInIntersectUnions
% 0.32/0.49 => ( intersectInPowersetIntersectUnions
% 0.32/0.49 => ( inComplementUnionImpNotIn1
% 0.32/0.49 => ( inComplementUnionImpInComplement1
% 0.32/0.49 => ( binunionTE
% 0.32/0.49 => ( binunionTEcontra
% 0.32/0.49 => ( demorgan2a1
% 0.32/0.49 => ( complementUnionInPowersetComplement
% 0.32/0.49 => ( demorgan2a2
% 0.32/0.49 => ( demorgan1a
% 0.32/0.49 => ( demorgan1b
% 0.32/0.49 => ( demorgan1
% 0.32/0.49 => ( demorgan2a
% 0.32/0.49 => ( demorgan2b2
% 0.32/0.49 => ( demorgan2b
% 0.32/0.49 => ( demorgan2
% 0.32/0.49 => ( woz13rule0
% 0.32/0.49 => ( woz13rule1
% 0.32/0.49 => ( woz13rule2
% 0.32/0.49 => ( woz13rule3
% 0.32/0.49 => ( woz13rule4
% 0.32/0.49 => ( woz1_1
% 0.32/0.49 => ( woz1_2
% 0.32/0.49 => ( woz1_3
% 0.32/0.49 => ( woz1_4
% 0.32/0.49 => ( woz1_5
% 0.32/0.49 => ( breln1all2
% 0.32/0.49 => ( breln1SetBreln1
% 0.32/0.49 => ( choice2fnsingleton
% 0.32/0.49 => ( setOfPairsIsBReln1
% 0.32/0.49 => ( breln1all1
% 0.32/0.49 => ( subbreln1
% 0.32/0.49 => ( eqbreln1
% 0.32/0.49 => ( breln1invprop
% 0.32/0.49 => ( breln1invI
% 0.32/0.49 => ( breln1invE
% 0.32/0.49 => ( breln1compprop
% 0.32/0.49 => ( breln1compI
% 0.32/0.49 => ( breln1compE
% 0.32/0.49 => ( breln1compEex
% 0.32/0.49 => ( breln1unionprop
% 0.32/0.49 => ( breln1unionIL
% 0.32/0.49 => ( breln1unionIR
% 0.32/0.49 => ( breln1unionI
% 0.32/0.49 => ( breln1unionE
% 0.32/0.49 => ( breln1unionEcases
% 0.32/0.49 => ( breln1unionCommutes
% 0.32/0.49 => ( woz2Ex
% 0.32/0.49 => ( woz2W
% 0.32/0.49 => ( woz2A
% 0.32/0.49 => ( woz2B
% 0.32/0.49 => ( image1Ex
% 0.32/0.49 => ( image1Ex1
% 0.32/0.49 => ( image1Equiv
% 0.32/0.49 => ( image1E
% 0.32/0.49 => ( image1I
% 0.32/0.49 => ( injFuncInInjFuncSet
% 0.32/0.49 => ( injFuncSetFuncIn
% 0.32/0.49 => ( injFuncSetFuncInj
% 0.32/0.49 => ( surjFuncSetFuncIn
% 0.32/0.49 => ( surjFuncSetFuncSurj
% 0.32/0.49 => ( leftInvIsSurj
% 0.32/0.49 => ( surjCantorThm
% 0.32/0.49 => ( foundation2
% 0.32/0.49 => ( notinself
% 0.32/0.49 => ( notinself2
% 0.32/0.49 => ( omegaSp
% 0.32/0.49 => ( omegaSclos
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ omega )
% 0.32/0.49 => ( ( omegaS @ Xx )
% 0.32/0.49 != emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 %------------------------------------------------------------------------------
% 0.32/0.49 ------------------------
% 3.63/1.81 ---- Embedded in THF ---
% 3.63/1.81 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.63/1.81 %%% Generated on Tue Feb 24 12:34:45 EST 2026
% 3.63/1.81 %%% using '$$dhol' embedding, version 1.3.0.
% 3.63/1.81 %%% Logic specification used:
% 3.63/1.81 %%% thf(spec, logic, $$dhol).
% 3.63/1.81
% 3.63/1.81 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.y5OtfSgMuG/DTF2DTF27360.p
% See solution above
% 3.63/1.81 ------------------------
% 3.63/1.89 ---- Cleaned THF ---
% 3.63/1.89 thf(in_type,type,
% 3.63/1.89 in: $i > $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(exu_type,type,
% 3.63/1.89 exu: ( $i > $o ) > $o ).
% 3.63/1.89
% 3.63/1.89 thf(exu,definition,
% 3.63/1.89 ( exu
% 3.63/1.89 = ( ^ [Xphi: $i > $o] :
% 3.63/1.89 ? [Xx: $i] :
% 3.63/1.89 ( ( Xphi @ Xx )
% 3.63/1.89 & ! [Xy: $i] :
% 3.63/1.89 ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setextAx_type,type,
% 3.63/1.89 setextAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setextAx,definition,
% 3.63/1.89 ( setextAx
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 <=> ( in @ Xx @ B ) )
% 3.63/1.89 => ( A = B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptyset_type,type,
% 3.63/1.89 emptyset: $i ).
% 3.63/1.89
% 3.63/1.89 thf(emptysetAx_type,type,
% 3.63/1.89 emptysetAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptysetAx,definition,
% 3.63/1.89 ( emptysetAx
% 3.63/1.89 = ( ! [Xx: $i] :
% 3.63/1.89 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoin_type,type,
% 3.63/1.89 setadjoin: $i > $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinAx_type,type,
% 3.63/1.89 setadjoinAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinAx,definition,
% 3.63/1.89 ( setadjoinAx
% 3.63/1.89 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.63/1.89 <=> ( ( Xy = Xx )
% 3.63/1.89 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(powerset_type,type,
% 3.63/1.89 powerset: $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(powersetAx_type,type,
% 3.63/1.89 powersetAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(powersetAx,definition,
% 3.63/1.89 ( powersetAx
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( in @ B @ ( powerset @ A ) )
% 3.63/1.89 <=> ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setunion_type,type,
% 3.63/1.89 setunion: $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(setunionAx_type,type,
% 3.63/1.89 setunionAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setunionAx,definition,
% 3.63/1.89 ( setunionAx
% 3.63/1.89 = ( ! [A: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( setunion @ A ) )
% 3.63/1.89 <=> ? [B: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 & ( in @ B @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(omega_type,type,
% 3.63/1.89 omega: $i ).
% 3.63/1.89
% 3.63/1.89 thf(omega0Ax_type,type,
% 3.63/1.89 omega0Ax: $o ).
% 3.63/1.89
% 3.63/1.89 thf(omega0Ax,definition,
% 3.63/1.89 ( omega0Ax
% 3.63/1.89 = ( in @ emptyset @ omega ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(omegaSAx_type,type,
% 3.63/1.89 omegaSAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(omegaSAx,definition,
% 3.63/1.89 ( omegaSAx
% 3.63/1.89 = ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ omega )
% 3.63/1.89 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(omegaIndAx_type,type,
% 3.63/1.89 omegaIndAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(omegaIndAx,definition,
% 3.63/1.89 ( omegaIndAx
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ( ( ( in @ emptyset @ A )
% 3.63/1.89 & ! [Xx: $i] :
% 3.63/1.89 ( ( ( in @ Xx @ omega )
% 3.63/1.89 & ( in @ Xx @ A ) )
% 3.63/1.89 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ omega )
% 3.63/1.89 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(replAx_type,type,
% 3.63/1.89 replAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(replAx,definition,
% 3.63/1.89 ( replAx
% 3.63/1.89 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( exu
% 3.63/1.89 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.63/1.89 => ? [B: $i] :
% 3.63/1.89 ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 <=> ? [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ A )
% 3.63/1.89 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(foundationAx_type,type,
% 3.63/1.89 foundationAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(foundationAx,definition,
% 3.63/1.89 ( foundationAx
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.63/1.89 => ? [B: $i] :
% 3.63/1.89 ( ( in @ B @ A )
% 3.63/1.89 & ~ ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(wellorderingAx_type,type,
% 3.63/1.89 wellorderingAx: $o ).
% 3.63/1.89
% 3.63/1.89 thf(wellorderingAx,definition,
% 3.63/1.89 ( wellorderingAx
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ? [B: $i] :
% 3.63/1.89 ( ! [C: $i] :
% 3.63/1.89 ( ( in @ C @ B )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ C )
% 3.63/1.89 => ( in @ Xx @ A ) ) )
% 3.63/1.89 & ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( ( in @ Xx @ A )
% 3.63/1.89 & ( in @ Xy @ A ) )
% 3.63/1.89 => ( ! [C: $i] :
% 3.63/1.89 ( ( in @ C @ B )
% 3.63/1.89 => ( ( in @ Xx @ C )
% 3.63/1.89 <=> ( in @ Xy @ C ) ) )
% 3.63/1.89 => ( Xx = Xy ) ) )
% 3.63/1.89 & ! [C: $i,D: $i] :
% 3.63/1.89 ( ( ( in @ C @ B )
% 3.63/1.89 & ( in @ D @ B ) )
% 3.63/1.89 => ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ C )
% 3.63/1.89 => ( in @ Xx @ D ) )
% 3.63/1.89 | ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ D )
% 3.63/1.89 => ( in @ Xx @ C ) ) ) )
% 3.63/1.89 & ! [C: $i] :
% 3.63/1.89 ( ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ C )
% 3.63/1.89 => ( in @ Xx @ A ) )
% 3.63/1.89 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.63/1.89 => ? [D: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ D @ B )
% 3.63/1.89 & ( in @ Xx @ C )
% 3.63/1.89 & ~ ? [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ D )
% 3.63/1.89 & ( in @ Xy @ C ) )
% 3.63/1.89 & ! [E: $i] :
% 3.63/1.89 ( ( in @ E @ B )
% 3.63/1.89 => ( ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ E )
% 3.63/1.89 => ( in @ Xy @ D ) )
% 3.63/1.89 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(descr_type,type,
% 3.63/1.89 descr: ( $i > $o ) > $i ).
% 3.63/1.89
% 3.63/1.89 thf(descrp_type,type,
% 3.63/1.89 descrp: $o ).
% 3.63/1.89
% 3.63/1.89 thf(descrp,definition,
% 3.63/1.89 ( descrp
% 3.63/1.89 = ( ! [Xphi: $i > $o] :
% 3.63/1.89 ( ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 => ( Xphi
% 3.63/1.89 @ ( descr
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(dsetconstr_type,type,
% 3.63/1.89 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.63/1.89
% 3.63/1.89 thf(dsetconstrI_type,type,
% 3.63/1.89 dsetconstrI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dsetconstrI,definition,
% 3.63/1.89 ( dsetconstrI
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( Xphi @ Xx )
% 3.63/1.89 => ( in @ Xx
% 3.63/1.89 @ ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(dsetconstrEL_type,type,
% 3.63/1.89 dsetconstrEL: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dsetconstrEL,definition,
% 3.63/1.89 ( dsetconstrEL
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx
% 3.63/1.89 @ ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.63/1.89 => ( in @ Xx @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(dsetconstrER_type,type,
% 3.63/1.89 dsetconstrER: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dsetconstrER,definition,
% 3.63/1.89 ( dsetconstrER
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx
% 3.63/1.89 @ ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.63/1.89 => ( Xphi @ Xx ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(exuE1_type,type,
% 3.63/1.89 exuE1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(exuE1,definition,
% 3.63/1.89 ( exuE1
% 3.63/1.89 = ( ! [Xphi: $i > $o] :
% 3.63/1.89 ( ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 => ? [Xx: $i] :
% 3.63/1.89 ( ( Xphi @ Xx )
% 3.63/1.89 & ! [Xy: $i] :
% 3.63/1.89 ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(prop2set_type,type,
% 3.63/1.89 prop2set: $o > $i ).
% 3.63/1.89
% 3.63/1.89 thf(prop2setE_type,type,
% 3.63/1.89 prop2setE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(prop2setE,definition,
% 3.63/1.89 ( prop2setE
% 3.63/1.89 = ( ! [Xphi: $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.63/1.89 => Xphi ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptysetE_type,type,
% 3.63/1.89 emptysetE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptysetE,definition,
% 3.63/1.89 ( emptysetE
% 3.63/1.89 = ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ emptyset )
% 3.63/1.89 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptysetimpfalse_type,type,
% 3.63/1.89 emptysetimpfalse: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptysetimpfalse,definition,
% 3.63/1.89 ( emptysetimpfalse
% 3.63/1.89 = ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ emptyset )
% 3.63/1.89 => $false ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(notinemptyset_type,type,
% 3.63/1.89 notinemptyset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(notinemptyset,definition,
% 3.63/1.89 ( notinemptyset
% 3.63/1.89 = ( ! [Xx: $i] :
% 3.63/1.89 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(exuE3e_type,type,
% 3.63/1.89 exuE3e: $o ).
% 3.63/1.89
% 3.63/1.89 thf(exuE3e,definition,
% 3.63/1.89 ( exuE3e
% 3.63/1.89 = ( ! [Xphi: $i > $o] :
% 3.63/1.89 ( ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setext_type,type,
% 3.63/1.89 setext: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setext,definition,
% 3.63/1.89 ( setext
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ Xx @ B ) )
% 3.63/1.89 => ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 => ( in @ Xx @ A ) )
% 3.63/1.89 => ( A = B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptyI_type,type,
% 3.63/1.89 emptyI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptyI,definition,
% 3.63/1.89 ( emptyI
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ~ ( in @ Xx @ A )
% 3.63/1.89 => ( A = emptyset ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(noeltsimpempty_type,type,
% 3.63/1.89 noeltsimpempty: $o ).
% 3.63/1.89
% 3.63/1.89 thf(noeltsimpempty,definition,
% 3.63/1.89 ( noeltsimpempty
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ~ ( in @ Xx @ A )
% 3.63/1.89 => ( A = emptyset ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setbeta_type,type,
% 3.63/1.89 setbeta: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setbeta,definition,
% 3.63/1.89 ( setbeta
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( in @ Xx
% 3.63/1.89 @ ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.63/1.89 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(nonempty_type,type,
% 3.63/1.89 nonempty: $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(nonempty,definition,
% 3.63/1.89 ( nonempty
% 3.63/1.89 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(nonemptyE1_type,type,
% 3.63/1.89 nonemptyE1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(nonemptyE1,definition,
% 3.63/1.89 ( nonemptyE1
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ( ( nonempty @ A )
% 3.63/1.89 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(nonemptyI_type,type,
% 3.63/1.89 nonemptyI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(nonemptyI,definition,
% 3.63/1.89 ( nonemptyI
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( Xphi @ Xx )
% 3.63/1.89 => ( nonempty
% 3.63/1.89 @ ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(nonemptyI1_type,type,
% 3.63/1.89 nonemptyI1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(nonemptyI1,definition,
% 3.63/1.89 ( nonemptyI1
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.63/1.89 => ( nonempty @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinIL_type,type,
% 3.63/1.89 setadjoinIL: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinIL,definition,
% 3.63/1.89 ( setadjoinIL
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptyinunitempty_type,type,
% 3.63/1.89 emptyinunitempty: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptyinunitempty,definition,
% 3.63/1.89 ( emptyinunitempty
% 3.63/1.89 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinIR_type,type,
% 3.63/1.89 setadjoinIR: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinIR,definition,
% 3.63/1.89 ( setadjoinIR
% 3.63/1.89 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ A )
% 3.63/1.89 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinE_type,type,
% 3.63/1.89 setadjoinE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinE,definition,
% 3.63/1.89 ( setadjoinE
% 3.63/1.89 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.63/1.89 => ! [Xphi: $o] :
% 3.63/1.89 ( ( ( Xy = Xx )
% 3.63/1.89 => Xphi )
% 3.63/1.89 => ( ( ( in @ Xy @ A )
% 3.63/1.89 => Xphi )
% 3.63/1.89 => Xphi ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinOr_type,type,
% 3.63/1.89 setadjoinOr: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinOr,definition,
% 3.63/1.89 ( setadjoinOr
% 3.63/1.89 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.63/1.89 => ( ( Xy = Xx )
% 3.63/1.89 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setoftrueEq_type,type,
% 3.63/1.89 setoftrueEq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setoftrueEq,definition,
% 3.63/1.89 ( setoftrueEq
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ( ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xx: $i] : $true )
% 3.63/1.89 = A ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(powersetI_type,type,
% 3.63/1.89 powersetI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(powersetI,definition,
% 3.63/1.89 ( powersetI
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 => ( in @ Xx @ A ) )
% 3.63/1.89 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptyinPowerset_type,type,
% 3.63/1.89 emptyinPowerset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptyinPowerset,definition,
% 3.63/1.89 ( emptyinPowerset
% 3.63/1.89 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptyInPowerset_type,type,
% 3.63/1.89 emptyInPowerset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptyInPowerset,definition,
% 3.63/1.89 ( emptyInPowerset
% 3.63/1.89 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(powersetE_type,type,
% 3.63/1.89 powersetE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(powersetE,definition,
% 3.63/1.89 ( powersetE
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ B @ ( powerset @ A ) )
% 3.63/1.89 => ( ( in @ Xx @ B )
% 3.63/1.89 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setunionI_type,type,
% 3.63/1.89 setunionI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setunionI,definition,
% 3.63/1.89 ( setunionI
% 3.63/1.89 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 => ( ( in @ B @ A )
% 3.63/1.89 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setunionE_type,type,
% 3.63/1.89 setunionE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setunionE,definition,
% 3.63/1.89 ( setunionE
% 3.63/1.89 = ( ! [A: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( setunion @ A ) )
% 3.63/1.89 => ! [Xphi: $o] :
% 3.63/1.89 ( ! [B: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 => ( ( in @ B @ A )
% 3.63/1.89 => Xphi ) )
% 3.63/1.89 => Xphi ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subPowSU_type,type,
% 3.63/1.89 subPowSU: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subPowSU,definition,
% 3.63/1.89 ( subPowSU
% 3.63/1.89 = ( ! [A: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(exuE2_type,type,
% 3.63/1.89 exuE2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(exuE2,definition,
% 3.63/1.89 ( exuE2
% 3.63/1.89 = ( ! [Xphi: $i > $o] :
% 3.63/1.89 ( ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 => ? [Xx: $i] :
% 3.63/1.89 ! [Xy: $i] :
% 3.63/1.89 ( ( Xphi @ Xy )
% 3.63/1.89 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(nonemptyImpWitness_type,type,
% 3.63/1.89 nonemptyImpWitness: $o ).
% 3.63/1.89
% 3.63/1.89 thf(nonemptyImpWitness,definition,
% 3.63/1.89 ( nonemptyImpWitness
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ( ( nonempty @ A )
% 3.63/1.89 => ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & $true ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(uniqinunit_type,type,
% 3.63/1.89 uniqinunit: $o ).
% 3.63/1.89
% 3.63/1.89 thf(uniqinunit,definition,
% 3.63/1.89 ( uniqinunit
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(notinsingleton_type,type,
% 3.63/1.89 notinsingleton: $o ).
% 3.63/1.89
% 3.63/1.89 thf(notinsingleton,definition,
% 3.63/1.89 ( notinsingleton
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xx != Xy )
% 3.63/1.89 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(eqinunit_type,type,
% 3.63/1.89 eqinunit: $o ).
% 3.63/1.89
% 3.63/1.89 thf(eqinunit,definition,
% 3.63/1.89 ( eqinunit
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xx = Xy )
% 3.63/1.89 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(singletonsswitch_type,type,
% 3.63/1.89 singletonsswitch: $o ).
% 3.63/1.89
% 3.63/1.89 thf(singletonsswitch,definition,
% 3.63/1.89 ( singletonsswitch
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.63/1.89 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(upairsetE_type,type,
% 3.63/1.89 upairsetE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(upairsetE,definition,
% 3.63/1.89 ( upairsetE
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.63/1.89 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.63/1.89 => ( ( Xz = Xx )
% 3.63/1.89 | ( Xz = Xy ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(upairsetIL_type,type,
% 3.63/1.89 upairsetIL: $o ).
% 3.63/1.89
% 3.63/1.89 thf(upairsetIL,definition,
% 3.63/1.89 ( upairsetIL
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(upairsetIR_type,type,
% 3.63/1.89 upairsetIR: $o ).
% 3.63/1.89
% 3.63/1.89 thf(upairsetIR,definition,
% 3.63/1.89 ( upairsetIR
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptyE1_type,type,
% 3.63/1.89 emptyE1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptyE1,definition,
% 3.63/1.89 ( emptyE1
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ( Xphi @ Xx ) )
% 3.63/1.89 => ( ( ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 = emptyset )
% 3.63/1.89 => $false ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(vacuousDall_type,type,
% 3.63/1.89 vacuousDall: $o ).
% 3.63/1.89
% 3.63/1.89 thf(vacuousDall,definition,
% 3.63/1.89 ( vacuousDall
% 3.63/1.89 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ emptyset )
% 3.63/1.89 => ( Xphi @ Xx ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(quantDeMorgan1_type,type,
% 3.63/1.89 quantDeMorgan1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(quantDeMorgan1,definition,
% 3.63/1.89 ( quantDeMorgan1
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ~ ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( Xphi @ Xx ) )
% 3.63/1.89 => ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(quantDeMorgan2_type,type,
% 3.63/1.89 quantDeMorgan2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(quantDeMorgan2,definition,
% 3.63/1.89 ( quantDeMorgan2
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ~ ( Xphi @ Xx ) )
% 3.63/1.89 => ~ ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(quantDeMorgan3_type,type,
% 3.63/1.89 quantDeMorgan3: $o ).
% 3.63/1.89
% 3.63/1.89 thf(quantDeMorgan3,definition,
% 3.63/1.89 ( quantDeMorgan3
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ~ ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ( Xphi @ Xx ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(quantDeMorgan4_type,type,
% 3.63/1.89 quantDeMorgan4: $o ).
% 3.63/1.89
% 3.63/1.89 thf(quantDeMorgan4,definition,
% 3.63/1.89 ( quantDeMorgan4
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ~ ( Xphi @ Xx ) )
% 3.63/1.89 => ~ ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(prop2setI_type,type,
% 3.63/1.89 prop2setI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(prop2setI,definition,
% 3.63/1.89 ( prop2setI
% 3.63/1.89 = ( ! [Xphi: $o] :
% 3.63/1.89 ( Xphi
% 3.63/1.89 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(set2prop_type,type,
% 3.63/1.89 set2prop: $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(prop2set2propI_type,type,
% 3.63/1.89 prop2set2propI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(prop2set2propI,definition,
% 3.63/1.89 ( prop2set2propI
% 3.63/1.89 = ( ! [Xphi: $o] :
% 3.63/1.89 ( Xphi
% 3.63/1.89 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(notdexE_type,type,
% 3.63/1.89 notdexE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(notdexE,definition,
% 3.63/1.89 ( notdexE
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ~ ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ( Xphi @ Xx ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(notdallE_type,type,
% 3.63/1.89 notdallE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(notdallE,definition,
% 3.63/1.89 ( notdallE
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ~ ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( Xphi @ Xx ) )
% 3.63/1.89 => ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(exuI1_type,type,
% 3.63/1.89 exuI1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(exuI1,definition,
% 3.63/1.89 ( exuI1
% 3.63/1.89 = ( ! [Xphi: $i > $o] :
% 3.63/1.89 ( ? [Xx: $i] :
% 3.63/1.89 ( ( Xphi @ Xx )
% 3.63/1.89 & ! [Xy: $i] :
% 3.63/1.89 ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xx = Xy ) ) )
% 3.63/1.89 => ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(exuI3_type,type,
% 3.63/1.89 exuI3: $o ).
% 3.63/1.89
% 3.63/1.89 thf(exuI3,definition,
% 3.63/1.89 ( exuI3
% 3.63/1.89 = ( ! [Xphi: $i > $o] :
% 3.63/1.89 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.63/1.89 => ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xphi @ Xx )
% 3.63/1.89 => ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xx = Xy ) ) )
% 3.63/1.89 => ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(exuI2_type,type,
% 3.63/1.89 exuI2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(exuI2,definition,
% 3.63/1.89 ( exuI2
% 3.63/1.89 = ( ! [Xphi: $i > $o] :
% 3.63/1.89 ( ? [Xx: $i] :
% 3.63/1.89 ! [Xy: $i] :
% 3.63/1.89 ( ( Xphi @ Xy )
% 3.63/1.89 <=> ( Xy = Xx ) )
% 3.63/1.89 => ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(inCongP_type,type,
% 3.63/1.89 inCongP: $o ).
% 3.63/1.89
% 3.63/1.89 thf(inCongP,definition,
% 3.63/1.89 ( inCongP
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( A = B )
% 3.63/1.89 => ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xx = Xy )
% 3.63/1.89 => ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(in__Cong_type,type,
% 3.63/1.89 in__Cong: $o ).
% 3.63/1.89
% 3.63/1.89 thf(in__Cong,definition,
% 3.63/1.89 ( in__Cong
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( A = B )
% 3.63/1.89 => ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xx = Xy )
% 3.63/1.89 => ( ( in @ Xx @ A )
% 3.63/1.89 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(exuE3u_type,type,
% 3.63/1.89 exuE3u: $o ).
% 3.63/1.89
% 3.63/1.89 thf(exuE3u,definition,
% 3.63/1.89 ( exuE3u
% 3.63/1.89 = ( ! [Xphi: $i > $o] :
% 3.63/1.89 ( ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 => ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xphi @ Xx )
% 3.63/1.89 => ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(exu__Cong_type,type,
% 3.63/1.89 exu__Cong: $o ).
% 3.63/1.89
% 3.63/1.89 thf(exu__Cong,definition,
% 3.63/1.89 ( exu__Cong
% 3.63/1.89 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.63/1.89 ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xx = Xy )
% 3.63/1.89 => ( ( Xphi @ Xx )
% 3.63/1.89 <=> ( Xpsi @ Xy ) ) )
% 3.63/1.89 => ( ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 <=> ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptyset__Cong_type,type,
% 3.63/1.89 emptyset__Cong: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptyset__Cong,definition,
% 3.63/1.89 ( emptyset__Cong
% 3.63/1.89 = ( emptyset = emptyset ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoin__Cong_type,type,
% 3.63/1.89 setadjoin__Cong: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoin__Cong,definition,
% 3.63/1.89 ( setadjoin__Cong
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xx = Xy )
% 3.63/1.89 => ! [Xz: $i,Xu: $i] :
% 3.63/1.89 ( ( Xz = Xu )
% 3.63/1.89 => ( ( setadjoin @ Xx @ Xz )
% 3.63/1.89 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(powerset__Cong_type,type,
% 3.63/1.89 powerset__Cong: $o ).
% 3.63/1.89
% 3.63/1.89 thf(powerset__Cong,definition,
% 3.63/1.89 ( powerset__Cong
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( A = B )
% 3.63/1.89 => ( ( powerset @ A )
% 3.63/1.89 = ( powerset @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setunion__Cong_type,type,
% 3.63/1.89 setunion__Cong: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setunion__Cong,definition,
% 3.63/1.89 ( setunion__Cong
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( A = B )
% 3.63/1.89 => ( ( setunion @ A )
% 3.63/1.89 = ( setunion @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(omega__Cong_type,type,
% 3.63/1.89 omega__Cong: $o ).
% 3.63/1.89
% 3.63/1.89 thf(omega__Cong,definition,
% 3.63/1.89 ( omega__Cong
% 3.63/1.89 = ( omega = omega ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(exuEu_type,type,
% 3.63/1.89 exuEu: $o ).
% 3.63/1.89
% 3.63/1.89 thf(exuEu,definition,
% 3.63/1.89 ( exuEu
% 3.63/1.89 = ( ! [Xphi: $i > $o] :
% 3.63/1.89 ( ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 => ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xphi @ Xx )
% 3.63/1.89 => ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(descr__Cong_type,type,
% 3.63/1.89 descr__Cong: $o ).
% 3.63/1.89
% 3.63/1.89 thf(descr__Cong,definition,
% 3.63/1.89 ( descr__Cong
% 3.63/1.89 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.63/1.89 ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xx = Xy )
% 3.63/1.89 => ( ( Xphi @ Xx )
% 3.63/1.89 <=> ( Xpsi @ Xy ) ) )
% 3.63/1.89 => ( ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 => ( ( exu
% 3.63/1.89 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.63/1.89 => ( ( descr
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 = ( descr
% 3.63/1.89 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(dsetconstr__Cong_type,type,
% 3.63/1.89 dsetconstr__Cong: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dsetconstr__Cong,definition,
% 3.63/1.89 ( dsetconstr__Cong
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( A = B )
% 3.63/1.89 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( Xx = Xy )
% 3.63/1.89 => ( ( Xphi @ Xx )
% 3.63/1.89 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.63/1.89 => ( ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 = ( dsetconstr @ B
% 3.63/1.89 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subset_type,type,
% 3.63/1.89 subset: $i > $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(disjoint_type,type,
% 3.63/1.89 disjoint: $i > $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(setsmeet_type,type,
% 3.63/1.89 setsmeet: $i > $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(subsetI1_type,type,
% 3.63/1.89 subsetI1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subsetI1,definition,
% 3.63/1.89 ( subsetI1
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ Xx @ B ) )
% 3.63/1.89 => ( subset @ A @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(eqimpsubset2_type,type,
% 3.63/1.89 eqimpsubset2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(eqimpsubset2,definition,
% 3.63/1.89 ( eqimpsubset2
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( A = B )
% 3.63/1.89 => ( subset @ B @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(eqimpsubset1_type,type,
% 3.63/1.89 eqimpsubset1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(eqimpsubset1,definition,
% 3.63/1.89 ( eqimpsubset1
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( A = B )
% 3.63/1.89 => ( subset @ A @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subsetI2_type,type,
% 3.63/1.89 subsetI2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subsetI2,definition,
% 3.63/1.89 ( subsetI2
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ Xx @ B ) )
% 3.63/1.89 => ( subset @ A @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(emptysetsubset_type,type,
% 3.63/1.89 emptysetsubset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(emptysetsubset,definition,
% 3.63/1.89 ( emptysetsubset
% 3.63/1.89 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subsetE_type,type,
% 3.63/1.89 subsetE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subsetE,definition,
% 3.63/1.89 ( subsetE
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( subset @ A @ B )
% 3.63/1.89 => ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subsetE2_type,type,
% 3.63/1.89 subsetE2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subsetE2,definition,
% 3.63/1.89 ( subsetE2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( subset @ A @ B )
% 3.63/1.89 => ( ~ ( in @ Xx @ B )
% 3.63/1.89 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(notsubsetI_type,type,
% 3.63/1.89 notsubsetI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(notsubsetI,definition,
% 3.63/1.89 ( notsubsetI
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ~ ( in @ Xx @ B )
% 3.63/1.89 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(notequalI1_type,type,
% 3.63/1.89 notequalI1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(notequalI1,definition,
% 3.63/1.89 ( notequalI1
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ~ ( subset @ A @ B )
% 3.63/1.89 => ( A != B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(notequalI2_type,type,
% 3.63/1.89 notequalI2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(notequalI2,definition,
% 3.63/1.89 ( notequalI2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ~ ( in @ Xx @ B )
% 3.63/1.89 => ( A != B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subsetRefl_type,type,
% 3.63/1.89 subsetRefl: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subsetRefl,definition,
% 3.63/1.89 ( subsetRefl
% 3.63/1.89 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subsetTrans_type,type,
% 3.63/1.89 subsetTrans: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subsetTrans,definition,
% 3.63/1.89 ( subsetTrans
% 3.63/1.89 = ( ! [A: $i,B: $i,C: $i] :
% 3.63/1.89 ( ( subset @ A @ B )
% 3.63/1.89 => ( ( subset @ B @ C )
% 3.63/1.89 => ( subset @ A @ C ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinSub_type,type,
% 3.63/1.89 setadjoinSub: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinSub,definition,
% 3.63/1.89 ( setadjoinSub
% 3.63/1.89 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinSub2_type,type,
% 3.63/1.89 setadjoinSub2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setadjoinSub2,definition,
% 3.63/1.89 ( setadjoinSub2
% 3.63/1.89 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.63/1.89 ( ( subset @ A @ B )
% 3.63/1.89 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subset2powerset_type,type,
% 3.63/1.89 subset2powerset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subset2powerset,definition,
% 3.63/1.89 ( subset2powerset
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( subset @ A @ B )
% 3.63/1.89 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setextsub_type,type,
% 3.63/1.89 setextsub: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setextsub,definition,
% 3.63/1.89 ( setextsub
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( subset @ A @ B )
% 3.63/1.89 => ( ( subset @ B @ A )
% 3.63/1.89 => ( A = B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subsetemptysetimpeq_type,type,
% 3.63/1.89 subsetemptysetimpeq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subsetemptysetimpeq,definition,
% 3.63/1.89 ( subsetemptysetimpeq
% 3.63/1.89 = ( ! [A: $i] :
% 3.63/1.89 ( ( subset @ A @ emptyset )
% 3.63/1.89 => ( A = emptyset ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(powersetI1_type,type,
% 3.63/1.89 powersetI1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(powersetI1,definition,
% 3.63/1.89 ( powersetI1
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( subset @ B @ A )
% 3.63/1.89 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(powersetE1_type,type,
% 3.63/1.89 powersetE1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(powersetE1,definition,
% 3.63/1.89 ( powersetE1
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( in @ B @ ( powerset @ A ) )
% 3.63/1.89 => ( subset @ B @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(inPowerset_type,type,
% 3.63/1.89 inPowerset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(inPowerset,definition,
% 3.63/1.89 ( inPowerset
% 3.63/1.89 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(powersetsubset_type,type,
% 3.63/1.89 powersetsubset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(powersetsubset,definition,
% 3.63/1.89 ( powersetsubset
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( subset @ A @ B )
% 3.63/1.89 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(sepInPowerset_type,type,
% 3.63/1.89 sepInPowerset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(sepInPowerset,definition,
% 3.63/1.89 ( sepInPowerset
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( in
% 3.63/1.89 @ ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 @ ( powerset @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(sepSubset_type,type,
% 3.63/1.89 sepSubset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(sepSubset,definition,
% 3.63/1.89 ( sepSubset
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( subset
% 3.63/1.89 @ ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 @ A ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binunion_type,type,
% 3.63/1.89 binunion: $i > $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(binunionIL_type,type,
% 3.63/1.89 binunionIL: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binunionIL,definition,
% 3.63/1.89 ( binunionIL
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(upairset2IR_type,type,
% 3.63/1.89 upairset2IR: $o ).
% 3.63/1.89
% 3.63/1.89 thf(upairset2IR,definition,
% 3.63/1.89 ( upairset2IR
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binunionIR_type,type,
% 3.63/1.89 binunionIR: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binunionIR,definition,
% 3.63/1.89 ( binunionIR
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binunionEcases_type,type,
% 3.63/1.89 binunionEcases: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binunionEcases,definition,
% 3.63/1.89 ( binunionEcases
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.63/1.89 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.63/1.89 => ( ( ( in @ Xx @ A )
% 3.63/1.89 => Xphi )
% 3.63/1.89 => ( ( ( in @ Xx @ B )
% 3.63/1.89 => Xphi )
% 3.63/1.89 => Xphi ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binunionE_type,type,
% 3.63/1.89 binunionE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binunionE,definition,
% 3.63/1.89 ( binunionE
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.63/1.89 => ( ( in @ Xx @ A )
% 3.63/1.89 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binunionLsub_type,type,
% 3.63/1.89 binunionLsub: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binunionLsub,definition,
% 3.63/1.89 ( binunionLsub
% 3.63/1.89 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binunionRsub_type,type,
% 3.63/1.89 binunionRsub: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binunionRsub,definition,
% 3.63/1.89 ( binunionRsub
% 3.63/1.89 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersect_type,type,
% 3.63/1.89 binintersect: $i > $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectI_type,type,
% 3.63/1.89 binintersectI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectI,definition,
% 3.63/1.89 ( binintersectI
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( in @ Xx @ B )
% 3.63/1.89 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset5_type,type,
% 3.63/1.89 binintersectSubset5: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset5,definition,
% 3.63/1.89 ( binintersectSubset5
% 3.63/1.89 = ( ! [A: $i,B: $i,C: $i] :
% 3.63/1.89 ( ( subset @ C @ A )
% 3.63/1.89 => ( ( subset @ C @ B )
% 3.63/1.89 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectEL_type,type,
% 3.63/1.89 binintersectEL: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectEL,definition,
% 3.63/1.89 ( binintersectEL
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.63/1.89 => ( in @ Xx @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectLsub_type,type,
% 3.63/1.89 binintersectLsub: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectLsub,definition,
% 3.63/1.89 ( binintersectLsub
% 3.63/1.89 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset2_type,type,
% 3.63/1.89 binintersectSubset2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset2,definition,
% 3.63/1.89 ( binintersectSubset2
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( subset @ A @ B )
% 3.63/1.89 => ( ( binintersect @ A @ B )
% 3.63/1.89 = A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset3_type,type,
% 3.63/1.89 binintersectSubset3: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset3,definition,
% 3.63/1.89 ( binintersectSubset3
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( ( binintersect @ A @ B )
% 3.63/1.89 = B )
% 3.63/1.89 => ( subset @ B @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectER_type,type,
% 3.63/1.89 binintersectER: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectER,definition,
% 3.63/1.89 ( binintersectER
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.63/1.89 => ( in @ Xx @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(disjointsetsI1_type,type,
% 3.63/1.89 disjointsetsI1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(disjointsetsI1,definition,
% 3.63/1.89 ( disjointsetsI1
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ~ ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ( in @ Xx @ B ) )
% 3.63/1.89 => ( ( binintersect @ A @ B )
% 3.63/1.89 = emptyset ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectRsub_type,type,
% 3.63/1.89 binintersectRsub: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectRsub,definition,
% 3.63/1.89 ( binintersectRsub
% 3.63/1.89 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset4_type,type,
% 3.63/1.89 binintersectSubset4: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset4,definition,
% 3.63/1.89 ( binintersectSubset4
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( subset @ B @ A )
% 3.63/1.89 => ( ( binintersect @ A @ B )
% 3.63/1.89 = B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset1_type,type,
% 3.63/1.89 binintersectSubset1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(binintersectSubset1,definition,
% 3.63/1.89 ( binintersectSubset1
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( ( binintersect @ A @ B )
% 3.63/1.89 = A )
% 3.63/1.89 => ( subset @ A @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(bs114d_type,type,
% 3.63/1.89 bs114d: $o ).
% 3.63/1.89
% 3.63/1.89 thf(bs114d,definition,
% 3.63/1.89 ( bs114d
% 3.63/1.89 = ( ! [A: $i,B: $i,C: $i] :
% 3.63/1.89 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.63/1.89 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(regular_type,type,
% 3.63/1.89 regular: $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminus_type,type,
% 3.63/1.89 setminus: $i > $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(setminusI_type,type,
% 3.63/1.89 setminusI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusI,definition,
% 3.63/1.89 ( setminusI
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ~ ( in @ Xx @ B )
% 3.63/1.89 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setminusEL_type,type,
% 3.63/1.89 setminusEL: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusEL,definition,
% 3.63/1.89 ( setminusEL
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.63/1.89 => ( in @ Xx @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setminusER_type,type,
% 3.63/1.89 setminusER: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusER,definition,
% 3.63/1.89 ( setminusER
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.63/1.89 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setminusSubset2_type,type,
% 3.63/1.89 setminusSubset2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusSubset2,definition,
% 3.63/1.89 ( setminusSubset2
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( subset @ A @ B )
% 3.63/1.89 => ( ( setminus @ A @ B )
% 3.63/1.89 = emptyset ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setminusERneg_type,type,
% 3.63/1.89 setminusERneg: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusERneg,definition,
% 3.63/1.89 ( setminusERneg
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.63/1.89 => ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setminusELneg_type,type,
% 3.63/1.89 setminusELneg: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusELneg,definition,
% 3.63/1.89 ( setminusELneg
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.63/1.89 => ( ~ ( in @ Xx @ B )
% 3.63/1.89 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setminusILneg_type,type,
% 3.63/1.89 setminusILneg: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusILneg,definition,
% 3.63/1.89 ( setminusILneg
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ~ ( in @ Xx @ A )
% 3.63/1.89 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setminusIRneg_type,type,
% 3.63/1.89 setminusIRneg: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusIRneg,definition,
% 3.63/1.89 ( setminusIRneg
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ B )
% 3.63/1.89 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setminusLsub_type,type,
% 3.63/1.89 setminusLsub: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusLsub,definition,
% 3.63/1.89 ( setminusLsub
% 3.63/1.89 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setminusSubset1_type,type,
% 3.63/1.89 setminusSubset1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setminusSubset1,definition,
% 3.63/1.89 ( setminusSubset1
% 3.63/1.89 = ( ! [A: $i,B: $i] :
% 3.63/1.89 ( ( ( setminus @ A @ B )
% 3.63/1.89 = emptyset )
% 3.63/1.89 => ( subset @ A @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(symdiff_type,type,
% 3.63/1.89 symdiff: $i > $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffE_type,type,
% 3.63/1.89 symdiffE: $o ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffE,definition,
% 3.63/1.89 ( symdiffE
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.63/1.89 => ! [Xphi: $o] :
% 3.63/1.89 ( ( ( in @ Xx @ A )
% 3.63/1.89 => ( ~ ( in @ Xx @ B )
% 3.63/1.89 => Xphi ) )
% 3.63/1.89 => ( ( ~ ( in @ Xx @ A )
% 3.63/1.89 => ( ( in @ Xx @ B )
% 3.63/1.89 => Xphi ) )
% 3.63/1.89 => Xphi ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffI1_type,type,
% 3.63/1.89 symdiffI1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffI1,definition,
% 3.63/1.89 ( symdiffI1
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ~ ( in @ Xx @ B )
% 3.63/1.89 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffI2_type,type,
% 3.63/1.89 symdiffI2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffI2,definition,
% 3.63/1.89 ( symdiffI2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ~ ( in @ Xx @ A )
% 3.63/1.89 => ( ( in @ Xx @ B )
% 3.63/1.89 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffIneg1_type,type,
% 3.63/1.89 symdiffIneg1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffIneg1,definition,
% 3.63/1.89 ( symdiffIneg1
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( in @ Xx @ B )
% 3.63/1.89 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffIneg2_type,type,
% 3.63/1.89 symdiffIneg2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(symdiffIneg2,definition,
% 3.63/1.89 ( symdiffIneg2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ~ ( in @ Xx @ A )
% 3.63/1.89 => ( ~ ( in @ Xx @ B )
% 3.63/1.89 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(iskpair_type,type,
% 3.63/1.89 iskpair: $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(secondinupair_type,type,
% 3.63/1.89 secondinupair: $o ).
% 3.63/1.89
% 3.63/1.89 thf(secondinupair,definition,
% 3.63/1.89 ( secondinupair
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairIL_type,type,
% 3.63/1.89 setukpairIL: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairIL,definition,
% 3.63/1.89 ( setukpairIL
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairIR_type,type,
% 3.63/1.89 setukpairIR: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairIR,definition,
% 3.63/1.89 ( setukpairIR
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(kpairiskpair_type,type,
% 3.63/1.89 kpairiskpair: $o ).
% 3.63/1.89
% 3.63/1.89 thf(kpairiskpair,definition,
% 3.63/1.89 ( kpairiskpair
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(kpair_type,type,
% 3.63/1.89 kpair: $i > $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(kpairp_type,type,
% 3.63/1.89 kpairp: $o ).
% 3.63/1.89
% 3.63/1.89 thf(kpairp,definition,
% 3.63/1.89 ( kpairp
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprod_type,type,
% 3.63/1.89 cartprod: $i > $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(singletonsubset_type,type,
% 3.63/1.89 singletonsubset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(singletonsubset,definition,
% 3.63/1.89 ( singletonsubset
% 3.63/1.89 = ( ! [A: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(singletoninpowerset_type,type,
% 3.63/1.89 singletoninpowerset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(singletoninpowerset,definition,
% 3.63/1.89 ( singletoninpowerset
% 3.63/1.89 = ( ! [A: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(singletoninpowunion_type,type,
% 3.63/1.89 singletoninpowunion: $o ).
% 3.63/1.89
% 3.63/1.89 thf(singletoninpowunion,definition,
% 3.63/1.89 ( singletoninpowunion
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(upairset2E_type,type,
% 3.63/1.89 upairset2E: $o ).
% 3.63/1.89
% 3.63/1.89 thf(upairset2E,definition,
% 3.63/1.89 ( upairset2E
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.63/1.89 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.63/1.89 => ( ( Xz = Xx )
% 3.63/1.89 | ( Xz = Xy ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(upairsubunion_type,type,
% 3.63/1.89 upairsubunion: $o ).
% 3.63/1.89
% 3.63/1.89 thf(upairsubunion,definition,
% 3.63/1.89 ( upairsubunion
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(upairinpowunion_type,type,
% 3.63/1.89 upairinpowunion: $o ).
% 3.63/1.89
% 3.63/1.89 thf(upairinpowunion,definition,
% 3.63/1.89 ( upairinpowunion
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ubforcartprodlem1_type,type,
% 3.63/1.89 ubforcartprodlem1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ubforcartprodlem1,definition,
% 3.63/1.89 ( ubforcartprodlem1
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ubforcartprodlem2_type,type,
% 3.63/1.89 ubforcartprodlem2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ubforcartprodlem2,definition,
% 3.63/1.89 ( ubforcartprodlem2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ubforcartprodlem3_type,type,
% 3.63/1.89 ubforcartprodlem3: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ubforcartprodlem3,definition,
% 3.63/1.89 ( ubforcartprodlem3
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodpairin_type,type,
% 3.63/1.89 cartprodpairin: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodpairin,definition,
% 3.63/1.89 ( cartprodpairin
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodmempair1_type,type,
% 3.63/1.89 cartprodmempair1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodmempair1,definition,
% 3.63/1.89 ( cartprodmempair1
% 3.63/1.89 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.63/1.89 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.63/1.89 => ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ? [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 & ( Xu
% 3.63/1.89 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodmempair_type,type,
% 3.63/1.89 cartprodmempair: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodmempair,definition,
% 3.63/1.89 ( cartprodmempair
% 3.63/1.89 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.63/1.89 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.63/1.89 => ( iskpair @ Xu ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setunionE2_type,type,
% 3.63/1.89 setunionE2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setunionE2,definition,
% 3.63/1.89 ( setunionE2
% 3.63/1.89 = ( ! [A: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ ( setunion @ A ) )
% 3.63/1.89 => ? [X: $i] :
% 3.63/1.89 ( ( in @ X @ A )
% 3.63/1.89 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setunionsingleton1_type,type,
% 3.63/1.89 setunionsingleton1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setunionsingleton1,definition,
% 3.63/1.89 ( setunionsingleton1
% 3.63/1.89 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setunionsingleton2_type,type,
% 3.63/1.89 setunionsingleton2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setunionsingleton2,definition,
% 3.63/1.89 ( setunionsingleton2
% 3.63/1.89 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setunionsingleton_type,type,
% 3.63/1.89 setunionsingleton: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setunionsingleton,definition,
% 3.63/1.89 ( setunionsingleton
% 3.63/1.89 = ( ! [Xx: $i] :
% 3.63/1.89 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.63/1.89 = Xx ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(singleton_type,type,
% 3.63/1.89 singleton: $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(singletonprop_type,type,
% 3.63/1.89 singletonprop: $o ).
% 3.63/1.89
% 3.63/1.89 thf(singletonprop,definition,
% 3.63/1.89 ( singletonprop
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ A )
% 3.63/1.89 => ( ( Xphi @ Xx )
% 3.63/1.89 => ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) )
% 3.63/1.89 => ( ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ( Xphi @ Xx ) )
% 3.63/1.89 => ( singleton
% 3.63/1.89 @ ( dsetconstr @ A
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ex1_type,type,
% 3.63/1.89 ex1: $i > ( $i > $o ) > $o ).
% 3.63/1.89
% 3.63/1.89 thf(ex1E1_type,type,
% 3.63/1.89 ex1E1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ex1E1,definition,
% 3.63/1.89 ( ex1E1
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ( ex1 @ A
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 => ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ex1I_type,type,
% 3.63/1.89 ex1I: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ex1I,definition,
% 3.63/1.89 ( ex1I
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( Xphi @ Xx )
% 3.63/1.89 => ( ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ A )
% 3.63/1.89 => ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xy = Xx ) ) )
% 3.63/1.89 => ( ex1 @ A
% 3.63/1.89 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ex1I2_type,type,
% 3.63/1.89 ex1I2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ex1I2,definition,
% 3.63/1.89 ( ex1I2
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ A )
% 3.63/1.89 => ( ( Xphi @ Xx )
% 3.63/1.89 => ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) )
% 3.63/1.89 => ( ? [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 & ( Xphi @ Xx ) )
% 3.63/1.89 => ( ex1 @ A
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(singletonsuniq_type,type,
% 3.63/1.89 singletonsuniq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(singletonsuniq,definition,
% 3.63/1.89 ( singletonsuniq
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( ( setadjoin @ Xx @ emptyset )
% 3.63/1.89 = ( setadjoin @ Xy @ emptyset ) )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(atmost1p_type,type,
% 3.63/1.89 atmost1p: $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(atleast2p_type,type,
% 3.63/1.89 atleast2p: $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(atmost2p_type,type,
% 3.63/1.89 atmost2p: $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(upairsetp_type,type,
% 3.63/1.89 upairsetp: $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjL1_type,type,
% 3.63/1.89 setukpairinjL1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjL1,definition,
% 3.63/1.89 ( setukpairinjL1
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.63/1.89 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.63/1.89 => ( Xx = Xz ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(kfstsingleton_type,type,
% 3.63/1.89 kfstsingleton: $o ).
% 3.63/1.89
% 3.63/1.89 thf(kfstsingleton,definition,
% 3.63/1.89 ( kfstsingleton
% 3.63/1.89 = ( ! [Xu: $i] :
% 3.63/1.89 ( ( iskpair @ Xu )
% 3.63/1.89 => ( singleton
% 3.63/1.89 @ ( dsetconstr @ ( setunion @ Xu )
% 3.63/1.89 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(theprop_type,type,
% 3.63/1.89 theprop: $o ).
% 3.63/1.89
% 3.63/1.89 thf(theprop,definition,
% 3.63/1.89 ( theprop
% 3.63/1.89 = ( ! [X: $i] :
% 3.63/1.89 ( ( singleton @ X )
% 3.63/1.89 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(kfst_type,type,
% 3.63/1.89 kfst: $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(kfstpairEq_type,type,
% 3.63/1.89 kfstpairEq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(kfstpairEq,definition,
% 3.63/1.89 ( kfstpairEq
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.63/1.89 = Xx ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodfstin_type,type,
% 3.63/1.89 cartprodfstin: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodfstin,definition,
% 3.63/1.89 ( cartprodfstin
% 3.63/1.89 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.63/1.89 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.63/1.89 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjL2_type,type,
% 3.63/1.89 setukpairinjL2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjL2,definition,
% 3.63/1.89 ( setukpairinjL2
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.63/1.89 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.63/1.89 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.63/1.89 => ( Xx = Xz ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjL_type,type,
% 3.63/1.89 setukpairinjL: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjL,definition,
% 3.63/1.89 ( setukpairinjL
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.63/1.89 ( ( ( kpair @ Xx @ Xy )
% 3.63/1.89 = ( kpair @ Xz @ Xu ) )
% 3.63/1.89 => ( Xx = Xz ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR11_type,type,
% 3.63/1.89 setukpairinjR11: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR11,definition,
% 3.63/1.89 ( setukpairinjR11
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xx = Xy )
% 3.63/1.89 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.63/1.89 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR12_type,type,
% 3.63/1.89 setukpairinjR12: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR12,definition,
% 3.63/1.89 ( setukpairinjR12
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xx = Xy )
% 3.63/1.89 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.63/1.89 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR1_type,type,
% 3.63/1.89 setukpairinjR1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR1,definition,
% 3.63/1.89 ( setukpairinjR1
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.63/1.89 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.63/1.89 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.63/1.89 => ( ( Xz = Xu )
% 3.63/1.89 => ( Xy = Xu ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(upairequniteq_type,type,
% 3.63/1.89 upairequniteq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(upairequniteq,definition,
% 3.63/1.89 ( upairequniteq
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.63/1.89 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.63/1.89 = ( setadjoin @ Xz @ emptyset ) )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR2_type,type,
% 3.63/1.89 setukpairinjR2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR2,definition,
% 3.63/1.89 ( setukpairinjR2
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.63/1.89 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.63/1.89 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.63/1.89 => ( Xy = Xu ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR_type,type,
% 3.63/1.89 setukpairinjR: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setukpairinjR,definition,
% 3.63/1.89 ( setukpairinjR
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.63/1.89 ( ( ( kpair @ Xx @ Xy )
% 3.63/1.89 = ( kpair @ Xz @ Xu ) )
% 3.63/1.89 => ( Xy = Xu ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ksndsingleton_type,type,
% 3.63/1.89 ksndsingleton: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ksndsingleton,definition,
% 3.63/1.89 ( ksndsingleton
% 3.63/1.89 = ( ! [Xu: $i] :
% 3.63/1.89 ( ( iskpair @ Xu )
% 3.63/1.89 => ( singleton
% 3.63/1.89 @ ( dsetconstr @ ( setunion @ Xu )
% 3.63/1.89 @ ^ [Xx: $i] :
% 3.63/1.89 ( Xu
% 3.63/1.89 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ksnd_type,type,
% 3.63/1.89 ksnd: $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(ksndpairEq_type,type,
% 3.63/1.89 ksndpairEq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ksndpairEq,definition,
% 3.63/1.89 ( ksndpairEq
% 3.63/1.89 = ( ! [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.63/1.89 = Xy ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(kpairsurjEq_type,type,
% 3.63/1.89 kpairsurjEq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(kpairsurjEq,definition,
% 3.63/1.89 ( kpairsurjEq
% 3.63/1.89 = ( ! [Xu: $i] :
% 3.63/1.89 ( ( iskpair @ Xu )
% 3.63/1.89 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.63/1.89 = Xu ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodsndin_type,type,
% 3.63/1.89 cartprodsndin: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodsndin,definition,
% 3.63/1.89 ( cartprodsndin
% 3.63/1.89 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.63/1.89 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.63/1.89 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodpairmemEL_type,type,
% 3.63/1.89 cartprodpairmemEL: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodpairmemEL,definition,
% 3.63/1.89 ( cartprodpairmemEL
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.63/1.89 => ( in @ Xx @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodpairmemER_type,type,
% 3.63/1.89 cartprodpairmemER: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodpairmemER,definition,
% 3.63/1.89 ( cartprodpairmemER
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.63/1.89 => ( in @ Xy @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodmempaircEq_type,type,
% 3.63/1.89 cartprodmempaircEq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodmempaircEq,definition,
% 3.63/1.89 ( cartprodmempaircEq
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( kpair @ Xx @ Xy )
% 3.63/1.89 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodfstpairEq_type,type,
% 3.63/1.89 cartprodfstpairEq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodfstpairEq,definition,
% 3.63/1.89 ( cartprodfstpairEq
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.63/1.89 = Xx ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodsndpairEq_type,type,
% 3.63/1.89 cartprodsndpairEq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodsndpairEq,definition,
% 3.63/1.89 ( cartprodsndpairEq
% 3.63/1.89 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.63/1.89 = Xy ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodpairsurjEq_type,type,
% 3.63/1.89 cartprodpairsurjEq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(cartprodpairsurjEq,definition,
% 3.63/1.89 ( cartprodpairsurjEq
% 3.63/1.89 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.63/1.89 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.63/1.89 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.63/1.89 = Xu ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(breln_type,type,
% 3.63/1.89 breln: $i > $i > $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstr_type,type,
% 3.63/1.89 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrI_type,type,
% 3.63/1.89 dpsetconstrI: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrI,definition,
% 3.63/1.89 ( dpsetconstrI
% 3.63/1.89 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( Xphi @ Xx @ Xy )
% 3.63/1.89 => ( in @ ( kpair @ Xx @ Xy )
% 3.63/1.89 @ ( dpsetconstr @ A @ B
% 3.63/1.89 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrSub_type,type,
% 3.63/1.89 dpsetconstrSub: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrSub,definition,
% 3.63/1.89 ( dpsetconstrSub
% 3.63/1.89 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.63/1.89 ( subset
% 3.63/1.89 @ ( dpsetconstr @ A @ B
% 3.63/1.89 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.63/1.89 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(setOfPairsIsBReln_type,type,
% 3.63/1.89 setOfPairsIsBReln: $o ).
% 3.63/1.89
% 3.63/1.89 thf(setOfPairsIsBReln,definition,
% 3.63/1.89 ( setOfPairsIsBReln
% 3.63/1.89 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.63/1.89 ( breln @ A @ B
% 3.63/1.89 @ ( dpsetconstr @ A @ B
% 3.63/1.89 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrERa_type,type,
% 3.63/1.89 dpsetconstrERa: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrERa,definition,
% 3.63/1.89 ( dpsetconstrERa
% 3.63/1.89 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.63/1.89 @ ( dpsetconstr @ A @ B
% 3.63/1.89 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.63/1.89 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrEL1_type,type,
% 3.63/1.89 dpsetconstrEL1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrEL1,definition,
% 3.63/1.89 ( dpsetconstrEL1
% 3.63/1.89 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ ( kpair @ Xx @ Xy )
% 3.63/1.89 @ ( dpsetconstr @ A @ B
% 3.63/1.89 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.63/1.89 => ( in @ Xx @ A ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrEL2_type,type,
% 3.63/1.89 dpsetconstrEL2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrEL2,definition,
% 3.63/1.89 ( dpsetconstrEL2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ ( kpair @ Xx @ Xy )
% 3.63/1.89 @ ( dpsetconstr @ A @ B
% 3.63/1.89 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.63/1.89 => ( in @ Xy @ B ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrER_type,type,
% 3.63/1.89 dpsetconstrER: $o ).
% 3.63/1.89
% 3.63/1.89 thf(dpsetconstrER,definition,
% 3.63/1.89 ( dpsetconstrER
% 3.63/1.89 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( in @ ( kpair @ Xx @ Xy )
% 3.63/1.89 @ ( dpsetconstr @ A @ B
% 3.63/1.89 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.63/1.89 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(func_type,type,
% 3.63/1.89 func: $i > $i > $i > $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcSet_type,type,
% 3.63/1.89 funcSet: $i > $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(funcImageSingleton_type,type,
% 3.63/1.89 funcImageSingleton: $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcImageSingleton,definition,
% 3.63/1.89 ( funcImageSingleton
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( singleton
% 3.63/1.89 @ ( dsetconstr @ B
% 3.63/1.89 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(apProp_type,type,
% 3.63/1.89 apProp: $o ).
% 3.63/1.89
% 3.63/1.89 thf(apProp,definition,
% 3.63/1.89 ( apProp
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in
% 3.63/1.89 @ ( setunion
% 3.63/1.89 @ ( dsetconstr @ B
% 3.63/1.89 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.63/1.89 @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ap_type,type,
% 3.63/1.89 ap: $i > $i > $i > $i > $i ).
% 3.63/1.89
% 3.63/1.89 thf(app_type,type,
% 3.63/1.89 app: $o ).
% 3.63/1.89
% 3.63/1.89 thf(app,definition,
% 3.63/1.89 ( app
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(infuncsetfunc_type,type,
% 3.63/1.89 infuncsetfunc: $o ).
% 3.63/1.89
% 3.63/1.89 thf(infuncsetfunc,definition,
% 3.63/1.89 ( infuncsetfunc
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.63/1.89 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ap2p_type,type,
% 3.63/1.89 ap2p: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ap2p,definition,
% 3.63/1.89 ( ap2p
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(funcinfuncset_type,type,
% 3.63/1.89 funcinfuncset: $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcinfuncset,definition,
% 3.63/1.89 ( funcinfuncset
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(lamProp_type,type,
% 3.63/1.89 lamProp: $o ).
% 3.63/1.89
% 3.63/1.89 thf(lamProp,definition,
% 3.63/1.89 ( lamProp
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.63/1.89 => ( func @ A @ B
% 3.63/1.89 @ ( dpsetconstr @ A @ B
% 3.63/1.89 @ ^ [Xx: $i,Xy: $i] :
% 3.63/1.89 ( ( Xf @ Xx )
% 3.63/1.89 = Xy ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(lam_type,type,
% 3.63/1.89 lam: $i > $i > ( $i > $i ) > $i ).
% 3.63/1.89
% 3.63/1.89 thf(lamp_type,type,
% 3.63/1.89 lamp: $o ).
% 3.63/1.89
% 3.63/1.89 thf(lamp,definition,
% 3.63/1.89 ( lamp
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.63/1.89 => ( func @ A @ B
% 3.63/1.89 @ ( lam @ A @ B
% 3.63/1.89 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(lam2p_type,type,
% 3.63/1.89 lam2p: $o ).
% 3.63/1.89
% 3.63/1.89 thf(lam2p,definition,
% 3.63/1.89 ( lam2p
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.63/1.89 => ( in
% 3.63/1.89 @ ( lam @ A @ B
% 3.63/1.89 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.63/1.89 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(brelnall1_type,type,
% 3.63/1.89 brelnall1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(brelnall1,definition,
% 3.63/1.89 ( brelnall1
% 3.63/1.89 = ( ! [A: $i,B: $i,R: $i] :
% 3.63/1.89 ( ( breln @ A @ B @ R )
% 3.63/1.89 => ! [Xphi: $i > $o] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.63/1.89 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ R )
% 3.63/1.89 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(brelnall2_type,type,
% 3.63/1.89 brelnall2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(brelnall2,definition,
% 3.63/1.89 ( brelnall2
% 3.63/1.89 = ( ! [A: $i,B: $i,R: $i] :
% 3.63/1.89 ( ( breln @ A @ B @ R )
% 3.63/1.89 => ! [Xphi: $i > $o] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.63/1.89 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ R )
% 3.63/1.89 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ex1E2_type,type,
% 3.63/1.89 ex1E2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ex1E2,definition,
% 3.63/1.89 ( ex1E2
% 3.63/1.89 = ( ! [A: $i,Xphi: $i > $o] :
% 3.63/1.89 ( ( ex1 @ A
% 3.63/1.89 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ A )
% 3.63/1.89 => ( ( Xphi @ Xx )
% 3.63/1.89 => ( ( Xphi @ Xy )
% 3.63/1.89 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(funcGraphProp1_type,type,
% 3.63/1.89 funcGraphProp1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcGraphProp1,definition,
% 3.63/1.89 ( funcGraphProp1
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(funcGraphProp3_type,type,
% 3.63/1.89 funcGraphProp3: $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcGraphProp3,definition,
% 3.63/1.89 ( funcGraphProp3
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(funcGraphProp2_type,type,
% 3.63/1.89 funcGraphProp2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcGraphProp2,definition,
% 3.63/1.89 ( funcGraphProp2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.63/1.89 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.63/1.89 = Xy ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(funcextLem_type,type,
% 3.63/1.89 funcextLem: $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcextLem,definition,
% 3.63/1.89 ( funcextLem
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ! [Xg: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xg )
% 3.63/1.89 => ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.63/1.89 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.63/1.89 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(funcGraphProp4_type,type,
% 3.63/1.89 funcGraphProp4: $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcGraphProp4,definition,
% 3.63/1.89 ( funcGraphProp4
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.63/1.89 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.63/1.89 = Xy ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(subbreln_type,type,
% 3.63/1.89 subbreln: $o ).
% 3.63/1.89
% 3.63/1.89 thf(subbreln,definition,
% 3.63/1.89 ( subbreln
% 3.63/1.89 = ( ! [A: $i,B: $i,R: $i] :
% 3.63/1.89 ( ( breln @ A @ B @ R )
% 3.63/1.89 => ! [S: $i] :
% 3.63/1.89 ( ( breln @ A @ B @ S )
% 3.63/1.89 => ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.63/1.89 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.63/1.89 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(eqbreln_type,type,
% 3.63/1.89 eqbreln: $o ).
% 3.63/1.89
% 3.63/1.89 thf(eqbreln,definition,
% 3.63/1.89 ( eqbreln
% 3.63/1.89 = ( ! [A: $i,B: $i,R: $i] :
% 3.63/1.89 ( ( breln @ A @ B @ R )
% 3.63/1.89 => ! [S: $i] :
% 3.63/1.89 ( ( breln @ A @ B @ S )
% 3.63/1.89 => ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.63/1.89 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.63/1.89 => ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ! [Xy: $i] :
% 3.63/1.89 ( ( in @ Xy @ B )
% 3.63/1.89 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.63/1.89 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.63/1.89 => ( R = S ) ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(funcext_type,type,
% 3.63/1.89 funcext: $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcext,definition,
% 3.63/1.89 ( funcext
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ! [Xg: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xg )
% 3.63/1.89 => ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.63/1.89 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.63/1.89 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(funcext2_type,type,
% 3.63/1.89 funcext2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(funcext2,definition,
% 3.63/1.89 ( funcext2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.63/1.89 => ! [Xg: $i] :
% 3.63/1.89 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.63/1.89 => ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.63/1.89 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.63/1.89 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ap2apEq1_type,type,
% 3.63/1.89 ap2apEq1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ap2apEq1,definition,
% 3.63/1.89 ( ap2apEq1
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.63/1.89 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(ap2apEq2_type,type,
% 3.63/1.89 ap2apEq2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(ap2apEq2,definition,
% 3.63/1.89 ( ap2apEq2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.63/1.89 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(beta1_type,type,
% 3.63/1.89 beta1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(beta1,definition,
% 3.63/1.89 ( beta1
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( ap @ A @ B
% 3.63/1.89 @ ( lam @ A @ B
% 3.63/1.89 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.63/1.89 @ Xx )
% 3.63/1.89 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(eta1_type,type,
% 3.63/1.89 eta1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(eta1,definition,
% 3.63/1.89 ( eta1
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( func @ A @ B @ Xf )
% 3.63/1.89 => ( ( lam @ A @ B
% 3.63/1.89 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.63/1.89 = Xf ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(lam2lamEq_type,type,
% 3.63/1.89 lam2lamEq: $o ).
% 3.63/1.89
% 3.63/1.89 thf(lam2lamEq,definition,
% 3.63/1.89 ( lam2lamEq
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.63/1.89 => ( ( lam @ A @ B
% 3.63/1.89 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.63/1.89 = ( lam @ A @ B
% 3.63/1.89 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(beta2_type,type,
% 3.63/1.89 beta2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(beta2,definition,
% 3.63/1.89 ( beta2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.63/1.89 ( ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.63/1.89 => ! [Xx: $i] :
% 3.63/1.89 ( ( in @ Xx @ A )
% 3.63/1.89 => ( ( ap @ A @ B
% 3.63/1.89 @ ( lam @ A @ B
% 3.63/1.89 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.63/1.89 @ Xx )
% 3.63/1.89 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(eta2_type,type,
% 3.63/1.89 eta2: $o ).
% 3.63/1.89
% 3.63/1.89 thf(eta2,definition,
% 3.63/1.89 ( eta2
% 3.63/1.89 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.63/1.89 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.63/1.89 => ( ( lam @ A @ B
% 3.63/1.89 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.63/1.89 = Xf ) ) ) ) ).
% 3.63/1.89
% 3.63/1.89 thf(iffalseProp1_type,type,
% 3.63/1.89 iffalseProp1: $o ).
% 3.63/1.89
% 3.63/1.89 thf(iffalseProp1,definition,
% 3.63/1.90 ( iffalseProp1
% 3.63/1.90 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ! [Xy: $i] :
% 3.63/1.90 ( ( in @ Xy @ A )
% 3.63/1.90 => ( ~ Xphi
% 3.63/1.90 => ( in @ Xy
% 3.63/1.90 @ ( dsetconstr @ A
% 3.63/1.90 @ ^ [Xz: $i] :
% 3.63/1.90 ( ( Xphi
% 3.63/1.90 & ( Xz = Xx ) )
% 3.63/1.90 | ( ~ Xphi
% 3.63/1.90 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(iffalseProp2_type,type,
% 3.63/1.90 iffalseProp2: $o ).
% 3.63/1.90
% 3.63/1.90 thf(iffalseProp2,definition,
% 3.63/1.90 ( iffalseProp2
% 3.63/1.90 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ! [Xy: $i] :
% 3.63/1.90 ( ( in @ Xy @ A )
% 3.63/1.90 => ( ~ Xphi
% 3.63/1.90 => ( ( dsetconstr @ A
% 3.63/1.90 @ ^ [Xz: $i] :
% 3.63/1.90 ( ( Xphi
% 3.63/1.90 & ( Xz = Xx ) )
% 3.63/1.90 | ( ~ Xphi
% 3.63/1.90 & ( Xz = Xy ) ) ) )
% 3.63/1.90 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(iftrueProp1_type,type,
% 3.63/1.90 iftrueProp1: $o ).
% 3.63/1.90
% 3.63/1.90 thf(iftrueProp1,definition,
% 3.63/1.90 ( iftrueProp1
% 3.63/1.90 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ! [Xy: $i] :
% 3.63/1.90 ( ( in @ Xy @ A )
% 3.63/1.90 => ( Xphi
% 3.63/1.90 => ( in @ Xx
% 3.63/1.90 @ ( dsetconstr @ A
% 3.63/1.90 @ ^ [Xz: $i] :
% 3.63/1.90 ( ( Xphi
% 3.63/1.90 & ( Xz = Xx ) )
% 3.63/1.90 | ( ~ Xphi
% 3.63/1.90 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(iftrueProp2_type,type,
% 3.63/1.90 iftrueProp2: $o ).
% 3.63/1.90
% 3.63/1.90 thf(iftrueProp2,definition,
% 3.63/1.90 ( iftrueProp2
% 3.63/1.90 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ! [Xy: $i] :
% 3.63/1.90 ( ( in @ Xy @ A )
% 3.63/1.90 => ( Xphi
% 3.63/1.90 => ( ( dsetconstr @ A
% 3.63/1.90 @ ^ [Xz: $i] :
% 3.63/1.90 ( ( Xphi
% 3.63/1.90 & ( Xz = Xx ) )
% 3.63/1.90 | ( ~ Xphi
% 3.63/1.90 & ( Xz = Xy ) ) ) )
% 3.63/1.90 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(ifSingleton_type,type,
% 3.63/1.90 ifSingleton: $o ).
% 3.63/1.90
% 3.63/1.90 thf(ifSingleton,definition,
% 3.63/1.90 ( ifSingleton
% 3.63/1.90 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ! [Xy: $i] :
% 3.63/1.90 ( ( in @ Xy @ A )
% 3.63/1.90 => ( singleton
% 3.63/1.90 @ ( dsetconstr @ A
% 3.63/1.90 @ ^ [Xz: $i] :
% 3.63/1.90 ( ( Xphi
% 3.63/1.90 & ( Xz = Xx ) )
% 3.63/1.90 | ( ~ Xphi
% 3.63/1.90 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(if_type,type,
% 3.63/1.90 if: $i > $o > $i > $i > $i ).
% 3.63/1.90
% 3.63/1.90 thf(ifp_type,type,
% 3.63/1.90 ifp: $o ).
% 3.63/1.90
% 3.63/1.90 thf(ifp,definition,
% 3.63/1.90 ( ifp
% 3.63/1.90 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ! [Xy: $i] :
% 3.63/1.90 ( ( in @ Xy @ A )
% 3.63/1.90 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(theeq_type,type,
% 3.63/1.90 theeq: $o ).
% 3.63/1.90
% 3.63/1.90 thf(theeq,definition,
% 3.63/1.90 ( theeq
% 3.63/1.90 = ( ! [X: $i] :
% 3.63/1.90 ( ( singleton @ X )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ X )
% 3.63/1.90 => ( ( setunion @ X )
% 3.63/1.90 = Xx ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(iftrue_type,type,
% 3.63/1.90 iftrue: $o ).
% 3.63/1.90
% 3.63/1.90 thf(iftrue,definition,
% 3.63/1.90 ( iftrue
% 3.63/1.90 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ! [Xy: $i] :
% 3.63/1.90 ( ( in @ Xy @ A )
% 3.63/1.90 => ( Xphi
% 3.63/1.90 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.63/1.90 = Xx ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(iffalse_type,type,
% 3.63/1.90 iffalse: $o ).
% 3.63/1.90
% 3.63/1.90 thf(iffalse,definition,
% 3.63/1.90 ( iffalse
% 3.63/1.90 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ! [Xy: $i] :
% 3.63/1.90 ( ( in @ Xy @ A )
% 3.63/1.90 => ( ~ Xphi
% 3.63/1.90 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.63/1.90 = Xy ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(iftrueorfalse_type,type,
% 3.63/1.90 iftrueorfalse: $o ).
% 3.63/1.90
% 3.63/1.90 thf(iftrueorfalse,definition,
% 3.63/1.90 ( iftrueorfalse
% 3.63/1.90 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ! [Xy: $i] :
% 3.63/1.90 ( ( in @ Xy @ A )
% 3.63/1.90 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(binintersectT_lem_type,type,
% 3.63/1.90 binintersectT_lem: $o ).
% 3.63/1.90
% 3.63/1.90 thf(binintersectT_lem,definition,
% 3.63/1.90 ( binintersectT_lem
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(binunionT_lem_type,type,
% 3.63/1.90 binunionT_lem: $o ).
% 3.63/1.90
% 3.63/1.90 thf(binunionT_lem,definition,
% 3.63/1.90 ( binunionT_lem
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(powersetT_lem_type,type,
% 3.63/1.90 powersetT_lem: $o ).
% 3.63/1.90
% 3.63/1.90 thf(powersetT_lem,definition,
% 3.63/1.90 ( powersetT_lem
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(setminusT_lem_type,type,
% 3.63/1.90 setminusT_lem: $o ).
% 3.63/1.90
% 3.63/1.90 thf(setminusT_lem,definition,
% 3.63/1.90 ( setminusT_lem
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(complementT_lem_type,type,
% 3.63/1.90 complementT_lem: $o ).
% 3.63/1.90
% 3.63/1.90 thf(complementT_lem,definition,
% 3.63/1.90 ( complementT_lem
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(setextT_type,type,
% 3.63/1.90 setextT: $o ).
% 3.63/1.90
% 3.63/1.90 thf(setextT,definition,
% 3.63/1.90 ( setextT
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ( ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( in @ Xx @ X )
% 3.63/1.90 => ( in @ Xx @ Y ) ) )
% 3.63/1.90 => ( ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( in @ Xx @ Y )
% 3.63/1.90 => ( in @ Xx @ X ) ) )
% 3.63/1.90 => ( X = Y ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(subsetTI_type,type,
% 3.63/1.90 subsetTI: $o ).
% 3.63/1.90
% 3.63/1.90 thf(subsetTI,definition,
% 3.63/1.90 ( subsetTI
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ( ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( in @ Xx @ X )
% 3.63/1.90 => ( in @ Xx @ Y ) ) )
% 3.63/1.90 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(powersetTI1_type,type,
% 3.63/1.90 powersetTI1: $o ).
% 3.63/1.90
% 3.63/1.90 thf(powersetTI1,definition,
% 3.63/1.90 ( powersetTI1
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ( ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( in @ Xx @ X )
% 3.63/1.90 => ( in @ Xx @ Y ) ) )
% 3.63/1.90 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(powersetTE1_type,type,
% 3.63/1.90 powersetTE1: $o ).
% 3.63/1.90
% 3.63/1.90 thf(powersetTE1,definition,
% 3.63/1.90 ( powersetTE1
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( in @ X @ ( powerset @ Y ) )
% 3.63/1.90 => ( ( in @ Xx @ X )
% 3.63/1.90 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(complementTI1_type,type,
% 3.63/1.90 complementTI1: $o ).
% 3.63/1.90
% 3.63/1.90 thf(complementTI1,definition,
% 3.63/1.90 ( complementTI1
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( in @ Xx @ X )
% 3.63/1.90 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(complementTE1_type,type,
% 3.63/1.90 complementTE1: $o ).
% 3.63/1.90
% 3.63/1.90 thf(complementTE1,definition,
% 3.63/1.90 ( complementTE1
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.63/1.90 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(binintersectTELcontra_type,type,
% 3.63/1.90 binintersectTELcontra: $o ).
% 3.63/1.90
% 3.63/1.90 thf(binintersectTELcontra,definition,
% 3.63/1.90 ( binintersectTELcontra
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ~ ( in @ Xx @ X )
% 3.63/1.90 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(binintersectTERcontra_type,type,
% 3.63/1.90 binintersectTERcontra: $o ).
% 3.63/1.90
% 3.63/1.90 thf(binintersectTERcontra,definition,
% 3.63/1.90 ( binintersectTERcontra
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ~ ( in @ Xx @ Y )
% 3.63/1.90 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(contrasubsetT_type,type,
% 3.63/1.90 contrasubsetT: $o ).
% 3.63/1.90
% 3.63/1.90 thf(contrasubsetT,definition,
% 3.63/1.90 ( contrasubsetT
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.63/1.90 => ( ( in @ Xx @ Y )
% 3.63/1.90 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(contrasubsetT1_type,type,
% 3.63/1.90 contrasubsetT1: $o ).
% 3.63/1.90
% 3.63/1.90 thf(contrasubsetT1,definition,
% 3.63/1.90 ( contrasubsetT1
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( subset @ X @ Y )
% 3.63/1.90 => ( ~ ( in @ Xx @ Y )
% 3.63/1.90 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(contrasubsetT2_type,type,
% 3.63/1.90 contrasubsetT2: $o ).
% 3.63/1.90
% 3.63/1.90 thf(contrasubsetT2,definition,
% 3.63/1.90 ( contrasubsetT2
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ( ( subset @ X @ Y )
% 3.63/1.90 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(contrasubsetT3_type,type,
% 3.63/1.90 contrasubsetT3: $o ).
% 3.63/1.90
% 3.63/1.90 thf(contrasubsetT3,definition,
% 3.63/1.90 ( contrasubsetT3
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Y: $i] :
% 3.63/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.63/1.90 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.63/1.90 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementI1_type,type,
% 3.63/1.90 doubleComplementI1: $o ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementI1,definition,
% 3.63/1.90 ( doubleComplementI1
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( in @ Xx @ X )
% 3.63/1.90 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementE1_type,type,
% 3.63/1.90 doubleComplementE1: $o ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementE1,definition,
% 3.63/1.90 ( doubleComplementE1
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ! [Xx: $i] :
% 3.63/1.90 ( ( in @ Xx @ A )
% 3.63/1.90 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.63/1.90 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementSub1_type,type,
% 3.63/1.90 doubleComplementSub1: $o ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementSub1,definition,
% 3.63/1.90 ( doubleComplementSub1
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementSub2_type,type,
% 3.63/1.90 doubleComplementSub2: $o ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementSub2,definition,
% 3.63/1.90 ( doubleComplementSub2
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementEq_type,type,
% 3.63/1.90 doubleComplementEq: $o ).
% 3.63/1.90
% 3.63/1.90 thf(doubleComplementEq,definition,
% 3.63/1.90 ( doubleComplementEq
% 3.63/1.90 = ( ! [A: $i,X: $i] :
% 3.63/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.63/1.90 => ( X
% 3.63/1.90 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.63/1.90
% 3.63/1.90 thf(complementTnotintersectT_type,type,
% 3.72/1.90 complementTnotintersectT: $o ).
% 3.72/1.90
% 3.72/1.90 thf(complementTnotintersectT,definition,
% 3.72/1.90 ( complementTnotintersectT
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.72/1.90 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(complementImpComplementIntersect_type,type,
% 3.72/1.90 complementImpComplementIntersect: $o ).
% 3.72/1.90
% 3.72/1.90 thf(complementImpComplementIntersect,definition,
% 3.72/1.90 ( complementImpComplementIntersect
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.72/1.90 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(complementSubsetComplementIntersect_type,type,
% 3.72/1.90 complementSubsetComplementIntersect: $o ).
% 3.72/1.90
% 3.72/1.90 thf(complementSubsetComplementIntersect,definition,
% 3.72/1.90 ( complementSubsetComplementIntersect
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(complementInPowersetComplementIntersect_type,type,
% 3.72/1.90 complementInPowersetComplementIntersect: $o ).
% 3.72/1.90
% 3.72/1.90 thf(complementInPowersetComplementIntersect,definition,
% 3.72/1.90 ( complementInPowersetComplementIntersect
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(contraSubsetComplement_type,type,
% 3.72/1.90 contraSubsetComplement: $o ).
% 3.72/1.90
% 3.72/1.90 thf(contraSubsetComplement,definition,
% 3.72/1.90 ( contraSubsetComplement
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ Y )
% 3.72/1.90 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(complementTcontraSubset_type,type,
% 3.72/1.90 complementTcontraSubset: $o ).
% 3.72/1.90
% 3.72/1.90 thf(complementTcontraSubset,definition,
% 3.72/1.90 ( complementTcontraSubset
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.72/1.90 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(binunionTILcontra_type,type,
% 3.72/1.90 binunionTILcontra: $o ).
% 3.72/1.90
% 3.72/1.90 thf(binunionTILcontra,definition,
% 3.72/1.90 ( binunionTILcontra
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.72/1.90 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(binunionTIRcontra_type,type,
% 3.72/1.90 binunionTIRcontra: $o ).
% 3.72/1.90
% 3.72/1.90 thf(binunionTIRcontra,definition,
% 3.72/1.90 ( binunionTIRcontra
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.72/1.90 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(inIntersectImpInUnion_type,type,
% 3.72/1.90 inIntersectImpInUnion: $o ).
% 3.72/1.90
% 3.72/1.90 thf(inIntersectImpInUnion,definition,
% 3.72/1.90 ( inIntersectImpInUnion
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.72/1.90 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(inIntersectImpInUnion2_type,type,
% 3.72/1.90 inIntersectImpInUnion2: $o ).
% 3.72/1.90
% 3.72/1.90 thf(inIntersectImpInUnion2,definition,
% 3.72/1.90 ( inIntersectImpInUnion2
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.72/1.90 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(inIntersectImpInIntersectUnions_type,type,
% 3.72/1.90 inIntersectImpInIntersectUnions: $o ).
% 3.72/1.90
% 3.72/1.90 thf(inIntersectImpInIntersectUnions,definition,
% 3.72/1.90 ( inIntersectImpInIntersectUnions
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.72/1.90 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(intersectInPowersetIntersectUnions_type,type,
% 3.72/1.90 intersectInPowersetIntersectUnions: $o ).
% 3.72/1.90
% 3.72/1.90 thf(intersectInPowersetIntersectUnions,definition,
% 3.72/1.90 ( intersectInPowersetIntersectUnions
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(inComplementUnionImpNotIn1_type,type,
% 3.72/1.90 inComplementUnionImpNotIn1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(inComplementUnionImpNotIn1,definition,
% 3.72/1.90 ( inComplementUnionImpNotIn1
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.72/1.90 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(inComplementUnionImpInComplement1_type,type,
% 3.72/1.90 inComplementUnionImpInComplement1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(inComplementUnionImpInComplement1,definition,
% 3.72/1.90 ( inComplementUnionImpInComplement1
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.72/1.90 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(binunionTE_type,type,
% 3.72/1.90 binunionTE: $o ).
% 3.72/1.90
% 3.72/1.90 thf(binunionTE,definition,
% 3.72/1.90 ( binunionTE
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xphi: $o,Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.72/1.90 => ( ( ( in @ Xx @ X )
% 3.72/1.90 => Xphi )
% 3.72/1.90 => ( ( ( in @ Xx @ Y )
% 3.72/1.90 => Xphi )
% 3.72/1.90 => Xphi ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(binunionTEcontra_type,type,
% 3.72/1.90 binunionTEcontra: $o ).
% 3.72/1.90
% 3.72/1.90 thf(binunionTEcontra,definition,
% 3.72/1.90 ( binunionTEcontra
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ~ ( in @ Xx @ X )
% 3.72/1.90 => ( ~ ( in @ Xx @ Y )
% 3.72/1.90 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2a1_type,type,
% 3.72/1.90 demorgan2a1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2a1,definition,
% 3.72/1.90 ( demorgan2a1
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.72/1.90 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(complementUnionInPowersetComplement_type,type,
% 3.72/1.90 complementUnionInPowersetComplement: $o ).
% 3.72/1.90
% 3.72/1.90 thf(complementUnionInPowersetComplement,definition,
% 3.72/1.90 ( complementUnionInPowersetComplement
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2a2_type,type,
% 3.72/1.90 demorgan2a2: $o ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2a2,definition,
% 3.72/1.90 ( demorgan2a2
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.72/1.90 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan1a_type,type,
% 3.72/1.90 demorgan1a: $o ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan1a,definition,
% 3.72/1.90 ( demorgan1a
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 3.72/1.90 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan1b_type,type,
% 3.72/1.90 demorgan1b: $o ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan1b,definition,
% 3.72/1.90 ( demorgan1b
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.72/1.90 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan1_type,type,
% 3.72/1.90 demorgan1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan1,definition,
% 3.72/1.90 ( demorgan1
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 3.72/1.90 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2a_type,type,
% 3.72/1.90 demorgan2a: $o ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2a,definition,
% 3.72/1.90 ( demorgan2a
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 3.72/1.90 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2b2_type,type,
% 3.72/1.90 demorgan2b2: $o ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2b2,definition,
% 3.72/1.90 ( demorgan2b2
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.72/1.90 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 3.72/1.90 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2b_type,type,
% 3.72/1.90 demorgan2b: $o ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2b,definition,
% 3.72/1.90 ( demorgan2b
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 3.72/1.90 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2_type,type,
% 3.72/1.90 demorgan2: $o ).
% 3.72/1.90
% 3.72/1.90 thf(demorgan2,definition,
% 3.72/1.90 ( demorgan2
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 3.72/1.90 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule0_type,type,
% 3.72/1.90 woz13rule0: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule0,definition,
% 3.72/1.90 ( woz13rule0
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.72/1.90 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule1_type,type,
% 3.72/1.90 woz13rule1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule1,definition,
% 3.72/1.90 ( woz13rule1
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ( ( subset @ X @ Z )
% 3.72/1.90 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule2_type,type,
% 3.72/1.90 woz13rule2: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule2,definition,
% 3.72/1.90 ( woz13rule2
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ( ( subset @ Y @ Z )
% 3.72/1.90 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule3_type,type,
% 3.72/1.90 woz13rule3: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule3,definition,
% 3.72/1.90 ( woz13rule3
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ( ( subset @ X @ Y )
% 3.72/1.90 => ( ( subset @ X @ Z )
% 3.72/1.90 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule4_type,type,
% 3.72/1.90 woz13rule4: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz13rule4,definition,
% 3.72/1.90 ( woz13rule4
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ! [W: $i] :
% 3.72/1.90 ( ( in @ W @ ( powerset @ A ) )
% 3.72/1.90 => ( ( subset @ X @ Z )
% 3.72/1.90 => ( ( subset @ Y @ W )
% 3.72/1.90 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_1_type,type,
% 3.72/1.90 woz1_1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_1,definition,
% 3.72/1.90 ( woz1_1
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_2_type,type,
% 3.72/1.90 woz1_2: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_2,definition,
% 3.72/1.90 ( woz1_2
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ! [W: $i] :
% 3.72/1.90 ( ( in @ W @ ( powerset @ A ) )
% 3.72/1.90 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 3.72/1.90 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_3_type,type,
% 3.72/1.90 woz1_3: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_3,definition,
% 3.72/1.90 ( woz1_3
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ! [Z: $i] :
% 3.72/1.90 ( ( in @ Z @ ( powerset @ A ) )
% 3.72/1.90 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_4_type,type,
% 3.72/1.90 woz1_4: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_4,definition,
% 3.72/1.90 ( woz1_4
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.72/1.90 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_5_type,type,
% 3.72/1.90 woz1_5: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz1_5,definition,
% 3.72/1.90 ( woz1_5
% 3.72/1.90 = ( ! [A: $i,X: $i] :
% 3.72/1.90 ( ( in @ X @ ( powerset @ A ) )
% 3.72/1.90 => ! [Y: $i] :
% 3.72/1.90 ( ( in @ Y @ ( powerset @ A ) )
% 3.72/1.90 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1_type,type,
% 3.72/1.90 breln1: $i > $i > $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1all2_type,type,
% 3.72/1.90 breln1all2: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1all2,definition,
% 3.72/1.90 ( breln1all2
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [Xphi: $i > $o] :
% 3.72/1.90 ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.72/1.90 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ R )
% 3.72/1.90 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1Set_type,type,
% 3.72/1.90 breln1Set: $i > $i ).
% 3.72/1.90
% 3.72/1.90 thf(breln1SetBreln1_type,type,
% 3.72/1.90 breln1SetBreln1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1SetBreln1,definition,
% 3.72/1.90 ( breln1SetBreln1
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( in @ R @ ( breln1Set @ A ) )
% 3.72/1.90 => ( breln1 @ A @ R ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(transitive_type,type,
% 3.72/1.90 transitive: $i > $i > $o ).
% 3.72/1.90
% 3.72/1.90 thf(antisymmetric_type,type,
% 3.72/1.90 antisymmetric: $i > $i > $o ).
% 3.72/1.90
% 3.72/1.90 thf(reflexive_type,type,
% 3.72/1.90 reflexive: $i > $i > $o ).
% 3.72/1.90
% 3.72/1.90 thf(refltransitive_type,type,
% 3.72/1.90 refltransitive: $i > $i > $o ).
% 3.72/1.90
% 3.72/1.90 thf(refllinearorder_type,type,
% 3.72/1.90 refllinearorder: $i > $i > $o ).
% 3.72/1.90
% 3.72/1.90 thf(reflwellordering_type,type,
% 3.72/1.90 reflwellordering: $i > $i > $o ).
% 3.72/1.90
% 3.72/1.90 thf(choice2fnsingleton_type,type,
% 3.72/1.90 choice2fnsingleton: $o ).
% 3.72/1.90
% 3.72/1.90 thf(choice2fnsingleton,definition,
% 3.72/1.90 ( choice2fnsingleton
% 3.72/1.90 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.72/1.90 ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ? [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ B )
% 3.72/1.90 & ( Xphi @ Xx @ Xy ) ) )
% 3.72/1.90 => ! [R: $i] :
% 3.72/1.90 ( ( in @ R @ ( breln1Set @ B ) )
% 3.72/1.90 => ( ( reflwellordering @ B @ R )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( singleton
% 3.72/1.90 @ ( dsetconstr @ B
% 3.72/1.90 @ ^ [Xy: $i] :
% 3.72/1.90 ( ( Xphi @ Xx @ Xy )
% 3.72/1.90 & ! [Xz: $i] :
% 3.72/1.90 ( ( in @ Xz @ B )
% 3.72/1.90 => ( ( Xphi @ Xx @ Xz )
% 3.72/1.90 => ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(setOfPairsIsBReln1_type,type,
% 3.72/1.90 setOfPairsIsBReln1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(setOfPairsIsBReln1,definition,
% 3.72/1.90 ( setOfPairsIsBReln1
% 3.72/1.90 = ( ! [A: $i,Xphi: $i > $i > $o] :
% 3.72/1.90 ( breln1 @ A
% 3.72/1.90 @ ( dpsetconstr @ A @ A
% 3.72/1.90 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1all1_type,type,
% 3.72/1.90 breln1all1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1all1,definition,
% 3.72/1.90 ( breln1all1
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [Xphi: $i > $o] :
% 3.72/1.90 ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.72/1.90 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ R )
% 3.72/1.90 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(subbreln1_type,type,
% 3.72/1.90 subbreln1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(subbreln1,definition,
% 3.72/1.90 ( subbreln1
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.72/1.90 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.72/1.90 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(eqbreln1_type,type,
% 3.72/1.90 eqbreln1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(eqbreln1,definition,
% 3.72/1.90 ( eqbreln1
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.72/1.90 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.72/1.90 => ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.72/1.90 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.72/1.90 => ( R = S ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1invset_type,type,
% 3.72/1.90 breln1invset: $i > $i > $i ).
% 3.72/1.90
% 3.72/1.90 thf(breln1invprop_type,type,
% 3.72/1.90 breln1invprop: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1invprop,definition,
% 3.72/1.90 ( breln1invprop
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1invI_type,type,
% 3.72/1.90 breln1invI: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1invI,definition,
% 3.72/1.90 ( breln1invI
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.72/1.90 => ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1invE_type,type,
% 3.72/1.90 breln1invE: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1invE,definition,
% 3.72/1.90 ( breln1invE
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
% 3.72/1.90 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1compset_type,type,
% 3.72/1.90 breln1compset: $i > $i > $i > $i ).
% 3.72/1.90
% 3.72/1.90 thf(breln1compprop_type,type,
% 3.72/1.90 breln1compprop: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1compprop,definition,
% 3.72/1.90 ( breln1compprop
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ( breln1 @ A @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1compI_type,type,
% 3.72/1.90 breln1compI: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1compI,definition,
% 3.72/1.90 ( breln1compI
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ! [Xz: $i] :
% 3.72/1.90 ( ( in @ Xz @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.72/1.90 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.72/1.90 => ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1compE_type,type,
% 3.72/1.90 breln1compE: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1compE,definition,
% 3.72/1.90 ( breln1compE
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.72/1.90 => ? [Xz: $i] :
% 3.72/1.90 ( ( in @ Xz @ A )
% 3.72/1.90 & ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.72/1.90 & ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1compEex_type,type,
% 3.72/1.90 breln1compEex: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1compEex,definition,
% 3.72/1.90 ( breln1compEex
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) )
% 3.72/1.90 => ! [Xphi: $o] :
% 3.72/1.90 ( ! [Xz: $i] :
% 3.72/1.90 ( ( in @ Xz @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xz ) @ R )
% 3.72/1.90 => ( ( in @ ( kpair @ Xz @ Xy ) @ S )
% 3.72/1.90 => Xphi ) ) )
% 3.72/1.90 => Xphi ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionprop_type,type,
% 3.72/1.90 breln1unionprop: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionprop,definition,
% 3.72/1.90 ( breln1unionprop
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionIL_type,type,
% 3.72/1.90 breln1unionIL: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionIL,definition,
% 3.72/1.90 ( breln1unionIL
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.72/1.90 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionIR_type,type,
% 3.72/1.90 breln1unionIR: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionIR,definition,
% 3.72/1.90 ( breln1unionIR
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.72/1.90 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionI_type,type,
% 3.72/1.90 breln1unionI: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionI,definition,
% 3.72/1.90 ( breln1unionI
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.72/1.90 | ( in @ ( kpair @ Xx @ Xy ) @ S ) )
% 3.72/1.90 => ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionE_type,type,
% 3.72/1.90 breln1unionE: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionE,definition,
% 3.72/1.90 ( breln1unionE
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.72/1.90 | ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionEcases_type,type,
% 3.72/1.90 breln1unionEcases: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionEcases,definition,
% 3.72/1.90 ( breln1unionEcases
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ! [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 => ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
% 3.72/1.90 => ! [Xphi: $o] :
% 3.72/1.90 ( ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.72/1.90 => Xphi )
% 3.72/1.90 => ( ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.72/1.90 => Xphi )
% 3.72/1.90 => Xphi ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionCommutes_type,type,
% 3.72/1.90 breln1unionCommutes: $o ).
% 3.72/1.90
% 3.72/1.90 thf(breln1unionCommutes,definition,
% 3.72/1.90 ( breln1unionCommutes
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ( ( binunion @ R @ S )
% 3.72/1.90 = ( binunion @ S @ R ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz2Ex_type,type,
% 3.72/1.90 woz2Ex: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz2Ex,definition,
% 3.72/1.90 ( woz2Ex
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ( R
% 3.72/1.90 = ( breln1invset @ A @ ( breln1invset @ A @ R ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz2W_type,type,
% 3.72/1.90 woz2W: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz2W,definition,
% 3.72/1.90 ( woz2W
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ( ( breln1invset @ A @ ( breln1compset @ A @ R @ S ) )
% 3.72/1.90 = ( breln1compset @ A @ ( breln1invset @ A @ S ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz2A_type,type,
% 3.72/1.90 woz2A: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz2A,definition,
% 3.72/1.90 ( woz2A
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [T: $i] :
% 3.72/1.90 ( ( breln1 @ A @ T )
% 3.72/1.90 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.72/1.90 = ( binunion @ ( breln1compset @ A @ R @ T ) @ ( breln1compset @ A @ S @ T ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(woz2B_type,type,
% 3.72/1.90 woz2B: $o ).
% 3.72/1.90
% 3.72/1.90 thf(woz2B,definition,
% 3.72/1.90 ( woz2B
% 3.72/1.90 = ( ! [A: $i,R: $i] :
% 3.72/1.90 ( ( breln1 @ A @ R )
% 3.72/1.90 => ! [S: $i] :
% 3.72/1.90 ( ( breln1 @ A @ S )
% 3.72/1.90 => ! [T: $i] :
% 3.72/1.90 ( ( breln1 @ A @ T )
% 3.72/1.90 => ( ( breln1compset @ A @ ( binunion @ R @ S ) @ T )
% 3.72/1.90 = ( binunion @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ S ) ) ) @ ( breln1invset @ A @ ( breln1compset @ A @ ( breln1invset @ A @ T ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(image1Ex_type,type,
% 3.72/1.90 image1Ex: $o ).
% 3.72/1.90
% 3.72/1.90 thf(image1Ex,definition,
% 3.72/1.90 ( image1Ex
% 3.72/1.90 = ( ! [A: $i,Xf: $i > $i] :
% 3.72/1.90 ? [B: $i] :
% 3.72/1.90 ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ B )
% 3.72/1.90 <=> ? [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 & ( Xx
% 3.72/1.90 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(image1Ex1_type,type,
% 3.72/1.90 image1Ex1: $o ).
% 3.72/1.90
% 3.72/1.90 thf(image1Ex1,definition,
% 3.72/1.90 ( image1Ex1
% 3.72/1.90 = ( ! [A: $i,Xf: $i > $i] :
% 3.72/1.90 ( exu
% 3.72/1.90 @ ^ [B: $i] :
% 3.72/1.90 ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ B )
% 3.72/1.90 <=> ? [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 & ( Xx
% 3.72/1.90 = ( Xf @ Xy ) ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(image1_type,type,
% 3.72/1.90 image1: $i > ( $i > $i ) > $i ).
% 3.72/1.90
% 3.72/1.90 thf(image1Equiv_type,type,
% 3.72/1.90 image1Equiv: $o ).
% 3.72/1.90
% 3.72/1.90 thf(image1Equiv,definition,
% 3.72/1.90 ( image1Equiv
% 3.72/1.90 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.72/1.90 ( ( in @ Xx
% 3.72/1.90 @ ( image1 @ A
% 3.72/1.90 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.72/1.90 <=> ? [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 & ( Xx
% 3.72/1.90 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(image1E_type,type,
% 3.72/1.90 image1E: $o ).
% 3.72/1.90
% 3.72/1.90 thf(image1E,definition,
% 3.72/1.90 ( image1E
% 3.72/1.90 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.72/1.90 ( ( in @ Xx
% 3.72/1.90 @ ( image1 @ A
% 3.72/1.90 @ ^ [Xy: $i] : ( Xf @ Xy ) ) )
% 3.72/1.90 => ? [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 & ( Xx
% 3.72/1.90 = ( Xf @ Xy ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(image1I_type,type,
% 3.72/1.90 image1I: $o ).
% 3.72/1.90
% 3.72/1.90 thf(image1I,definition,
% 3.72/1.90 ( image1I
% 3.72/1.90 = ( ! [A: $i,Xf: $i > $i,Xx: $i] :
% 3.72/1.90 ( ? [Xy: $i] :
% 3.72/1.90 ( ( in @ Xy @ A )
% 3.72/1.90 & ( Xx
% 3.72/1.90 = ( Xf @ Xy ) ) )
% 3.72/1.90 => ( in @ Xx
% 3.72/1.90 @ ( image1 @ A
% 3.72/1.90 @ ^ [Xy: $i] : ( Xf @ Xy ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(injective_type,type,
% 3.72/1.90 injective: $i > $i > $i > $o ).
% 3.72/1.90
% 3.72/1.90 thf(injFuncSet_type,type,
% 3.72/1.90 injFuncSet: $i > $i > $i ).
% 3.72/1.90
% 3.72/1.90 thf(injFuncInInjFuncSet_type,type,
% 3.72/1.90 injFuncInInjFuncSet: $o ).
% 3.72/1.90
% 3.72/1.90 thf(injFuncInInjFuncSet,definition,
% 3.72/1.90 ( injFuncInInjFuncSet
% 3.72/1.90 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.72/1.90 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.72/1.90 => ( ( injective @ A @ B @ Xf )
% 3.72/1.90 => ( in @ Xf @ ( injFuncSet @ A @ B ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(injFuncSetFuncIn_type,type,
% 3.72/1.90 injFuncSetFuncIn: $o ).
% 3.72/1.90
% 3.72/1.90 thf(injFuncSetFuncIn,definition,
% 3.72/1.90 ( injFuncSetFuncIn
% 3.72/1.90 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.72/1.90 ( ( in @ Xf @ ( injFuncSet @ A @ B ) )
% 3.72/1.90 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(injFuncSetFuncInj_type,type,
% 3.72/1.90 injFuncSetFuncInj: $o ).
% 3.72/1.90
% 3.72/1.90 thf(injFuncSetFuncInj,definition,
% 3.72/1.90 ( injFuncSetFuncInj
% 3.72/1.90 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.72/1.90 ( ( in @ Xf @ ( injFuncSet @ Xx @ Xy ) )
% 3.72/1.90 => ( injective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(surjective_type,type,
% 3.72/1.90 surjective: $i > $i > $i > $o ).
% 3.72/1.90
% 3.72/1.90 thf(surjFuncSet_type,type,
% 3.72/1.90 surjFuncSet: $i > $i > $i ).
% 3.72/1.90
% 3.72/1.90 thf(surjFuncSetFuncIn_type,type,
% 3.72/1.90 surjFuncSetFuncIn: $o ).
% 3.72/1.90
% 3.72/1.90 thf(surjFuncSetFuncIn,definition,
% 3.72/1.90 ( surjFuncSetFuncIn
% 3.72/1.90 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.72/1.90 ( ( in @ Xf @ ( surjFuncSet @ A @ B ) )
% 3.72/1.90 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(surjFuncSetFuncSurj_type,type,
% 3.72/1.90 surjFuncSetFuncSurj: $o ).
% 3.72/1.90
% 3.72/1.90 thf(surjFuncSetFuncSurj,definition,
% 3.72/1.90 ( surjFuncSetFuncSurj
% 3.72/1.90 = ( ! [Xx: $i,Xy: $i,Xf: $i] :
% 3.72/1.90 ( ( in @ Xf @ ( surjFuncSet @ Xx @ Xy ) )
% 3.72/1.90 => ( surjective @ Xx @ Xy @ Xf ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(leftInvIsSurj_type,type,
% 3.72/1.90 leftInvIsSurj: $o ).
% 3.72/1.90
% 3.72/1.90 thf(leftInvIsSurj,definition,
% 3.72/1.90 ( leftInvIsSurj
% 3.72/1.90 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.72/1.90 ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.72/1.90 => ! [Xg: $i] :
% 3.72/1.90 ( ( in @ Xg @ ( funcSet @ B @ A ) )
% 3.72/1.90 => ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ A )
% 3.72/1.90 => ( ( ap @ B @ A @ Xg @ ( Xf @ Xx ) )
% 3.72/1.90 = Xx ) )
% 3.72/1.90 => ( surjective @ B @ A @ Xg ) ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(surjCantorThm_type,type,
% 3.72/1.90 surjCantorThm: $o ).
% 3.72/1.90
% 3.72/1.90 thf(surjCantorThm,definition,
% 3.72/1.90 ( surjCantorThm
% 3.72/1.90 = ( ! [A: $i,Xf: $i] :
% 3.72/1.90 ( ( in @ Xf @ ( funcSet @ A @ ( powerset @ A ) ) )
% 3.72/1.90 => ~ ( surjective @ A @ ( powerset @ A ) @ Xf ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(foundation2_type,type,
% 3.72/1.90 foundation2: $o ).
% 3.72/1.90
% 3.72/1.90 thf(foundation2,definition,
% 3.72/1.90 ( foundation2
% 3.72/1.90 = ( ! [A: $i] :
% 3.72/1.90 ( ( nonempty @ A )
% 3.72/1.90 => ? [X: $i] :
% 3.72/1.90 ( ( in @ X @ A )
% 3.72/1.90 & ( ( binintersect @ X @ A )
% 3.72/1.90 = emptyset ) ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(notinself_type,type,
% 3.72/1.90 notinself: $o ).
% 3.72/1.90
% 3.72/1.90 thf(notinself,definition,
% 3.72/1.90 ( notinself
% 3.72/1.90 = ( ! [A: $i] :
% 3.72/1.90 ~ ( in @ A @ A ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(notinself2_type,type,
% 3.72/1.90 notinself2: $o ).
% 3.72/1.90
% 3.72/1.90 thf(notinself2,definition,
% 3.72/1.90 ( notinself2
% 3.72/1.90 = ( ! [A: $i,B: $i] :
% 3.72/1.90 ( ( in @ A @ B )
% 3.72/1.90 => ~ ( in @ B @ A ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(omegaS_type,type,
% 3.72/1.90 omegaS: $i > $i ).
% 3.72/1.90
% 3.72/1.90 thf(omegaS,definition,
% 3.72/1.90 ( omegaS
% 3.72/1.90 = ( ^ [Xx: $i] : ( setadjoin @ Xx @ Xx ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(omegaSp_type,type,
% 3.72/1.90 omegaSp: $o ).
% 3.72/1.90
% 3.72/1.90 thf(omegaSp,definition,
% 3.72/1.90 ( omegaSp
% 3.72/1.90 = ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ omega )
% 3.72/1.90 => ( in @ ( omegaS @ Xx ) @ omega ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(omegaSclos_type,type,
% 3.72/1.90 omegaSclos: $o ).
% 3.72/1.90
% 3.72/1.90 thf(omegaSclos,definition,
% 3.72/1.90 ( omegaSclos
% 3.72/1.90 = ( ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ omega )
% 3.72/1.90 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.72/1.90
% 3.72/1.90 thf(peano0notS,conjecture,
% 3.72/1.90 ( setextAx
% 3.72/1.90 => ( emptysetAx
% 3.72/1.90 => ( setadjoinAx
% 3.72/1.90 => ( powersetAx
% 3.72/1.90 => ( setunionAx
% 3.72/1.90 => ( omega0Ax
% 3.72/1.90 => ( omegaSAx
% 3.72/1.90 => ( omegaIndAx
% 3.72/1.90 => ( replAx
% 3.72/1.90 => ( foundationAx
% 3.72/1.90 => ( wellorderingAx
% 3.72/1.90 => ( descrp
% 3.72/1.90 => ( dsetconstrI
% 3.72/1.90 => ( dsetconstrEL
% 3.72/1.90 => ( dsetconstrER
% 3.72/1.90 => ( exuE1
% 3.72/1.90 => ( prop2setE
% 3.72/1.90 => ( emptysetE
% 3.72/1.90 => ( emptysetimpfalse
% 3.72/1.90 => ( notinemptyset
% 3.72/1.90 => ( exuE3e
% 3.72/1.90 => ( setext
% 3.72/1.90 => ( emptyI
% 3.72/1.90 => ( noeltsimpempty
% 3.72/1.90 => ( setbeta
% 3.72/1.90 => ( nonemptyE1
% 3.72/1.90 => ( nonemptyI
% 3.72/1.90 => ( nonemptyI1
% 3.72/1.90 => ( setadjoinIL
% 3.72/1.90 => ( emptyinunitempty
% 3.72/1.90 => ( setadjoinIR
% 3.72/1.90 => ( setadjoinE
% 3.72/1.90 => ( setadjoinOr
% 3.72/1.90 => ( setoftrueEq
% 3.72/1.90 => ( powersetI
% 3.72/1.90 => ( emptyinPowerset
% 3.72/1.90 => ( emptyInPowerset
% 3.72/1.90 => ( powersetE
% 3.72/1.90 => ( setunionI
% 3.72/1.90 => ( setunionE
% 3.72/1.90 => ( subPowSU
% 3.72/1.90 => ( exuE2
% 3.72/1.90 => ( nonemptyImpWitness
% 3.72/1.90 => ( uniqinunit
% 3.72/1.90 => ( notinsingleton
% 3.72/1.90 => ( eqinunit
% 3.72/1.90 => ( singletonsswitch
% 3.72/1.90 => ( upairsetE
% 3.72/1.90 => ( upairsetIL
% 3.72/1.90 => ( upairsetIR
% 3.72/1.90 => ( emptyE1
% 3.72/1.90 => ( vacuousDall
% 3.72/1.90 => ( quantDeMorgan1
% 3.72/1.90 => ( quantDeMorgan2
% 3.72/1.90 => ( quantDeMorgan3
% 3.72/1.90 => ( quantDeMorgan4
% 3.72/1.90 => ( prop2setI
% 3.72/1.90 => ( prop2set2propI
% 3.72/1.90 => ( notdexE
% 3.72/1.90 => ( notdallE
% 3.72/1.90 => ( exuI1
% 3.72/1.90 => ( exuI3
% 3.72/1.90 => ( exuI2
% 3.72/1.90 => ( inCongP
% 3.72/1.90 => ( in__Cong
% 3.72/1.90 => ( exuE3u
% 3.72/1.90 => ( exu__Cong
% 3.72/1.90 => ( emptyset__Cong
% 3.72/1.90 => ( setadjoin__Cong
% 3.72/1.90 => ( powerset__Cong
% 3.72/1.90 => ( setunion__Cong
% 3.72/1.90 => ( omega__Cong
% 3.72/1.90 => ( exuEu
% 3.72/1.90 => ( descr__Cong
% 3.72/1.90 => ( dsetconstr__Cong
% 3.72/1.90 => ( subsetI1
% 3.72/1.90 => ( eqimpsubset2
% 3.72/1.90 => ( eqimpsubset1
% 3.72/1.90 => ( subsetI2
% 3.72/1.90 => ( emptysetsubset
% 3.72/1.90 => ( subsetE
% 3.72/1.90 => ( subsetE2
% 3.72/1.90 => ( notsubsetI
% 3.72/1.90 => ( notequalI1
% 3.72/1.90 => ( notequalI2
% 3.72/1.90 => ( subsetRefl
% 3.72/1.90 => ( subsetTrans
% 3.72/1.90 => ( setadjoinSub
% 3.72/1.90 => ( setadjoinSub2
% 3.72/1.90 => ( subset2powerset
% 3.72/1.90 => ( setextsub
% 3.72/1.90 => ( subsetemptysetimpeq
% 3.72/1.90 => ( powersetI1
% 3.72/1.90 => ( powersetE1
% 3.72/1.90 => ( inPowerset
% 3.72/1.90 => ( powersetsubset
% 3.72/1.90 => ( sepInPowerset
% 3.72/1.90 => ( sepSubset
% 3.72/1.90 => ( binunionIL
% 3.72/1.90 => ( upairset2IR
% 3.72/1.90 => ( binunionIR
% 3.72/1.90 => ( binunionEcases
% 3.72/1.90 => ( binunionE
% 3.72/1.90 => ( binunionLsub
% 3.72/1.90 => ( binunionRsub
% 3.72/1.90 => ( binintersectI
% 3.72/1.90 => ( binintersectSubset5
% 3.72/1.90 => ( binintersectEL
% 3.72/1.90 => ( binintersectLsub
% 3.72/1.90 => ( binintersectSubset2
% 3.72/1.90 => ( binintersectSubset3
% 3.72/1.90 => ( binintersectER
% 3.72/1.90 => ( disjointsetsI1
% 3.72/1.90 => ( binintersectRsub
% 3.72/1.90 => ( binintersectSubset4
% 3.72/1.90 => ( binintersectSubset1
% 3.72/1.90 => ( bs114d
% 3.72/1.90 => ( setminusI
% 3.72/1.90 => ( setminusEL
% 3.72/1.90 => ( setminusER
% 3.72/1.90 => ( setminusSubset2
% 3.72/1.90 => ( setminusERneg
% 3.72/1.90 => ( setminusELneg
% 3.72/1.90 => ( setminusILneg
% 3.72/1.90 => ( setminusIRneg
% 3.72/1.90 => ( setminusLsub
% 3.72/1.90 => ( setminusSubset1
% 3.72/1.90 => ( symdiffE
% 3.72/1.90 => ( symdiffI1
% 3.72/1.90 => ( symdiffI2
% 3.72/1.90 => ( symdiffIneg1
% 3.72/1.90 => ( symdiffIneg2
% 3.72/1.90 => ( secondinupair
% 3.72/1.90 => ( setukpairIL
% 3.72/1.90 => ( setukpairIR
% 3.72/1.90 => ( kpairiskpair
% 3.72/1.90 => ( kpairp
% 3.72/1.90 => ( singletonsubset
% 3.72/1.90 => ( singletoninpowerset
% 3.72/1.90 => ( singletoninpowunion
% 3.72/1.90 => ( upairset2E
% 3.72/1.90 => ( upairsubunion
% 3.72/1.90 => ( upairinpowunion
% 3.72/1.90 => ( ubforcartprodlem1
% 3.72/1.90 => ( ubforcartprodlem2
% 3.72/1.90 => ( ubforcartprodlem3
% 3.72/1.90 => ( cartprodpairin
% 3.72/1.90 => ( cartprodmempair1
% 3.72/1.90 => ( cartprodmempair
% 3.72/1.90 => ( setunionE2
% 3.72/1.90 => ( setunionsingleton1
% 3.72/1.90 => ( setunionsingleton2
% 3.72/1.90 => ( setunionsingleton
% 3.72/1.90 => ( singletonprop
% 3.72/1.90 => ( ex1E1
% 3.72/1.90 => ( ex1I
% 3.72/1.90 => ( ex1I2
% 3.72/1.90 => ( singletonsuniq
% 3.72/1.90 => ( setukpairinjL1
% 3.72/1.90 => ( kfstsingleton
% 3.72/1.90 => ( theprop
% 3.72/1.90 => ( kfstpairEq
% 3.72/1.90 => ( cartprodfstin
% 3.72/1.90 => ( setukpairinjL2
% 3.72/1.90 => ( setukpairinjL
% 3.72/1.90 => ( setukpairinjR11
% 3.72/1.90 => ( setukpairinjR12
% 3.72/1.90 => ( setukpairinjR1
% 3.72/1.90 => ( upairequniteq
% 3.72/1.90 => ( setukpairinjR2
% 3.72/1.90 => ( setukpairinjR
% 3.72/1.90 => ( ksndsingleton
% 3.72/1.90 => ( ksndpairEq
% 3.72/1.90 => ( kpairsurjEq
% 3.72/1.90 => ( cartprodsndin
% 3.72/1.90 => ( cartprodpairmemEL
% 3.72/1.90 => ( cartprodpairmemER
% 3.72/1.90 => ( cartprodmempaircEq
% 3.72/1.90 => ( cartprodfstpairEq
% 3.72/1.90 => ( cartprodsndpairEq
% 3.72/1.90 => ( cartprodpairsurjEq
% 3.72/1.90 => ( dpsetconstrI
% 3.72/1.90 => ( dpsetconstrSub
% 3.72/1.90 => ( setOfPairsIsBReln
% 3.72/1.90 => ( dpsetconstrERa
% 3.72/1.90 => ( dpsetconstrEL1
% 3.72/1.90 => ( dpsetconstrEL2
% 3.72/1.90 => ( dpsetconstrER
% 3.72/1.90 => ( funcImageSingleton
% 3.72/1.90 => ( apProp
% 3.72/1.90 => ( app
% 3.72/1.90 => ( infuncsetfunc
% 3.72/1.90 => ( ap2p
% 3.72/1.90 => ( funcinfuncset
% 3.72/1.90 => ( lamProp
% 3.72/1.90 => ( lamp
% 3.72/1.90 => ( lam2p
% 3.72/1.90 => ( brelnall1
% 3.72/1.90 => ( brelnall2
% 3.72/1.90 => ( ex1E2
% 3.72/1.90 => ( funcGraphProp1
% 3.72/1.90 => ( funcGraphProp3
% 3.72/1.90 => ( funcGraphProp2
% 3.72/1.90 => ( funcextLem
% 3.72/1.90 => ( funcGraphProp4
% 3.72/1.90 => ( subbreln
% 3.72/1.90 => ( eqbreln
% 3.72/1.90 => ( funcext
% 3.72/1.90 => ( funcext2
% 3.72/1.90 => ( ap2apEq1
% 3.72/1.90 => ( ap2apEq2
% 3.72/1.90 => ( beta1
% 3.72/1.90 => ( eta1
% 3.72/1.90 => ( lam2lamEq
% 3.72/1.90 => ( beta2
% 3.72/1.90 => ( eta2
% 3.72/1.90 => ( iffalseProp1
% 3.72/1.90 => ( iffalseProp2
% 3.72/1.90 => ( iftrueProp1
% 3.72/1.90 => ( iftrueProp2
% 3.72/1.90 => ( ifSingleton
% 3.72/1.90 => ( ifp
% 3.72/1.90 => ( theeq
% 3.72/1.90 => ( iftrue
% 3.72/1.90 => ( iffalse
% 3.72/1.90 => ( iftrueorfalse
% 3.72/1.90 => ( binintersectT_lem
% 3.72/1.90 => ( binunionT_lem
% 3.72/1.90 => ( powersetT_lem
% 3.72/1.90 => ( setminusT_lem
% 3.72/1.90 => ( complementT_lem
% 3.72/1.90 => ( setextT
% 3.72/1.90 => ( subsetTI
% 3.72/1.90 => ( powersetTI1
% 3.72/1.90 => ( powersetTE1
% 3.72/1.90 => ( complementTI1
% 3.72/1.90 => ( complementTE1
% 3.72/1.90 => ( binintersectTELcontra
% 3.72/1.90 => ( binintersectTERcontra
% 3.72/1.90 => ( contrasubsetT
% 3.72/1.90 => ( contrasubsetT1
% 3.72/1.90 => ( contrasubsetT2
% 3.72/1.90 => ( contrasubsetT3
% 3.72/1.90 => ( doubleComplementI1
% 3.72/1.90 => ( doubleComplementE1
% 3.72/1.90 => ( doubleComplementSub1
% 3.72/1.90 => ( doubleComplementSub2
% 3.72/1.90 => ( doubleComplementEq
% 3.72/1.90 => ( complementTnotintersectT
% 3.72/1.90 => ( complementImpComplementIntersect
% 3.72/1.90 => ( complementSubsetComplementIntersect
% 3.72/1.90 => ( complementInPowersetComplementIntersect
% 3.72/1.90 => ( contraSubsetComplement
% 3.72/1.90 => ( complementTcontraSubset
% 3.72/1.90 => ( binunionTILcontra
% 3.72/1.90 => ( binunionTIRcontra
% 3.72/1.90 => ( inIntersectImpInUnion
% 3.72/1.90 => ( inIntersectImpInUnion2
% 3.72/1.90 => ( inIntersectImpInIntersectUnions
% 3.72/1.90 => ( intersectInPowersetIntersectUnions
% 3.72/1.90 => ( inComplementUnionImpNotIn1
% 3.72/1.90 => ( inComplementUnionImpInComplement1
% 3.72/1.90 => ( binunionTE
% 3.72/1.90 => ( binunionTEcontra
% 3.72/1.90 => ( demorgan2a1
% 3.72/1.90 => ( complementUnionInPowersetComplement
% 3.72/1.90 => ( demorgan2a2
% 3.72/1.90 => ( demorgan1a
% 3.72/1.90 => ( demorgan1b
% 3.72/1.90 => ( demorgan1
% 3.72/1.90 => ( demorgan2a
% 3.72/1.90 => ( demorgan2b2
% 3.72/1.90 => ( demorgan2b
% 3.72/1.90 => ( demorgan2
% 3.72/1.90 => ( woz13rule0
% 3.72/1.90 => ( woz13rule1
% 3.72/1.90 => ( woz13rule2
% 3.72/1.90 => ( woz13rule3
% 3.72/1.90 => ( woz13rule4
% 3.72/1.90 => ( woz1_1
% 3.72/1.90 => ( woz1_2
% 3.72/1.90 => ( woz1_3
% 3.72/1.90 => ( woz1_4
% 3.72/1.90 => ( woz1_5
% 3.72/1.90 => ( breln1all2
% 3.72/1.90 => ( breln1SetBreln1
% 3.72/1.90 => ( choice2fnsingleton
% 3.72/1.90 => ( setOfPairsIsBReln1
% 3.72/1.90 => ( breln1all1
% 3.72/1.90 => ( subbreln1
% 3.72/1.90 => ( eqbreln1
% 3.72/1.90 => ( breln1invprop
% 3.72/1.90 => ( breln1invI
% 3.72/1.90 => ( breln1invE
% 3.72/1.90 => ( breln1compprop
% 3.72/1.90 => ( breln1compI
% 3.72/1.90 => ( breln1compE
% 3.72/1.90 => ( breln1compEex
% 3.72/1.90 => ( breln1unionprop
% 3.72/1.90 => ( breln1unionIL
% 3.72/1.90 => ( breln1unionIR
% 3.72/1.90 => ( breln1unionI
% 3.72/1.90 => ( breln1unionE
% 3.72/1.90 => ( breln1unionEcases
% 3.72/1.90 => ( breln1unionCommutes
% 3.72/1.90 => ( woz2Ex
% 3.72/1.90 => ( woz2W
% 3.72/1.90 => ( woz2A
% 3.72/1.90 => ( woz2B
% 3.72/1.90 => ( image1Ex
% 3.72/1.90 => ( image1Ex1
% 3.72/1.90 => ( image1Equiv
% 3.72/1.90 => ( image1E
% 3.72/1.90 => ( image1I
% 3.72/1.90 => ( injFuncInInjFuncSet
% 3.72/1.90 => ( injFuncSetFuncIn
% 3.72/1.90 => ( injFuncSetFuncInj
% 3.72/1.90 => ( surjFuncSetFuncIn
% 3.72/1.90 => ( surjFuncSetFuncSurj
% 3.72/1.90 => ( leftInvIsSurj
% 3.72/1.90 => ( surjCantorThm
% 3.72/1.90 => ( foundation2
% 3.72/1.90 => ( notinself
% 3.72/1.90 => ( notinself2
% 3.72/1.90 => ( omegaSp
% 3.72/1.90 => ( omegaSclos
% 3.72/1.90 => ! [Xx: $i] :
% 3.72/1.90 ( ( in @ Xx @ omega )
% 3.72/1.90 => ( ( omegaS @ Xx )
% 3.72/1.90 != emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.72/1.90 ------------------------
% 3.72/1.95 % (27463)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_27360 for (2999ds/2Mi)
% 3.72/1.95 % (27465)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/18Mi)
% 3.72/1.95 % (27459)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/183Mi)
% 3.72/1.95 % (27462)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/2Mi)
% 3.72/1.95 % (27464)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_27360 for (2999ds/275Mi)
% 3.72/1.95 % (27461)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_27360 for (2999ds/27Mi)
% 3.72/1.95 % (27460)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_27360 for (2999ds/4Mi)
% 3.72/1.95 % (27462)Instruction limit reached!
% 3.72/1.95 % (27462)------------------------------
% 3.72/1.95 % (27462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.95 % (27462)Termination reason: Unknown
% 3.72/1.95 % (27462)Termination phase: shuffling
% 3.72/1.95 % (27463)Instruction limit reached!
% 3.72/1.95 % (27463)------------------------------
% 3.72/1.95 % (27463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.95 % (27463)Termination reason: Unknown
% 3.72/1.95 % (27463)Termination phase: shuffling
% 3.72/1.95
% 3.72/1.95 % (27463)Memory used [KB]: 1663
% 3.72/1.95 % (27463)Time elapsed: 0.003 s
% 3.72/1.95 % (27463)Instructions burned: 3 (million)
% 3.72/1.95 % (27463)------------------------------
% 3.72/1.95 % (27463)------------------------------
% 3.72/1.95
% 3.72/1.95 % (27462)Memory used [KB]: 1663
% 3.72/1.95 % (27462)Time elapsed: 0.003 s
% 3.72/1.95 % (27462)Instructions burned: 3 (million)
% 3.72/1.95 % (27462)------------------------------
% 3.72/1.95 % (27462)------------------------------
% 3.72/1.95 % (27460)Instruction limit reached!
% 3.72/1.95 % (27460)------------------------------
% 3.72/1.95 % (27460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.95 % (27460)Termination reason: Unknown
% 3.72/1.95 % (27460)Termination phase: shuffling
% 3.72/1.95
% 3.72/1.95 % (27460)Memory used [KB]: 1663
% 3.72/1.95 % (27460)Time elapsed: 0.004 s
% 3.72/1.95 % (27460)Instructions burned: 5 (million)
% 3.72/1.95 % (27460)------------------------------
% 3.72/1.95 % (27460)------------------------------
% 3.72/1.96 % (27465)Instruction limit reached!
% 3.72/1.96 % (27465)------------------------------
% 3.72/1.96 % (27465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.96 % (27465)Termination reason: Unknown
% 3.72/1.96 % (27465)Termination phase: shuffling
% 3.72/1.96
% 3.72/1.96 % (27465)Memory used [KB]: 1918
% 3.72/1.96 % (27465)Time elapsed: 0.011 s
% 3.72/1.96 % (27465)Instructions burned: 18 (million)
% 3.72/1.96 % (27465)------------------------------
% 3.72/1.96 % (27465)------------------------------
% 3.72/1.96 % (27461)Instruction limit reached!
% 3.72/1.96 % (27461)------------------------------
% 3.72/1.96 % (27461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.96 % (27461)Termination reason: Unknown
% 3.72/1.96 % (27461)Termination phase: shuffling
% 3.72/1.96
% 3.72/1.96 % (27461)Memory used [KB]: 2046
% 3.72/1.96 % (27461)Time elapsed: 0.014 s
% 3.72/1.96 % (27461)Instructions burned: 27 (million)
% 3.72/1.96 % (27461)------------------------------
% 3.72/1.96 % (27461)------------------------------
% 3.72/1.97 % (27466)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/3Mi)
% 3.72/1.97 % (27468)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_27360 for (2999ds/15Mi)
% 3.72/1.97 % (27467)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_27360 for (2999ds/37Mi)
% 3.72/1.97 % (27466)Instruction limit reached!
% 3.72/1.97 % (27466)------------------------------
% 3.72/1.97 % (27466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.97 % (27466)Termination reason: Unknown
% 3.72/1.97 % (27466)Termination phase: shuffling
% 3.72/1.97
% 3.72/1.97 % (27466)Memory used [KB]: 1663
% 3.72/1.97 % (27466)Time elapsed: 0.003 s
% 3.72/1.97 % (27466)Instructions burned: 3 (million)
% 3.72/1.97 % (27466)------------------------------
% 3.72/1.97 % (27466)------------------------------
% 3.72/1.98 % (27469)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_27360 for (2999ds/3Mi)
% 3.72/1.98 % (27469)Instruction limit reached!
% 3.72/1.98 % (27469)------------------------------
% 3.72/1.98 % (27469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.98 % (27469)Termination reason: Unknown
% 3.72/1.98 % (27469)Termination phase: shuffling
% 3.72/1.98
% 3.72/1.98 % (27469)Memory used [KB]: 1663
% 3.72/1.98 % (27469)Time elapsed: 0.003 s
% 3.72/1.98 % (27469)Instructions burned: 3 (million)
% 3.72/1.98 % (27469)------------------------------
% 3.72/1.98 % (27469)------------------------------
% 3.72/1.98 % (27468)Instruction limit reached!
% 3.72/1.98 % (27468)------------------------------
% 3.72/1.98 % (27468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.98 % (27468)Termination reason: Unknown
% 3.72/1.98 % (27468)Termination phase: shuffling
% 3.72/1.98
% 3.72/1.98 % (27468)Memory used [KB]: 1918
% 3.72/1.98 % (27468)Time elapsed: 0.010 s
% 3.72/1.98 % (27468)Instructions burned: 16 (million)
% 3.72/1.98 % (27468)------------------------------
% 3.72/1.98 % (27468)------------------------------
% 3.72/1.98 % (27470)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_27360 for (2999ds/1041Mi)
% 3.72/1.99 % (27467)Instruction limit reached!
% 3.72/1.99 % (27467)------------------------------
% 3.72/1.99 % (27467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.99 % (27467)Termination reason: Unknown
% 3.72/1.99 % (27467)Termination phase: shuffling
% 3.72/1.99
% 3.72/1.99 % (27467)Memory used [KB]: 2430
% 3.72/1.99 % (27467)Time elapsed: 0.020 s
% 3.72/1.99 % (27467)Instructions burned: 38 (million)
% 3.72/1.99 % (27467)------------------------------
% 3.72/1.99 % (27467)------------------------------
% 3.72/1.99 % (27471)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/7Mi)
% 3.72/1.99 % (27471)Instruction limit reached!
% 3.72/1.99 % (27471)------------------------------
% 3.72/1.99 % (27471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.99 % (27471)Termination reason: Unknown
% 3.72/1.99 % (27471)Termination phase: shuffling
% 3.72/1.99
% 3.72/1.99 % (27471)Memory used [KB]: 1663
% 3.72/1.99 % (27471)Time elapsed: 0.005 s
% 3.72/1.99 % (27471)Instructions burned: 7 (million)
% 3.72/1.99 % (27471)------------------------------
% 3.72/1.99 % (27471)------------------------------
% 3.72/1.99 % (27473)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_27360 for (2999ds/3Mi)
% 3.72/1.99 % (27472)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/16Mi)
% 3.72/1.99 % (27473)Instruction limit reached!
% 3.72/1.99 % (27473)------------------------------
% 3.72/1.99 % (27473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/1.99 % (27473)Termination reason: Unknown
% 3.72/1.99 % (27473)Termination phase: shuffling
% 3.72/1.99
% 3.72/1.99 % (27473)Memory used [KB]: 1663
% 3.72/1.99 % (27473)Time elapsed: 0.003 s
% 3.72/1.99 % (27473)Instructions burned: 3 (million)
% 3.72/1.99 % (27473)------------------------------
% 3.72/1.99 % (27473)------------------------------
% 3.72/2.00 % (27472)Instruction limit reached!
% 3.72/2.00 % (27472)------------------------------
% 3.72/2.00 % (27472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/2.00 % (27472)Termination reason: Unknown
% 3.72/2.00 % (27472)Termination phase: shuffling
% 3.72/2.00
% 3.72/2.00 % (27472)Memory used [KB]: 1918
% 3.72/2.00 % (27472)Time elapsed: 0.010 s
% 3.72/2.00 % (27472)Instructions burned: 17 (million)
% 3.72/2.00 % (27472)------------------------------
% 3.72/2.00 % (27472)------------------------------
% 3.72/2.00 % (27474)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_27360 for (2999ds/3Mi)
% 3.72/2.01 % (27474)Instruction limit reached!
% 3.72/2.01 % (27474)------------------------------
% 3.72/2.01 % (27474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/2.01 % (27474)Termination reason: Unknown
% 3.72/2.01 % (27474)Termination phase: shuffling
% 3.72/2.01
% 3.72/2.01 % (27474)Memory used [KB]: 1663
% 3.72/2.01 % (27474)Time elapsed: 0.004 s
% 3.72/2.01 % (27474)Instructions burned: 3 (million)
% 3.72/2.01 % (27474)------------------------------
% 3.72/2.01 % (27474)------------------------------
% 3.72/2.01 % (27475)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/7Mi)
% 3.72/2.01 % (27476)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/3Mi)
% 3.72/2.01 % (27475)Instruction limit reached!
% 3.72/2.01 % (27475)------------------------------
% 3.72/2.01 % (27475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/2.01 % (27475)Termination reason: Unknown
% 3.72/2.01 % (27475)Termination phase: shuffling
% 3.72/2.01
% 3.72/2.01 % (27475)Memory used [KB]: 1663
% 3.72/2.01 % (27475)Time elapsed: 0.006 s
% 3.72/2.01 % (27475)Instructions burned: 9 (million)
% 3.72/2.01 % (27475)------------------------------
% 3.72/2.01 % (27475)------------------------------
% 3.72/2.01 % (27476)Instruction limit reached!
% 3.72/2.01 % (27476)------------------------------
% 3.72/2.01 % (27476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/2.01 % (27476)Termination reason: Unknown
% 3.72/2.01 % (27476)Termination phase: shuffling
% 3.72/2.01
% 3.72/2.01 % (27476)Memory used [KB]: 1663
% 3.72/2.01 % (27476)Time elapsed: 0.004 s
% 3.72/2.01 % (27476)Instructions burned: 4 (million)
% 3.72/2.01 % (27476)------------------------------
% 3.72/2.01 % (27476)------------------------------
% 3.72/2.02 % (27477)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/4Mi)
% 3.72/2.02 % (27477)Instruction limit reached!
% 3.72/2.02 % (27477)------------------------------
% 3.72/2.02 % (27477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.72/2.02 % (27477)Termination reason: Unknown
% 3.72/2.02 % (27477)Termination phase: shuffling
% 3.72/2.02
% 3.72/2.02 % (27477)Memory used [KB]: 1663
% 3.72/2.02 % (27477)Time elapsed: 0.004 s
% 3.72/2.02 % (27477)Instructions burned: 4 (million)
% 3.72/2.02 % (27477)------------------------------
% 3.72/2.02 % (27477)------------------------------
% 3.72/2.02 % (27478)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/18Mi)
% 4.29/2.03 % (27479)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/710Mi)
% 4.29/2.03 % (27480)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_27360 for (2999ds/6Mi)
% 4.29/2.03 % (27478)Instruction limit reached!
% 4.29/2.03 % (27478)------------------------------
% 4.29/2.03 % (27478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.29/2.03 % (27478)Termination reason: Unknown
% 4.29/2.03 % (27478)Termination phase: shuffling
% 4.29/2.03
% 4.29/2.03 % (27478)Memory used [KB]: 1918
% 4.29/2.03 % (27478)Time elapsed: 0.010 s
% 4.29/2.03 % (27478)Instructions burned: 18 (million)
% 4.29/2.03 % (27478)------------------------------
% 4.29/2.03 % (27478)------------------------------
% 4.29/2.03 % (27459)Instruction limit reached!
% 4.29/2.03 % (27459)------------------------------
% 4.29/2.03 % (27459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.29/2.03 % (27459)Termination reason: Unknown
% 4.29/2.03 % (27459)Termination phase: Property scanning
% 4.29/2.03
% 4.29/2.03 % (27459)Memory used [KB]: 3198
% 4.29/2.03 % (27459)Time elapsed: 0.082 s
% 4.29/2.03 % (27459)Instructions burned: 184 (million)
% 4.29/2.03 % (27459)------------------------------
% 4.29/2.03 % (27459)------------------------------
% 4.29/2.03 % (27480)Instruction limit reached!
% 4.29/2.03 % (27480)------------------------------
% 4.29/2.03 % (27480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.29/2.03 % (27480)Termination reason: Unknown
% 4.29/2.03 % (27480)Termination phase: shuffling
% 4.29/2.03
% 4.29/2.03 % (27480)Memory used [KB]: 1663
% 4.29/2.03 % (27480)Time elapsed: 0.005 s
% 4.29/2.03 % (27480)Instructions burned: 7 (million)
% 4.29/2.03 % (27480)------------------------------
% 4.29/2.03 % (27480)------------------------------
% 4.29/2.04 % (27481)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_27360 for (2998ds/902Mi)
% 4.29/2.05 % (27482)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_27360 for (2998ds/21Mi)
% 4.29/2.05 % (27483)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_27360 for (2998ds/5Mi)
% 4.29/2.05 % (27484)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_27360 for (2998ds/6Mi)
% 4.29/2.05 % (27483)Instruction limit reached!
% 4.29/2.05 % (27483)------------------------------
% 4.29/2.05 % (27483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.29/2.05 % (27483)Termination reason: Unknown
% 4.29/2.05 % (27483)Termination phase: shuffling
% 4.29/2.05
% 4.29/2.05 % (27483)Memory used [KB]: 1663
% 4.29/2.05 % (27483)Time elapsed: 0.005 s
% 4.29/2.05 % (27483)Instructions burned: 6 (million)
% 4.29/2.05 % (27483)------------------------------
% 4.29/2.05 % (27483)------------------------------
% 4.29/2.05 % (27484)Instruction limit reached!
% 4.29/2.05 % (27484)------------------------------
% 4.29/2.05 % (27484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.29/2.05 % (27484)Termination reason: Unknown
% 4.29/2.05 % (27484)Termination phase: shuffling
% 4.29/2.05
% 4.29/2.05 % (27484)Memory used [KB]: 1663
% 4.29/2.05 % (27484)Time elapsed: 0.005 s
% 4.29/2.05 % (27484)Instructions burned: 7 (million)
% 4.29/2.05 % (27484)------------------------------
% 4.29/2.05 % (27484)------------------------------
% 4.29/2.06 % (27482)Instruction limit reached!
% 4.29/2.06 % (27482)------------------------------
% 4.29/2.06 % (27482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.29/2.06 % (27482)Termination reason: Unknown
% 4.29/2.06 % (27482)Termination phase: shuffling
% 4.29/2.06
% 4.29/2.06 % (27482)Memory used [KB]: 2046
% 4.29/2.06 % (27482)Time elapsed: 0.011 s
% 4.29/2.06 % (27482)Instructions burned: 21 (million)
% 4.29/2.06 % (27482)------------------------------
% 4.29/2.06 % (27482)------------------------------
% 4.29/2.07 % (27485)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_27360 for (2998ds/377Mi)
% 4.29/2.07 % (27486)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_27360 for (2998ds/779Mi)
% 4.29/2.07 % (27487)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_27360 for (2998ds/19Mi)
% 4.29/2.08 % (27464)Instruction limit reached!
% 4.29/2.08 % (27464)------------------------------
% 4.29/2.08 % (27464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.29/2.08 % (27464)Termination reason: Unknown
% 4.29/2.08 % (27464)Termination phase: Property scanning
% 4.29/2.08
% 4.29/2.08 % (27464)Memory used [KB]: 6012
% 4.29/2.08 % (27464)Time elapsed: 0.131 s
% 4.29/2.08 % (27464)Instructions burned: 275 (million)
% 4.29/2.08 % (27464)------------------------------
% 4.29/2.08 % (27464)------------------------------
% 4.29/2.08 % (27487)Instruction limit reached!
% 4.29/2.08 % (27487)------------------------------
% 4.29/2.08 % (27487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.29/2.08 % (27487)Termination reason: Unknown
% 4.29/2.08 % (27487)Termination phase: shuffling
% 4.29/2.08
% 4.29/2.08 % (27487)Memory used [KB]: 1918
% 4.29/2.08 % (27487)Time elapsed: 0.011 s
% 4.29/2.08 % (27487)Instructions burned: 19 (million)
% 4.29/2.08 % (27487)------------------------------
% 4.29/2.08 % (27487)------------------------------
% 4.29/2.10 % (27488)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_27360 for (2998ds/879Mi)
% 4.29/2.10 % (27489)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_27360 for (2998ds/17Mi)
% 4.29/2.11 % (27489)Instruction limit reached!
% 4.29/2.11 % (27489)------------------------------
% 4.29/2.11 % (27489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.29/2.11 % (27489)Termination reason: Unknown
% 4.29/2.11 % (27489)Termination phase: shuffling
% 4.29/2.11
% 4.29/2.11 % (27489)Memory used [KB]: 1918
% 4.29/2.11 % (27489)Time elapsed: 0.010 s
% 4.29/2.11 % (27489)Instructions burned: 18 (million)
% 4.29/2.11 % (27489)------------------------------
% 4.29/2.11 % (27489)------------------------------
% 4.64/2.13 % (27490)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_27360 for (2997ds/3Mi)
% 4.64/2.13 % (27490)Instruction limit reached!
% 4.64/2.13 % (27490)------------------------------
% 4.64/2.13 % (27490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.64/2.13 % (27490)Termination reason: Unknown
% 4.64/2.13 % (27490)Termination phase: shuffling
% 4.64/2.13
% 4.64/2.13 % (27490)Memory used [KB]: 1663
% 4.64/2.13 % (27490)Time elapsed: 0.005 s
% 4.64/2.13 % (27490)Instructions burned: 4 (million)
% 4.64/2.13 % (27490)------------------------------
% 4.64/2.13 % (27490)------------------------------
% 4.64/2.15 % (27491)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_27360 for (2997ds/30Mi)
% 4.64/2.16 % (27491)Instruction limit reached!
% 4.64/2.16 % (27491)------------------------------
% 4.64/2.16 % (27491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.64/2.16 % (27491)Termination reason: Unknown
% 4.64/2.16 % (27491)Termination phase: shuffling
% 4.64/2.16
% 4.64/2.16 % (27491)Memory used [KB]: 2302
% 4.64/2.16 % (27491)Time elapsed: 0.016 s
% 4.64/2.16 % (27491)Instructions burned: 31 (million)
% 4.64/2.16 % (27491)------------------------------
% 4.64/2.16 % (27491)------------------------------
% 4.64/2.18 % (27492)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_27360 for (2997ds/127Mi)
% 4.77/2.23 % (27492)Instruction limit reached!
% 4.77/2.23 % (27492)------------------------------
% 4.77/2.23 % (27492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.77/2.23 % (27492)Termination reason: Unknown
% 4.77/2.23 % (27492)Termination phase: Property scanning
% 4.77/2.23
% 4.77/2.23 % (27492)Memory used [KB]: 3070
% 4.77/2.23 % (27492)Time elapsed: 0.057 s
% 4.77/2.23 % (27492)Instructions burned: 127 (million)
% 4.77/2.23 % (27492)------------------------------
% 4.77/2.23 % (27492)------------------------------
% 4.77/2.24 % (27485)Instruction limit reached!
% 4.77/2.24 % (27485)------------------------------
% 4.77/2.24 % (27485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.77/2.24 % (27485)Termination reason: Unknown
% 4.77/2.24 % (27485)Termination phase: Saturation
% 4.77/2.24
% 4.77/2.24 % (27485)Memory used [KB]: 9466
% 4.77/2.24 % (27485)Time elapsed: 0.171 s
% 4.77/2.24 % (27485)Instructions burned: 377 (million)
% 4.77/2.24 % (27485)------------------------------
% 4.77/2.24 % (27485)------------------------------
% 4.77/2.25 % (27493)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_27360 for (2996ds/100Mi)
% 4.77/2.25 % (27494)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_27360 for (2996ds/3Mi)
% 4.77/2.25 % (27494)Instruction limit reached!
% 4.77/2.25 % (27494)------------------------------
% 4.77/2.25 % (27494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.77/2.25 % (27494)Termination reason: Unknown
% 4.77/2.25 % (27494)Termination phase: shuffling
% 4.77/2.25
% 4.77/2.25 % (27494)Memory used [KB]: 1663
% 4.77/2.25 % (27494)Time elapsed: 0.004 s
% 4.77/2.25 % (27494)Instructions burned: 3 (million)
% 4.77/2.25 % (27494)------------------------------
% 4.77/2.25 % (27494)------------------------------
% 4.77/2.27 % (27495)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_27360 for (2996ds/20Mi)
% 4.77/2.28 % (27495)Instruction limit reached!
% 4.77/2.28 % (27495)------------------------------
% 4.77/2.28 % (27495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.77/2.28 % (27495)Termination reason: Unknown
% 4.77/2.28 % (27495)Termination phase: shuffling
% 4.77/2.28
% 4.77/2.28 % (27495)Memory used [KB]: 2046
% 4.77/2.28 % (27495)Time elapsed: 0.012 s
% 4.77/2.28 % (27495)Instructions burned: 21 (million)
% 4.77/2.28 % (27495)------------------------------
% 4.77/2.28 % (27495)------------------------------
% 5.16/2.30 % (27493)Instruction limit reached!
% 5.16/2.30 % (27493)------------------------------
% 5.16/2.30 % (27493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.30 % (27493)Termination reason: Unknown
% 5.16/2.30 % (27493)Termination phase: Preprocessing 1
% 5.16/2.30
% 5.16/2.30 % (27493)Memory used [KB]: 2814
% 5.16/2.30 % (27493)Time elapsed: 0.048 s
% 5.16/2.30 % (27493)Instructions burned: 100 (million)
% 5.16/2.30 % (27493)------------------------------
% 5.16/2.30 % (27493)------------------------------
% 5.16/2.30 % (27496)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_27360 for (2996ds/86Mi)
% 5.16/2.32 % (27497)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_27360 for (2996ds/107Mi)
% 5.16/2.34 % (27496)Instruction limit reached!
% 5.16/2.34 % (27496)------------------------------
% 5.16/2.34 % (27496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.34 % (27496)Termination reason: Unknown
% 5.16/2.34 % (27496)Termination phase: Property scanning
% 5.16/2.34
% 5.16/2.34 % (27496)Memory used [KB]: 2814
% 5.16/2.34 % (27496)Time elapsed: 0.041 s
% 5.16/2.34 % (27496)Instructions burned: 87 (million)
% 5.16/2.34 % (27496)------------------------------
% 5.16/2.34 % (27496)------------------------------
% 5.16/2.36 % (27498)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_27360 for (2995ds/39Mi)
% 5.16/2.36 % (27497)Instruction limit reached!
% 5.16/2.36 % (27497)------------------------------
% 5.16/2.36 % (27497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.36 % (27497)Termination reason: Unknown
% 5.16/2.36 % (27497)Termination phase: SInE selection
% 5.16/2.36
% 5.16/2.36 % (27497)Memory used [KB]: 2814
% 5.16/2.36 % (27497)Time elapsed: 0.051 s
% 5.16/2.36 % (27497)Instructions burned: 108 (million)
% 5.16/2.36 % (27497)------------------------------
% 5.16/2.36 % (27497)------------------------------
% 5.16/2.38 % (27498)Instruction limit reached!
% 5.16/2.38 % (27498)------------------------------
% 5.16/2.38 % (27498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.38 % (27498)Termination reason: Unknown
% 5.16/2.38 % (27498)Termination phase: shuffling
% 5.16/2.38
% 5.16/2.38 % (27498)Memory used [KB]: 2430
% 5.16/2.38 % (27498)Time elapsed: 0.020 s
% 5.16/2.38 % (27498)Instructions burned: 39 (million)
% 5.16/2.38 % (27498)------------------------------
% 5.16/2.38 % (27498)------------------------------
% 5.16/2.38 % (27486)Instruction limit reached!
% 5.16/2.38 % (27486)------------------------------
% 5.16/2.38 % (27486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.38 % (27486)Termination reason: Unknown
% 5.16/2.38 % (27486)Termination phase: Saturation
% 5.16/2.38
% 5.16/2.38 % (27486)Memory used [KB]: 7931
% 5.16/2.38 % (27486)Time elapsed: 0.316 s
% 5.16/2.38 % (27486)Instructions burned: 780 (million)
% 5.16/2.38 % (27486)------------------------------
% 5.16/2.38 % (27486)------------------------------
% 5.16/2.38 % (27499)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_27360 for (2995ds/448Mi)
% 5.16/2.39 % (27479)Instruction limit reached!
% 5.16/2.39 % (27479)------------------------------
% 5.16/2.39 % (27479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.39 % (27479)Termination reason: Unknown
% 5.16/2.39 % (27479)Termination phase: Saturation
% 5.16/2.39
% 5.16/2.39 % (27479)Memory used [KB]: 11769
% 5.16/2.39 % (27479)Time elapsed: 0.362 s
% 5.16/2.39 % (27479)Instructions burned: 711 (million)
% 5.16/2.39 % (27479)------------------------------
% 5.16/2.39 % (27479)------------------------------
% 5.16/2.39 % (27500)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_27360 for (2995ds/46Mi)
% 5.16/2.39 % (27470)Instruction limit reached!
% 5.16/2.39 % (27470)------------------------------
% 5.16/2.39 % (27470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.16/2.40 % (27470)Termination reason: Unknown
% 5.16/2.40 % (27470)Termination phase: Saturation
% 5.16/2.40
% 5.16/2.40 % (27470)Memory used [KB]: 18933
% 5.16/2.40 % (27470)Time elapsed: 0.417 s
% 5.16/2.40 % (27470)Instructions burned: 1041 (million)
% 5.16/2.40 % (27470)------------------------------
% 5.16/2.40 % (27470)------------------------------
% 5.77/2.40 % (27501)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_27360 for (2995ds/98Mi)
% 5.77/2.41 % (27502)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_27360 for (2995ds/507Mi)
% 5.77/2.41 % (27503)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_27360 for (2995ds/149Mi)
% 5.77/2.41 % (27500)Instruction limit reached!
% 5.77/2.41 % (27500)------------------------------
% 5.77/2.41 % (27500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.77/2.41 % (27500)Termination reason: Unknown
% 5.77/2.41 % (27500)Termination phase: shuffling
% 5.77/2.41
% 5.77/2.41 % (27500)Memory used [KB]: 2430
% 5.77/2.42 % (27500)Time elapsed: 0.024 s
% 5.77/2.42 % (27500)Instructions burned: 47 (million)
% 5.77/2.42 % (27500)------------------------------
% 5.77/2.42 % (27500)------------------------------
% 5.77/2.43 % (27504)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_27360 for (2994ds/39Mi)
% 5.77/2.44 % (27501)Instruction limit reached!
% 5.77/2.44 % (27501)------------------------------
% 5.77/2.44 % (27501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.77/2.44 % (27501)Termination reason: Unknown
% 5.77/2.44 % (27501)Termination phase: Property scanning
% 5.77/2.44
% 5.77/2.44 % (27501)Memory used [KB]: 2814
% 5.77/2.44 % (27501)Time elapsed: 0.044 s
% 5.77/2.44 % (27501)Instructions burned: 98 (million)
% 5.77/2.44 % (27501)------------------------------
% 5.77/2.44 % (27501)------------------------------
% 5.77/2.45 % (27504)Instruction limit reached!
% 5.77/2.45 % (27504)------------------------------
% 5.77/2.45 % (27504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.77/2.45 % (27504)Termination reason: Unknown
% 5.77/2.45 % (27504)Termination phase: shuffling
% 5.77/2.45
% 5.77/2.45 % (27504)Memory used [KB]: 2430
% 5.77/2.45 % (27504)Time elapsed: 0.020 s
% 5.77/2.45 % (27504)Instructions burned: 40 (million)
% 5.77/2.45 % (27504)------------------------------
% 5.77/2.45 % (27504)------------------------------
% 5.77/2.46 % (27505)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_27360 for (2994ds/6Mi)
% 5.77/2.46 % (27505)Instruction limit reached!
% 5.77/2.46 % (27505)------------------------------
% 5.77/2.46 % (27505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.77/2.46 % (27505)Termination reason: Unknown
% 5.77/2.46 % (27505)Termination phase: shuffling
% 5.77/2.46
% 5.77/2.46 % (27505)Memory used [KB]: 1663
% 5.77/2.46 % (27505)Time elapsed: 0.005 s
% 5.77/2.46 % (27505)Instructions burned: 6 (million)
% 5.77/2.46 % (27505)------------------------------
% 5.77/2.46 % (27505)------------------------------
% 5.77/2.47 % (27506)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_27360 for (2994ds/1087Mi)
% 5.77/2.48 % (27507)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_27360 for (2994ds/47Mi)
% 5.77/2.48 % (27503)Instruction limit reached!
% 5.77/2.48 % (27503)------------------------------
% 5.77/2.48 % (27503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.77/2.48 % (27503)Termination reason: Unknown
% 5.77/2.48 % (27503)Termination phase: Function definition elimination
% 5.77/2.48
% 5.77/2.48 % (27503)Memory used [KB]: 2942
% 5.77/2.48 % (27503)Time elapsed: 0.068 s
% 5.77/2.48 % (27503)Instructions burned: 149 (million)
% 5.77/2.48 % (27503)------------------------------
% 5.77/2.48 % (27503)------------------------------
% 5.77/2.48 % (27481)Instruction limit reached!
% 5.77/2.48 % (27481)------------------------------
% 5.77/2.48 % (27481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.77/2.48 % (27481)Termination reason: Unknown
% 5.77/2.48 % (27481)Termination phase: Saturation
% 5.77/2.48
% 5.77/2.48 % (27481)Memory used [KB]: 13176
% 5.77/2.48 % (27481)Time elapsed: 0.466 s
% 5.77/2.48 % (27481)Instructions burned: 902 (million)
% 5.77/2.48 % (27481)------------------------------
% 5.77/2.48 % (27481)------------------------------
% 6.90/2.50 % (27508)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_27360 for (2994ds/44Mi)
% 6.90/2.50 % (27509)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_27360 for (2994ds/432Mi)
% 6.90/2.50 % (27507)Instruction limit reached!
% 6.90/2.50 % (27507)------------------------------
% 6.90/2.50 % (27507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.90/2.50 % (27507)Termination reason: Unknown
% 6.90/2.50 % (27507)Termination phase: shuffling
% 6.90/2.50
% 6.90/2.50 % (27507)Memory used [KB]: 2558
% 6.90/2.50 % (27507)Time elapsed: 0.024 s
% 6.90/2.50 % (27507)Instructions burned: 48 (million)
% 6.90/2.50 % (27507)------------------------------
% 6.90/2.50 % (27507)------------------------------
% 6.90/2.51 % (27510)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_27360 for (2994ds/62Mi)
% 6.90/2.52 % (27508)Instruction limit reached!
% 6.90/2.52 % (27508)------------------------------
% 6.90/2.52 % (27508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.90/2.52 % (27508)Termination reason: Unknown
% 6.90/2.52 % (27508)Termination phase: shuffling
% 6.90/2.52
% 6.90/2.52 % (27508)Memory used [KB]: 2430
% 6.90/2.52 % (27508)Time elapsed: 0.022 s
% 6.90/2.52 % (27508)Instructions burned: 45 (million)
% 6.90/2.52 % (27508)------------------------------
% 6.90/2.52 % (27508)------------------------------
% 6.90/2.52 % (27488)Instruction limit reached!
% 6.90/2.52 % (27488)------------------------------
% 6.90/2.52 % (27488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.90/2.52 % (27488)Termination reason: Unknown
% 6.90/2.52 % (27488)Termination phase: Saturation
% 6.90/2.52
% 6.90/2.52 % (27488)Memory used [KB]: 10874
% 6.90/2.52 % (27488)Time elapsed: 0.422 s
% 6.90/2.52 % (27488)Instructions burned: 879 (million)
% 6.90/2.52 % (27488)------------------------------
% 6.90/2.52 % (27488)------------------------------
% 6.90/2.53 % (27511)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_27360 for (2993ds/31Mi)
% 6.90/2.53 % (27512)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_27360 for (2993ds/96Mi)
% 6.90/2.54 % (27510)Instruction limit reached!
% 6.90/2.54 % (27510)------------------------------
% 6.90/2.54 % (27510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.90/2.54 % (27510)Termination reason: Unknown
% 6.90/2.54 % (27510)Termination phase: Property scanning
% 6.90/2.54
% 6.90/2.54 % (27510)Memory used [KB]: 2814
% 6.90/2.54 % (27510)Time elapsed: 0.030 s
% 6.90/2.54 % (27510)Instructions burned: 64 (million)
% 6.90/2.54 % (27510)------------------------------
% 6.90/2.54 % (27510)------------------------------
% 6.90/2.55 % (27511)Instruction limit reached!
% 6.90/2.55 % (27511)------------------------------
% 6.90/2.55 % (27511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.90/2.55 % (27511)Termination reason: Unknown
% 6.90/2.55 % (27511)Termination phase: shuffling
% 6.90/2.55
% 6.90/2.55 % (27511)Memory used [KB]: 2302
% 6.90/2.55 % (27511)Time elapsed: 0.017 s
% 6.90/2.55 % (27511)Instructions burned: 32 (million)
% 6.90/2.55 % (27511)------------------------------
% 6.90/2.55 % (27511)------------------------------
% 6.90/2.55 % (27502)First to succeed.
% 6.90/2.56 % (27502)Refutation found. Thanks to Tanya!
% 6.90/2.56 % SZS status Theorem for DTF2THF_27360
% 6.90/2.56 % SZS output start Proof for DTF2THF_27360
% 6.90/2.56 thf(func_def_0, type, in: $i > $i > $o).
% 6.90/2.56 thf(func_def_1, type, exu: ($i > $o) > $o).
% 6.90/2.56 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 6.90/2.56 thf(func_def_8, type, powerset: $i > $i).
% 6.90/2.56 thf(func_def_10, type, setunion: $i > $i).
% 6.90/2.56 thf(func_def_19, type, descr: ($i > $o) > $i).
% 6.90/2.56 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 6.90/2.56 thf(func_def_26, type, prop2set: $o > $i).
% 6.90/2.56 thf(func_def_36, type, nonempty: $i > $o).
% 6.90/2.56 thf(func_def_69, type, set2prop: $i > $o).
% 6.90/2.56 thf(func_def_88, type, subset: $i > $i > $o).
% 6.90/2.56 thf(func_def_89, type, disjoint: $i > $i > $o).
% 6.90/2.56 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 6.90/2.56 thf(func_def_114, type, binunion: $i > $i > $i).
% 6.90/2.56 thf(func_def_122, type, binintersect: $i > $i > $i).
% 6.90/2.56 thf(func_def_135, type, regular: $i > $o).
% 6.90/2.56 thf(func_def_136, type, setminus: $i > $i > $i).
% 6.90/2.56 thf(func_def_147, type, symdiff: $i > $i > $i).
% 6.90/2.56 thf(func_def_153, type, iskpair: $i > $o).
% 6.90/2.56 thf(func_def_158, type, kpair: $i > $i > $i).
% 6.90/2.56 thf(func_def_160, type, cartprod: $i > $i > $i).
% 6.90/2.56 thf(func_def_177, type, singleton: $i > $o).
% 6.90/2.56 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 6.90/2.56 thf(func_def_184, type, atmost1p: $i > $o).
% 6.90/2.56 thf(func_def_185, type, atleast2p: $i > $o).
% 6.90/2.56 thf(func_def_186, type, atmost2p: $i > $o).
% 6.90/2.56 thf(func_def_187, type, upairsetp: $i > $o).
% 6.90/2.56 thf(func_def_191, type, kfst: $i > $i).
% 6.90/2.56 thf(func_def_203, type, ksnd: $i > $i).
% 6.90/2.56 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 6.90/2.56 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 6.90/2.56 thf(func_def_222, type, func: $i > $i > $i > $o).
% 6.90/2.56 thf(func_def_223, type, funcSet: $i > $i > $i).
% 6.90/2.56 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 6.90/2.56 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 6.90/2.56 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 6.90/2.56 thf(func_def_323, type, breln1: $i > $i > $o).
% 6.90/2.56 thf(func_def_325, type, breln1Set: $i > $i).
% 6.90/2.56 thf(func_def_327, type, transitive: $i > $i > $o).
% 6.90/2.56 thf(func_def_328, type, antisymmetric: $i > $i > $o).
% 6.90/2.56 thf(func_def_329, type, reflexive: $i > $i > $o).
% 6.90/2.56 thf(func_def_330, type, refltransitive: $i > $i > $o).
% 6.90/2.56 thf(func_def_331, type, refllinearorder: $i > $i > $o).
% 6.90/2.56 thf(func_def_332, type, reflwellordering: $i > $i > $o).
% 6.90/2.56 thf(func_def_338, type, breln1invset: $i > $i > $i).
% 6.90/2.56 thf(func_def_342, type, breln1compset: $i > $i > $i > $i).
% 6.90/2.56 thf(func_def_360, type, image1: $i > ($i > $i) > $i).
% 6.90/2.56 thf(func_def_364, type, injective: $i > $i > $i > $o).
% 6.90/2.56 thf(func_def_365, type, injFuncSet: $i > $i > $i).
% 6.90/2.56 thf(func_def_369, type, surjective: $i > $i > $i > $o).
% 6.90/2.56 thf(func_def_370, type, surjFuncSet: $i > $i > $i).
% 6.90/2.56 thf(func_def_378, type, omegaS: $i > $i).
% 6.90/2.56 thf(func_def_399, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 6.90/2.56 thf(func_def_400, type, sK1: !>[X0: $tType]:((X0 > $o) > X0)).
% 6.90/2.56 thf(func_def_403, type, ph4: !>[X0: $tType]:(X0)).
% 6.90/2.56 thf(f2411,plain,(
% 6.90/2.56 $false),
% 6.90/2.56 inference(trivial_inequality_removal,[status(thm)],[f2409])).
% 6.90/2.56 thf(f2409,plain,(
% 6.90/2.56 ($true = $false)),
% 6.90/2.56 inference(superposition,[status(thm)],[f2205,f2400])).
% 6.90/2.56 thf(f2400,plain,(
% 6.90/2.56 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (setadjoin @ X1 @ X0)))) )),
% 6.90/2.56 inference(beta_eta_normalization,[status(thm)],[f2399])).
% 6.90/2.56 thf(f2399,plain,(
% 6.90/2.56 ( ! [X0 : $i,X1 : $i] : (($true = ((^[Y0 : $i]: (in @ Y0 @ (setadjoin @ Y0 @ X0))) @ X1))) )),
% 6.90/2.56 inference(trivial_inequality_removal,[status(thm)],[f2396])).
% 6.90/2.56 thf(f2396,plain,(
% 6.90/2.56 ( ! [X0 : $i,X1 : $i] : (($true = $false) | ($true = ((^[Y0 : $i]: (in @ Y0 @ (setadjoin @ Y0 @ X0))) @ X1))) )),
% 6.90/2.56 inference(superposition,[status(thm)],[f994,f2219])).
% 6.90/2.56 thf(f2219,plain,(
% 6.90/2.56 ( ! [X0 : $i] : (((!! @ $i @ (^[Y0 : $i]: (in @ Y0 @ (setadjoin @ Y0 @ X0)))) = $true)) )),
% 6.90/2.56 inference(beta_eta_normalization,[status(thm)],[f2218])).
% 6.90/2.56 thf(f2218,plain,(
% 6.90/2.56 ( ! [X0 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y1 @ Y0))))) @ X0))) )),
% 6.90/2.56 inference(trivial_inequality_removal,[status(thm)],[f2214])).
% 6.90/2.56 thf(f2214,plain,(
% 6.90/2.56 ( ! [X0 : $i] : (($true = ((^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y1 @ Y0))))) @ X0)) | ($true = $false)) )),
% 6.90/2.56 inference(superposition,[status(thm)],[f1868,f994])).
% 6.90/2.56 thf(f1868,plain,(
% 6.90/2.56 ((!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y1 @ Y0)))))) = $true)),
% 6.90/2.56 inference(definition_unfolding,[status(thm)],[f1521,f1338])).
% 6.90/2.56 thf(f1338,plain,(
% 6.90/2.56 (setadjoinIL = $true)),
% 6.90/2.56 inference(cnf_transformation,[status(thm)],[f1009])).
% 6.90/2.56 thf(f1009,plain,(
% 6.90/2.56 (subsetRefl = $true) & (symdiffIneg2 = $true) & (funcGraphProp4 = $true) & (complementInPowersetComplementIntersect = $true) & (subsetemptysetimpeq = $true) & (singletonsuniq = $true) & (dpsetconstrERa = $true) & (breln1all1 = $true) & (($true = (in @ sK2 @ omega)) & (emptyset = (omegaS @ sK2))) & (binunionLsub = $true) & (notsubsetI = $true) & (binintersectRsub = $true) & (setukpairinjR11 = $true) & (notinself = $true) & (ex1E2 = $true) & (subsetE2 = $true) & (powersetE = $true) & (binunionTILcontra = $true) & (binintersectT_lem = $true) & (singletonprop = $true) & (ubforcartprodlem1 = $true) & (kfstpairEq = $true) & (setextT = $true) & (quantDeMorgan1 = $true) & (symdiffIneg1 = $true) & (contraSubsetComplement = $true) & (upairinpowunion = $true) & (symdiffE = $true) & (kpairsurjEq = $true) & (notequalI2 = $true) & (lam2lamEq = $true) & (binunionTEcontra = $true) & (exuE3u = $true) & (complementImpComplementIntersect = $true) & (woz2B = $true) & (setunion__Cong = $true) & (prop2set2propI = $true) & (notequalI1 = $true) & (omegaSAx = $true) & (eta1 = $true) & (breln1compprop = $true) & (app = $true) & (ap2p = $true) & (powersetTE1 = $true) & (dsetconstrI = $true) & (in__Cong = $true) & (image1Ex1 = $true) & (quantDeMorgan2 = $true) & (leftInvIsSurj = $true) & (singletonsubset = $true) & (setminusIRneg = $true) & (notinsingleton = $true) & (dpsetconstrSub = $true) & (symdiffI2 = $true) & (woz2W = $true) & (emptyInPowerset = $true) & (setunionAx = $true) & (cartprodpairmemER = $true) & (image1Equiv = $true) & (ksndpairEq = $true) & (contrasubsetT1 = $true) & (nonemptyE1 = $true) & (ap2apEq2 = $true) & (eqbreln = $true) & (setminusEL = $true) & (symdiffI1 = $true) & (singletoninpowerset = $true) & (nonemptyI1 = $true) & (omegaSclos = $true) & (binintersectSubset5 = $true) & (inCongP = $true) & (sepSubset = $true) & (setadjoinIL = $true) & (brelnall1 = $true) & (setukpairinjR2 = $true) & (subbreln1 = $true) & (contrasubsetT = $true) & (funcGraphProp2 = $true) & (powersetI = $true) & (ubforcartprodlem3 = $true) & (dsetconstrEL = $true) & (setukpairinjL1 = $true) & (woz2Ex = $true) & (woz2A = $true) & (quantDeMorgan3 = $true) & (beta1 = $true) & (setukpairinjR12 = $true) & (kfstsingleton = $true) & (ex1I = $true) & (secondinupair = $true) & (inIntersectImpInUnion2 = $true) & (kpairiskpair = $true) & (emptyinunitempty = $true) & (inIntersectImpInUnion = $true) & (breln1invE = $true) & (bs114d = $true) & (setminusSubset2 = $true) & (iffalse = $true) & (setunionsingleton1 = $true) & (omega0Ax = $true) & (setextsub = $true) & (lam2p = $true) & (complementTcontraSubset = $true) & (setunionsingleton2 = $true) & (emptysetAx = $true) & (complementSubsetComplementIntersect = $true) & (binunionIR = $true) & (cartprodpairmemEL = $true) & (dsetconstr__Cong = $true) & (doubleComplementI1 = $true) & (woz13rule1 = $true) & (breln1unionprop = $true) & (vacuousDall = $true) & (binintersectI = $true) & (funcinfuncset = $true) & (ifp = $true) & (infuncsetfunc = $true) & (exuE1 = $true) & (doubleComplementSub2 = $true) & (funcImageSingleton = $true) & (binunionTIRcontra = $true) & (binunionRsub = $true) & (demorgan2 = $true) & (funcext = $true) & (binunionEcases = $true) & (subsetI2 = $true) & (demorgan2b2 = $true) & (setminusERneg = $true) & (ksndsingleton = $true) & (woz1_4 = $true) & (setOfPairsIsBReln1 = $true) & (powersetT_lem = $true) & (breln1unionIL = $true) & (omegaIndAx = $true) & (setukpairinjR1 = $true) & (setadjoinE = $true) & (cartprodpairin = $true) & (complementUnionInPowersetComplement = $true) & (subsetTI = $true) & (setadjoinIR = $true) & (image1E = $true) & (setukpairIL = $true) & (demorgan2a = $true) & (subbreln = $true) & (binunionTE = $true) & (eqinunit = $true) & (injFuncSetFuncInj = $true) & (wellorderingAx = $true) & (setminusER = $true) & (prop2setE = $true) & (complementTI1 = $true) & (complementTE1 = $true) & (setadjoinSub = $true) & (cartprodfstpairEq = $true) & (surjCantorThm = $true) & (setunionE = $true) & (subPowSU = $true) & (funcext2 = $true) & (cartprodfstin = $true) & (surjFuncSetFuncSurj = $true) & (woz1_2 = $true) & (subset2powerset = $true) & (ex1E1 = $true) & (subsetE = $true) & (lamProp = $true) & (woz1_3 = $true) & (powersetE1 = $true) & (breln1all2 = $true) & (binintersectSubset1 = $true) & (inComplementUnionImpNotIn1 = $true) & (setukpairIR = $true) & (setminusI = $true) & (cartprodpairsurjEq = $true) & (notdexE = $true) & (singletoninpowunion = $true) & (upairsetIL = $true) & (exu__Cong = $true) & (funcGraphProp1 = $true) & (binintersectSubset4 = $true) & (exuE2 = $true) & (exuI3 = $true) & (demorgan1a = $true) & (disjointsetsI1 = $true) & (complementTnotintersectT = $true) & (binintersectSubset2 = $true) & (breln1invI = $true) & (powerset__Cong = $true) & (binintersectEL = $true) & (woz1_1 = $true) & (descr__Cong = $true) & (setukpairinjL = $true) & (image1Ex = $true) & (breln1unionIR = $true) & (emptyI = $true) & (cartprodmempaircEq = $true) & (singletonsswitch = $true) & (notinself2 = $true) & (ex1I2 = $true) & (powersetI1 = $true) & (demorgan1b = $true) & (setoftrueEq = $true) & (powersetsubset = $true) & (subsetTrans = $true) & (woz13rule4 = $true) & (woz13rule0 = $true) & (ubforcartprodlem2 = $true) & (eqbreln1 = $true) & (binunionT_lem = $true) & (omegaSp = $true) & (binunionIL = $true) & (exuE3e = $true) & (dpsetconstrI = $true) & (noeltsimpempty = $true) & (intersectInPowersetIntersectUnions = $true) & (eta2 = $true) & (binintersectTERcontra = $true) & (beta2 = $true) & (setOfPairsIsBReln = $true) & (iftrue = $true) & (demorgan2a1 = $true) & (dpsetconstrER = $true) & (binintersectLsub = $true) & (lamp = $true) & (complementT_lem = $true) & (iftrueorfalse = $true) & (demorgan2b = $true) & (cartprodsndin = $true) & (image1I = $true) & (iffalseProp1 = $true) & (upairsetIR = $true) & (omega__Cong = $true) & (replAx = $true) & (sepInPowerset = $true) & (nonemptyImpWitness = $true) & (setextAx = $true) & (iftrueProp2 = $true) & (binintersectTELcontra = $true) & (upairset2E = $true) & (doubleComplementEq = $true) & (surjFuncSetFuncIn = $true) & (eqimpsubset1 = $true) & (emptyset__Cong = $true) & (setunionI = $true) & (emptysetsubset = $true) & (cartprodmempair = $true) & (exuI1 = $true) & (woz1_5 = $true) & (prop2setI = $true) & (emptyE1 = $true) & (breln1compE = $true) & (breln1unionCommutes = $true) & (demorgan1 = $true) & (injFuncInInjFuncSet = $true) & (funcextLem = $true) & (setminusT_lem = $true) & (emptysetE = $true) & (theprop = $true) & (doubleComplementSub1 = $true) & (binintersectSubset3 = $true) & (cartprodsndpairEq = $true) & (contrasubsetT2 = $true) & (cartprodmempair1 = $true) & (descrp = $true) & (breln1compI = $true) & (powersetAx = $true) & (injFuncSetFuncIn = $true) & (funcGraphProp3 = $true) & (upairsubunion = $true) & (uniqinunit = $true) & (iftrueProp1 = $true) & (woz13rule3 = $true) & (dsetconstrER = $true) & (setbeta = $true) & (setext = $true) & (setminusELneg = $true) & (setukpairinjL2 = $true) & (upairsetE = $true) & (contrasubsetT3 = $true) & (foundationAx = $true) & (notinemptyset = $true) & (setadjoinAx = $true) & (upairequniteq = $true) & (setadjoinSub2 = $true) & (breln1unionE = $true) & (choice2fnsingleton = $true) & (upairset2IR = $true) & (eqimpsubset2 = $true) & (exuI2 = $true) & (quantDeMorgan4 = $true) & (brelnall2 = $true) & (powersetTI1 = $true) & (binunionE = $true) & (binintersectER = $true) & (setunionE2 = $true) & (breln1SetBreln1 = $true) & (inComplementUnionImpInComplement1 = $true) & (dpsetconstrEL1 = $true) & (ifSingleton = $true) & (apProp = $true) & (breln1unionI = $true) & (emptyinPowerset = $true) & (setminusLsub = $true) & (setukpairinjR = $true) & (inPowerset = $true) & (dpsetconstrEL2 = $true) & (iffalseProp2 = $true) & (notdallE = $true) & (setminusSubset1 = $true) & (breln1compEex = $true) & (breln1unionEcases = $true) & (subsetI1 = $true) & (kpairp = $true) & (doubleComplementE1 = $true) & (ap2apEq1 = $true) & (setminusILneg = $true) & (setunionsingleton = $true) & (demorgan2a2 = $true) & (breln1invprop = $true) & (nonemptyI = $true) & (woz13rule2 = $true) & (emptysetimpfalse = $true) & (setadjoin__Cong = $true) & (inIntersectImpInIntersectUnions = $true) & (theeq = $true) & (foundation2 = $true) & (setadjoinOr = $true) & (exuEu = $true)),
% 6.90/2.56 inference(skolemisation,[status(esa),new_symbols(skolem,[sK2])],[f1007,f1008])).
% 6.90/2.56 thf(f1008,plain,(
% 6.90/2.56 ? [X0 : $i] : (($true = (in @ X0 @ omega)) & (emptyset = (omegaS @ X0))) => (($true = (in @ sK2 @ omega)) & (emptyset = (omegaS @ sK2)))),
% 6.90/2.56 introduced(definition,[],[choice_axiom])).
% 6.90/2.56 thf(f1007,plain,(
% 6.90/2.56 (subsetRefl = $true) & (symdiffIneg2 = $true) & (funcGraphProp4 = $true) & (complementInPowersetComplementIntersect = $true) & (subsetemptysetimpeq = $true) & (singletonsuniq = $true) & (dpsetconstrERa = $true) & (breln1all1 = $true) & ? [X0 : $i] : (($true = (in @ X0 @ omega)) & (emptyset = (omegaS @ X0))) & (binunionLsub = $true) & (notsubsetI = $true) & (binintersectRsub = $true) & (setukpairinjR11 = $true) & (notinself = $true) & (ex1E2 = $true) & (subsetE2 = $true) & (powersetE = $true) & (binunionTILcontra = $true) & (binintersectT_lem = $true) & (singletonprop = $true) & (ubforcartprodlem1 = $true) & (kfstpairEq = $true) & (setextT = $true) & (quantDeMorgan1 = $true) & (symdiffIneg1 = $true) & (contraSubsetComplement = $true) & (upairinpowunion = $true) & (symdiffE = $true) & (kpairsurjEq = $true) & (notequalI2 = $true) & (lam2lamEq = $true) & (binunionTEcontra = $true) & (exuE3u = $true) & (complementImpComplementIntersect = $true) & (woz2B = $true) & (setunion__Cong = $true) & (prop2set2propI = $true) & (notequalI1 = $true) & (omegaSAx = $true) & (eta1 = $true) & (breln1compprop = $true) & (app = $true) & (ap2p = $true) & (powersetTE1 = $true) & (dsetconstrI = $true) & (in__Cong = $true) & (image1Ex1 = $true) & (quantDeMorgan2 = $true) & (leftInvIsSurj = $true) & (singletonsubset = $true) & (setminusIRneg = $true) & (notinsingleton = $true) & (dpsetconstrSub = $true) & (symdiffI2 = $true) & (woz2W = $true) & (emptyInPowerset = $true) & (setunionAx = $true) & (cartprodpairmemER = $true) & (image1Equiv = $true) & (ksndpairEq = $true) & (contrasubsetT1 = $true) & (nonemptyE1 = $true) & (ap2apEq2 = $true) & (eqbreln = $true) & (setminusEL = $true) & (symdiffI1 = $true) & (singletoninpowerset = $true) & (nonemptyI1 = $true) & (omegaSclos = $true) & (binintersectSubset5 = $true) & (inCongP = $true) & (sepSubset = $true) & (setadjoinIL = $true) & (brelnall1 = $true) & (setukpairinjR2 = $true) & (subbreln1 = $true) & (contrasubsetT = $true) & (funcGraphProp2 = $true) & (powersetI = $true) & (ubforcartprodlem3 = $true) & (dsetconstrEL = $true) & (setukpairinjL1 = $true) & (woz2Ex = $true) & (woz2A = $true) & (quantDeMorgan3 = $true) & (beta1 = $true) & (setukpairinjR12 = $true) & (kfstsingleton = $true) & (ex1I = $true) & (secondinupair = $true) & (inIntersectImpInUnion2 = $true) & (kpairiskpair = $true) & (emptyinunitempty = $true) & (inIntersectImpInUnion = $true) & (breln1invE = $true) & (bs114d = $true) & (setminusSubset2 = $true) & (iffalse = $true) & (setunionsingleton1 = $true) & (omega0Ax = $true) & (setextsub = $true) & (lam2p = $true) & (complementTcontraSubset = $true) & (setunionsingleton2 = $true) & (emptysetAx = $true) & (complementSubsetComplementIntersect = $true) & (binunionIR = $true) & (cartprodpairmemEL = $true) & (dsetconstr__Cong = $true) & (doubleComplementI1 = $true) & (woz13rule1 = $true) & (breln1unionprop = $true) & (vacuousDall = $true) & (binintersectI = $true) & (funcinfuncset = $true) & (ifp = $true) & (infuncsetfunc = $true) & (exuE1 = $true) & (doubleComplementSub2 = $true) & (funcImageSingleton = $true) & (binunionTIRcontra = $true) & (binunionRsub = $true) & (demorgan2 = $true) & (funcext = $true) & (binunionEcases = $true) & (subsetI2 = $true) & (demorgan2b2 = $true) & (setminusERneg = $true) & (ksndsingleton = $true) & (woz1_4 = $true) & (setOfPairsIsBReln1 = $true) & (powersetT_lem = $true) & (breln1unionIL = $true) & (omegaIndAx = $true) & (setukpairinjR1 = $true) & (setadjoinE = $true) & (cartprodpairin = $true) & (complementUnionInPowersetComplement = $true) & (subsetTI = $true) & (setadjoinIR = $true) & (image1E = $true) & (setukpairIL = $true) & (demorgan2a = $true) & (subbreln = $true) & (binunionTE = $true) & (eqinunit = $true) & (injFuncSetFuncInj = $true) & (wellorderingAx = $true) & (setminusER = $true) & (prop2setE = $true) & (complementTI1 = $true) & (complementTE1 = $true) & (setadjoinSub = $true) & (cartprodfstpairEq = $true) & (surjCantorThm = $true) & (setunionE = $true) & (subPowSU = $true) & (funcext2 = $true) & (cartprodfstin = $true) & (surjFuncSetFuncSurj = $true) & (woz1_2 = $true) & (subset2powerset = $true) & (ex1E1 = $true) & (subsetE = $true) & (lamProp = $true) & (woz1_3 = $true) & (powersetE1 = $true) & (breln1all2 = $true) & (binintersectSubset1 = $true) & (inComplementUnionImpNotIn1 = $true) & (setukpairIR = $true) & (setminusI = $true) & (cartprodpairsurjEq = $true) & (notdexE = $true) & (singletoninpowunion = $true) & (upairsetIL = $true) & (exu__Cong = $true) & (funcGraphProp1 = $true) & (binintersectSubset4 = $true) & (exuE2 = $true) & (exuI3 = $true) & (demorgan1a = $true) & (disjointsetsI1 = $true) & (complementTnotintersectT = $true) & (binintersectSubset2 = $true) & (breln1invI = $true) & (powerset__Cong = $true) & (binintersectEL = $true) & (woz1_1 = $true) & (descr__Cong = $true) & (setukpairinjL = $true) & (image1Ex = $true) & (breln1unionIR = $true) & (emptyI = $true) & (cartprodmempaircEq = $true) & (singletonsswitch = $true) & (notinself2 = $true) & (ex1I2 = $true) & (powersetI1 = $true) & (demorgan1b = $true) & (setoftrueEq = $true) & (powersetsubset = $true) & (subsetTrans = $true) & (woz13rule4 = $true) & (woz13rule0 = $true) & (ubforcartprodlem2 = $true) & (eqbreln1 = $true) & (binunionT_lem = $true) & (omegaSp = $true) & (binunionIL = $true) & (exuE3e = $true) & (dpsetconstrI = $true) & (noeltsimpempty = $true) & (intersectInPowersetIntersectUnions = $true) & (eta2 = $true) & (binintersectTERcontra = $true) & (beta2 = $true) & (setOfPairsIsBReln = $true) & (iftrue = $true) & (demorgan2a1 = $true) & (dpsetconstrER = $true) & (binintersectLsub = $true) & (lamp = $true) & (complementT_lem = $true) & (iftrueorfalse = $true) & (demorgan2b = $true) & (cartprodsndin = $true) & (image1I = $true) & (iffalseProp1 = $true) & (upairsetIR = $true) & (omega__Cong = $true) & (replAx = $true) & (sepInPowerset = $true) & (nonemptyImpWitness = $true) & (setextAx = $true) & (iftrueProp2 = $true) & (binintersectTELcontra = $true) & (upairset2E = $true) & (doubleComplementEq = $true) & (surjFuncSetFuncIn = $true) & (eqimpsubset1 = $true) & (emptyset__Cong = $true) & (setunionI = $true) & (emptysetsubset = $true) & (cartprodmempair = $true) & (exuI1 = $true) & (woz1_5 = $true) & (prop2setI = $true) & (emptyE1 = $true) & (breln1compE = $true) & (breln1unionCommutes = $true) & (demorgan1 = $true) & (injFuncInInjFuncSet = $true) & (funcextLem = $true) & (setminusT_lem = $true) & (emptysetE = $true) & (theprop = $true) & (doubleComplementSub1 = $true) & (binintersectSubset3 = $true) & (cartprodsndpairEq = $true) & (contrasubsetT2 = $true) & (cartprodmempair1 = $true) & (descrp = $true) & (breln1compI = $true) & (powersetAx = $true) & (injFuncSetFuncIn = $true) & (funcGraphProp3 = $true) & (upairsubunion = $true) & (uniqinunit = $true) & (iftrueProp1 = $true) & (woz13rule3 = $true) & (dsetconstrER = $true) & (setbeta = $true) & (setext = $true) & (setminusELneg = $true) & (setukpairinjL2 = $true) & (upairsetE = $true) & (contrasubsetT3 = $true) & (foundationAx = $true) & (notinemptyset = $true) & (setadjoinAx = $true) & (upairequniteq = $true) & (setadjoinSub2 = $true) & (breln1unionE = $true) & (choice2fnsingleton = $true) & (upairset2IR = $true) & (eqimpsubset2 = $true) & (exuI2 = $true) & (quantDeMorgan4 = $true) & (brelnall2 = $true) & (powersetTI1 = $true) & (binunionE = $true) & (binintersectER = $true) & (setunionE2 = $true) & (breln1SetBreln1 = $true) & (inComplementUnionImpInComplement1 = $true) & (dpsetconstrEL1 = $true) & (ifSingleton = $true) & (apProp = $true) & (breln1unionI = $true) & (emptyinPowerset = $true) & (setminusLsub = $true) & (setukpairinjR = $true) & (inPowerset = $true) & (dpsetconstrEL2 = $true) & (iffalseProp2 = $true) & (notdallE = $true) & (setminusSubset1 = $true) & (breln1compEex = $true) & (breln1unionEcases = $true) & (subsetI1 = $true) & (kpairp = $true) & (doubleComplementE1 = $true) & (ap2apEq1 = $true) & (setminusILneg = $true) & (setunionsingleton = $true) & (demorgan2a2 = $true) & (breln1invprop = $true) & (nonemptyI = $true) & (woz13rule2 = $true) & (emptysetimpfalse = $true) & (setadjoin__Cong = $true) & (inIntersectImpInIntersectUnions = $true) & (theeq = $true) & (foundation2 = $true) & (setadjoinOr = $true) & (exuEu = $true)),
% 6.90/2.56 inference(flattening,[status(thm)],[f1006])).
% 6.90/2.56 thf(f1006,plain,(
% 6.90/2.56 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i] : (($true = (in @ X0 @ omega)) & (emptyset = (omegaS @ X0))) & (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)),
% 6.90/2.56 inference(ennf_transformation,[status(thm)],[f702])).
% 6.90/2.56 thf(f702,plain,(
% 6.90/2.56 ~((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) => ! [X0 : $i] : (($true = (in @ X0 @ omega)) => (emptyset != (omegaS @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.90/2.56 inference(fool_elimination,[status(thm)],[f701])).
% 6.90/2.56 thf(f701,plain,(
% 6.90/2.56 ~(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 => ! [X0 : $i] : ((in @ X0 @ omega) => (emptyset != (omegaS @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.90/2.56 inference(rectify,[status(thm)],[f331])).
% 6.90/2.56 thf(f331,negated_conjecture,(
% 6.90/2.56 ~(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 => ! [X1 : $i] : ((in @ X1 @ omega) => (emptyset != (omegaS @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.90/2.56 inference(negated_conjecture,[status(cth)],[f330])).
% 6.90/2.56 thf(f330,conjecture,(
% 6.90/2.56 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 => ! [X1 : $i] : ((in @ X1 @ omega) => (emptyset != (omegaS @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 6.90/2.56 file('/export/starexec/sandbox/tmp/tmp.y5OtfSgMuG/DTF2THF_27360.p',peano0notS)).
% 6.90/2.56 thf(f1521,plain,(
% 6.90/2.56 (setadjoinIL = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y1 @ Y0)))))))),
% 6.90/2.56 inference(cnf_transformation,[status(thm)],[f432])).
% 6.90/2.56 thf(f432,plain,(
% 6.90/2.56 (setadjoinIL = (!! @ $i @ (^[Y0 : $i]: (!! @ $i @ (^[Y1 : $i]: (in @ Y1 @ (setadjoin @ Y1 @ Y0)))))))),
% 6.90/2.56 inference(fool_elimination,[status(thm)],[f431])).
% 6.90/2.56 thf(f431,plain,(
% 6.90/2.56 (setadjoinIL = ! [X0 : $i,X1 : $i] : (in @ X0 @ (setadjoin @ X0 @ X1)))),
% 6.90/2.56 inference(rectify,[status(thm)],[f31])).
% 6.90/2.56 thf(f31,axiom,(
% 6.90/2.56 (setadjoinIL = ! [X1 : $i,X2 : $i] : (in @ X1 @ (setadjoin @ X1 @ X2)))),
% 6.90/2.56 file('/export/starexec/sandbox/tmp/tmp.y5OtfSgMuG/DTF2THF_27360.p',setadjoinIL)).
% 6.90/2.56 thf(f994,plain,(
% 6.90/2.56 ( ! [X0 : $tType,X2 : X0,X1 : X0 > $o] : (($false = (!! @ X0 @ X1)) | ((X1 @ X2) = $true)) )),
% 6.90/2.56 introduced(definition,[],[pi_proxy_axiom])).
% 6.90/2.56 thf(f2205,plain,(
% 6.90/2.56 ( ! [X0 : $i] : (((in @ X0 @ (setadjoin @ sK2 @ sK2)) = $false)) )),
% 6.90/2.56 inference(trivial_inequality_removal,[status(thm)],[f2202])).
% 6.90/2.56 thf(f2202,plain,(
% 6.90/2.56 ( ! [X0 : $i] : (($true = $false) | ((in @ X0 @ (setadjoin @ sK2 @ sK2)) = $false)) )),
% 6.90/2.56 inference(superposition,[status(thm)],[f2157,f992])).
% 6.90/2.56 thf(f992,plain,(
% 6.90/2.56 ( ! [X1 : $o] : (($false = (~ X1)) | ($false = X1)) )),
% 6.90/2.56 introduced(definition,[],[logical_not_proxy_axiom])).
% 6.90/2.56 thf(f2157,plain,(
% 6.90/2.56 ( ! [X0 : $i] : (($true = (~ (in @ X0 @ (setadjoin @ sK2 @ sK2))))) )),
% 6.90/2.56 inference(beta_eta_normalization,[status(thm)],[f2156])).
% 6.90/2.56 thf(f2156,plain,(
% 6.90/2.56 ( ! [X0 : $i] : (($true = ((^[Y0 : $i]: (~ (in @ Y0 @ (setadjoin @ sK2 @ sK2)))) @ X0))) )),
% 6.90/2.56 inference(trivial_inequality_removal,[status(thm)],[f2152])).
% 6.90/2.56 thf(f2152,plain,(
% 6.90/2.56 ( ! [X0 : $i] : (($true = ((^[Y0 : $i]: (~ (in @ Y0 @ (setadjoin @ sK2 @ sK2)))) @ X0)) | ($true = $false)) )),
% 6.90/2.56 inference(superposition,[status(thm)],[f2065,f994])).
% 6.90/2.56 thf(f2065,plain,(
% 6.90/2.56 ((!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ (setadjoin @ sK2 @ sK2))))) = $true)),
% 6.90/2.56 inference(beta_eta_normalization,[status(thm)],[f1773])).
% 6.90/2.56 thf(f1773,plain,(
% 6.90/2.56 ((!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ ((^[Y1 : $i]: (setadjoin @ Y1 @ Y1)) @ sK2))))) = $true)),
% 6.90/2.56 inference(definition_unfolding,[status(thm)],[f1425,f1306,f1667])).
% 6.90/2.56 thf(f1667,plain,(
% 6.90/2.56 (emptyset = ((^[Y0 : $i]: (setadjoin @ Y0 @ Y0)) @ sK2))),
% 6.90/2.56 inference(definition_unfolding,[status(thm)],[f1402,f1560])).
% 6.90/2.56 thf(f1560,plain,(
% 6.90/2.56 (omegaS = (^[Y0 : $i]: (setadjoin @ Y0 @ Y0)))),
% 6.90/2.56 inference(cnf_transformation,[status(thm)],[f764])).
% 6.90/2.56 thf(f764,plain,(
% 6.90/2.56 (omegaS = (^[Y0 : $i]: (setadjoin @ Y0 @ Y0)))),
% 6.90/2.56 inference(fool_elimination,[status(thm)],[f763])).
% 6.90/2.56 thf(f763,plain,(
% 6.90/2.56 (omegaS = (^[X0 : $i] : (setadjoin @ X0 @ X0)))),
% 6.90/2.56 inference(rectify,[status(thm)],[f327])).
% 6.90/2.56 thf(f327,axiom,(
% 6.90/2.56 (omegaS = (^[X1 : $i] : (setadjoin @ X1 @ X1)))),
% 6.90/2.56 file('/export/starexec/sandbox/tmp/tmp.y5OtfSgMuG/DTF2THF_27360.p',omegaS)).
% 6.90/2.56 thf(f1402,plain,(
% 6.90/2.56 (emptyset = (omegaS @ sK2))),
% 6.90/2.56 inference(cnf_transformation,[status(thm)],[f1009])).
% 6.90/2.56 thf(f1306,plain,(
% 6.90/2.56 (emptysetAx = $true)),
% 6.90/2.56 inference(cnf_transformation,[status(thm)],[f1009])).
% 6.90/2.56 thf(f1425,plain,(
% 6.90/2.56 (emptysetAx = (!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ emptyset)))))),
% 6.90/2.56 inference(cnf_transformation,[status(thm)],[f676])).
% 6.90/2.56 thf(f676,plain,(
% 6.90/2.56 (emptysetAx = (!! @ $i @ (^[Y0 : $i]: (~ (in @ Y0 @ emptyset)))))),
% 6.90/2.56 inference(fool_elimination,[status(thm)],[f675])).
% 6.90/2.56 thf(f675,plain,(
% 6.90/2.56 (! [X0 : $i] : ~(in @ X0 @ emptyset) = emptysetAx)),
% 6.90/2.56 inference(rectify,[status(thm)],[f3])).
% 6.90/2.56 thf(f3,axiom,(
% 6.90/2.56 (! [X1 : $i] : ~(in @ X1 @ emptyset) = emptysetAx)),
% 6.90/2.56 file('/export/starexec/sandbox/tmp/tmp.y5OtfSgMuG/DTF2THF_27360.p',emptysetAx)).
% 6.90/2.56 % SZS output end Proof for DTF2THF_27360
% 6.90/2.56 % (27502)------------------------------
% 6.90/2.56 % (27502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.90/2.56 % (27502)Termination reason: Refutation
% 6.90/2.56
% 6.90/2.56 % (27502)Memory used [KB]: 8315
% 6.90/2.56 % (27502)Time elapsed: 0.151 s
% 6.90/2.56 % (27502)Instructions burned: 306 (million)
% 6.90/2.56 % (27502)------------------------------
% 6.90/2.56 % (27502)------------------------------
% 6.90/2.56 % (27458)Success in time 0.654 s
%------------------------------------------------------------------------------