%------------------------------------------------------------------------------
% File : Zipperpin---2.1.9999
% Problem : SEU777^1 : TPTP v8.1.2. Released v3.7.0.
% Transfm : NO INFORMATION
% Format : NO INFORMATION
% Command : python3 /export/starexec/sandbox/solver/bin/portfolio.lams.parallel.py %s %d /export/starexec/sandbox/tmp/tmp.Z6OcAbSkpJ true
% Computer : n026.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 : Thu Aug 31 19:17:29 EDT 2023
% Result : Theorem 132.10s 17.65s
% Output : Refutation 132.57s
% Verified :
% SZS Type : Refutation
% Derivation depth : 4
% Number of leaves : 952
% Syntax : Number of formulae : 966 ( 608 unt; 345 typ; 0 def)
% Number of atoms : 6076 (1301 equ; 0 cnn)
% Maximal formula atoms : 905 ( 9 avg)
% Number of connectives : 19447 ( 386 ~; 64 |; 260 &;14306 @)
% ( 76 <=>;4355 =>; 0 <=; 0 <~>)
% Maximal formula depth : 311 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 450 ( 450 >; 0 *; 0 +; 0 <<)
% Number of symbols : 342 ( 339 usr; 301 con; 0-4 aty)
% Number of variables : 5239 ( 358 ^;4661 !; 220 ?;5239 :)
% Comments :
%------------------------------------------------------------------------------
thf(setadjoinAx_type,type,
setadjoinAx: $o ).
thf(cartprodfstpairEq_type,type,
cartprodfstpairEq: $o ).
thf(notdexE_type,type,
notdexE: $o ).
thf(setunionAx_type,type,
setunionAx: $o ).
thf(setminusELneg_type,type,
setminusELneg: $o ).
thf(funcGraphProp1_type,type,
funcGraphProp1: $o ).
thf(beta1_type,type,
beta1: $o ).
thf(powersetE_type,type,
powersetE: $o ).
thf(doubleComplementE1_type,type,
doubleComplementE1: $o ).
thf(omegaSAx_type,type,
omegaSAx: $o ).
thf(zip_tseitin_0_type,type,
zip_tseitin_0: $i > $i > $i > $o ).
thf(doubleComplementI1_type,type,
doubleComplementI1: $o ).
thf(exuI1_type,type,
exuI1: $o ).
thf(binunionTEcontra_type,type,
binunionTEcontra: $o ).
thf(exuI2_type,type,
exuI2: $o ).
thf(notequalI2_type,type,
notequalI2: $o ).
thf(kfstsingleton_type,type,
kfstsingleton: $o ).
thf(demorgan1b_type,type,
demorgan1b: $o ).
thf(kpairiskpair_type,type,
kpairiskpair: $o ).
thf(demorgan2a_type,type,
demorgan2a: $o ).
thf(subsetTI_type,type,
subsetTI: $o ).
thf(upairsubunion_type,type,
upairsubunion: $o ).
thf(binintersectT_lem_type,type,
binintersectT_lem: $o ).
thf(funcGraphProp4_type,type,
funcGraphProp4: $o ).
thf(powerset__Cong_type,type,
powerset__Cong: $o ).
thf(complementImpComplementIntersect_type,type,
complementImpComplementIntersect: $o ).
thf(setukpairinjR2_type,type,
setukpairinjR2: $o ).
thf(singletonsuniq_type,type,
singletonsuniq: $o ).
thf(binunionT_lem_type,type,
binunionT_lem: $o ).
thf(inIntersectImpInUnion2_type,type,
inIntersectImpInUnion2: $o ).
thf(setminusILneg_type,type,
setminusILneg: $o ).
thf(complementTcontraSubset_type,type,
complementTcontraSubset: $o ).
thf(funcinfuncset_type,type,
funcinfuncset: $o ).
thf(wellorderingAx_type,type,
wellorderingAx: $o ).
thf(kpairsurjEq_type,type,
kpairsurjEq: $o ).
thf(setadjoinSub2_type,type,
setadjoinSub2: $o ).
thf(binintersectSubset5_type,type,
binintersectSubset5: $o ).
thf(brelnall1_type,type,
brelnall1: $o ).
thf(upairinpowunion_type,type,
upairinpowunion: $o ).
thf(doubleComplementEq_type,type,
doubleComplementEq: $o ).
thf(setoftrueEq_type,type,
setoftrueEq: $o ).
thf(complementT_lem_type,type,
complementT_lem: $o ).
thf(funcextLem_type,type,
funcextLem: $o ).
thf(iftrueorfalse_type,type,
iftrueorfalse: $o ).
thf(emptyset__Cong_type,type,
emptyset__Cong: $o ).
thf(funcImageSingleton_type,type,
funcImageSingleton: $o ).
thf(cartprodmempaircEq_type,type,
cartprodmempaircEq: $o ).
thf(demorgan2b2_type,type,
demorgan2b2: $o ).
thf(zip_tseitin_3_type,type,
zip_tseitin_3: $i > $i > $i > ( $i > $i > $o ) > $o ).
thf(demorgan2b_type,type,
demorgan2b: $o ).
thf(dpsetconstrI_type,type,
dpsetconstrI: $o ).
thf(ex1E1_type,type,
ex1E1: $o ).
thf(sepInPowerset_type,type,
sepInPowerset: $o ).
thf(setukpairIR_type,type,
setukpairIR: $o ).
thf(kpair_type,type,
kpair: $i > $i > $i ).
thf(binunionE_type,type,
binunionE: $o ).
thf(setext_type,type,
setext: $o ).
thf(lam2p_type,type,
lam2p: $o ).
thf(emptyinPowerset_type,type,
emptyinPowerset: $o ).
thf(breln1_type,type,
breln1: $i > $i > $o ).
thf(ubforcartprodlem1_type,type,
ubforcartprodlem1: $o ).
thf(binintersectTERcontra_type,type,
binintersectTERcontra: $o ).
thf(theeq_type,type,
theeq: $o ).
thf(descr__Cong_type,type,
descr__Cong: $o ).
thf(theprop_type,type,
theprop: $o ).
thf(complementInPowersetComplementIntersect_type,type,
complementInPowersetComplementIntersect: $o ).
thf('#l_lift13430_type',type,
'#l_lift13430': $i > $i > $i > $o ).
thf(complementTnotintersectT_type,type,
complementTnotintersectT: $o ).
thf(binintersectSubset1_type,type,
binintersectSubset1: $o ).
thf(infuncsetfunc_type,type,
infuncsetfunc: $o ).
thf(in_type,type,
in: $i > $i > $o ).
thf(lamProp_type,type,
lamProp: $o ).
thf(cartprodsndpairEq_type,type,
cartprodsndpairEq: $o ).
thf(quantDeMorgan1_type,type,
quantDeMorgan1: $o ).
thf(setextT_type,type,
setextT: $o ).
thf(setukpairinjL2_type,type,
setukpairinjL2: $o ).
thf(binunion_type,type,
binunion: $i > $i > $i ).
thf(setOfPairsIsBReln_type,type,
setOfPairsIsBReln: $o ).
thf(setadjoinIR_type,type,
setadjoinIR: $o ).
thf(funcGraphProp3_type,type,
funcGraphProp3: $o ).
thf(ap2p_type,type,
ap2p: $o ).
thf(eqbreln_type,type,
eqbreln: $o ).
thf(setadjoin__Cong_type,type,
setadjoin__Cong: $o ).
thf(ksndpairEq_type,type,
ksndpairEq: $o ).
thf(ex1I2_type,type,
ex1I2: $o ).
thf(dpsetconstrEL2_type,type,
dpsetconstrEL2: $o ).
thf(woz1_1_type,type,
woz1_1: $o ).
thf(demorgan1_type,type,
demorgan1: $o ).
thf(binintersectTELcontra_type,type,
binintersectTELcontra: $o ).
thf(symdiffI2_type,type,
symdiffI2: $o ).
thf(upairset2IR_type,type,
upairset2IR: $o ).
thf(ap_type,type,
ap: $i > $i > $i > $i > $i ).
thf(subsetE_type,type,
subsetE: $o ).
thf(prop2set2propI_type,type,
prop2set2propI: $o ).
thf(doubleComplementSub1_type,type,
doubleComplementSub1: $o ).
thf(dsetconstrEL_type,type,
dsetconstrEL: $o ).
thf(setbeta_type,type,
setbeta: $o ).
thf(binunionTE_type,type,
binunionTE: $o ).
thf(eqinunit_type,type,
eqinunit: $o ).
thf(upairsetIL_type,type,
upairsetIL: $o ).
thf(powersetI1_type,type,
powersetI1: $o ).
thf(cartprodfstin_type,type,
cartprodfstin: $o ).
thf(iftrueProp1_type,type,
iftrueProp1: $o ).
thf(powersetTE1_type,type,
powersetTE1: $o ).
thf(binintersectSubset3_type,type,
binintersectSubset3: $o ).
thf(app_type,type,
app: $o ).
thf(ksndsingleton_type,type,
ksndsingleton: $o ).
thf(setminusIRneg_type,type,
setminusIRneg: $o ).
thf(zip_tseitin_4_type,type,
zip_tseitin_4: $i > $i > ( $i > $o ) > $o ).
thf(inComplementUnionImpInComplement1_type,type,
inComplementUnionImpInComplement1: $o ).
thf(powersetAx_type,type,
powersetAx: $o ).
thf(lamp_type,type,
lamp: $o ).
thf(binintersectSubset4_type,type,
binintersectSubset4: $o ).
thf(ifSingleton_type,type,
ifSingleton: $o ).
thf(iffalse_type,type,
iffalse: $o ).
thf(singletoninpowunion_type,type,
singletoninpowunion: $o ).
thf(inIntersectImpInUnion_type,type,
inIntersectImpInUnion: $o ).
thf(powersetI_type,type,
powersetI: $o ).
thf(cartprodmempair_type,type,
cartprodmempair: $o ).
thf(upairsetIR_type,type,
upairsetIR: $o ).
thf(cartprodpairsurjEq_type,type,
cartprodpairsurjEq: $o ).
thf(cartprodpairmemEL_type,type,
cartprodpairmemEL: $o ).
thf(funcext2_type,type,
funcext2: $o ).
thf(inCongP_type,type,
inCongP: $o ).
thf(setukpairinjR11_type,type,
setukpairinjR11: $o ).
thf(setadjoinSub_type,type,
setadjoinSub: $o ).
thf(exuE3e_type,type,
exuE3e: $o ).
thf(emptysetE_type,type,
emptysetE: $o ).
thf(binintersectSubset2_type,type,
binintersectSubset2: $o ).
thf(zip_tseitin_1_type,type,
zip_tseitin_1: $i > $i > $i > $o ).
thf(breln1invprop_type,type,
breln1invprop: $o ).
thf(setunionI_type,type,
setunionI: $o ).
thf(symdiff_type,type,
symdiff: $i > $i > $i ).
thf(reflwellordering_type,type,
reflwellordering: $i > $i > $o ).
thf(setunion__Cong_type,type,
setunion__Cong: $o ).
thf(prop2setI_type,type,
prop2setI: $o ).
thf(nonemptyImpWitness_type,type,
nonemptyImpWitness: $o ).
thf(setminusER_type,type,
setminusER: $o ).
thf(exuE2_type,type,
exuE2: $o ).
thf(dpsetconstrEL1_type,type,
dpsetconstrEL1: $o ).
thf(emptyset_type,type,
emptyset: $i ).
thf(setukpairinjR12_type,type,
setukpairinjR12: $o ).
thf(binunionTILcontra_type,type,
binunionTILcontra: $o ).
thf(complementTI1_type,type,
complementTI1: $o ).
thf(inComplementUnionImpNotIn1_type,type,
inComplementUnionImpNotIn1: $o ).
thf(breln_type,type,
breln: $i > $i > $i > $o ).
thf(dpsetconstrER_type,type,
dpsetconstrER: $o ).
thf(setminus_type,type,
setminus: $i > $i > $i ).
thf(subbreln1_type,type,
subbreln1: $o ).
thf(powersetsubset_type,type,
powersetsubset: $o ).
thf(upairequniteq_type,type,
upairequniteq: $o ).
thf(woz1_4_type,type,
woz1_4: $o ).
thf(subPowSU_type,type,
subPowSU: $o ).
thf(notequalI1_type,type,
notequalI1: $o ).
thf(notinemptyset_type,type,
notinemptyset: $o ).
thf(nonemptyI1_type,type,
nonemptyI1: $o ).
thf(sk__1001_type,type,
sk__1001: $i ).
thf(subsetTrans_type,type,
subsetTrans: $o ).
thf(setukpairinjR_type,type,
setukpairinjR: $o ).
thf(kpairp_type,type,
kpairp: $o ).
thf(setukpairinjR1_type,type,
setukpairinjR1: $o ).
thf(setminusI_type,type,
setminusI: $o ).
thf(eta1_type,type,
eta1: $o ).
thf(binunionIL_type,type,
binunionIL: $o ).
thf(setukpairinjL1_type,type,
setukpairinjL1: $o ).
thf(setunionsingleton_type,type,
setunionsingleton: $o ).
thf(contrasubsetT1_type,type,
contrasubsetT1: $o ).
thf(breln1invset_type,type,
breln1invset: $i > $i > $i ).
thf(subsetE2_type,type,
subsetE2: $o ).
thf(singletonsswitch_type,type,
singletonsswitch: $o ).
thf(ubforcartprodlem3_type,type,
ubforcartprodlem3: $o ).
thf(eta2_type,type,
eta2: $o ).
thf(contrasubsetT3_type,type,
contrasubsetT3: $o ).
thf(setextsub_type,type,
setextsub: $o ).
thf(binunionLsub_type,type,
binunionLsub: $o ).
thf(setunionsingleton1_type,type,
setunionsingleton1: $o ).
thf(binintersect_type,type,
binintersect: $i > $i > $i ).
thf(emptysetsubset_type,type,
emptysetsubset: $o ).
thf(quantDeMorgan3_type,type,
quantDeMorgan3: $o ).
thf(setunionE_type,type,
setunionE: $o ).
thf(ksnd_type,type,
ksnd: $i > $i ).
thf(setminusSubset1_type,type,
setminusSubset1: $o ).
thf(funcSet_type,type,
funcSet: $i > $i > $i ).
thf(setminusT_lem_type,type,
setminusT_lem: $o ).
thf(emptyinunitempty_type,type,
emptyinunitempty: $o ).
thf(binunionIR_type,type,
binunionIR: $o ).
thf(iftrue_type,type,
iftrue: $o ).
thf(setadjoinE_type,type,
setadjoinE: $o ).
thf(doubleComplementSub2_type,type,
doubleComplementSub2: $o ).
thf(woz1_5_type,type,
woz1_5: $o ).
thf(descr_type,type,
descr: ( $i > $o ) > $i ).
thf(cartprodpairmemER_type,type,
cartprodpairmemER: $o ).
thf(ex1_type,type,
ex1: $i > ( $i > $o ) > $o ).
thf(beta2_type,type,
beta2: $o ).
thf(complementSubsetComplementIntersect_type,type,
complementSubsetComplementIntersect: $o ).
thf(zip_tseitin_2_type,type,
zip_tseitin_2: $i > $i > $i > $o ).
thf(brelnall2_type,type,
brelnall2: $o ).
thf(setadjoinIL_type,type,
setadjoinIL: $o ).
thf(exuE1_type,type,
exuE1: $o ).
thf(ubforcartprodlem2_type,type,
ubforcartprodlem2: $o ).
thf(binintersectRsub_type,type,
binintersectRsub: $o ).
thf(ifp_type,type,
ifp: $o ).
thf(setminusLsub_type,type,
setminusLsub: $o ).
thf(prop2set_type,type,
prop2set: $o > $i ).
thf(woz1_3_type,type,
woz1_3: $o ).
thf(notinsingleton_type,type,
notinsingleton: $o ).
thf(dpsetconstr_type,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(eqimpsubset2_type,type,
eqimpsubset2: $o ).
thf(exuEu_type,type,
exuEu: $o ).
thf(emptysetimpfalse_type,type,
emptysetimpfalse: $o ).
thf(upairsetE_type,type,
upairsetE: $o ).
thf(omega0Ax_type,type,
omega0Ax: $o ).
thf(complementTE1_type,type,
complementTE1: $o ).
thf(contrasubsetT2_type,type,
contrasubsetT2: $o ).
thf(notsubsetI_type,type,
notsubsetI: $o ).
thf(intersectInPowersetIntersectUnions_type,type,
intersectInPowersetIntersectUnions: $o ).
thf(sk__999_type,type,
sk__999: $i ).
thf(quantDeMorgan4_type,type,
quantDeMorgan4: $o ).
thf(setextAx_type,type,
setextAx: $o ).
thf(setOfPairsIsBReln1_type,type,
setOfPairsIsBReln1: $o ).
thf(breln1SetBreln1_type,type,
breln1SetBreln1: $o ).
thf(powersetE1_type,type,
powersetE1: $o ).
thf(bs114d_type,type,
bs114d: $o ).
thf(breln1all2_type,type,
breln1all2: $o ).
thf(woz13rule0_type,type,
woz13rule0: $o ).
thf(noeltsimpempty_type,type,
noeltsimpempty: $o ).
thf(contraSubsetComplement_type,type,
contraSubsetComplement: $o ).
thf(setunion_type,type,
setunion: $i > $i ).
thf(woz1_2_type,type,
woz1_2: $o ).
thf(iskpair_type,type,
iskpair: $i > $o ).
thf(sk__1002_type,type,
sk__1002: $i ).
thf(binintersectLsub_type,type,
binintersectLsub: $o ).
thf(subsetI1_type,type,
subsetI1: $o ).
thf(symdiffE_type,type,
symdiffE: $o ).
thf(descrp_type,type,
descrp: $o ).
thf(dsetconstr__Cong_type,type,
dsetconstr__Cong: $o ).
thf(foundationAx_type,type,
foundationAx: $o ).
thf(emptysetAx_type,type,
emptysetAx: $o ).
thf(emptyI_type,type,
emptyI: $o ).
thf(setadjoinOr_type,type,
setadjoinOr: $o ).
thf(breln1invI_type,type,
breln1invI: $o ).
thf(binintersectEL_type,type,
binintersectEL: $o ).
thf(emptyE1_type,type,
emptyE1: $o ).
thf(emptyInPowerset_type,type,
emptyInPowerset: $o ).
thf(demorgan2_type,type,
demorgan2: $o ).
thf(vacuousDall_type,type,
vacuousDall: $o ).
thf(dsetconstr_type,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(dsetconstrER_type,type,
dsetconstrER: $o ).
thf(funcGraphProp2_type,type,
funcGraphProp2: $o ).
thf(ex1E2_type,type,
ex1E2: $o ).
thf(exu_type,type,
exu: ( $i > $o ) > $o ).
thf(binintersectI_type,type,
binintersectI: $o ).
thf(woz13rule3_type,type,
woz13rule3: $o ).
thf(iffalseProp2_type,type,
iffalseProp2: $o ).
thf(woz13rule2_type,type,
woz13rule2: $o ).
thf(cartprodpairin_type,type,
cartprodpairin: $o ).
thf(lam2lamEq_type,type,
lam2lamEq: $o ).
thf(choice2fnsingleton_type,type,
choice2fnsingleton: $o ).
thf(apProp_type,type,
apProp: $o ).
thf(dpsetconstrSub_type,type,
dpsetconstrSub: $o ).
thf(cartprodmempair1_type,type,
cartprodmempair1: $o ).
thf(iffalseProp1_type,type,
iffalseProp1: $o ).
thf(demorgan2a2_type,type,
demorgan2a2: $o ).
thf(powersetT_lem_type,type,
powersetT_lem: $o ).
thf(singleton_type,type,
singleton: $i > $o ).
thf(if_type,type,
if: $i > $o > $i > $i > $i ).
thf(symdiffI1_type,type,
symdiffI1: $o ).
thf(breln1Set_type,type,
breln1Set: $i > $i ).
thf(kfstpairEq_type,type,
kfstpairEq: $o ).
thf(subsetI2_type,type,
subsetI2: $o ).
thf(zip_tseitin_5_type,type,
zip_tseitin_5: $i > ( $i > $o ) > $o ).
thf(symdiffIneg1_type,type,
symdiffIneg1: $o ).
thf(setminusERneg_type,type,
setminusERneg: $o ).
thf(symdiffIneg2_type,type,
symdiffIneg2: $o ).
thf(omega__Cong_type,type,
omega__Cong: $o ).
thf(subsetRefl_type,type,
subsetRefl: $o ).
thf(sk__1000_type,type,
sk__1000: $i ).
thf(uniqinunit_type,type,
uniqinunit: $o ).
thf(setminusSubset2_type,type,
setminusSubset2: $o ).
thf(prop2setE_type,type,
prop2setE: $o ).
thf(binunionEcases_type,type,
binunionEcases: $o ).
thf(omega_type,type,
omega: $i ).
thf(inIntersectImpInIntersectUnions_type,type,
inIntersectImpInIntersectUnions: $o ).
thf(nonemptyI_type,type,
nonemptyI: $o ).
thf(binunionTIRcontra_type,type,
binunionTIRcontra: $o ).
thf(disjointsetsI1_type,type,
disjointsetsI1: $o ).
thf(exuE3u_type,type,
exuE3u: $o ).
thf(setminusEL_type,type,
setminusEL: $o ).
thf(woz13rule4_type,type,
woz13rule4: $o ).
thf(breln1all1_type,type,
breln1all1: $o ).
thf(setunionsingleton2_type,type,
setunionsingleton2: $o ).
thf(sepSubset_type,type,
sepSubset: $o ).
thf(upairset2E_type,type,
upairset2E: $o ).
thf(eqimpsubset1_type,type,
eqimpsubset1: $o ).
thf(eqbreln1_type,type,
eqbreln1: $o ).
thf(woz13rule1_type,type,
woz13rule1: $o ).
thf(exuI3_type,type,
exuI3: $o ).
thf(notdallE_type,type,
notdallE: $o ).
thf(demorgan1a_type,type,
demorgan1a: $o ).
thf(binintersectER_type,type,
binintersectER: $o ).
thf(powerset_type,type,
powerset: $i > $i ).
thf(ap2apEq2_type,type,
ap2apEq2: $o ).
thf(kfst_type,type,
kfst: $i > $i ).
thf(dsetconstrI_type,type,
dsetconstrI: $o ).
thf(nonempty_type,type,
nonempty: $i > $o ).
thf(set2prop_type,type,
set2prop: $i > $o ).
thf(lam_type,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_type,type,
func: $i > $i > $i > $o ).
thf(subbreln_type,type,
subbreln: $o ).
thf(setadjoin_type,type,
setadjoin: $i > $i > $i ).
thf(exu__Cong_type,type,
exu__Cong: $o ).
thf(iftrueProp2_type,type,
iftrueProp2: $o ).
thf(cartprodsndin_type,type,
cartprodsndin: $o ).
thf(contrasubsetT_type,type,
contrasubsetT: $o ).
thf(powersetTI1_type,type,
powersetTI1: $o ).
thf(funcext_type,type,
funcext: $o ).
thf(quantDeMorgan2_type,type,
quantDeMorgan2: $o ).
thf(ex1I_type,type,
ex1I: $o ).
thf(secondinupair_type,type,
secondinupair: $o ).
thf(singletonprop_type,type,
singletonprop: $o ).
thf(singletonsubset_type,type,
singletonsubset: $o ).
thf(singletoninpowerset_type,type,
singletoninpowerset: $o ).
thf(subsetemptysetimpeq_type,type,
subsetemptysetimpeq: $o ).
thf(replAx_type,type,
replAx: $o ).
thf(nonemptyE1_type,type,
nonemptyE1: $o ).
thf(omegaIndAx_type,type,
omegaIndAx: $o ).
thf(setukpairIL_type,type,
setukpairIL: $o ).
thf(setunionE2_type,type,
setunionE2: $o ).
thf(inPowerset_type,type,
inPowerset: $o ).
thf(demorgan2a1_type,type,
demorgan2a1: $o ).
thf(cartprod_type,type,
cartprod: $i > $i > $i ).
thf(subset_type,type,
subset: $i > $i > $o ).
thf(complementUnionInPowersetComplement_type,type,
complementUnionInPowersetComplement: $o ).
thf(in__Cong_type,type,
in__Cong: $o ).
thf(dpsetconstrERa_type,type,
dpsetconstrERa: $o ).
thf(setukpairinjL_type,type,
setukpairinjL: $o ).
thf(binunionRsub_type,type,
binunionRsub: $o ).
thf(subset2powerset_type,type,
subset2powerset: $o ).
thf(ap2apEq1_type,type,
ap2apEq1: $o ).
thf(breln1invI,axiom,
( breln1invI
= ( ! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) ) ) ) ) ) ) ) ).
thf('0',plain,
( breln1invI
= ( ! [X4: $i,X6: $i] :
( ( breln1 @ X4 @ X6 )
=> ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ ( kpair @ X8 @ X10 ) @ X6 )
=> ( in @ ( kpair @ X10 @ X8 ) @ ( breln1invset @ X4 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(breln1invprop,axiom,
( breln1invprop
= ( ! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ( breln1 @ A @ ( breln1invset @ A @ R ) ) ) ) ) ).
thf('1',plain,
( breln1invprop
= ( ! [X4: $i,X6: $i] :
( ( breln1 @ X4 @ X6 )
=> ( breln1 @ X4 @ ( breln1invset @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(breln1invset,axiom,
( breln1invset
= ( ^ [A: $i,R: $i] :
( dpsetconstr @ A @ A
@ ^ [Xx: $i,Xy: $i] : ( in @ ( kpair @ Xy @ Xx ) @ R ) ) ) ) ).
thf('2',plain,
( breln1invset
= ( ^ [A: $i,R: $i] :
( dpsetconstr @ A @ A
@ ^ [Xx: $i,Xy: $i] : ( in @ ( kpair @ Xy @ Xx ) @ R ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[breln1invset]) ).
thf('3',plain,
( breln1invset
= ( ^ [V_1: $i,V_2: $i] :
( dpsetconstr @ V_1 @ V_1
@ ^ [V_3: $i,V_4: $i] : ( in @ ( kpair @ V_4 @ V_3 ) @ V_2 ) ) ) ),
define([status(thm)]) ).
thf(eqbreln1,axiom,
( eqbreln1
= ( ! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ S )
=> ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
=> ( R = S ) ) ) ) ) ) ) ).
thf('4',plain,
( eqbreln1
= ( ! [X4: $i,X6: $i] :
( ( breln1 @ X4 @ X6 )
=> ! [X8: $i] :
( ( breln1 @ X4 @ X8 )
=> ( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( in @ ( kpair @ X10 @ X12 ) @ X6 )
=> ( in @ ( kpair @ X10 @ X12 ) @ X8 ) ) ) )
=> ( ! [X14: $i] :
( ( in @ X14 @ X4 )
=> ! [X16: $i] :
( ( in @ X16 @ X4 )
=> ( ( in @ ( kpair @ X14 @ X16 ) @ X8 )
=> ( in @ ( kpair @ X14 @ X16 ) @ X6 ) ) ) )
=> ( X6 = X8 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(subbreln1,axiom,
( subbreln1
= ( ! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [S: $i] :
( ( breln1 @ A @ S )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
=> ( subset @ R @ S ) ) ) ) ) ) ).
thf('5',plain,
( subbreln1
= ( ! [X4: $i,X6: $i] :
( ( breln1 @ X4 @ X6 )
=> ! [X8: $i] :
( ( breln1 @ X4 @ X8 )
=> ( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( in @ ( kpair @ X10 @ X12 ) @ X6 )
=> ( in @ ( kpair @ X10 @ X12 ) @ X8 ) ) ) )
=> ( subset @ X6 @ X8 ) ) ) ) ) ),
define([status(thm)]) ).
thf(breln1all1,axiom,
( breln1all1
= ( ! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ R )
=> ( Xphi @ Xx ) ) ) ) ) ) ).
thf('6',plain,
( breln1all1
= ( ! [X4: $i,X6: $i] :
( ( breln1 @ X4 @ X6 )
=> ! [X8: $i > $o] :
( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( in @ ( kpair @ X10 @ X12 ) @ X6 )
=> ( X8 @ ( kpair @ X10 @ X12 ) ) ) ) )
=> ! [X14: $i] :
( ( in @ X14 @ X6 )
=> ( X8 @ X14 ) ) ) ) ) ),
define([status(thm)]) ).
thf(setOfPairsIsBReln1,axiom,
( setOfPairsIsBReln1
= ( ! [A: $i,Xphi: $i > $i > $o] :
( breln1 @ A
@ ( dpsetconstr @ A @ A
@ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
thf('7',plain,
( setOfPairsIsBReln1
= ( ! [X4: $i,X6: $i > $i > $o] :
( breln1 @ X4
@ ( dpsetconstr @ X4 @ X4
@ ^ [V_1: $i,V_2: $i] : ( X6 @ V_1 @ V_2 ) ) ) ) ),
define([status(thm)]) ).
thf(choice2fnsingleton,axiom,
( choice2fnsingleton
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ? [Xy: $i] :
( ( Xphi @ Xx @ Xy )
& ( in @ Xy @ B ) ) )
=> ! [R: $i] :
( ( in @ R @ ( breln1Set @ B ) )
=> ( ( reflwellordering @ B @ R )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( singleton
@ ( dsetconstr @ B
@ ^ [Xy: $i] :
( ( Xphi @ Xx @ Xy )
& ! [Xz: $i] :
( ( in @ Xz @ B )
=> ( ( Xphi @ Xx @ Xz )
=> ( in @ ( kpair @ Xy @ Xz ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ).
thf('8',plain,
( choice2fnsingleton
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o] :
( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ? [X12: $i] :
( ( X8 @ X10 @ X12 )
& ( in @ X12 @ X6 ) ) )
=> ! [X14: $i] :
( ( in @ X14 @ ( breln1Set @ X6 ) )
=> ( ( reflwellordering @ X6 @ X14 )
=> ! [X16: $i] :
( ( in @ X16 @ X4 )
=> ( singleton
@ ( dsetconstr @ X6
@ ^ [V_1: $i] :
( ( X8 @ X16 @ V_1 )
& ! [X18: $i] :
( ( in @ X18 @ X6 )
=> ( ( X8 @ X16 @ X18 )
=> ( in @ ( kpair @ V_1 @ X18 ) @ X14 ) ) ) ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(breln1SetBreln1,axiom,
( breln1SetBreln1
= ( ! [A: $i,R: $i] :
( ( in @ R @ ( breln1Set @ A ) )
=> ( breln1 @ A @ R ) ) ) ) ).
thf('9',plain,
( breln1SetBreln1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( breln1Set @ X4 ) )
=> ( breln1 @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(breln1all2,axiom,
( breln1all2
= ( ! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ R )
=> ( Xphi @ Xx ) ) ) ) ) ) ).
thf('10',plain,
( breln1all2
= ( ! [X4: $i,X6: $i] :
( ( breln1 @ X4 @ X6 )
=> ! [X8: $i > $o] :
( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( in @ ( kpair @ X10 @ X12 ) @ X6 )
=> ( X8 @ ( kpair @ X10 @ X12 ) ) ) ) )
=> ! [X14: $i] :
( ( in @ X14 @ X6 )
=> ( X8 @ X14 ) ) ) ) ) ),
define([status(thm)]) ).
thf(breln1,axiom,
( breln1
= ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ) ).
thf(breln,axiom,
( breln
= ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
thf('11',plain,
( breln
= ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[breln]) ).
thf('12',plain,
( breln
= ( ^ [V_1: $i,V_2: $i,V_3: $i] : ( subset @ V_3 @ ( cartprod @ V_1 @ V_2 ) ) ) ),
define([status(thm)]) ).
thf('13',plain,
( breln1
= ( ^ [A: $i,R: $i] : ( breln @ A @ A @ R ) ) ),
inference(simplify_rw_rule,[status(thm)],[breln1,'12']) ).
thf('14',plain,
( breln1
= ( ^ [V_1: $i,V_2: $i] : ( breln @ V_1 @ V_1 @ V_2 ) ) ),
define([status(thm)]) ).
thf(woz1_5,axiom,
( woz1_5
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
thf('15',plain,
( woz1_5
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( in @ ( setminus @ X4 @ ( binunion @ X6 @ X8 ) ) @ ( powerset @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(woz1_4,axiom,
( woz1_4
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ X @ ( setminus @ A @ Y ) )
=> ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
thf('16',plain,
( woz1_4
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ( subset @ X6 @ ( setminus @ X4 @ X8 ) )
=> ( subset @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(woz1_3,axiom,
( woz1_3
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
thf('17',plain,
( woz1_3
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ( in @ ( binintersect @ X6 @ X8 ) @ ( powerset @ ( binintersect @ ( binunion @ X6 @ X10 ) @ ( binunion @ X8 @ X10 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(woz1_2,axiom,
( woz1_2
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [W: $i] :
( ( in @ W @ ( powerset @ A ) )
=> ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
= ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
thf('18',plain,
( woz1_2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ! [X12: $i] :
( ( in @ X12 @ ( powerset @ X4 ) )
=> ( ( setminus @ X4 @ ( binintersect @ ( binunion @ X6 @ X8 ) @ ( binunion @ X10 @ X12 ) ) )
= ( binunion @ ( binintersect @ ( setminus @ X4 @ X6 ) @ ( setminus @ X4 @ X8 ) ) @ ( binintersect @ ( setminus @ X4 @ X10 ) @ ( setminus @ X4 @ X12 ) ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(woz1_1,axiom,
( woz1_1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
thf('19',plain,
( woz1_1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( in @ ( setminus @ X4 @ X6 ) @ ( powerset @ ( setminus @ X4 @ ( binintersect @ X6 @ X8 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(woz13rule4,axiom,
( woz13rule4
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [W: $i] :
( ( in @ W @ ( powerset @ A ) )
=> ( ( subset @ X @ Z )
=> ( ( subset @ Y @ W )
=> ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
thf('20',plain,
( woz13rule4
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ! [X12: $i] :
( ( in @ X12 @ ( powerset @ X4 ) )
=> ( ( subset @ X6 @ X10 )
=> ( ( subset @ X8 @ X12 )
=> ( subset @ ( binintersect @ X6 @ X8 ) @ ( binintersect @ X10 @ X12 ) ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(woz13rule3,axiom,
( woz13rule3
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( ( subset @ X @ Y )
=> ( ( subset @ X @ Z )
=> ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
thf('21',plain,
( woz13rule3
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ( ( subset @ X6 @ X8 )
=> ( ( subset @ X6 @ X10 )
=> ( subset @ X6 @ ( binintersect @ X8 @ X10 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(woz13rule2,axiom,
( woz13rule2
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( ( subset @ Y @ Z )
=> ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
thf('22',plain,
( woz13rule2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ( ( subset @ X8 @ X10 )
=> ( subset @ ( binintersect @ X6 @ X8 ) @ X10 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(woz13rule1,axiom,
( woz13rule1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( ( subset @ X @ Z )
=> ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
thf('23',plain,
( woz13rule1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ( ( subset @ X6 @ X10 )
=> ( subset @ ( binintersect @ X6 @ X8 ) @ X10 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(woz13rule0,axiom,
( woz13rule0
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ ( binintersect @ X @ Y ) )
=> ( in @ Xx @ A ) ) ) ) ) ) ).
thf('24',plain,
( woz13rule0
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( binintersect @ X6 @ X8 ) )
=> ( in @ X10 @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(demorgan2,axiom,
( demorgan2
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( setminus @ A @ ( binunion @ X @ Y ) )
= ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
thf('25',plain,
( demorgan2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ( setminus @ X4 @ ( binunion @ X6 @ X8 ) )
= ( binintersect @ ( setminus @ X4 @ X6 ) @ ( setminus @ X4 @ X8 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(demorgan2b,axiom,
( demorgan2b
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
thf('26',plain,
( demorgan2b
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( binintersect @ ( setminus @ X4 @ X6 ) @ ( setminus @ X4 @ X8 ) ) )
=> ( in @ X10 @ ( setminus @ X4 @ ( binunion @ X6 @ X8 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(demorgan2b2,axiom,
( demorgan2b2
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ X ) )
=> ( ( in @ Xx @ ( setminus @ A @ Y ) )
=> ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
thf('27',plain,
( demorgan2b2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( setminus @ X4 @ X6 ) )
=> ( ( in @ X10 @ ( setminus @ X4 @ X8 ) )
=> ( in @ X10 @ ( setminus @ X4 @ ( binunion @ X6 @ X8 ) ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(demorgan2a,axiom,
( demorgan2a
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
thf('28',plain,
( demorgan2a
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( setminus @ X4 @ ( binunion @ X6 @ X8 ) ) )
=> ( in @ X10 @ ( binintersect @ ( setminus @ X4 @ X6 ) @ ( setminus @ X4 @ X8 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(demorgan1,axiom,
( demorgan1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( setminus @ A @ ( binintersect @ X @ Y ) )
= ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
thf('29',plain,
( demorgan1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ( setminus @ X4 @ ( binintersect @ X6 @ X8 ) )
= ( binunion @ ( setminus @ X4 @ X6 ) @ ( setminus @ X4 @ X8 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(demorgan1b,axiom,
( demorgan1b
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
thf('30',plain,
( demorgan1b
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( binunion @ ( setminus @ X4 @ X6 ) @ ( setminus @ X4 @ X8 ) ) )
=> ( in @ X10 @ ( setminus @ X4 @ ( binintersect @ X6 @ X8 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(demorgan1a,axiom,
( demorgan1a
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
=> ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
thf('31',plain,
( demorgan1a
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( setminus @ X4 @ ( binintersect @ X6 @ X8 ) ) )
=> ( in @ X10 @ ( binunion @ ( setminus @ X4 @ X6 ) @ ( setminus @ X4 @ X8 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(demorgan2a2,axiom,
( demorgan2a2
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
thf('32',plain,
( demorgan2a2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( setminus @ X4 @ ( binunion @ X6 @ X8 ) ) )
=> ( in @ X10 @ ( setminus @ X4 @ X8 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(complementUnionInPowersetComplement,axiom,
( complementUnionInPowersetComplement
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
thf('33',plain,
( complementUnionInPowersetComplement
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( in @ ( setminus @ X4 @ ( binunion @ X6 @ X8 ) ) @ ( powerset @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(demorgan2a1,axiom,
( demorgan2a1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
thf('34',plain,
( demorgan2a1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( setminus @ X4 @ ( binunion @ X6 @ X8 ) ) )
=> ( in @ X10 @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(binunionTEcontra,axiom,
( binunionTEcontra
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ X )
=> ( ~ ( in @ Xx @ Y )
=> ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
thf('35',plain,
( binunionTEcontra
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ~ ( in @ X10 @ X6 )
=> ( ~ ( in @ X10 @ X8 )
=> ~ ( in @ X10 @ ( binunion @ X6 @ X8 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(binunionTE,axiom,
( binunionTE
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binunion @ X @ Y ) )
=> ( ( ( in @ Xx @ X )
=> Xphi )
=> ( ( ( in @ Xx @ Y )
=> Xphi )
=> Xphi ) ) ) ) ) ) ) ) ).
thf('36',plain,
( binunionTE
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $o,X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( in @ X12 @ ( binunion @ X6 @ X8 ) )
=> ( ( ( in @ X12 @ X6 )
=> X10 )
=> ( ( ( in @ X12 @ X8 )
=> X10 )
=> X10 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(inComplementUnionImpInComplement1,axiom,
( inComplementUnionImpInComplement1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
thf('37',plain,
( inComplementUnionImpInComplement1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( setminus @ X4 @ ( binunion @ X6 @ X8 ) ) )
=> ( in @ X10 @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(inComplementUnionImpNotIn1,axiom,
( inComplementUnionImpNotIn1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
=> ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
thf('38',plain,
( inComplementUnionImpNotIn1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( setminus @ X4 @ ( binunion @ X6 @ X8 ) ) )
=> ~ ( in @ X10 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(intersectInPowersetIntersectUnions,axiom,
( intersectInPowersetIntersectUnions
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
thf('39',plain,
( intersectInPowersetIntersectUnions
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ( in @ ( binintersect @ X6 @ X8 ) @ ( powerset @ ( binintersect @ ( binunion @ X6 @ X10 ) @ ( binunion @ X8 @ X10 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(inIntersectImpInIntersectUnions,axiom,
( inIntersectImpInIntersectUnions
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binintersect @ X @ Y ) )
=> ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
thf('40',plain,
( inIntersectImpInIntersectUnions
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( in @ X12 @ ( binintersect @ X6 @ X8 ) )
=> ( in @ X12 @ ( binintersect @ ( binunion @ X6 @ X10 ) @ ( binunion @ X8 @ X10 ) ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(inIntersectImpInUnion2,axiom,
( inIntersectImpInUnion2
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binintersect @ X @ Y ) )
=> ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
thf('41',plain,
( inIntersectImpInUnion2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( in @ X12 @ ( binintersect @ X6 @ X8 ) )
=> ( in @ X12 @ ( binunion @ X8 @ X10 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(inIntersectImpInUnion,axiom,
( inIntersectImpInUnion
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Z: $i] :
( ( in @ Z @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( binintersect @ X @ Y ) )
=> ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
thf('42',plain,
( inIntersectImpInUnion
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( powerset @ X4 ) )
=> ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( in @ X12 @ ( binintersect @ X6 @ X8 ) )
=> ( in @ X12 @ ( binunion @ X6 @ X10 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(binunionTIRcontra,axiom,
( binunionTIRcontra
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
=> ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
thf('43',plain,
( binunionTIRcontra
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ~ ( in @ X10 @ ( binunion @ X6 @ X8 ) )
=> ~ ( in @ X10 @ X8 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(binunionTILcontra,axiom,
( binunionTILcontra
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
=> ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
thf('44',plain,
( binunionTILcontra
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ~ ( in @ X10 @ ( binunion @ X6 @ X8 ) )
=> ~ ( in @ X10 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(complementTcontraSubset,axiom,
( complementTcontraSubset
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ X @ ( setminus @ A @ Y ) )
=> ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
thf('45',plain,
( complementTcontraSubset
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ( subset @ X6 @ ( setminus @ X4 @ X8 ) )
=> ( subset @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(contraSubsetComplement,axiom,
( contraSubsetComplement
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ X @ ( setminus @ A @ Y ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ Y )
=> ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
thf('46',plain,
( contraSubsetComplement
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ( subset @ X6 @ ( setminus @ X4 @ X8 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ X8 )
=> ( in @ X10 @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(complementInPowersetComplementIntersect,axiom,
( complementInPowersetComplementIntersect
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
thf('47',plain,
( complementInPowersetComplementIntersect
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( in @ ( setminus @ X4 @ X6 ) @ ( powerset @ ( setminus @ X4 @ ( binintersect @ X6 @ X8 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(complementSubsetComplementIntersect,axiom,
( complementSubsetComplementIntersect
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
thf('48',plain,
( complementSubsetComplementIntersect
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( subset @ ( setminus @ X4 @ X6 ) @ ( setminus @ X4 @ ( binintersect @ X6 @ X8 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(complementImpComplementIntersect,axiom,
( complementImpComplementIntersect
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ X ) )
=> ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
thf('49',plain,
( complementImpComplementIntersect
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( setminus @ X4 @ X6 ) )
=> ( in @ X10 @ ( setminus @ X4 @ ( binintersect @ X6 @ X8 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(complementTnotintersectT,axiom,
( complementTnotintersectT
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ X ) )
=> ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
thf('50',plain,
( complementTnotintersectT
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ ( setminus @ X4 @ X6 ) )
=> ~ ( in @ X10 @ ( binintersect @ X6 @ X8 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(doubleComplementEq,axiom,
( doubleComplementEq
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( X
= ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
thf('51',plain,
( doubleComplementEq
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( X6
= ( setminus @ X4 @ ( setminus @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(doubleComplementSub2,axiom,
( doubleComplementSub2
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
thf('52',plain,
( doubleComplementSub2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( subset @ ( setminus @ X4 @ ( setminus @ X4 @ X6 ) ) @ X6 ) ) ) ),
define([status(thm)]) ).
thf(doubleComplementSub1,axiom,
( doubleComplementSub1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
thf('53',plain,
( doubleComplementSub1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( subset @ X6 @ ( setminus @ X4 @ ( setminus @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(doubleComplementE1,axiom,
( doubleComplementE1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
=> ( in @ Xx @ X ) ) ) ) ) ) ).
thf('54',plain,
( doubleComplementE1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( in @ X8 @ ( setminus @ X4 @ ( setminus @ X4 @ X6 ) ) )
=> ( in @ X8 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(doubleComplementI1,axiom,
( doubleComplementI1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
thf('55',plain,
( doubleComplementI1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> ( in @ X8 @ ( setminus @ X4 @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(contrasubsetT3,axiom,
( contrasubsetT3
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
=> ( subset @ X @ Y ) ) ) ) ) ) ).
thf('56',plain,
( contrasubsetT3
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ( subset @ ( setminus @ X4 @ X8 ) @ ( setminus @ X4 @ X6 ) )
=> ( subset @ X6 @ X8 ) ) ) ) ) ),
define([status(thm)]) ).
thf(contrasubsetT2,axiom,
( contrasubsetT2
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ( subset @ X @ Y )
=> ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
thf('57',plain,
( contrasubsetT2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ( subset @ X6 @ X8 )
=> ( subset @ ( setminus @ X4 @ X8 ) @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(contrasubsetT1,axiom,
( contrasubsetT1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( subset @ X @ Y )
=> ( ~ ( in @ Xx @ Y )
=> ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
thf('58',plain,
( contrasubsetT1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( subset @ X6 @ X8 )
=> ( ~ ( in @ X10 @ X8 )
=> ~ ( in @ X10 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(contrasubsetT,axiom,
( contrasubsetT
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( subset @ X @ ( setminus @ A @ Y ) )
=> ( ( in @ Xx @ Y )
=> ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
thf('59',plain,
( contrasubsetT
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( subset @ X6 @ ( setminus @ X4 @ X8 ) )
=> ( ( in @ X10 @ X8 )
=> ~ ( in @ X10 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(binintersectTERcontra,axiom,
( binintersectTERcontra
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ Y )
=> ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
thf('60',plain,
( binintersectTERcontra
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ~ ( in @ X10 @ X8 )
=> ~ ( in @ X10 @ ( binintersect @ X6 @ X8 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(binintersectTELcontra,axiom,
( binintersectTELcontra
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ X )
=> ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
thf('61',plain,
( binintersectTELcontra
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ~ ( in @ X10 @ X6 )
=> ~ ( in @ X10 @ ( binintersect @ X6 @ X8 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(complementTE1,axiom,
( complementTE1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
=> ( in @ Xx @ X ) ) ) ) ) ) ).
thf('62',plain,
( complementTE1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) )
=> ( in @ X8 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(complementTI1,axiom,
( complementTI1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
thf('63',plain,
( complementTI1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(powersetTE1,axiom,
( powersetTE1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ X @ ( powerset @ Y ) )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
thf('64',plain,
( powersetTE1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X6 @ ( powerset @ X8 ) )
=> ( ( in @ X10 @ X6 )
=> ( in @ X10 @ X8 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(powersetTI1,axiom,
( powersetTI1
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) )
=> ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
thf('65',plain,
( powersetTI1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ X6 )
=> ( in @ X10 @ X8 ) ) )
=> ( in @ X6 @ ( powerset @ X8 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(subsetTI,axiom,
( subsetTI
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) )
=> ( subset @ X @ Y ) ) ) ) ) ) ).
thf('66',plain,
( subsetTI
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ X6 )
=> ( in @ X10 @ X8 ) ) )
=> ( subset @ X6 @ X8 ) ) ) ) ) ),
define([status(thm)]) ).
thf(setextT,axiom,
( setextT
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ X )
=> ( in @ Xx @ Y ) ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ Y )
=> ( in @ Xx @ X ) ) )
=> ( X = Y ) ) ) ) ) ) ) ).
thf('67',plain,
( setextT
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( in @ X10 @ X6 )
=> ( in @ X10 @ X8 ) ) )
=> ( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( in @ X12 @ X8 )
=> ( in @ X12 @ X6 ) ) )
=> ( X6 = X8 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(complementT_lem,axiom,
( complementT_lem
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
thf('68',plain,
( complementT_lem
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( setminus @ X4 @ X6 ) @ ( powerset @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusT_lem,axiom,
( setminusT_lem
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
thf('69',plain,
( setminusT_lem
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( in @ ( setminus @ X6 @ X8 ) @ ( powerset @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(powersetT_lem,axiom,
( powersetT_lem
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
thf('70',plain,
( powersetT_lem
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( in @ ( powerset @ X6 ) @ ( powerset @ ( powerset @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(binunionT_lem,axiom,
( binunionT_lem
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
thf('71',plain,
( binunionT_lem
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( in @ ( binunion @ X6 @ X8 ) @ ( powerset @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(binintersectT_lem,axiom,
( binintersectT_lem
= ( ! [A: $i,X: $i] :
( ( in @ X @ ( powerset @ A ) )
=> ! [Y: $i] :
( ( in @ Y @ ( powerset @ A ) )
=> ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
thf('72',plain,
( binintersectT_lem
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ! [X8: $i] :
( ( in @ X8 @ ( powerset @ X4 ) )
=> ( in @ ( binintersect @ X6 @ X8 ) @ ( powerset @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(iftrueorfalse,axiom,
( iftrueorfalse
= ( ! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
thf('73',plain,
( iftrueorfalse
= ( ! [X4: $i,X6: $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( if @ X4 @ X6 @ X8 @ X10 ) @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(iffalse,axiom,
( iffalse
= ( ! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ~ Xphi
=> ( ( if @ A @ Xphi @ Xx @ Xy )
= Xy ) ) ) ) ) ) ).
thf('74',plain,
( iffalse
= ( ! [X4: $i,X6: $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ~ X6
=> ( ( if @ X4 @ X6 @ X8 @ X10 )
= X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(iftrue,axiom,
( iftrue
= ( ! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( Xphi
=> ( ( if @ A @ Xphi @ Xx @ Xy )
= Xx ) ) ) ) ) ) ).
thf('75',plain,
( iftrue
= ( ! [X4: $i,X6: $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( X6
=> ( ( if @ X4 @ X6 @ X8 @ X10 )
= X8 ) ) ) ) ) ),
define([status(thm)]) ).
thf(theeq,axiom,
( theeq
= ( ! [X: $i] :
( ( singleton @ X )
=> ! [Xx: $i] :
( ( in @ Xx @ X )
=> ( ( setunion @ X )
= Xx ) ) ) ) ) ).
thf('76',plain,
( theeq
= ( ! [X4: $i] :
( ( singleton @ X4 )
=> ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ( ( setunion @ X4 )
= X6 ) ) ) ) ),
define([status(thm)]) ).
thf(ifp,axiom,
( ifp
= ( ! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
thf('77',plain,
( ifp
= ( ! [X4: $i,X6: $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( if @ X4 @ X6 @ X8 @ X10 ) @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(ifSingleton,axiom,
( ifSingleton
= ( ! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( singleton
@ ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
thf('78',plain,
( ifSingleton
= ( ! [X4: $i,X6: $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( singleton
@ ( dsetconstr @ X4
@ ^ [V_1: $i] :
( ( X6
& ( V_1 = X8 ) )
| ( ~ X6
& ( V_1 = X10 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(iftrueProp2,axiom,
( iftrueProp2
= ( ! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( Xphi
=> ( ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) )
= ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
thf('79',plain,
( iftrueProp2
= ( ! [X4: $i,X6: $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( X6
=> ( ( dsetconstr @ X4
@ ^ [V_1: $i] :
( ( X6
& ( V_1 = X8 ) )
| ( ~ X6
& ( V_1 = X10 ) ) ) )
= ( setadjoin @ X8 @ emptyset ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(iftrueProp1,axiom,
( iftrueProp1
= ( ! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( Xphi
=> ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
thf('80',plain,
( iftrueProp1
= ( ! [X4: $i,X6: $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( X6
=> ( in @ X8
@ ( dsetconstr @ X4
@ ^ [V_1: $i] :
( ( X6
& ( V_1 = X8 ) )
| ( ~ X6
& ( V_1 = X10 ) ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(iffalseProp2,axiom,
( iffalseProp2
= ( ! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ~ Xphi
=> ( ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) )
= ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
thf('81',plain,
( iffalseProp2
= ( ! [X4: $i,X6: $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ~ X6
=> ( ( dsetconstr @ X4
@ ^ [V_1: $i] :
( ( X6
& ( V_1 = X8 ) )
| ( ~ X6
& ( V_1 = X10 ) ) ) )
= ( setadjoin @ X10 @ emptyset ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(iffalseProp1,axiom,
( iffalseProp1
= ( ! [A: $i,Xphi: $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ~ Xphi
=> ( in @ Xy
@ ( dsetconstr @ A
@ ^ [Xz: $i] :
( ( Xphi
& ( Xz = Xx ) )
| ( ~ Xphi
& ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
thf('82',plain,
( iffalseProp1
= ( ! [X4: $i,X6: $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ~ X6
=> ( in @ X10
@ ( dsetconstr @ X4
@ ^ [V_1: $i] :
( ( X6
& ( V_1 = X8 ) )
| ( ~ X6
& ( V_1 = X10 ) ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(eta2,axiom,
( eta2
= ( ! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ( ( lam @ A @ B
@ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
= Xf ) ) ) ) ).
thf('83',plain,
( eta2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( funcSet @ X4 @ X6 ) )
=> ( ( lam @ X4 @ X6
@ ^ [V_1: $i] : ( ap @ X4 @ X6 @ X8 @ V_1 ) )
= X8 ) ) ) ),
define([status(thm)]) ).
thf(beta2,axiom,
( beta2
= ( ! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B
@ ( lam @ A @ B
@ ^ [Xy: $i] : ( Xf @ Xy ) )
@ Xx )
= ( Xf @ Xx ) ) ) ) ) ) ).
thf('84',plain,
( beta2
= ( ! [X4: $i,X6: $i,X8: $i > $i] :
( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( X8 @ X10 ) @ X6 ) )
=> ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( ap @ X4 @ X6
@ ( lam @ X4 @ X6
@ ^ [V_1: $i] : ( X8 @ V_1 ) )
@ X12 )
= ( X8 @ X12 ) ) ) ) ) ),
define([status(thm)]) ).
thf(lam2lamEq,axiom,
( lam2lamEq
= ( ! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ( ( lam @ A @ B
@ ^ [Xx: $i] : ( Xf @ Xx ) )
= ( lam @ A @ B
@ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
thf('85',plain,
( lam2lamEq
= ( ! [X4: $i,X6: $i,X8: $i > $i] :
( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( X8 @ X10 ) @ X6 ) )
=> ( ( lam @ X4 @ X6
@ ^ [V_1: $i] : ( X8 @ V_1 ) )
= ( lam @ X4 @ X6
@ ^ [V_2: $i] : ( X8 @ V_2 ) ) ) ) ) ),
define([status(thm)]) ).
thf(eta1,axiom,
( eta1
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ( ( lam @ A @ B
@ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
= Xf ) ) ) ) ).
thf('86',plain,
( eta1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ( ( lam @ X4 @ X6
@ ^ [V_1: $i] : ( ap @ X4 @ X6 @ X8 @ V_1 ) )
= X8 ) ) ) ),
define([status(thm)]) ).
thf(beta1,axiom,
( beta1
= ( ! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B
@ ( lam @ A @ B
@ ^ [Xy: $i] : ( Xf @ Xy ) )
@ Xx )
= ( Xf @ Xx ) ) ) ) ) ) ).
thf('87',plain,
( beta1
= ( ! [X4: $i,X6: $i,X8: $i > $i] :
( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( X8 @ X10 ) @ X6 ) )
=> ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( ap @ X4 @ X6
@ ( lam @ X4 @ X6
@ ^ [V_1: $i] : ( X8 @ V_1 ) )
@ X12 )
= ( X8 @ X12 ) ) ) ) ) ),
define([status(thm)]) ).
thf(ap2apEq2,axiom,
( ap2apEq2
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
thf('88',plain,
( ap2apEq2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( ap @ X4 @ X6 @ X8 @ X10 )
= ( ap @ X4 @ X6 @ X8 @ X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(ap2apEq1,axiom,
( ap2apEq1
= ( ! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
thf('89',plain,
( ap2apEq1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( funcSet @ X4 @ X6 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( ap @ X4 @ X6 @ X8 @ X10 )
= ( ap @ X4 @ X6 @ X8 @ X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(funcext2,axiom,
( funcext2
= ( ! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xg: $i] :
( ( in @ Xg @ ( funcSet @ A @ B ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xg @ Xx ) ) )
=> ( Xf = Xg ) ) ) ) ) ) ).
thf('90',plain,
( funcext2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( funcSet @ X4 @ X6 ) )
=> ! [X10: $i] :
( ( in @ X10 @ ( funcSet @ X4 @ X6 ) )
=> ( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( ap @ X4 @ X6 @ X8 @ X12 )
= ( ap @ X4 @ X6 @ X10 @ X12 ) ) )
=> ( X8 = X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(funcext,axiom,
( funcext
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xg: $i] :
( ( func @ A @ B @ Xg )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xg @ Xx ) ) )
=> ( Xf = Xg ) ) ) ) ) ) ).
thf('91',plain,
( funcext
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( func @ X4 @ X6 @ X10 )
=> ( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( ap @ X4 @ X6 @ X8 @ X12 )
= ( ap @ X4 @ X6 @ X10 @ X12 ) ) )
=> ( X8 = X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(eqbreln,axiom,
( eqbreln
= ( ! [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
=> ! [S: $i] :
( ( breln @ A @ B @ S )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ S )
=> ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
=> ( R = S ) ) ) ) ) ) ) ).
thf('92',plain,
( eqbreln
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( breln @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( breln @ X4 @ X6 @ X10 )
=> ( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ! [X14: $i] :
( ( in @ X14 @ X6 )
=> ( ( in @ ( kpair @ X12 @ X14 ) @ X8 )
=> ( in @ ( kpair @ X12 @ X14 ) @ X10 ) ) ) )
=> ( ! [X16: $i] :
( ( in @ X16 @ X4 )
=> ! [X18: $i] :
( ( in @ X18 @ X6 )
=> ( ( in @ ( kpair @ X16 @ X18 ) @ X10 )
=> ( in @ ( kpair @ X16 @ X18 ) @ X8 ) ) ) )
=> ( X8 = X10 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(subbreln,axiom,
( subbreln
= ( ! [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
=> ! [S: $i] :
( ( breln @ A @ B @ S )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
=> ( subset @ R @ S ) ) ) ) ) ) ).
thf('93',plain,
( subbreln
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( breln @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( breln @ X4 @ X6 @ X10 )
=> ( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ! [X14: $i] :
( ( in @ X14 @ X6 )
=> ( ( in @ ( kpair @ X12 @ X14 ) @ X8 )
=> ( in @ ( kpair @ X12 @ X14 ) @ X10 ) ) ) )
=> ( subset @ X8 @ X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(funcGraphProp4,axiom,
( funcGraphProp4
= ( ! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
=> ( ( ap @ A @ B @ Xf @ Xx )
= Xy ) ) ) ) ) ) ) ).
thf('94',plain,
( funcGraphProp4
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( funcSet @ X4 @ X6 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X6 )
=> ( ( in @ ( kpair @ X10 @ X12 ) @ X8 )
=> ( ( ap @ X4 @ X6 @ X8 @ X10 )
= X12 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(funcextLem,axiom,
( funcextLem
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xg: $i] :
( ( func @ A @ B @ Xg )
=> ( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ( ap @ A @ B @ Xf @ Xx )
= ( ap @ A @ B @ Xg @ Xx ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
=> ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
thf('95',plain,
( funcextLem
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( func @ X4 @ X6 @ X10 )
=> ( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ( ( ap @ X4 @ X6 @ X8 @ X12 )
= ( ap @ X4 @ X6 @ X10 @ X12 ) ) )
=> ! [X14: $i] :
( ( in @ X14 @ X4 )
=> ! [X16: $i] :
( ( in @ X16 @ X6 )
=> ( ( in @ ( kpair @ X14 @ X16 ) @ X10 )
=> ( in @ ( kpair @ X14 @ X16 ) @ X8 ) ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(funcGraphProp2,axiom,
( funcGraphProp2
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
=> ( ( ap @ A @ B @ Xf @ Xx )
= Xy ) ) ) ) ) ) ) ).
thf('96',plain,
( funcGraphProp2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X6 )
=> ( ( in @ ( kpair @ X10 @ X12 ) @ X8 )
=> ( ( ap @ X4 @ X6 @ X8 @ X10 )
= X12 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(funcGraphProp3,axiom,
( funcGraphProp3
= ( ! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
thf('97',plain,
( funcGraphProp3
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( funcSet @ X4 @ X6 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( kpair @ X10 @ ( ap @ X4 @ X6 @ X8 @ X10 ) ) @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(funcGraphProp1,axiom,
( funcGraphProp1
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
thf('98',plain,
( funcGraphProp1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( kpair @ X10 @ ( ap @ X4 @ X6 @ X8 @ X10 ) ) @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(ex1E2,axiom,
( ex1E2
= ( ! [A: $i,Xphi: $i > $o] :
( ( ex1 @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ) ) ) ) ).
thf('99',plain,
( ex1E2
= ( ! [X4: $i,X6: $i > $o] :
( ( ex1 @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) )
=> ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( X6 @ X8 )
=> ( ( X6 @ X10 )
=> ( X8 = X10 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(brelnall2,axiom,
( brelnall2
= ( ! [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
=> ! [Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ R )
=> ( Xphi @ Xx ) ) ) ) ) ) ).
thf('100',plain,
( brelnall2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( breln @ X4 @ X6 @ X8 )
=> ! [X10: $i > $o] :
( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ! [X14: $i] :
( ( in @ X14 @ X6 )
=> ( ( in @ ( kpair @ X12 @ X14 ) @ X8 )
=> ( X10 @ ( kpair @ X12 @ X14 ) ) ) ) )
=> ! [X16: $i] :
( ( in @ X16 @ X8 )
=> ( X10 @ X16 ) ) ) ) ) ),
define([status(thm)]) ).
thf(brelnall1,axiom,
( brelnall1
= ( ! [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
=> ! [Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ R )
=> ( Xphi @ Xx ) ) ) ) ) ) ).
thf('101',plain,
( brelnall1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( breln @ X4 @ X6 @ X8 )
=> ! [X10: $i > $o] :
( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ! [X14: $i] :
( ( in @ X14 @ X6 )
=> ( ( in @ ( kpair @ X12 @ X14 ) @ X8 )
=> ( X10 @ ( kpair @ X12 @ X14 ) ) ) ) )
=> ! [X16: $i] :
( ( in @ X16 @ X8 )
=> ( X10 @ X16 ) ) ) ) ) ),
define([status(thm)]) ).
thf(lam2p,axiom,
( lam2p
= ( ! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ( in
@ ( lam @ A @ B
@ ^ [Xx: $i] : ( Xf @ Xx ) )
@ ( funcSet @ A @ B ) ) ) ) ) ).
thf('102',plain,
( lam2p
= ( ! [X4: $i,X6: $i,X8: $i > $i] :
( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( X8 @ X10 ) @ X6 ) )
=> ( in
@ ( lam @ X4 @ X6
@ ^ [V_1: $i] : ( X8 @ V_1 ) )
@ ( funcSet @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(lamp,axiom,
( lamp
= ( ! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ( func @ A @ B
@ ( lam @ A @ B
@ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
thf('103',plain,
( lamp
= ( ! [X4: $i,X6: $i,X8: $i > $i] :
( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( X8 @ X10 ) @ X6 ) )
=> ( func @ X4 @ X6
@ ( lam @ X4 @ X6
@ ^ [V_1: $i] : ( X8 @ V_1 ) ) ) ) ) ),
define([status(thm)]) ).
thf(lamProp,axiom,
( lamProp
= ( ! [A: $i,B: $i,Xf: $i > $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( Xf @ Xx ) @ B ) )
=> ( func @ A @ B
@ ( dpsetconstr @ A @ B
@ ^ [Xx: $i,Xy: $i] :
( ( Xf @ Xx )
= Xy ) ) ) ) ) ) ).
thf('104',plain,
( lamProp
= ( ! [X4: $i,X6: $i,X8: $i > $i] :
( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( X8 @ X10 ) @ X6 ) )
=> ( func @ X4 @ X6
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] :
( ( X8 @ V_1 )
= V_2 ) ) ) ) ) ),
define([status(thm)]) ).
thf(funcinfuncset,axiom,
( funcinfuncset
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
thf('105',plain,
( funcinfuncset
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ( in @ X8 @ ( funcSet @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(ap2p,axiom,
( ap2p
= ( ! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
thf('106',plain,
( ap2p
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( funcSet @ X4 @ X6 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( ap @ X4 @ X6 @ X8 @ X10 ) @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(infuncsetfunc,axiom,
( infuncsetfunc
= ( ! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ( func @ A @ B @ Xf ) ) ) ) ).
thf('107',plain,
( infuncsetfunc
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( funcSet @ X4 @ X6 ) )
=> ( func @ X4 @ X6 @ X8 ) ) ) ),
define([status(thm)]) ).
thf(app,axiom,
( app
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
thf('108',plain,
( app
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( ap @ X4 @ X6 @ X8 @ X10 ) @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(apProp,axiom,
( apProp
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in
@ ( setunion
@ ( dsetconstr @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
@ B ) ) ) ) ) ).
thf('109',plain,
( apProp
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X6
@ ^ [V_1: $i] : ( in @ ( kpair @ X10 @ V_1 ) @ X8 ) ) )
@ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(funcImageSingleton,axiom,
( funcImageSingleton
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( singleton
@ ( dsetconstr @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
thf('110',plain,
( funcImageSingleton
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( singleton
@ ( dsetconstr @ X6
@ ^ [V_1: $i] : ( in @ ( kpair @ X10 @ V_1 ) @ X8 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrER,axiom,
( dpsetconstrER
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( Xphi @ Xx @ Xy ) ) ) ) ).
thf('111',plain,
( dpsetconstrER
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i,X12: $i] :
( ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) )
=> ( X8 @ X10 @ X12 ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrEL2,axiom,
( dpsetconstrEL2
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( in @ Xy @ B ) ) ) ) ).
thf('112',plain,
( dpsetconstrEL2
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i,X12: $i] :
( ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) )
=> ( in @ X12 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrEL1,axiom,
( dpsetconstrEL1
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('113',plain,
( dpsetconstrEL1
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i,X12: $i] :
( ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) )
=> ( in @ X10 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrERa,axiom,
( dpsetconstrERa
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
thf('114',plain,
( dpsetconstrERa
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X6 )
=> ( ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) )
=> ( X8 @ X10 @ X12 ) ) ) ) ) ),
define([status(thm)]) ).
thf(setOfPairsIsBReln,axiom,
( setOfPairsIsBReln
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
( breln @ A @ B
@ ( dpsetconstr @ A @ B
@ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
thf('115',plain,
( setOfPairsIsBReln
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o] :
( breln @ X4 @ X6
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrSub,axiom,
( dpsetconstrSub
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
( subset
@ ( dpsetconstr @ A @ B
@ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
@ ( cartprod @ A @ B ) ) ) ) ).
thf('116',plain,
( dpsetconstrSub
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) )
@ ( cartprod @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrI,axiom,
( dpsetconstrI
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( Xphi @ Xx @ Xy )
=> ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
thf('117',plain,
( dpsetconstrI
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X6 )
=> ( ( X8 @ X10 @ X12 )
=> ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodpairsurjEq,axiom,
( cartprodpairsurjEq
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
= Xu ) ) ) ) ).
thf('118',plain,
( cartprodpairsurjEq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ( ( kpair @ ( kfst @ X8 ) @ ( ksnd @ X8 ) )
= X8 ) ) ) ),
define([status(thm)]) ).
thf(cartprodsndpairEq,axiom,
( cartprodsndpairEq
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( ksnd @ ( kpair @ Xx @ Xy ) )
= Xy ) ) ) ) ) ).
thf('119',plain,
( cartprodsndpairEq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( ( ksnd @ ( kpair @ X8 @ X10 ) )
= X10 ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodfstpairEq,axiom,
( cartprodfstpairEq
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( kfst @ ( kpair @ Xx @ Xy ) )
= Xx ) ) ) ) ) ).
thf('120',plain,
( cartprodfstpairEq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( ( kfst @ ( kpair @ X8 @ X10 ) )
= X8 ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodmempaircEq,axiom,
( cartprodmempaircEq
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( kpair @ Xx @ Xy )
= ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
thf('121',plain,
( cartprodmempaircEq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( ( kpair @ X8 @ X10 )
= ( kpair @ X8 @ X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodpairmemER,axiom,
( cartprodpairmemER
= ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
=> ( in @ Xy @ B ) ) ) ) ).
thf('122',plain,
( cartprodpairmemER
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( in @ ( kpair @ X8 @ X10 ) @ ( cartprod @ X4 @ X6 ) )
=> ( in @ X10 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(cartprodpairmemEL,axiom,
( cartprodpairmemEL
= ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('123',plain,
( cartprodpairmemEL
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( in @ ( kpair @ X8 @ X10 ) @ ( cartprod @ X4 @ X6 ) )
=> ( in @ X8 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(cartprodsndin,axiom,
( cartprodsndin
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
thf('124',plain,
( cartprodsndin
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ( in @ ( ksnd @ X8 ) @ X6 ) ) ) ),
define([status(thm)]) ).
thf(kpairsurjEq,axiom,
( kpairsurjEq
= ( ! [Xu: $i] :
( ( iskpair @ Xu )
=> ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
= Xu ) ) ) ) ).
thf('125',plain,
( kpairsurjEq
= ( ! [X4: $i] :
( ( iskpair @ X4 )
=> ( ( kpair @ ( kfst @ X4 ) @ ( ksnd @ X4 ) )
= X4 ) ) ) ),
define([status(thm)]) ).
thf(ksndpairEq,axiom,
( ksndpairEq
= ( ! [Xx: $i,Xy: $i] :
( ( ksnd @ ( kpair @ Xx @ Xy ) )
= Xy ) ) ) ).
thf('126',plain,
( ksndpairEq
= ( ! [X4: $i,X6: $i] :
( ( ksnd @ ( kpair @ X4 @ X6 ) )
= X6 ) ) ),
define([status(thm)]) ).
thf(ksndsingleton,axiom,
( ksndsingleton
= ( ! [Xu: $i] :
( ( iskpair @ Xu )
=> ( singleton
@ ( dsetconstr @ ( setunion @ Xu )
@ ^ [Xx: $i] :
( Xu
= ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
thf('127',plain,
( ksndsingleton
= ( ! [X4: $i] :
( ( iskpair @ X4 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X4 )
@ ^ [V_1: $i] :
( X4
= ( kpair @ ( kfst @ X4 ) @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR,axiom,
( setukpairinjR
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( kpair @ Xx @ Xy )
= ( kpair @ Xz @ Xu ) )
=> ( Xy = Xu ) ) ) ) ).
thf('128',plain,
( setukpairinjR
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( kpair @ X4 @ X6 )
= ( kpair @ X8 @ X10 ) )
=> ( X6 = X10 ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR2,axiom,
( setukpairinjR2
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
=> ( Xy = Xu ) ) ) ) ).
thf('129',plain,
( setukpairinjR2
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) )
=> ( X6 = X10 ) ) ) ),
define([status(thm)]) ).
thf(upairequniteq,axiom,
( upairequniteq
= ( ! [Xx: $i,Xy: $i,Xz: $i] :
( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
= ( setadjoin @ Xz @ emptyset ) )
=> ( Xx = Xy ) ) ) ) ).
thf('130',plain,
( upairequniteq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) )
= ( setadjoin @ X8 @ emptyset ) )
=> ( X4 = X6 ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR1,axiom,
( setukpairinjR1
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
=> ( ( Xz = Xu )
=> ( Xy = Xu ) ) ) ) ) ).
thf('131',plain,
( setukpairinjR1
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X8 = X10 )
=> ( X6 = X10 ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR12,axiom,
( setukpairinjR12
= ( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
thf('132',plain,
( setukpairinjR12
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR11,axiom,
( setukpairinjR11
= ( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
= ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
thf('133',plain,
( setukpairinjR11
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjL,axiom,
( setukpairinjL
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( kpair @ Xx @ Xy )
= ( kpair @ Xz @ Xu ) )
=> ( Xx = Xz ) ) ) ) ).
thf('134',plain,
( setukpairinjL
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( kpair @ X4 @ X6 )
= ( kpair @ X8 @ X10 ) )
=> ( X4 = X8 ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjL2,axiom,
( setukpairinjL2
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
=> ( Xx = Xz ) ) ) ) ).
thf('135',plain,
( setukpairinjL2
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) )
=> ( X4 = X8 ) ) ) ),
define([status(thm)]) ).
thf(cartprodfstin,axiom,
( cartprodfstin
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
thf('136',plain,
( cartprodfstin
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ( in @ ( kfst @ X8 ) @ X4 ) ) ) ),
define([status(thm)]) ).
thf(kfstpairEq,axiom,
( kfstpairEq
= ( ! [Xx: $i,Xy: $i] :
( ( kfst @ ( kpair @ Xx @ Xy ) )
= Xx ) ) ) ).
thf('137',plain,
( kfstpairEq
= ( ! [X4: $i,X6: $i] :
( ( kfst @ ( kpair @ X4 @ X6 ) )
= X4 ) ) ),
define([status(thm)]) ).
thf(theprop,axiom,
( theprop
= ( ! [X: $i] :
( ( singleton @ X )
=> ( in @ ( setunion @ X ) @ X ) ) ) ) ).
thf('138',plain,
( theprop
= ( ! [X4: $i] :
( ( singleton @ X4 )
=> ( in @ ( setunion @ X4 ) @ X4 ) ) ) ),
define([status(thm)]) ).
thf(kfstsingleton,axiom,
( kfstsingleton
= ( ! [Xu: $i] :
( ( iskpair @ Xu )
=> ( singleton
@ ( dsetconstr @ ( setunion @ Xu )
@ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
thf('139',plain,
( kfstsingleton
= ( ! [X4: $i] :
( ( iskpair @ X4 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X4 )
@ ^ [V_1: $i] : ( in @ ( setadjoin @ V_1 @ emptyset ) @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjL1,axiom,
( setukpairinjL1
= ( ! [Xx: $i,Xy: $i,Xz: $i] :
( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
=> ( Xx = Xz ) ) ) ) ).
thf('140',plain,
( setukpairinjL1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) )
=> ( X4 = X8 ) ) ) ),
define([status(thm)]) ).
thf(singletonsuniq,axiom,
( singletonsuniq
= ( ! [Xx: $i,Xy: $i] :
( ( ( setadjoin @ Xx @ emptyset )
= ( setadjoin @ Xy @ emptyset ) )
=> ( Xx = Xy ) ) ) ) ).
thf('141',plain,
( singletonsuniq
= ( ! [X4: $i,X6: $i] :
( ( ( setadjoin @ X4 @ emptyset )
= ( setadjoin @ X6 @ emptyset ) )
=> ( X4 = X6 ) ) ) ),
define([status(thm)]) ).
thf(ex1I2,axiom,
( ex1I2
= ( ! [A: $i,Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) )
=> ( ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ( ex1 @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
thf('142',plain,
( ex1I2
= ( ! [X4: $i,X6: $i > $o] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( X6 @ X8 )
=> ( ( X6 @ X10 )
=> ( X8 = X10 ) ) ) ) )
=> ( ? [X12: $i] :
( ( X6 @ X12 )
& ( in @ X12 @ X4 ) )
=> ( ex1 @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ),
define([status(thm)]) ).
thf(ex1I,axiom,
( ex1I
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xy )
=> ( Xy = Xx ) ) )
=> ( ex1 @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
thf('143',plain,
( ex1I
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( X6 @ X8 )
=> ( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( X6 @ X10 )
=> ( X10 = X8 ) ) )
=> ( ex1 @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(ex1E1,axiom,
( ex1E1
= ( ! [A: $i,Xphi: $i > $o] :
( ( ex1 @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('144',plain,
( ex1E1
= ( ! [X4: $i,X6: $i > $o] :
( ( ex1 @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) )
=> ? [X8: $i] :
( ( X6 @ X8 )
& ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(singletonprop,axiom,
( singletonprop
= ( ! [A: $i,Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) )
=> ( ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ( singleton
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
thf('145',plain,
( singletonprop
= ( ! [X4: $i,X6: $i > $o] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( X6 @ X8 )
=> ( ( X6 @ X10 )
=> ( X8 = X10 ) ) ) ) )
=> ( ? [X12: $i] :
( ( X6 @ X12 )
& ( in @ X12 @ X4 ) )
=> ( singleton
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(setunionsingleton,axiom,
( setunionsingleton
= ( ! [Xx: $i] :
( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
= Xx ) ) ) ).
thf('146',plain,
( setunionsingleton
= ( ! [X4: $i] :
( ( setunion @ ( setadjoin @ X4 @ emptyset ) )
= X4 ) ) ),
define([status(thm)]) ).
thf(setunionsingleton2,axiom,
( setunionsingleton2
= ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
thf('147',plain,
( setunionsingleton2
= ( ! [X4: $i] : ( subset @ X4 @ ( setunion @ ( setadjoin @ X4 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(setunionsingleton1,axiom,
( setunionsingleton1
= ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
thf('148',plain,
( setunionsingleton1
= ( ! [X4: $i] : ( subset @ ( setunion @ ( setadjoin @ X4 @ emptyset ) ) @ X4 ) ) ),
define([status(thm)]) ).
thf(setunionE2,axiom,
( setunionE2
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ ( setunion @ A ) )
=> ? [X: $i] :
( ( in @ Xx @ X )
& ( in @ X @ A ) ) ) ) ) ).
thf('149',plain,
( setunionE2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( setunion @ X4 ) )
=> ? [X8: $i] :
( ( in @ X6 @ X8 )
& ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodmempair,axiom,
( cartprodmempair
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( iskpair @ Xu ) ) ) ) ).
thf('150',plain,
( cartprodmempair
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ( iskpair @ X8 ) ) ) ),
define([status(thm)]) ).
thf(cartprodmempair1,axiom,
( cartprodmempair1
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ? [Xx: $i] :
( ? [Xy: $i] :
( ( Xu
= ( kpair @ Xx @ Xy ) )
& ( in @ Xy @ B ) )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('151',plain,
( cartprodmempair1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ? [X10: $i] :
( ? [X12: $i] :
( ( X8
= ( kpair @ X10 @ X12 ) )
& ( in @ X12 @ X6 ) )
& ( in @ X10 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodpairin,axiom,
( cartprodpairin
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
thf('152',plain,
( cartprodpairin
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ ( kpair @ X8 @ X10 ) @ ( cartprod @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(ubforcartprodlem3,axiom,
( ubforcartprodlem3
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
thf('153',plain,
( ubforcartprodlem3
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ ( kpair @ X8 @ X10 ) @ ( powerset @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(ubforcartprodlem2,axiom,
( ubforcartprodlem2
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
thf('154',plain,
( ubforcartprodlem2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(ubforcartprodlem1,axiom,
( ubforcartprodlem1
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
thf('155',plain,
( ubforcartprodlem1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(upairinpowunion,axiom,
( upairinpowunion
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
thf('156',plain,
( upairinpowunion
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(upairsubunion,axiom,
( upairsubunion
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
thf('157',plain,
( upairsubunion
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( subset @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ ( binunion @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(upairset2E,axiom,
( upairset2E
= ( ! [Xx: $i,Xy: $i,Xz: $i] :
( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
=> ( ( Xz = Xx )
| ( Xz = Xy ) ) ) ) ) ).
thf('158',plain,
( upairset2E
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) )
=> ( ( X8 = X4 )
| ( X8 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(singletoninpowunion,axiom,
( singletoninpowunion
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
thf('159',plain,
( singletoninpowunion
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ ( setadjoin @ X8 @ emptyset ) @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(singletoninpowerset,axiom,
( singletoninpowerset
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
thf('160',plain,
( singletoninpowerset
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ X4 )
=> ( in @ ( setadjoin @ X6 @ emptyset ) @ ( powerset @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(singletonsubset,axiom,
( singletonsubset
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
thf('161',plain,
( singletonsubset
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ X4 )
=> ( subset @ ( setadjoin @ X6 @ emptyset ) @ X4 ) ) ) ),
define([status(thm)]) ).
thf(kpairp,axiom,
( kpairp
= ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
thf('162',plain,
( kpairp
= ( ! [X4: $i,X6: $i] : ( iskpair @ ( kpair @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(kpairiskpair,axiom,
( kpairiskpair
= ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
thf('163',plain,
( kpairiskpair
= ( ! [X4: $i,X6: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairIR,axiom,
( setukpairIR
= ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf('164',plain,
( setukpairIR
= ( ! [X4: $i,X6: $i] : ( in @ X6 @ ( setunion @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairIL,axiom,
( setukpairIL
= ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf('165',plain,
( setukpairIL
= ( ! [X4: $i,X6: $i] : ( in @ X4 @ ( setunion @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) ) ) ),
define([status(thm)]) ).
thf(secondinupair,axiom,
( secondinupair
= ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('166',plain,
( secondinupair
= ( ! [X4: $i,X6: $i] : ( in @ X6 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffIneg2,axiom,
( symdiffIneg2
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf('167',plain,
( symdiffIneg2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffIneg1,axiom,
( symdiffIneg1
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf('168',plain,
( symdiffIneg1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffI2,axiom,
( symdiffI2
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf('169',plain,
( symdiffI2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffI1,axiom,
( symdiffI1
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf('170',plain,
( symdiffI1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffE,axiom,
( symdiffE
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( symdiff @ A @ B ) )
=> ! [Xphi: $o] :
( ( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> Xphi ) )
=> ( ( ~ ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> Xphi ) )
=> Xphi ) ) ) ) ) ).
thf('171',plain,
( symdiffE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( symdiff @ X4 @ X6 ) )
=> ! [X10: $o] :
( ( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> X10 ) )
=> ( ( ~ ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> X10 ) )
=> X10 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusSubset1,axiom,
( setminusSubset1
= ( ! [A: $i,B: $i] :
( ( ( setminus @ A @ B )
= emptyset )
=> ( subset @ A @ B ) ) ) ) ).
thf('172',plain,
( setminusSubset1
= ( ! [X4: $i,X6: $i] :
( ( ( setminus @ X4 @ X6 )
= emptyset )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(setminusLsub,axiom,
( setminusLsub
= ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
thf('173',plain,
( setminusLsub
= ( ! [X4: $i,X6: $i] : ( subset @ ( setminus @ X4 @ X6 ) @ X4 ) ) ),
define([status(thm)]) ).
thf(setminusIRneg,axiom,
( setminusIRneg
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ B )
=> ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
thf('174',plain,
( setminusIRneg
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusILneg,axiom,
( setminusILneg
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ A )
=> ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
thf('175',plain,
( setminusILneg
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ X4 )
=> ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusELneg,axiom,
( setminusELneg
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ ( setminus @ A @ B ) )
=> ( ~ ( in @ Xx @ B )
=> ~ ( in @ Xx @ A ) ) ) ) ) ).
thf('176',plain,
( setminusELneg
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) )
=> ( ~ ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusERneg,axiom,
( setminusERneg
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ ( setminus @ A @ B ) )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ B ) ) ) ) ) ).
thf('177',plain,
( setminusERneg
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) )
=> ( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusSubset2,axiom,
( setminusSubset2
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( setminus @ A @ B )
= emptyset ) ) ) ) ).
thf('178',plain,
( setminusSubset2
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( ( setminus @ X4 @ X6 )
= emptyset ) ) ) ),
define([status(thm)]) ).
thf(setminusER,axiom,
( setminusER
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( setminus @ A @ B ) )
=> ~ ( in @ Xx @ B ) ) ) ) ).
thf('179',plain,
( setminusER
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setminus @ X4 @ X6 ) )
=> ~ ( in @ X8 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(setminusEL,axiom,
( setminusEL
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( setminus @ A @ B ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('180',plain,
( setminusEL
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setminus @ X4 @ X6 ) )
=> ( in @ X8 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(setminusI,axiom,
( setminusI
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
thf('181',plain,
( setminusI
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(bs114d,axiom,
( bs114d
= ( ! [A: $i,B: $i,C: $i] :
( ( binintersect @ A @ ( binunion @ B @ C ) )
= ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
thf('182',plain,
( bs114d
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( binintersect @ X4 @ ( binunion @ X6 @ X8 ) )
= ( binunion @ ( binintersect @ X4 @ X6 ) @ ( binintersect @ X4 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset1,axiom,
( binintersectSubset1
= ( ! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= A )
=> ( subset @ A @ B ) ) ) ) ).
thf('183',plain,
( binintersectSubset1
= ( ! [X4: $i,X6: $i] :
( ( ( binintersect @ X4 @ X6 )
= X4 )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset4,axiom,
( binintersectSubset4
= ( ! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( ( binintersect @ A @ B )
= B ) ) ) ) ).
thf('184',plain,
( binintersectSubset4
= ( ! [X4: $i,X6: $i] :
( ( subset @ X6 @ X4 )
=> ( ( binintersect @ X4 @ X6 )
= X6 ) ) ) ),
define([status(thm)]) ).
thf(binintersectRsub,axiom,
( binintersectRsub
= ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
thf('185',plain,
( binintersectRsub
= ( ! [X4: $i,X6: $i] : ( subset @ ( binintersect @ X4 @ X6 ) @ X6 ) ) ),
define([status(thm)]) ).
thf(disjointsetsI1,axiom,
( disjointsetsI1
= ( ! [A: $i,B: $i] :
( ~ ? [Xx: $i] :
( ( in @ Xx @ B )
& ( in @ Xx @ A ) )
=> ( ( binintersect @ A @ B )
= emptyset ) ) ) ) ).
thf('186',plain,
( disjointsetsI1
= ( ! [X4: $i,X6: $i] :
( ~ ? [X8: $i] :
( ( in @ X8 @ X6 )
& ( in @ X8 @ X4 ) )
=> ( ( binintersect @ X4 @ X6 )
= emptyset ) ) ) ),
define([status(thm)]) ).
thf(binintersectER,axiom,
( binintersectER
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( binintersect @ A @ B ) )
=> ( in @ Xx @ B ) ) ) ) ).
thf('187',plain,
( binintersectER
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( binintersect @ X4 @ X6 ) )
=> ( in @ X8 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset3,axiom,
( binintersectSubset3
= ( ! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= B )
=> ( subset @ B @ A ) ) ) ) ).
thf('188',plain,
( binintersectSubset3
= ( ! [X4: $i,X6: $i] :
( ( ( binintersect @ X4 @ X6 )
= X6 )
=> ( subset @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset2,axiom,
( binintersectSubset2
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( binintersect @ A @ B )
= A ) ) ) ) ).
thf('189',plain,
( binintersectSubset2
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( ( binintersect @ X4 @ X6 )
= X4 ) ) ) ),
define([status(thm)]) ).
thf(binintersectLsub,axiom,
( binintersectLsub
= ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
thf('190',plain,
( binintersectLsub
= ( ! [X4: $i,X6: $i] : ( subset @ ( binintersect @ X4 @ X6 ) @ X4 ) ) ),
define([status(thm)]) ).
thf(binintersectEL,axiom,
( binintersectEL
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( binintersect @ A @ B ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('191',plain,
( binintersectEL
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( binintersect @ X4 @ X6 ) )
=> ( in @ X8 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset5,axiom,
( binintersectSubset5
= ( ! [A: $i,B: $i,C: $i] :
( ( subset @ C @ A )
=> ( ( subset @ C @ B )
=> ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
thf('192',plain,
( binintersectSubset5
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X8 @ X4 )
=> ( ( subset @ X8 @ X6 )
=> ( subset @ X8 @ ( binintersect @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(binintersectI,axiom,
( binintersectI
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
thf('193',plain,
( binintersectI
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> ( in @ X8 @ ( binintersect @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(binunionRsub,axiom,
( binunionRsub
= ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
thf('194',plain,
( binunionRsub
= ( ! [X4: $i,X6: $i] : ( subset @ X6 @ ( binunion @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(binunionLsub,axiom,
( binunionLsub
= ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
thf('195',plain,
( binunionLsub
= ( ! [X4: $i,X6: $i] : ( subset @ X4 @ ( binunion @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(binunionE,axiom,
( binunionE
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( binunion @ A @ B ) )
=> ( ( in @ Xx @ A )
| ( in @ Xx @ B ) ) ) ) ) ).
thf('196',plain,
( binunionE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( binunion @ X4 @ X6 ) )
=> ( ( in @ X8 @ X4 )
| ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(binunionEcases,axiom,
( binunionEcases
= ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
( ( in @ Xx @ ( binunion @ A @ B ) )
=> ( ( ( in @ Xx @ A )
=> Xphi )
=> ( ( ( in @ Xx @ B )
=> Xphi )
=> Xphi ) ) ) ) ) ).
thf('197',plain,
( binunionEcases
= ( ! [X4: $i,X6: $i,X8: $i,X10: $o] :
( ( in @ X8 @ ( binunion @ X4 @ X6 ) )
=> ( ( ( in @ X8 @ X4 )
=> X10 )
=> ( ( ( in @ X8 @ X6 )
=> X10 )
=> X10 ) ) ) ) ),
define([status(thm)]) ).
thf(binunionIR,axiom,
( binunionIR
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
thf('198',plain,
( binunionIR
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X6 )
=> ( in @ X8 @ ( binunion @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(upairset2IR,axiom,
( upairset2IR
= ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('199',plain,
( upairset2IR
= ( ! [X4: $i,X6: $i] : ( in @ X6 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(binunionIL,axiom,
( binunionIL
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
thf('200',plain,
( binunionIL
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ X8 @ ( binunion @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(sepSubset,axiom,
( sepSubset
= ( ! [A: $i,Xphi: $i > $o] :
( subset
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
@ A ) ) ) ).
thf('201',plain,
( sepSubset
= ( ! [X4: $i,X6: $i > $o] :
( subset
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) )
@ X4 ) ) ),
define([status(thm)]) ).
thf(sepInPowerset,axiom,
( sepInPowerset
= ( ! [A: $i,Xphi: $i > $o] :
( in
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
@ ( powerset @ A ) ) ) ) ).
thf('202',plain,
( sepInPowerset
= ( ! [X4: $i,X6: $i > $o] :
( in
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) )
@ ( powerset @ X4 ) ) ) ),
define([status(thm)]) ).
thf(powersetsubset,axiom,
( powersetsubset
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
thf('203',plain,
( powersetsubset
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( subset @ ( powerset @ X4 ) @ ( powerset @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(inPowerset,axiom,
( inPowerset
= ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
thf('204',plain,
( inPowerset
= ( ! [X4: $i] : ( in @ X4 @ ( powerset @ X4 ) ) ) ),
define([status(thm)]) ).
thf(powersetE1,axiom,
( powersetE1
= ( ! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( subset @ B @ A ) ) ) ) ).
thf('205',plain,
( powersetE1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( subset @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(powersetI1,axiom,
( powersetI1
= ( ! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( in @ B @ ( powerset @ A ) ) ) ) ) ).
thf('206',plain,
( powersetI1
= ( ! [X4: $i,X6: $i] :
( ( subset @ X6 @ X4 )
=> ( in @ X6 @ ( powerset @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(subsetemptysetimpeq,axiom,
( subsetemptysetimpeq
= ( ! [A: $i] :
( ( subset @ A @ emptyset )
=> ( A = emptyset ) ) ) ) ).
thf('207',plain,
( subsetemptysetimpeq
= ( ! [X4: $i] :
( ( subset @ X4 @ emptyset )
=> ( X4 = emptyset ) ) ) ),
define([status(thm)]) ).
thf(setextsub,axiom,
( setextsub
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ A )
=> ( A = B ) ) ) ) ) ).
thf('208',plain,
( setextsub
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( ( subset @ X6 @ X4 )
=> ( X4 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(subset2powerset,axiom,
( subset2powerset
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( in @ A @ ( powerset @ B ) ) ) ) ) ).
thf('209',plain,
( subset2powerset
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( in @ X4 @ ( powerset @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinSub2,axiom,
( setadjoinSub2
= ( ! [A: $i,Xx: $i,B: $i] :
( ( subset @ A @ B )
=> ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
thf('210',plain,
( setadjoinSub2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X4 @ X8 )
=> ( subset @ X4 @ ( setadjoin @ X6 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinSub,axiom,
( setadjoinSub
= ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
thf('211',plain,
( setadjoinSub
= ( ! [X4: $i,X6: $i] : ( subset @ X6 @ ( setadjoin @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(subsetTrans,axiom,
( subsetTrans
= ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ C )
=> ( subset @ A @ C ) ) ) ) ) ).
thf('212',plain,
( subsetTrans
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X4 @ X6 )
=> ( ( subset @ X6 @ X8 )
=> ( subset @ X4 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(subsetRefl,axiom,
( subsetRefl
= ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
thf('213',plain,
( subsetRefl
= ( ! [X4: $i] : ( subset @ X4 @ X4 ) ) ),
define([status(thm)]) ).
thf(notequalI2,axiom,
( notequalI2
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ( A != B ) ) ) ) ) ).
thf('214',plain,
( notequalI2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ( X4 != X6 ) ) ) ) ),
define([status(thm)]) ).
thf(notequalI1,axiom,
( notequalI1
= ( ! [A: $i,B: $i] :
( ~ ( subset @ A @ B )
=> ( A != B ) ) ) ) ).
thf('215',plain,
( notequalI1
= ( ! [X4: $i,X6: $i] :
( ~ ( subset @ X4 @ X6 )
=> ( X4 != X6 ) ) ) ),
define([status(thm)]) ).
thf(notsubsetI,axiom,
( notsubsetI
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ~ ( subset @ A @ B ) ) ) ) ) ).
thf('216',plain,
( notsubsetI
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ~ ( subset @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(subsetE2,axiom,
( subsetE2
= ( ! [A: $i,B: $i,Xx: $i] :
( ( subset @ A @ B )
=> ( ~ ( in @ Xx @ B )
=> ~ ( in @ Xx @ A ) ) ) ) ) ).
thf('217',plain,
( subsetE2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X4 @ X6 )
=> ( ~ ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(subsetE,axiom,
( subsetE
= ( ! [A: $i,B: $i,Xx: $i] :
( ( subset @ A @ B )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ B ) ) ) ) ) ).
thf('218',plain,
( subsetE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X4 @ X6 )
=> ( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(emptysetsubset,axiom,
( emptysetsubset
= ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
thf('219',plain,
( emptysetsubset
= ( ! [X4: $i] : ( subset @ emptyset @ X4 ) ) ),
define([status(thm)]) ).
thf(subsetI2,axiom,
( subsetI2
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( subset @ A @ B ) ) ) ) ).
thf('220',plain,
( subsetI2
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(eqimpsubset1,axiom,
( eqimpsubset1
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ A @ B ) ) ) ) ).
thf('221',plain,
( eqimpsubset1
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(eqimpsubset2,axiom,
( eqimpsubset2
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ B @ A ) ) ) ) ).
thf('222',plain,
( eqimpsubset2
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( subset @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(subsetI1,axiom,
( subsetI1
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( subset @ A @ B ) ) ) ) ).
thf('223',plain,
( subsetI1
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(dsetconstr__Cong,axiom,
( dsetconstr__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [Xphi: $i > $o,Xpsi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( Xx = Xy )
=> ( ( Xphi @ Xx )
<=> ( Xpsi @ Xy ) ) ) ) )
=> ( ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
= ( dsetconstr @ B
@ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
thf('224',plain,
( dsetconstr__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ! [X8: $i > $o,X10: $i > $o] :
( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ! [X14: $i] :
( ( in @ X14 @ X6 )
=> ( ( X12 = X14 )
=> ( ( X8 @ X12 )
<=> ( X10 @ X14 ) ) ) ) )
=> ( ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X8 @ V_1 ) )
= ( dsetconstr @ X6
@ ^ [V_2: $i] : ( X10 @ V_2 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(descr__Cong,axiom,
( descr__Cong
= ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( Xphi @ Xx )
<=> ( Xpsi @ Xy ) ) )
=> ( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ( ( exu
@ ^ [Xx: $i] : ( Xpsi @ Xx ) )
=> ( ( descr
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
= ( descr
@ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
thf('225',plain,
( descr__Cong
= ( ! [X4: $i > $o,X6: $i > $o] :
( ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( X4 @ X8 )
<=> ( X6 @ X10 ) ) )
=> ( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ( ( exu
@ ^ [V_2: $i] : ( X6 @ V_2 ) )
=> ( ( descr
@ ^ [V_3: $i] : ( X4 @ V_3 ) )
= ( descr
@ ^ [V_4: $i] : ( X6 @ V_4 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(exuEu,axiom,
( exuEu
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ! [Xx: $i,Xy: $i] :
( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ) ) ).
thf('226',plain,
( exuEu
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ! [X6: $i,X8: $i] :
( ( X4 @ X6 )
=> ( ( X4 @ X8 )
=> ( X6 = X8 ) ) ) ) ) ),
define([status(thm)]) ).
thf(omega__Cong,axiom,
( omega__Cong
= ( omega = omega ) ) ).
thf('227',plain,
( omega__Cong
= ( omega = omega ) ),
define([status(thm)]) ).
thf(setunion__Cong,axiom,
( setunion__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( setunion @ A )
= ( setunion @ B ) ) ) ) ) ).
thf('228',plain,
( setunion__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( ( setunion @ X4 )
= ( setunion @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(powerset__Cong,axiom,
( powerset__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( powerset @ A )
= ( powerset @ B ) ) ) ) ) ).
thf('229',plain,
( powerset__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( ( powerset @ X4 )
= ( powerset @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoin__Cong,axiom,
( setadjoin__Cong
= ( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ! [Xz: $i,Xu: $i] :
( ( Xz = Xu )
=> ( ( setadjoin @ Xx @ Xz )
= ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
thf('230',plain,
( setadjoin__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( setadjoin @ X4 @ X8 )
= ( setadjoin @ X6 @ X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(emptyset__Cong,axiom,
( emptyset__Cong
= ( emptyset = emptyset ) ) ).
thf('231',plain,
( emptyset__Cong
= ( emptyset = emptyset ) ),
define([status(thm)]) ).
thf(exu__Cong,axiom,
( exu__Cong
= ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( Xphi @ Xx )
<=> ( Xpsi @ Xy ) ) )
=> ( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
<=> ( exu
@ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
thf('232',plain,
( exu__Cong
= ( ! [X4: $i > $o,X6: $i > $o] :
( ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( X4 @ X8 )
<=> ( X6 @ X10 ) ) )
=> ( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
<=> ( exu
@ ^ [V_2: $i] : ( X6 @ V_2 ) ) ) ) ) ),
define([status(thm)]) ).
thf(exuE3u,axiom,
( exuE3u
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ! [Xx: $i,Xy: $i] :
( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ) ) ).
thf('233',plain,
( exuE3u
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ! [X6: $i,X8: $i] :
( ( X4 @ X6 )
=> ( ( X4 @ X8 )
=> ( X6 = X8 ) ) ) ) ) ),
define([status(thm)]) ).
thf(in__Cong,axiom,
( in__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( in @ Xx @ A )
<=> ( in @ Xy @ B ) ) ) ) ) ) ).
thf('234',plain,
( in__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( in @ X8 @ X4 )
<=> ( in @ X10 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(inCongP,axiom,
( inCongP
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( in @ Xx @ A )
=> ( in @ Xy @ B ) ) ) ) ) ) ).
thf('235',plain,
( inCongP
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( in @ X8 @ X4 )
=> ( in @ X10 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(exuI2,axiom,
( exuI2
= ( ! [Xphi: $i > $o] :
( ? [Xx: $i] :
! [Xy: $i] :
( ( Xphi @ Xy )
<=> ( Xy = Xx ) )
=> ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
thf('236',plain,
( exuI2
= ( ! [X4: $i > $o] :
( ? [X6: $i] :
! [X8: $i] :
( ( X4 @ X8 )
<=> ( X8 = X6 ) )
=> ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) ) ) ) ),
define([status(thm)]) ).
thf(exuI3,axiom,
( exuI3
= ( ! [Xphi: $i > $o] :
( ? [Xx: $i] : ( Xphi @ Xx )
=> ( ! [Xx: $i,Xy: $i] :
( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) )
=> ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
thf('237',plain,
( exuI3
= ( ! [X4: $i > $o] :
( ? [X6: $i] : ( X4 @ X6 )
=> ( ! [X8: $i,X10: $i] :
( ( X4 @ X8 )
=> ( ( X4 @ X10 )
=> ( X8 = X10 ) ) )
=> ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) ) ) ) ) ),
define([status(thm)]) ).
thf(exuI1,axiom,
( exuI1
= ( ! [Xphi: $i > $o] :
( ? [Xx: $i] :
( ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) )
& ( Xphi @ Xx ) )
=> ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
thf('238',plain,
( exuI1
= ( ! [X4: $i > $o] :
( ? [X6: $i] :
( ! [X8: $i] :
( ( X4 @ X8 )
=> ( X6 = X8 ) )
& ( X4 @ X6 ) )
=> ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) ) ) ) ),
define([status(thm)]) ).
thf(notdallE,axiom,
( notdallE
= ( ! [A: $i,Xphi: $i > $o] :
( ~ ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ~ ( Xphi @ Xx )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('239',plain,
( notdallE
= ( ! [X4: $i,X6: $i > $o] :
( ~ ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( X6 @ X8 ) )
=> ? [X10: $i] :
( ~ ( X6 @ X10 )
& ( in @ X10 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(notdexE,axiom,
( notdexE
= ( ! [A: $i,Xphi: $i > $o] :
( ~ ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ~ ( Xphi @ Xx ) ) ) ) ) ).
thf('240',plain,
( notdexE
= ( ! [X4: $i,X6: $i > $o] :
( ~ ? [X8: $i] :
( ( X6 @ X8 )
& ( in @ X8 @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ~ ( X6 @ X10 ) ) ) ) ),
define([status(thm)]) ).
thf(prop2set2propI,axiom,
( prop2set2propI
= ( ! [Xphi: $o] :
( Xphi
=> ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
thf('241',plain,
( prop2set2propI
= ( ! [X4: $o] :
( X4
=> ( set2prop @ ( prop2set @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(prop2setI,axiom,
( prop2setI
= ( ! [Xphi: $o] :
( Xphi
=> ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
thf('242',plain,
( prop2setI
= ( ! [X4: $o] :
( X4
=> ( in @ emptyset @ ( prop2set @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(quantDeMorgan4,axiom,
( quantDeMorgan4
= ( ! [A: $i,Xphi: $i > $o] :
( ? [Xx: $i] :
( ~ ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ~ ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( Xphi @ Xx ) ) ) ) ) ).
thf('243',plain,
( quantDeMorgan4
= ( ! [X4: $i,X6: $i > $o] :
( ? [X8: $i] :
( ~ ( X6 @ X8 )
& ( in @ X8 @ X4 ) )
=> ~ ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( X6 @ X10 ) ) ) ) ),
define([status(thm)]) ).
thf(quantDeMorgan3,axiom,
( quantDeMorgan3
= ( ! [A: $i,Xphi: $i > $o] :
( ~ ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ~ ( Xphi @ Xx ) ) ) ) ) ).
thf('244',plain,
( quantDeMorgan3
= ( ! [X4: $i,X6: $i > $o] :
( ~ ? [X8: $i] :
( ( X6 @ X8 )
& ( in @ X8 @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ~ ( X6 @ X10 ) ) ) ) ),
define([status(thm)]) ).
thf(quantDeMorgan2,axiom,
( quantDeMorgan2
= ( ! [A: $i,Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ~ ( Xphi @ Xx ) )
=> ~ ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('245',plain,
( quantDeMorgan2
= ( ! [X4: $i,X6: $i > $o] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ~ ( X6 @ X8 ) )
=> ~ ? [X10: $i] :
( ( X6 @ X10 )
& ( in @ X10 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(quantDeMorgan1,axiom,
( quantDeMorgan1
= ( ! [A: $i,Xphi: $i > $o] :
( ~ ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ~ ( Xphi @ Xx )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('246',plain,
( quantDeMorgan1
= ( ! [X4: $i,X6: $i > $o] :
( ~ ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( X6 @ X8 ) )
=> ? [X10: $i] :
( ~ ( X6 @ X10 )
& ( in @ X10 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(vacuousDall,axiom,
( vacuousDall
= ( ! [Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ emptyset )
=> ( Xphi @ Xx ) ) ) ) ).
thf('247',plain,
( vacuousDall
= ( ! [X4: $i > $o,X6: $i] :
( ( in @ X6 @ emptyset )
=> ( X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(emptyE1,axiom,
( emptyE1
= ( ! [A: $i,Xphi: $i > $o] :
( ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ( ( ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
= emptyset )
=> $false ) ) ) ) ).
thf('248',plain,
( emptyE1
= ( ! [X4: $i,X6: $i > $o] :
( ? [X8: $i] :
( ( X6 @ X8 )
& ( in @ X8 @ X4 ) )
=> ( ( ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) )
= emptyset )
=> $false ) ) ) ),
define([status(thm)]) ).
thf(upairsetIR,axiom,
( upairsetIR
= ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('249',plain,
( upairsetIR
= ( ! [X4: $i,X6: $i] : ( in @ X6 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(upairsetIL,axiom,
( upairsetIL
= ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('250',plain,
( upairsetIL
= ( ! [X4: $i,X6: $i] : ( in @ X4 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(upairsetE,axiom,
( upairsetE
= ( ! [Xx: $i,Xy: $i,Xz: $i] :
( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
=> ( ( Xz = Xx )
| ( Xz = Xy ) ) ) ) ) ).
thf('251',plain,
( upairsetE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) )
=> ( ( X8 = X4 )
| ( X8 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(singletonsswitch,axiom,
( singletonsswitch
= ( ! [Xx: $i,Xy: $i] :
( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
=> ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
thf('252',plain,
( singletonsswitch
= ( ! [X4: $i,X6: $i] :
( ( in @ X4 @ ( setadjoin @ X6 @ emptyset ) )
=> ( in @ X6 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(eqinunit,axiom,
( eqinunit
= ( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('253',plain,
( eqinunit
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( in @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(notinsingleton,axiom,
( notinsingleton
= ( ! [Xx: $i,Xy: $i] :
( ( Xx != Xy )
=> ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
thf('254',plain,
( notinsingleton
= ( ! [X4: $i,X6: $i] :
( ( X4 != X6 )
=> ~ ( in @ X6 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(uniqinunit,axiom,
( uniqinunit
= ( ! [Xx: $i,Xy: $i] :
( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
=> ( Xx = Xy ) ) ) ) ).
thf('255',plain,
( uniqinunit
= ( ! [X4: $i,X6: $i] :
( ( in @ X4 @ ( setadjoin @ X6 @ emptyset ) )
=> ( X4 = X6 ) ) ) ),
define([status(thm)]) ).
thf(nonemptyImpWitness,axiom,
( nonemptyImpWitness
= ( ! [A: $i] :
( ( nonempty @ A )
=> ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
thf('256',plain,
( nonemptyImpWitness
= ( ! [X4: $i] :
( ( nonempty @ X4 )
=> ? [X6: $i] : ( in @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(exuE2,axiom,
( exuE2
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] :
! [Xy: $i] :
( ( Xphi @ Xy )
<=> ( Xy = Xx ) ) ) ) ) ).
thf('257',plain,
( exuE2
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ? [X6: $i] :
! [X8: $i] :
( ( X4 @ X8 )
<=> ( X8 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(subPowSU,axiom,
( subPowSU
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
thf('258',plain,
( subPowSU
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ X4 )
=> ( in @ X6 @ ( powerset @ ( setunion @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(setunionE,axiom,
( setunionE
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ ( setunion @ A ) )
=> ! [Xphi: $o] :
( ! [B: $i] :
( ( in @ Xx @ B )
=> ( ( in @ B @ A )
=> Xphi ) )
=> Xphi ) ) ) ) ).
thf('259',plain,
( setunionE
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( setunion @ X4 ) )
=> ! [X8: $o] :
( ! [X10: $i] :
( ( in @ X6 @ X10 )
=> ( ( in @ X10 @ X4 )
=> X8 ) )
=> X8 ) ) ) ),
define([status(thm)]) ).
thf(setunionI,axiom,
( setunionI
= ( ! [A: $i,Xx: $i,B: $i] :
( ( in @ Xx @ B )
=> ( ( in @ B @ A )
=> ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
thf('260',plain,
( setunionI
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X6 @ X8 )
=> ( ( in @ X8 @ X4 )
=> ( in @ X6 @ ( setunion @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(powersetE,axiom,
( powersetE
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( ( in @ Xx @ B )
=> ( in @ Xx @ A ) ) ) ) ) ).
thf('261',plain,
( powersetE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( ( in @ X8 @ X6 )
=> ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(emptyInPowerset,axiom,
( emptyInPowerset
= ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
thf('262',plain,
( emptyInPowerset
= ( ! [X4: $i] : ( in @ emptyset @ ( powerset @ X4 ) ) ) ),
define([status(thm)]) ).
thf(emptyinPowerset,axiom,
( emptyinPowerset
= ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
thf('263',plain,
( emptyinPowerset
= ( ! [X4: $i] : ( in @ emptyset @ ( powerset @ X4 ) ) ) ),
define([status(thm)]) ).
thf(powersetI,axiom,
( powersetI
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) )
=> ( in @ B @ ( powerset @ A ) ) ) ) ) ).
thf('264',plain,
( powersetI
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X6 )
=> ( in @ X8 @ X4 ) )
=> ( in @ X6 @ ( powerset @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(setoftrueEq,axiom,
( setoftrueEq
= ( ! [A: $i] :
( ( dsetconstr @ A
@ ^ [Xx: $i] : $true )
= A ) ) ) ).
thf('265',plain,
( setoftrueEq
= ( ! [X4: $i] :
( ( dsetconstr @ X4
@ ^ [V_1: $i] : $true )
= X4 ) ) ),
define([status(thm)]) ).
thf(setadjoinOr,axiom,
( setadjoinOr
= ( ! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
=> ( ( Xy = Xx )
| ( in @ Xy @ A ) ) ) ) ) ).
thf('266',plain,
( setadjoinOr
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ X6 ) )
=> ( ( X8 = X4 )
| ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinE,axiom,
( setadjoinE
= ( ! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
=> ! [Xphi: $o] :
( ( ( Xy = Xx )
=> Xphi )
=> ( ( ( in @ Xy @ A )
=> Xphi )
=> Xphi ) ) ) ) ) ).
thf('267',plain,
( setadjoinE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ X6 ) )
=> ! [X10: $o] :
( ( ( X8 = X4 )
=> X10 )
=> ( ( ( in @ X8 @ X6 )
=> X10 )
=> X10 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinIR,axiom,
( setadjoinIR
= ( ! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ A )
=> ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
thf('268',plain,
( setadjoinIR
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X6 )
=> ( in @ X8 @ ( setadjoin @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(emptyinunitempty,axiom,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
thf('269',plain,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ),
inference(simplify_rw_rule,[status(thm)],[emptyinunitempty]) ).
thf('270',plain,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ),
define([status(thm)]) ).
thf(setadjoinIL,axiom,
( setadjoinIL
= ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
thf('271',plain,
( setadjoinIL
= ( ! [X4: $i,X6: $i] : ( in @ X4 @ ( setadjoin @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(nonemptyI1,axiom,
( nonemptyI1
= ( ! [A: $i] :
( ? [Xx: $i] : ( in @ Xx @ A )
=> ( nonempty @ A ) ) ) ) ).
thf('272',plain,
( nonemptyI1
= ( ! [X4: $i] :
( ? [X6: $i] : ( in @ X6 @ X4 )
=> ( nonempty @ X4 ) ) ) ),
define([status(thm)]) ).
thf(nonemptyI,axiom,
( nonemptyI
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( nonempty
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
thf('273',plain,
( nonemptyI
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( X6 @ X8 )
=> ( nonempty
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(nonemptyE1,axiom,
( nonemptyE1
= ( ! [A: $i] :
( ( nonempty @ A )
=> ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
thf('274',plain,
( nonemptyE1
= ( ! [X4: $i] :
( ( nonempty @ X4 )
=> ? [X6: $i] : ( in @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(nonempty,axiom,
( nonempty
= ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
thf('275',plain,
( nonempty
= ( ^ [Xx: $i] : ( Xx != emptyset ) ) ),
inference(simplify_rw_rule,[status(thm)],[nonempty]) ).
thf('276',plain,
( nonempty
= ( ^ [V_1: $i] : ( V_1 != emptyset ) ) ),
define([status(thm)]) ).
thf(setbeta,axiom,
( setbeta
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
<=> ( Xphi @ Xx ) ) ) ) ) ).
thf('277',plain,
( setbeta
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( in @ X8
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) )
<=> ( X6 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(noeltsimpempty,axiom,
( noeltsimpempty
= ( ! [A: $i] :
( ! [Xx: $i] :
~ ( in @ Xx @ A )
=> ( A = emptyset ) ) ) ) ).
thf('278',plain,
( noeltsimpempty
= ( ! [X4: $i] :
( ! [X6: $i] :
~ ( in @ X6 @ X4 )
=> ( X4 = emptyset ) ) ) ),
define([status(thm)]) ).
thf(emptyI,axiom,
( emptyI
= ( ! [A: $i] :
( ! [Xx: $i] :
~ ( in @ Xx @ A )
=> ( A = emptyset ) ) ) ) ).
thf('279',plain,
( emptyI
= ( ! [X4: $i] :
( ! [X6: $i] :
~ ( in @ X6 @ X4 )
=> ( X4 = emptyset ) ) ) ),
define([status(thm)]) ).
thf(setext,axiom,
( setext
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) )
=> ( A = B ) ) ) ) ) ).
thf('280',plain,
( setext
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) )
=> ( ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ X10 @ X4 ) )
=> ( X4 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(exuE3e,axiom,
( exuE3e
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
thf('281',plain,
( exuE3e
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ? [X6: $i] : ( X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(notinemptyset,axiom,
( notinemptyset
= ( ! [Xx: $i] :
~ ( in @ Xx @ emptyset ) ) ) ).
thf('282',plain,
( notinemptyset
= ( ! [X4: $i] :
~ ( in @ X4 @ emptyset ) ) ),
define([status(thm)]) ).
thf(emptysetimpfalse,axiom,
( emptysetimpfalse
= ( ! [Xx: $i] :
( ( in @ Xx @ emptyset )
=> $false ) ) ) ).
thf('283',plain,
( emptysetimpfalse
= ( ! [X4: $i] :
( ( in @ X4 @ emptyset )
=> $false ) ) ),
define([status(thm)]) ).
thf(emptysetE,axiom,
( emptysetE
= ( ! [Xx: $i] :
( ( in @ Xx @ emptyset )
=> ! [Xphi: $o] : Xphi ) ) ) ).
thf('284',plain,
( emptysetE
= ( ! [X4: $i] :
( ( in @ X4 @ emptyset )
=> ! [X6: $o] : X6 ) ) ),
define([status(thm)]) ).
thf(prop2setE,axiom,
( prop2setE
= ( ! [Xphi: $o,Xx: $i] :
( ( in @ Xx @ ( prop2set @ Xphi ) )
=> Xphi ) ) ) ).
thf('285',plain,
( prop2setE
= ( ! [X4: $o,X6: $i] :
( ( in @ X6 @ ( prop2set @ X4 ) )
=> X4 ) ) ),
define([status(thm)]) ).
thf(exuE1,axiom,
( exuE1
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) )
& ( Xphi @ Xx ) ) ) ) ) ).
thf('286',plain,
( exuE1
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ? [X6: $i] :
( ! [X8: $i] :
( ( X4 @ X8 )
=> ( X6 = X8 ) )
& ( X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(dsetconstrER,axiom,
( dsetconstrER
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
=> ( Xphi @ Xx ) ) ) ) ).
thf('287',plain,
( dsetconstrER
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) )
=> ( X6 @ X8 ) ) ) ),
define([status(thm)]) ).
thf(dsetconstrEL,axiom,
( dsetconstrEL
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('288',plain,
( dsetconstrEL
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) )
=> ( in @ X8 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(dsetconstrI,axiom,
( dsetconstrI
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
thf('289',plain,
( dsetconstrI
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( X6 @ X8 )
=> ( in @ X8
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(descrp,axiom,
( descrp
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ( Xphi
@ ( descr
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
thf('290',plain,
( descrp
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ( X4
@ ( descr
@ ^ [V_2: $i] : ( X4 @ V_2 ) ) ) ) ) ),
define([status(thm)]) ).
thf(wellorderingAx,axiom,
( wellorderingAx
= ( ! [A: $i] :
? [B: $i] :
( ! [C: $i] :
( ( ! [Xx: $i] :
( ( in @ Xx @ C )
=> ( in @ Xx @ A ) )
& ? [Xx: $i] : ( in @ Xx @ C ) )
=> ? [D: $i,Xx: $i] :
( ! [E: $i] :
( ( in @ E @ B )
=> ( ! [Xy: $i] :
( ( in @ Xy @ E )
=> ( in @ Xy @ D ) )
| ( in @ Xx @ E ) ) )
& ~ ? [Xy: $i] :
( ( in @ Xy @ C )
& ( in @ Xy @ D ) )
& ( in @ Xx @ C )
& ( in @ D @ B ) ) )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ C )
=> ( in @ Xx @ D ) )
| ! [Xx: $i] :
( ( in @ Xx @ D )
=> ( in @ Xx @ C ) ) ) )
& ! [Xx: $i,Xy: $i] :
( ( ( in @ Xx @ A )
& ( in @ Xy @ A ) )
=> ( ! [C: $i] :
( ( in @ C @ B )
=> ( ( in @ Xx @ C )
<=> ( in @ Xy @ C ) ) )
=> ( Xx = Xy ) ) )
& ! [C: $i] :
( ( in @ C @ B )
=> ! [Xx: $i] :
( ( in @ Xx @ C )
=> ( in @ Xx @ A ) ) ) ) ) ) ).
thf('291',plain,
( wellorderingAx
= ( ! [X4: $i] :
? [X6: $i] :
( ! [X8: $i] :
( ( ! [X10: $i] :
( ( in @ X10 @ X8 )
=> ( in @ X10 @ X4 ) )
& ? [X12: $i] : ( in @ X12 @ X8 ) )
=> ? [X14: $i,X16: $i] :
( ! [X18: $i] :
( ( in @ X18 @ X6 )
=> ( ! [X20: $i] :
( ( in @ X20 @ X18 )
=> ( in @ X20 @ X14 ) )
| ( in @ X16 @ X18 ) ) )
& ~ ? [X22: $i] :
( ( in @ X22 @ X8 )
& ( in @ X22 @ X14 ) )
& ( in @ X16 @ X8 )
& ( in @ X14 @ X6 ) ) )
& ! [X24: $i,X26: $i] :
( ( ( in @ X24 @ X6 )
& ( in @ X26 @ X6 ) )
=> ( ! [X28: $i] :
( ( in @ X28 @ X24 )
=> ( in @ X28 @ X26 ) )
| ! [X30: $i] :
( ( in @ X30 @ X26 )
=> ( in @ X30 @ X24 ) ) ) )
& ! [X32: $i,X34: $i] :
( ( ( in @ X32 @ X4 )
& ( in @ X34 @ X4 ) )
=> ( ! [X36: $i] :
( ( in @ X36 @ X6 )
=> ( ( in @ X32 @ X36 )
<=> ( in @ X34 @ X36 ) ) )
=> ( X32 = X34 ) ) )
& ! [X38: $i] :
( ( in @ X38 @ X6 )
=> ! [X40: $i] :
( ( in @ X40 @ X38 )
=> ( in @ X40 @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(foundationAx,axiom,
( foundationAx
= ( ! [A: $i] :
( ? [Xx: $i] : ( in @ Xx @ A )
=> ? [B: $i] :
( ~ ? [Xx: $i] :
( ( in @ Xx @ A )
& ( in @ Xx @ B ) )
& ( in @ B @ A ) ) ) ) ) ).
thf('292',plain,
( foundationAx
= ( ! [X4: $i] :
( ? [X6: $i] : ( in @ X6 @ X4 )
=> ? [X8: $i] :
( ~ ? [X10: $i] :
( ( in @ X10 @ X4 )
& ( in @ X10 @ X8 ) )
& ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(replAx,axiom,
( replAx
= ( ! [Xphi: $i > $i > $o,A: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( exu
@ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
=> ? [B: $i] :
! [Xx: $i] :
( ( in @ Xx @ B )
<=> ? [Xy: $i] :
( ( Xphi @ Xy @ Xx )
& ( in @ Xy @ A ) ) ) ) ) ) ).
thf('293',plain,
( replAx
= ( ! [X4: $i > $i > $o,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X6 )
=> ( exu
@ ^ [V_1: $i] : ( X4 @ X8 @ V_1 ) ) )
=> ? [X10: $i] :
! [X12: $i] :
( ( in @ X12 @ X10 )
<=> ? [X14: $i] :
( ( X4 @ X14 @ X12 )
& ( in @ X14 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(omegaIndAx,axiom,
( omegaIndAx
= ( ! [A: $i] :
( ( ( in @ emptyset @ A )
& ! [Xx: $i] :
( ( ( in @ Xx @ omega )
& ( in @ Xx @ A ) )
=> ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ omega )
=> ( in @ Xx @ A ) ) ) ) ) ).
thf('294',plain,
( omegaIndAx
= ( ! [X4: $i] :
( ( ( in @ emptyset @ X4 )
& ! [X6: $i] :
( ( ( in @ X6 @ omega )
& ( in @ X6 @ X4 ) )
=> ( in @ ( setadjoin @ X6 @ X6 ) @ X4 ) ) )
=> ! [X8: $i] :
( ( in @ X8 @ omega )
=> ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(omegaSAx,axiom,
( omegaSAx
= ( ! [Xx: $i] :
( ( in @ Xx @ omega )
=> ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
thf('295',plain,
( omegaSAx
= ( ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( in @ ( setadjoin @ X4 @ X4 ) @ omega ) ) ) ),
define([status(thm)]) ).
thf(omega0Ax,axiom,
( omega0Ax
= ( in @ emptyset @ omega ) ) ).
thf('296',plain,
( omega0Ax
= ( in @ emptyset @ omega ) ),
inference(simplify_rw_rule,[status(thm)],[omega0Ax]) ).
thf('297',plain,
( omega0Ax
= ( in @ emptyset @ omega ) ),
define([status(thm)]) ).
thf(setunionAx,axiom,
( setunionAx
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ ( setunion @ A ) )
<=> ? [B: $i] :
( ( in @ B @ A )
& ( in @ Xx @ B ) ) ) ) ) ).
thf('298',plain,
( setunionAx
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( setunion @ X4 ) )
<=> ? [X8: $i] :
( ( in @ X8 @ X4 )
& ( in @ X6 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(powersetAx,axiom,
( powersetAx
= ( ! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
<=> ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) ) ) ) ) ).
thf('299',plain,
( powersetAx
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
<=> ! [X8: $i] :
( ( in @ X8 @ X6 )
=> ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinAx,axiom,
( setadjoinAx
= ( ! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
<=> ( ( Xy = Xx )
| ( in @ Xy @ A ) ) ) ) ) ).
thf('300',plain,
( setadjoinAx
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ X6 ) )
<=> ( ( X8 = X4 )
| ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(emptysetAx,axiom,
( emptysetAx
= ( ! [Xx: $i] :
~ ( in @ Xx @ emptyset ) ) ) ).
thf('301',plain,
( emptysetAx
= ( ! [X4: $i] :
~ ( in @ X4 @ emptyset ) ) ),
define([status(thm)]) ).
thf(setextAx,axiom,
( setextAx
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
<=> ( in @ Xx @ B ) )
=> ( A = B ) ) ) ) ).
thf('302',plain,
( setextAx
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
<=> ( in @ X8 @ X6 ) )
=> ( X4 = X6 ) ) ) ),
define([status(thm)]) ).
thf(exu,axiom,
( exu
= ( ^ [Xphi: $i > $o] :
? [Xx: $i] :
( ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) )
& ( Xphi @ Xx ) ) ) ) ).
thf('303',plain,
( exu
= ( ^ [Xphi: $i > $o] :
? [Xx: $i] :
( ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) )
& ( Xphi @ Xx ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[exu]) ).
thf('304',plain,
( exu
= ( ^ [V_1: $i > $o] :
? [X4: $i] :
( ! [X6: $i] :
( ( V_1 @ X6 )
=> ( X4 = X6 ) )
& ( V_1 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(breln1invE,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
=> ! [A: $i,R: $i] :
( ( breln1 @ A @ R )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xy @ Xx ) @ ( breln1invset @ A @ R ) )
=> ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(zf_stmt_0,conjecture,
( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
<=> ( in @ X8 @ X6 ) )
=> ( X4 = X6 ) )
=> ( ! [X10: $i] :
~ ( in @ X10 @ emptyset )
=> ( ! [X12: $i,X14: $i,X16: $i] :
( ( in @ X16 @ ( setadjoin @ X12 @ X14 ) )
<=> ( ( in @ X16 @ X14 )
| ( X16 = X12 ) ) )
=> ( ! [X18: $i,X20: $i] :
( ( in @ X20 @ ( powerset @ X18 ) )
<=> ! [X22: $i] :
( ( in @ X22 @ X20 )
=> ( in @ X22 @ X18 ) ) )
=> ( ! [X24: $i,X26: $i] :
( ( in @ X26 @ ( setunion @ X24 ) )
<=> ? [X28: $i] :
( ( in @ X26 @ X28 )
& ( in @ X28 @ X24 ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X30: $i] :
( ( in @ X30 @ omega )
=> ( in @ ( setadjoin @ X30 @ X30 ) @ omega ) )
=> ( ! [X32: $i] :
( ( ! [X34: $i] :
( ( ( in @ X34 @ X32 )
& ( in @ X34 @ omega ) )
=> ( in @ ( setadjoin @ X34 @ X34 ) @ X32 ) )
& ( in @ emptyset @ X32 ) )
=> ! [X36: $i] :
( ( in @ X36 @ omega )
=> ( in @ X36 @ X32 ) ) )
=> ( ! [X38: $i > $i > $o,X40: $i] :
( ! [X42: $i] :
( ( in @ X42 @ X40 )
=> ? [X44: $i] :
( ( X38 @ X42 @ X44 )
& ! [X46: $i] :
( ( X38 @ X42 @ X46 )
=> ( X44 = X46 ) ) ) )
=> ? [X48: $i] :
! [X50: $i] :
( ( in @ X50 @ X48 )
<=> ? [X52: $i] :
( ( in @ X52 @ X40 )
& ( X38 @ X52 @ X50 ) ) ) )
=> ( ! [X54: $i] :
( ? [X56: $i] : ( in @ X56 @ X54 )
=> ? [X58: $i] :
( ( in @ X58 @ X54 )
& ~ ? [X60: $i] :
( ( in @ X60 @ X58 )
& ( in @ X60 @ X54 ) ) ) )
=> ( ! [X62: $i] :
? [X64: $i] :
( ! [X96: $i] :
( ( in @ X96 @ X64 )
=> ! [X98: $i] :
( ( in @ X98 @ X96 )
=> ( in @ X98 @ X62 ) ) )
& ! [X90: $i,X92: $i] :
( ( ( in @ X92 @ X62 )
& ( in @ X90 @ X62 ) )
=> ( ! [X94: $i] :
( ( in @ X94 @ X64 )
=> ( ( in @ X90 @ X94 )
<=> ( in @ X92 @ X94 ) ) )
=> ( X90 = X92 ) ) )
& ! [X82: $i,X84: $i] :
( ( ( in @ X84 @ X64 )
& ( in @ X82 @ X64 ) )
=> ( ! [X88: $i] :
( ( in @ X88 @ X84 )
=> ( in @ X88 @ X82 ) )
| ! [X86: $i] :
( ( in @ X86 @ X82 )
=> ( in @ X86 @ X84 ) ) ) )
& ! [X66: $i] :
( ( ? [X70: $i] : ( in @ X70 @ X66 )
& ! [X68: $i] :
( ( in @ X68 @ X66 )
=> ( in @ X68 @ X62 ) ) )
=> ? [X72: $i,X74: $i] :
( ( in @ X72 @ X64 )
& ( in @ X74 @ X66 )
& ~ ? [X80: $i] :
( ( in @ X80 @ X72 )
& ( in @ X80 @ X66 ) )
& ! [X76: $i] :
( ( in @ X76 @ X64 )
=> ( ( in @ X74 @ X76 )
| ! [X78: $i] :
( ( in @ X78 @ X76 )
=> ( in @ X78 @ X72 ) ) ) ) ) ) )
=> ( ! [X100: $i > $o] :
( ? [X102: $i] :
( ( X100 @ X102 )
& ! [X104: $i] :
( ( X100 @ X104 )
=> ( X102 = X104 ) ) )
=> ( X100
@ ( descr
@ ^ [V_1: $i] : ( X100 @ V_1 ) ) ) )
=> ( ! [X106: $i,X108: $i > $o,X110: $i] :
( ( in @ X110 @ X106 )
=> ( ( X108 @ X110 )
=> ( in @ X110
@ ( dsetconstr @ X106
@ ^ [V_2: $i] : ( X108 @ V_2 ) ) ) ) )
=> ( ! [X112: $i,X114: $i > $o,X116: $i] :
( ( in @ X116
@ ( dsetconstr @ X112
@ ^ [V_3: $i] : ( X114 @ V_3 ) ) )
=> ( in @ X116 @ X112 ) )
=> ( ! [X118: $i,X120: $i > $o,X122: $i] :
( ( in @ X122
@ ( dsetconstr @ X118
@ ^ [V_4: $i] : ( X120 @ V_4 ) ) )
=> ( X120 @ X122 ) )
=> ( ! [X124: $i > $o] :
( ? [X126: $i] :
( ( X124 @ X126 )
& ! [X128: $i] :
( ( X124 @ X128 )
=> ( X126 = X128 ) ) )
=> ? [X130: $i] :
( ( X124 @ X130 )
& ! [X132: $i] :
( ( X124 @ X132 )
=> ( X130 = X132 ) ) ) )
=> ( ! [X134: $o,X136: $i] :
( ( in @ X136 @ ( prop2set @ X134 ) )
=> X134 )
=> ( ! [X138: $i] :
( ( in @ X138 @ emptyset )
=> ! [X140: $o] : X140 )
=> ( ! [X142: $i] :
( ( in @ X142 @ emptyset )
=> $false )
=> ( ! [X144: $i] :
~ ( in @ X144 @ emptyset )
=> ( ! [X146: $i > $o] :
( ? [X148: $i] :
( ( X146 @ X148 )
& ! [X150: $i] :
( ( X146 @ X150 )
=> ( X148 = X150 ) ) )
=> ? [X152: $i] : ( X146 @ X152 ) )
=> ( ! [X154: $i,X156: $i] :
( ! [X158: $i] :
( ( in @ X158 @ X154 )
=> ( in @ X158 @ X156 ) )
=> ( ! [X160: $i] :
( ( in @ X160 @ X156 )
=> ( in @ X160 @ X154 ) )
=> ( X154 = X156 ) ) )
=> ( ! [X162: $i] :
( ! [X164: $i] :
~ ( in @ X164 @ X162 )
=> ( X162 = emptyset ) )
=> ( ! [X166: $i] :
( ! [X168: $i] :
~ ( in @ X168 @ X166 )
=> ( X166 = emptyset ) )
=> ( ! [X170: $i,X172: $i > $o,X174: $i] :
( ( in @ X174 @ X170 )
=> ( ( in @ X174
@ ( dsetconstr @ X170
@ ^ [V_5: $i] : ( X172 @ V_5 ) ) )
<=> ( X172 @ X174 ) ) )
=> ( ! [X176: $i] :
( ( X176 != emptyset )
=> ? [X178: $i] : ( in @ X178 @ X176 ) )
=> ( ! [X180: $i,X182: $i > $o,X184: $i] :
( ( in @ X184 @ X180 )
=> ( ( X182 @ X184 )
=> ( ( dsetconstr @ X180
@ ^ [V_6: $i] : ( X182 @ V_6 ) )
!= emptyset ) ) )
=> ( ! [X186: $i] :
( ? [X188: $i] : ( in @ X188 @ X186 )
=> ( X186 != emptyset ) )
=> ( ! [X190: $i,X192: $i] : ( in @ X190 @ ( setadjoin @ X190 @ X192 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X194: $i,X196: $i,X198: $i] :
( ( in @ X198 @ X196 )
=> ( in @ X198 @ ( setadjoin @ X194 @ X196 ) ) )
=> ( ! [X200: $i,X202: $i,X204: $i] :
( ( in @ X204 @ ( setadjoin @ X200 @ X202 ) )
=> ! [X206: $o] :
( ( ( X204 = X200 )
=> X206 )
=> ( ( ( in @ X204 @ X202 )
=> X206 )
=> X206 ) ) )
=> ( ! [X208: $i,X210: $i,X212: $i] :
( ( in @ X212 @ ( setadjoin @ X208 @ X210 ) )
=> ( ( in @ X212 @ X210 )
| ( X212 = X208 ) ) )
=> ( ! [X214: $i] :
( ( dsetconstr @ X214
@ ^ [V_7: $i] : $true )
= X214 )
=> ( ! [X216: $i,X218: $i] :
( ! [X220: $i] :
( ( in @ X220 @ X218 )
=> ( in @ X220 @ X216 ) )
=> ( in @ X218 @ ( powerset @ X216 ) ) )
=> ( ! [X222: $i] : ( in @ emptyset @ ( powerset @ X222 ) )
=> ( ! [X224: $i] : ( in @ emptyset @ ( powerset @ X224 ) )
=> ( ! [X226: $i,X228: $i,X230: $i] :
( ( in @ X228 @ ( powerset @ X226 ) )
=> ( ( in @ X230 @ X228 )
=> ( in @ X230 @ X226 ) ) )
=> ( ! [X232: $i,X234: $i,X236: $i] :
( ( in @ X234 @ X236 )
=> ( ( in @ X236 @ X232 )
=> ( in @ X234 @ ( setunion @ X232 ) ) ) )
=> ( ! [X238: $i,X240: $i] :
( ( in @ X240 @ ( setunion @ X238 ) )
=> ! [X242: $o] :
( ! [X244: $i] :
( ( in @ X240 @ X244 )
=> ( ( in @ X244 @ X238 )
=> X242 ) )
=> X242 ) )
=> ( ! [X246: $i,X248: $i] :
( ( in @ X248 @ X246 )
=> ( in @ X248 @ ( powerset @ ( setunion @ X246 ) ) ) )
=> ( ! [X250: $i > $o] :
( ? [X252: $i] :
( ( X250 @ X252 )
& ! [X254: $i] :
( ( X250 @ X254 )
=> ( X252 = X254 ) ) )
=> ? [X256: $i] :
! [X258: $i] :
( ( X250 @ X258 )
<=> ( X258 = X256 ) ) )
=> ( ! [X260: $i] :
( ( X260 != emptyset )
=> ? [X262: $i] : ( in @ X262 @ X260 ) )
=> ( ! [X264: $i,X266: $i] :
( ( in @ X264 @ ( setadjoin @ X266 @ emptyset ) )
=> ( X264 = X266 ) )
=> ( ! [X268: $i,X270: $i] :
( ( X268 != X270 )
=> ~ ( in @ X270 @ ( setadjoin @ X268 @ emptyset ) ) )
=> ( ! [X272: $i,X274: $i] :
( ( X272 = X274 )
=> ( in @ X272 @ ( setadjoin @ X274 @ emptyset ) ) )
=> ( ! [X276: $i,X278: $i] :
( ( in @ X276 @ ( setadjoin @ X278 @ emptyset ) )
=> ( in @ X278 @ ( setadjoin @ X276 @ emptyset ) ) )
=> ( ! [X280: $i,X282: $i,X284: $i] :
( ( in @ X284 @ ( setadjoin @ X280 @ ( setadjoin @ X282 @ emptyset ) ) )
=> ( ( X284 = X282 )
| ( X284 = X280 ) ) )
=> ( ! [X286: $i,X288: $i] : ( in @ X286 @ ( setadjoin @ X286 @ ( setadjoin @ X288 @ emptyset ) ) )
=> ( ! [X290: $i,X292: $i] : ( in @ X292 @ ( setadjoin @ X290 @ ( setadjoin @ X292 @ emptyset ) ) )
=> ( ! [X294: $i,X296: $i > $o] :
( ? [X298: $i] :
( ( in @ X298 @ X294 )
& ( X296 @ X298 ) )
=> ( ( ( dsetconstr @ X294
@ ^ [V_8: $i] : ( X296 @ V_8 ) )
= emptyset )
=> $false ) )
=> ( ! [X300: $i > $o,X302: $i] :
( ( in @ X302 @ emptyset )
=> ( X300 @ X302 ) )
=> ( ! [X304: $i,X306: $i > $o] :
( ~ ! [X308: $i] :
( ( in @ X308 @ X304 )
=> ( X306 @ X308 ) )
=> ? [X310: $i] :
( ( in @ X310 @ X304 )
& ~ ( X306 @ X310 ) ) )
=> ( ! [X312: $i,X314: $i > $o] :
( ! [X316: $i] :
( ( in @ X316 @ X312 )
=> ~ ( X314 @ X316 ) )
=> ~ ? [X318: $i] :
( ( in @ X318 @ X312 )
& ( X314 @ X318 ) ) )
=> ( ! [X320: $i,X322: $i > $o] :
( ~ ? [X324: $i] :
( ( in @ X324 @ X320 )
& ( X322 @ X324 ) )
=> ! [X326: $i] :
( ( in @ X326 @ X320 )
=> ~ ( X322 @ X326 ) ) )
=> ( ! [X328: $i,X330: $i > $o] :
( ? [X332: $i] :
( ( in @ X332 @ X328 )
& ~ ( X330 @ X332 ) )
=> ~ ! [X334: $i] :
( ( in @ X334 @ X328 )
=> ( X330 @ X334 ) ) )
=> ( ! [X336: $o] :
( X336
=> ( in @ emptyset @ ( prop2set @ X336 ) ) )
=> ( ! [X338: $o] :
( X338
=> ( set2prop @ ( prop2set @ X338 ) ) )
=> ( ! [X340: $i,X342: $i > $o] :
( ~ ? [X344: $i] :
( ( in @ X344 @ X340 )
& ( X342 @ X344 ) )
=> ! [X346: $i] :
( ( in @ X346 @ X340 )
=> ~ ( X342 @ X346 ) ) )
=> ( ! [X348: $i,X350: $i > $o] :
( ~ ! [X352: $i] :
( ( in @ X352 @ X348 )
=> ( X350 @ X352 ) )
=> ? [X354: $i] :
( ( in @ X354 @ X348 )
& ~ ( X350 @ X354 ) ) )
=> ( ! [X356: $i > $o] :
( ? [X358: $i] :
( ( X356 @ X358 )
& ! [X360: $i] :
( ( X356 @ X360 )
=> ( X358 = X360 ) ) )
=> ? [X362: $i] :
( ( X356 @ X362 )
& ! [X364: $i] :
( ( X356 @ X364 )
=> ( X362 = X364 ) ) ) )
=> ( ! [X366: $i > $o] :
( ? [X368: $i] : ( X366 @ X368 )
=> ( ! [X370: $i,X372: $i] :
( ( X366 @ X370 )
=> ( ( X366 @ X372 )
=> ( X370 = X372 ) ) )
=> ? [X374: $i] :
( ( X366 @ X374 )
& ! [X376: $i] :
( ( X366 @ X376 )
=> ( X374 = X376 ) ) ) ) )
=> ( ! [X378: $i > $o] :
( ? [X380: $i] :
! [X382: $i] :
( ( X378 @ X382 )
<=> ( X382 = X380 ) )
=> ? [X384: $i] :
( ( X378 @ X384 )
& ! [X386: $i] :
( ( X378 @ X386 )
=> ( X384 = X386 ) ) ) )
=> ( ! [X388: $i,X390: $i] :
( ( X388 = X390 )
=> ! [X392: $i,X394: $i] :
( ( X392 = X394 )
=> ( ( in @ X392 @ X388 )
=> ( in @ X394 @ X390 ) ) ) )
=> ( ! [X396: $i,X398: $i] :
( ( X396 = X398 )
=> ! [X400: $i,X402: $i] :
( ( X400 = X402 )
=> ( ( in @ X400 @ X396 )
<=> ( in @ X402 @ X398 ) ) ) )
=> ( ! [X404: $i > $o] :
( ? [X406: $i] :
( ( X404 @ X406 )
& ! [X408: $i] :
( ( X404 @ X408 )
=> ( X406 = X408 ) ) )
=> ! [X410: $i,X412: $i] :
( ( X404 @ X410 )
=> ( ( X404 @ X412 )
=> ( X410 = X412 ) ) ) )
=> ( ! [X414: $i > $o,X416: $i > $o] :
( ! [X418: $i,X420: $i] :
( ( X418 = X420 )
=> ( ( X414 @ X418 )
<=> ( X416 @ X420 ) ) )
=> ( ? [X422: $i] :
( ( X414 @ X422 )
& ! [X424: $i] :
( ( X414 @ X424 )
=> ( X422 = X424 ) ) )
<=> ? [X426: $i] :
( ( X416 @ X426 )
& ! [X428: $i] :
( ( X416 @ X428 )
=> ( X426 = X428 ) ) ) ) )
=> ( ( emptyset = emptyset )
=> ( ! [X430: $i,X432: $i] :
( ( X430 = X432 )
=> ! [X434: $i,X436: $i] :
( ( X434 = X436 )
=> ( ( setadjoin @ X430 @ X434 )
= ( setadjoin @ X432 @ X436 ) ) ) )
=> ( ! [X438: $i,X440: $i] :
( ( X438 = X440 )
=> ( ( powerset @ X438 )
= ( powerset @ X440 ) ) )
=> ( ! [X442: $i,X444: $i] :
( ( X442 = X444 )
=> ( ( setunion @ X442 )
= ( setunion @ X444 ) ) )
=> ( ( omega = omega )
=> ( ! [X446: $i > $o] :
( ? [X448: $i] :
( ( X446 @ X448 )
& ! [X450: $i] :
( ( X446 @ X450 )
=> ( X448 = X450 ) ) )
=> ! [X452: $i,X454: $i] :
( ( X446 @ X452 )
=> ( ( X446 @ X454 )
=> ( X452 = X454 ) ) ) )
=> ( ! [X456: $i > $o,X458: $i > $o] :
( ! [X460: $i,X462: $i] :
( ( X460 = X462 )
=> ( ( X456 @ X460 )
<=> ( X458 @ X462 ) ) )
=> ( ? [X464: $i] :
( ( X456 @ X464 )
& ! [X466: $i] :
( ( X456 @ X466 )
=> ( X464 = X466 ) ) )
=> ( ? [X468: $i] :
( ( X458 @ X468 )
& ! [X470: $i] :
( ( X458 @ X470 )
=> ( X468 = X470 ) ) )
=> ( ( descr
@ ^ [V_9: $i] : ( X456 @ V_9 ) )
= ( descr
@ ^ [V_10: $i] : ( X458 @ V_10 ) ) ) ) ) )
=> ( ! [X472: $i,X474: $i] :
( ( X472 = X474 )
=> ! [X476: $i > $o,X478: $i > $o] :
( ! [X480: $i] :
( ( in @ X480 @ X472 )
=> ! [X482: $i] :
( ( in @ X482 @ X474 )
=> ( ( X480 = X482 )
=> ( ( X476 @ X480 )
<=> ( X478 @ X482 ) ) ) ) )
=> ( ( dsetconstr @ X472
@ ^ [V_11: $i] : ( X476 @ V_11 ) )
= ( dsetconstr @ X474
@ ^ [V_12: $i] : ( X478 @ V_12 ) ) ) ) )
=> ( ! [X484: $i,X486: $i] :
( ! [X488: $i] :
( ( in @ X488 @ X484 )
=> ( in @ X488 @ X486 ) )
=> ( subset @ X484 @ X486 ) )
=> ( ! [X490: $i,X492: $i] :
( ( X490 = X492 )
=> ( subset @ X492 @ X490 ) )
=> ( ! [X494: $i,X496: $i] :
( ( X494 = X496 )
=> ( subset @ X494 @ X496 ) )
=> ( ! [X498: $i,X500: $i] :
( ! [X502: $i] :
( ( in @ X502 @ X498 )
=> ( in @ X502 @ X500 ) )
=> ( subset @ X498 @ X500 ) )
=> ( ! [X504: $i] : ( subset @ emptyset @ X504 )
=> ( ! [X506: $i,X508: $i,X510: $i] :
( ( subset @ X506 @ X508 )
=> ( ( in @ X510 @ X506 )
=> ( in @ X510 @ X508 ) ) )
=> ( ! [X512: $i,X514: $i,X516: $i] :
( ( subset @ X512 @ X514 )
=> ( ~ ( in @ X516 @ X514 )
=> ~ ( in @ X516 @ X512 ) ) )
=> ( ! [X518: $i,X520: $i,X522: $i] :
( ( in @ X522 @ X518 )
=> ( ~ ( in @ X522 @ X520 )
=> ~ ( subset @ X518 @ X520 ) ) )
=> ( ! [X524: $i,X526: $i] :
( ~ ( subset @ X524 @ X526 )
=> ( X524 != X526 ) )
=> ( ! [X528: $i,X530: $i,X532: $i] :
( ( in @ X532 @ X528 )
=> ( ~ ( in @ X532 @ X530 )
=> ( X528 != X530 ) ) )
=> ( ! [X534: $i] : ( subset @ X534 @ X534 )
=> ( ! [X536: $i,X538: $i,X540: $i] :
( ( subset @ X536 @ X538 )
=> ( ( subset @ X538 @ X540 )
=> ( subset @ X536 @ X540 ) ) )
=> ( ! [X542: $i,X544: $i] : ( subset @ X544 @ ( setadjoin @ X542 @ X544 ) )
=> ( ! [X546: $i,X548: $i,X550: $i] :
( ( subset @ X546 @ X550 )
=> ( subset @ X546 @ ( setadjoin @ X548 @ X550 ) ) )
=> ( ! [X552: $i,X554: $i] :
( ( subset @ X552 @ X554 )
=> ( in @ X552 @ ( powerset @ X554 ) ) )
=> ( ! [X556: $i,X558: $i] :
( ( subset @ X556 @ X558 )
=> ( ( subset @ X558 @ X556 )
=> ( X556 = X558 ) ) )
=> ( ! [X560: $i] :
( ( subset @ X560 @ emptyset )
=> ( X560 = emptyset ) )
=> ( ! [X562: $i,X564: $i] :
( ( subset @ X564 @ X562 )
=> ( in @ X564 @ ( powerset @ X562 ) ) )
=> ( ! [X566: $i,X568: $i] :
( ( in @ X568 @ ( powerset @ X566 ) )
=> ( subset @ X568 @ X566 ) )
=> ( ! [X570: $i] : ( in @ X570 @ ( powerset @ X570 ) )
=> ( ! [X572: $i,X574: $i] :
( ( subset @ X572 @ X574 )
=> ( subset @ ( powerset @ X572 ) @ ( powerset @ X574 ) ) )
=> ( ! [X576: $i,X578: $i > $o] :
( in
@ ( dsetconstr @ X576
@ ^ [V_13: $i] : ( X578 @ V_13 ) )
@ ( powerset @ X576 ) )
=> ( ! [X580: $i,X582: $i > $o] :
( subset
@ ( dsetconstr @ X580
@ ^ [V_14: $i] : ( X582 @ V_14 ) )
@ X580 )
=> ( ! [X584: $i,X586: $i,X588: $i] :
( ( in @ X588 @ X584 )
=> ( in @ X588 @ ( binunion @ X584 @ X586 ) ) )
=> ( ! [X590: $i,X592: $i] : ( in @ X592 @ ( setadjoin @ X590 @ ( setadjoin @ X592 @ emptyset ) ) )
=> ( ! [X594: $i,X596: $i,X598: $i] :
( ( in @ X598 @ X596 )
=> ( in @ X598 @ ( binunion @ X594 @ X596 ) ) )
=> ( ! [X600: $i,X602: $i,X604: $i,X606: $o] :
( ( in @ X604 @ ( binunion @ X600 @ X602 ) )
=> ( ( ( in @ X604 @ X600 )
=> X606 )
=> ( ( ( in @ X604 @ X602 )
=> X606 )
=> X606 ) ) )
=> ( ! [X608: $i,X610: $i,X612: $i] :
( ( in @ X612 @ ( binunion @ X608 @ X610 ) )
=> ( ( in @ X612 @ X610 )
| ( in @ X612 @ X608 ) ) )
=> ( ! [X614: $i,X616: $i] : ( subset @ X614 @ ( binunion @ X614 @ X616 ) )
=> ( ! [X618: $i,X620: $i] : ( subset @ X620 @ ( binunion @ X618 @ X620 ) )
=> ( ! [X622: $i,X624: $i,X626: $i] :
( ( in @ X626 @ X622 )
=> ( ( in @ X626 @ X624 )
=> ( in @ X626 @ ( binintersect @ X622 @ X624 ) ) ) )
=> ( ! [X628: $i,X630: $i,X632: $i] :
( ( subset @ X632 @ X628 )
=> ( ( subset @ X632 @ X630 )
=> ( subset @ X632 @ ( binintersect @ X628 @ X630 ) ) ) )
=> ( ! [X634: $i,X636: $i,X638: $i] :
( ( in @ X638 @ ( binintersect @ X634 @ X636 ) )
=> ( in @ X638 @ X634 ) )
=> ( ! [X640: $i,X642: $i] : ( subset @ ( binintersect @ X640 @ X642 ) @ X640 )
=> ( ! [X644: $i,X646: $i] :
( ( subset @ X644 @ X646 )
=> ( ( binintersect @ X644 @ X646 )
= X644 ) )
=> ( ! [X648: $i,X650: $i] :
( ( ( binintersect @ X648 @ X650 )
= X650 )
=> ( subset @ X650 @ X648 ) )
=> ( ! [X652: $i,X654: $i,X656: $i] :
( ( in @ X656 @ ( binintersect @ X652 @ X654 ) )
=> ( in @ X656 @ X654 ) )
=> ( ! [X658: $i,X660: $i] :
( ~ ? [X662: $i] :
( ( in @ X662 @ X658 )
& ( in @ X662 @ X660 ) )
=> ( ( binintersect @ X658 @ X660 )
= emptyset ) )
=> ( ! [X664: $i,X666: $i] : ( subset @ ( binintersect @ X664 @ X666 ) @ X666 )
=> ( ! [X668: $i,X670: $i] :
( ( subset @ X670 @ X668 )
=> ( ( binintersect @ X668 @ X670 )
= X670 ) )
=> ( ! [X672: $i,X674: $i] :
( ( ( binintersect @ X672 @ X674 )
= X672 )
=> ( subset @ X672 @ X674 ) )
=> ( ! [X676: $i,X678: $i,X680: $i] :
( ( binintersect @ X676 @ ( binunion @ X678 @ X680 ) )
= ( binunion @ ( binintersect @ X676 @ X678 ) @ ( binintersect @ X676 @ X680 ) ) )
=> ( ! [X682: $i,X684: $i,X686: $i] :
( ( in @ X686 @ X682 )
=> ( ~ ( in @ X686 @ X684 )
=> ( in @ X686 @ ( setminus @ X682 @ X684 ) ) ) )
=> ( ! [X688: $i,X690: $i,X692: $i] :
( ( in @ X692 @ ( setminus @ X688 @ X690 ) )
=> ( in @ X692 @ X688 ) )
=> ( ! [X694: $i,X696: $i,X698: $i] :
( ( in @ X698 @ ( setminus @ X694 @ X696 ) )
=> ~ ( in @ X698 @ X696 ) )
=> ( ! [X700: $i,X702: $i] :
( ( subset @ X700 @ X702 )
=> ( ( setminus @ X700 @ X702 )
= emptyset ) )
=> ( ! [X704: $i,X706: $i,X708: $i] :
( ~ ( in @ X708 @ ( setminus @ X704 @ X706 ) )
=> ( ( in @ X708 @ X704 )
=> ( in @ X708 @ X706 ) ) )
=> ( ! [X710: $i,X712: $i,X714: $i] :
( ~ ( in @ X714 @ ( setminus @ X710 @ X712 ) )
=> ( ~ ( in @ X714 @ X712 )
=> ~ ( in @ X714 @ X710 ) ) )
=> ( ! [X716: $i,X718: $i,X720: $i] :
( ~ ( in @ X720 @ X716 )
=> ~ ( in @ X720 @ ( setminus @ X716 @ X718 ) ) )
=> ( ! [X722: $i,X724: $i,X726: $i] :
( ( in @ X726 @ X724 )
=> ~ ( in @ X726 @ ( setminus @ X722 @ X724 ) ) )
=> ( ! [X728: $i,X730: $i] : ( subset @ ( setminus @ X728 @ X730 ) @ X728 )
=> ( ! [X732: $i,X734: $i] :
( ( ( setminus @ X732 @ X734 )
= emptyset )
=> ( subset @ X732 @ X734 ) )
=> ( ! [X736: $i,X738: $i,X740: $i] :
( ( in @ X740 @ ( symdiff @ X736 @ X738 ) )
=> ! [X742: $o] :
( ( ( in @ X740 @ X736 )
=> ( ~ ( in @ X740 @ X738 )
=> X742 ) )
=> ( ( ~ ( in @ X740 @ X736 )
=> ( ( in @ X740 @ X738 )
=> X742 ) )
=> X742 ) ) )
=> ( ! [X744: $i,X746: $i,X748: $i] :
( ( in @ X748 @ X744 )
=> ( ~ ( in @ X748 @ X746 )
=> ( in @ X748 @ ( symdiff @ X744 @ X746 ) ) ) )
=> ( ! [X750: $i,X752: $i,X754: $i] :
( ~ ( in @ X754 @ X750 )
=> ( ( in @ X754 @ X752 )
=> ( in @ X754 @ ( symdiff @ X750 @ X752 ) ) ) )
=> ( ! [X756: $i,X758: $i,X760: $i] :
( ( in @ X760 @ X756 )
=> ( ( in @ X760 @ X758 )
=> ~ ( in @ X760 @ ( symdiff @ X756 @ X758 ) ) ) )
=> ( ! [X762: $i,X764: $i,X766: $i] :
( ~ ( in @ X766 @ X762 )
=> ( ~ ( in @ X766 @ X764 )
=> ~ ( in @ X766 @ ( symdiff @ X762 @ X764 ) ) ) )
=> ( ! [X768: $i,X770: $i] : ( in @ X770 @ ( setadjoin @ X768 @ ( setadjoin @ X770 @ emptyset ) ) )
=> ( ! [X772: $i,X774: $i] : ( in @ X772 @ ( setunion @ ( setadjoin @ ( setadjoin @ X772 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X772 @ ( setadjoin @ X774 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X776: $i,X778: $i] : ( in @ X778 @ ( setunion @ ( setadjoin @ ( setadjoin @ X776 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X776 @ ( setadjoin @ X778 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X780: $i,X782: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X780 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X780 @ ( setadjoin @ X782 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X784: $i,X786: $i] : ( iskpair @ ( kpair @ X784 @ X786 ) )
=> ( ! [X788: $i,X790: $i] :
( ( in @ X790 @ X788 )
=> ( subset @ ( setadjoin @ X790 @ emptyset ) @ X788 ) )
=> ( ! [X792: $i,X794: $i] :
( ( in @ X794 @ X792 )
=> ( in @ ( setadjoin @ X794 @ emptyset ) @ ( powerset @ X792 ) ) )
=> ( ! [X796: $i,X798: $i,X800: $i] :
( ( in @ X800 @ X796 )
=> ( in @ ( setadjoin @ X800 @ emptyset ) @ ( powerset @ ( binunion @ X796 @ X798 ) ) ) )
=> ( ! [X802: $i,X804: $i,X806: $i] :
( ( in @ X806 @ ( setadjoin @ X802 @ ( setadjoin @ X804 @ emptyset ) ) )
=> ( ( X806 = X804 )
| ( X806 = X802 ) ) )
=> ( ! [X808: $i,X810: $i,X812: $i] :
( ( in @ X812 @ X808 )
=> ! [X814: $i] :
( ( in @ X814 @ X810 )
=> ( subset @ ( setadjoin @ X812 @ ( setadjoin @ X814 @ emptyset ) ) @ ( binunion @ X808 @ X810 ) ) ) )
=> ( ! [X816: $i,X818: $i,X820: $i] :
( ( in @ X820 @ X816 )
=> ! [X822: $i] :
( ( in @ X822 @ X818 )
=> ( in @ ( setadjoin @ X820 @ ( setadjoin @ X822 @ emptyset ) ) @ ( powerset @ ( binunion @ X816 @ X818 ) ) ) ) )
=> ( ! [X824: $i,X826: $i,X828: $i] :
( ( in @ X828 @ X824 )
=> ! [X830: $i] :
( ( in @ X830 @ X826 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X828 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X828 @ ( setadjoin @ X830 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X824 @ X826 ) ) ) ) )
=> ( ! [X832: $i,X834: $i,X836: $i] :
( ( in @ X836 @ X832 )
=> ! [X838: $i] :
( ( in @ X838 @ X834 )
=> ( in @ ( setadjoin @ ( setadjoin @ X836 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X836 @ ( setadjoin @ X838 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X832 @ X834 ) ) ) ) ) )
=> ( ! [X840: $i,X842: $i,X844: $i] :
( ( in @ X844 @ X840 )
=> ! [X846: $i] :
( ( in @ X846 @ X842 )
=> ( in @ ( kpair @ X844 @ X846 ) @ ( powerset @ ( powerset @ ( binunion @ X840 @ X842 ) ) ) ) ) )
=> ( ! [X848: $i,X850: $i,X852: $i] :
( ( in @ X852 @ X848 )
=> ! [X854: $i] :
( ( in @ X854 @ X850 )
=> ( in @ ( kpair @ X852 @ X854 ) @ ( cartprod @ X848 @ X850 ) ) ) )
=> ( ! [X856: $i,X858: $i,X860: $i] :
( ( in @ X860 @ ( cartprod @ X856 @ X858 ) )
=> ? [X862: $i] :
( ( in @ X862 @ X856 )
& ? [X864: $i] :
( ( in @ X864 @ X858 )
& ( X860
= ( kpair @ X862 @ X864 ) ) ) ) )
=> ( ! [X866: $i,X868: $i,X870: $i] :
( ( in @ X870 @ ( cartprod @ X866 @ X868 ) )
=> ( iskpair @ X870 ) )
=> ( ! [X872: $i,X874: $i] :
( ( in @ X874 @ ( setunion @ X872 ) )
=> ? [X876: $i] :
( ( in @ X876 @ X872 )
& ( in @ X874 @ X876 ) ) )
=> ( ! [X878: $i] : ( subset @ ( setunion @ ( setadjoin @ X878 @ emptyset ) ) @ X878 )
=> ( ! [X880: $i] : ( subset @ X880 @ ( setunion @ ( setadjoin @ X880 @ emptyset ) ) )
=> ( ! [X882: $i] :
( ( setunion @ ( setadjoin @ X882 @ emptyset ) )
= X882 )
=> ( ! [X884: $i,X886: $i > $o] :
( ! [X888: $i] :
( ( in @ X888 @ X884 )
=> ! [X890: $i] :
( ( in @ X890 @ X884 )
=> ( ( X886 @ X888 )
=> ( ( X886 @ X890 )
=> ( X888 = X890 ) ) ) ) )
=> ( ? [X892: $i] :
( ( in @ X892 @ X884 )
& ( X886 @ X892 ) )
=> ( singleton
@ ( dsetconstr @ X884
@ ^ [V_15: $i] : ( X886 @ V_15 ) ) ) ) )
=> ( ! [X894: $i,X896: $i > $o] :
( ( ex1 @ X894
@ ^ [V_16: $i] : ( X896 @ V_16 ) )
=> ? [X898: $i] :
( ( in @ X898 @ X894 )
& ( X896 @ X898 ) ) )
=> ( ! [X900: $i,X902: $i > $o,X904: $i] :
( ( in @ X904 @ X900 )
=> ( ( X902 @ X904 )
=> ( ! [X906: $i] :
( ( in @ X906 @ X900 )
=> ( ( X902 @ X906 )
=> ( X906 = X904 ) ) )
=> ( ex1 @ X900
@ ^ [V_17: $i] : ( X902 @ V_17 ) ) ) ) )
=> ( ! [X908: $i,X910: $i > $o] :
( ! [X912: $i] :
( ( in @ X912 @ X908 )
=> ! [X914: $i] :
( ( in @ X914 @ X908 )
=> ( ( X910 @ X912 )
=> ( ( X910 @ X914 )
=> ( X912 = X914 ) ) ) ) )
=> ( ? [X916: $i] :
( ( in @ X916 @ X908 )
& ( X910 @ X916 ) )
=> ( ex1 @ X908
@ ^ [V_18: $i] : ( X910 @ V_18 ) ) ) )
=> ( ! [X918: $i,X920: $i] :
( ( ( setadjoin @ X918 @ emptyset )
= ( setadjoin @ X920 @ emptyset ) )
=> ( X918 = X920 ) )
=> ( ! [X922: $i,X924: $i,X926: $i] :
( ( in @ ( setadjoin @ X926 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X922 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X922 @ ( setadjoin @ X924 @ emptyset ) ) @ emptyset ) ) )
=> ( X922 = X926 ) )
=> ( ! [X928: $i] :
( ( iskpair @ X928 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X928 )
@ ^ [V_19: $i] : ( in @ ( setadjoin @ V_19 @ emptyset ) @ X928 ) ) ) )
=> ( ! [X930: $i] :
( ( singleton @ X930 )
=> ( in @ ( setunion @ X930 ) @ X930 ) )
=> ( ! [X932: $i,X934: $i] :
( ( kfst @ ( kpair @ X932 @ X934 ) )
= X932 )
=> ( ! [X936: $i,X938: $i,X940: $i] :
( ( in @ X940 @ ( cartprod @ X936 @ X938 ) )
=> ( in @ ( kfst @ X940 ) @ X936 ) )
=> ( ! [X942: $i,X944: $i,X946: $i,X948: $i] :
( ( ( setadjoin @ ( setadjoin @ X942 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X942 @ ( setadjoin @ X944 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X946 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X946 @ ( setadjoin @ X948 @ emptyset ) ) @ emptyset ) ) )
=> ( X942 = X946 ) )
=> ( ! [X950: $i,X952: $i,X954: $i,X956: $i] :
( ( ( kpair @ X950 @ X952 )
= ( kpair @ X954 @ X956 ) )
=> ( X950 = X954 ) )
=> ( ! [X958: $i,X960: $i] :
( ( X958 = X960 )
=> ( ( setadjoin @ X958 @ ( setadjoin @ X960 @ emptyset ) )
= ( setadjoin @ X958 @ emptyset ) ) )
=> ( ! [X962: $i,X964: $i] :
( ( X962 = X964 )
=> ( ( setadjoin @ ( setadjoin @ X962 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X962 @ ( setadjoin @ X964 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X962 @ emptyset ) @ emptyset ) ) )
=> ( ! [X966: $i,X968: $i,X970: $i,X972: $i] :
( ( ( setadjoin @ ( setadjoin @ X966 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X966 @ ( setadjoin @ X968 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X970 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X970 @ ( setadjoin @ X972 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X970 = X972 )
=> ( X968 = X972 ) ) )
=> ( ! [X974: $i,X976: $i,X978: $i] :
( ( ( setadjoin @ X974 @ ( setadjoin @ X976 @ emptyset ) )
= ( setadjoin @ X978 @ emptyset ) )
=> ( X974 = X976 ) )
=> ( ! [X980: $i,X982: $i,X984: $i,X986: $i] :
( ( ( setadjoin @ ( setadjoin @ X980 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X980 @ ( setadjoin @ X982 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X984 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X984 @ ( setadjoin @ X986 @ emptyset ) ) @ emptyset ) ) )
=> ( X982 = X986 ) )
=> ( ! [X988: $i,X990: $i,X992: $i,X994: $i] :
( ( ( kpair @ X988 @ X990 )
= ( kpair @ X992 @ X994 ) )
=> ( X990 = X994 ) )
=> ( ! [X996: $i] :
( ( iskpair @ X996 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X996 )
@ ^ [V_20: $i] :
( X996
= ( kpair @ ( kfst @ X996 ) @ V_20 ) ) ) ) )
=> ( ! [X998: $i,X1000: $i] :
( ( ksnd @ ( kpair @ X998 @ X1000 ) )
= X1000 )
=> ( ! [X1002: $i] :
( ( iskpair @ X1002 )
=> ( ( kpair @ ( kfst @ X1002 ) @ ( ksnd @ X1002 ) )
= X1002 ) )
=> ( ! [X1004: $i,X1006: $i,X1008: $i] :
( ( in @ X1008 @ ( cartprod @ X1004 @ X1006 ) )
=> ( in @ ( ksnd @ X1008 ) @ X1006 ) )
=> ( ! [X1010: $i,X1012: $i,X1014: $i,X1016: $i] :
( ( in @ ( kpair @ X1014 @ X1016 ) @ ( cartprod @ X1010 @ X1012 ) )
=> ( in @ X1014 @ X1010 ) )
=> ( ! [X1018: $i,X1020: $i,X1022: $i,X1024: $i] :
( ( in @ ( kpair @ X1022 @ X1024 ) @ ( cartprod @ X1018 @ X1020 ) )
=> ( in @ X1024 @ X1020 ) )
=> ( ! [X1026: $i,X1028: $i,X1030: $i] :
( ( in @ X1030 @ X1026 )
=> ! [X1032: $i] :
( ( in @ X1032 @ X1028 )
=> ( ( kpair @ X1030 @ X1032 )
= ( kpair @ X1030 @ X1032 ) ) ) )
=> ( ! [X1034: $i,X1036: $i,X1038: $i] :
( ( in @ X1038 @ X1034 )
=> ! [X1040: $i] :
( ( in @ X1040 @ X1036 )
=> ( ( kfst @ ( kpair @ X1038 @ X1040 ) )
= X1038 ) ) )
=> ( ! [X1042: $i,X1044: $i,X1046: $i] :
( ( in @ X1046 @ X1042 )
=> ! [X1048: $i] :
( ( in @ X1048 @ X1044 )
=> ( ( ksnd @ ( kpair @ X1046 @ X1048 ) )
= X1048 ) ) )
=> ( ! [X1050: $i,X1052: $i,X1054: $i] :
( ( in @ X1054 @ ( cartprod @ X1050 @ X1052 ) )
=> ( ( kpair @ ( kfst @ X1054 ) @ ( ksnd @ X1054 ) )
= X1054 ) )
=> ( ! [X1056: $i,X1058: $i,X1060: $i > $i > $o,X1062: $i] :
( ( in @ X1062 @ X1056 )
=> ! [X1064: $i] :
( ( in @ X1064 @ X1058 )
=> ( ( X1060 @ X1062 @ X1064 )
=> ( in @ ( kpair @ X1062 @ X1064 )
@ ( dpsetconstr @ X1056 @ X1058
@ ^ [V_21: $i,V_22: $i] : ( X1060 @ V_21 @ V_22 ) ) ) ) ) )
=> ( ! [X1066: $i,X1068: $i,X1070: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1066 @ X1068
@ ^ [V_23: $i,V_24: $i] : ( X1070 @ V_23 @ V_24 ) )
@ ( cartprod @ X1066 @ X1068 ) )
=> ( ! [X1072: $i,X1074: $i,X1076: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1072 @ X1074
@ ^ [V_25: $i,V_26: $i] : ( X1076 @ V_25 @ V_26 ) )
@ ( cartprod @ X1072 @ X1074 ) )
=> ( ! [X1078: $i,X1080: $i,X1082: $i > $i > $o,X1084: $i] :
( ( in @ X1084 @ X1078 )
=> ! [X1086: $i] :
( ( in @ X1086 @ X1080 )
=> ( ( in @ ( kpair @ X1084 @ X1086 )
@ ( dpsetconstr @ X1078 @ X1080
@ ^ [V_27: $i,V_28: $i] : ( X1082 @ V_27 @ V_28 ) ) )
=> ( X1082 @ X1084 @ X1086 ) ) ) )
=> ( ! [X1088: $i,X1090: $i,X1092: $i > $i > $o,X1094: $i,X1096: $i] :
( ( in @ ( kpair @ X1094 @ X1096 )
@ ( dpsetconstr @ X1088 @ X1090
@ ^ [V_29: $i,V_30: $i] : ( X1092 @ V_29 @ V_30 ) ) )
=> ( in @ X1094 @ X1088 ) )
=> ( ! [X1098: $i,X1100: $i,X1102: $i > $i > $o,X1104: $i,X1106: $i] :
( ( in @ ( kpair @ X1104 @ X1106 )
@ ( dpsetconstr @ X1098 @ X1100
@ ^ [V_31: $i,V_32: $i] : ( X1102 @ V_31 @ V_32 ) ) )
=> ( in @ X1106 @ X1100 ) )
=> ( ! [X1108: $i,X1110: $i,X1112: $i > $i > $o,X1114: $i,X1116: $i] :
( ( in @ ( kpair @ X1114 @ X1116 )
@ ( dpsetconstr @ X1108 @ X1110
@ ^ [V_33: $i,V_34: $i] : ( X1112 @ V_33 @ V_34 ) ) )
=> ( X1112 @ X1114 @ X1116 ) )
=> ( ! [X1118: $i,X1120: $i,X1122: $i] :
( ( func @ X1118 @ X1120 @ X1122 )
=> ! [X1124: $i] :
( ( in @ X1124 @ X1118 )
=> ( singleton
@ ( dsetconstr @ X1120
@ ^ [V_35: $i] : ( in @ ( kpair @ X1124 @ V_35 ) @ X1122 ) ) ) ) )
=> ( ! [X1126: $i,X1128: $i,X1130: $i] :
( ( func @ X1126 @ X1128 @ X1130 )
=> ! [X1132: $i] :
( ( in @ X1132 @ X1126 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X1128
@ ^ [V_36: $i] : ( in @ ( kpair @ X1132 @ V_36 ) @ X1130 ) ) )
@ X1128 ) ) )
=> ( ! [X1134: $i,X1136: $i,X1138: $i] :
( ( func @ X1134 @ X1136 @ X1138 )
=> ! [X1140: $i] :
( ( in @ X1140 @ X1134 )
=> ( in @ ( ap @ X1134 @ X1136 @ X1138 @ X1140 ) @ X1136 ) ) )
=> ( ! [X1142: $i,X1144: $i,X1146: $i] :
( ( in @ X1146 @ ( funcSet @ X1142 @ X1144 ) )
=> ( func @ X1142 @ X1144 @ X1146 ) )
=> ( ! [X1148: $i,X1150: $i,X1152: $i] :
( ( in @ X1152 @ ( funcSet @ X1148 @ X1150 ) )
=> ! [X1154: $i] :
( ( in @ X1154 @ X1148 )
=> ( in @ ( ap @ X1148 @ X1150 @ X1152 @ X1154 ) @ X1150 ) ) )
=> ( ! [X1156: $i,X1158: $i,X1160: $i] :
( ( func @ X1156 @ X1158 @ X1160 )
=> ( in @ X1160 @ ( funcSet @ X1156 @ X1158 ) ) )
=> ( ! [X1162: $i,X1164: $i,X1166: $i > $i] :
( ! [X1168: $i] :
( ( in @ X1168 @ X1162 )
=> ( in @ ( X1166 @ X1168 ) @ X1164 ) )
=> ( func @ X1162 @ X1164
@ ( dpsetconstr @ X1162 @ X1164
@ ^ [V_37: $i,V_38: $i] :
( ( X1166 @ V_37 )
= V_38 ) ) ) )
=> ( ! [X1170: $i,X1172: $i,X1174: $i > $i] :
( ! [X1176: $i] :
( ( in @ X1176 @ X1170 )
=> ( in @ ( X1174 @ X1176 ) @ X1172 ) )
=> ( func @ X1170 @ X1172
@ ( lam @ X1170 @ X1172
@ ^ [V_39: $i] : ( X1174 @ V_39 ) ) ) )
=> ( ! [X1178: $i,X1180: $i,X1182: $i > $i] :
( ! [X1184: $i] :
( ( in @ X1184 @ X1178 )
=> ( in @ ( X1182 @ X1184 ) @ X1180 ) )
=> ( in
@ ( lam @ X1178 @ X1180
@ ^ [V_40: $i] : ( X1182 @ V_40 ) )
@ ( funcSet @ X1178 @ X1180 ) ) )
=> ( ! [X1186: $i,X1188: $i,X1190: $i] :
( ( subset @ X1190 @ ( cartprod @ X1186 @ X1188 ) )
=> ! [X1192: $i > $o] :
( ! [X1194: $i] :
( ( in @ X1194 @ X1186 )
=> ! [X1196: $i] :
( ( in @ X1196 @ X1188 )
=> ( ( in @ ( kpair @ X1194 @ X1196 ) @ X1190 )
=> ( X1192 @ ( kpair @ X1194 @ X1196 ) ) ) ) )
=> ! [X1198: $i] :
( ( in @ X1198 @ X1190 )
=> ( X1192 @ X1198 ) ) ) )
=> ( ! [X1200: $i,X1202: $i,X1204: $i] :
( ( subset @ X1204 @ ( cartprod @ X1200 @ X1202 ) )
=> ! [X1206: $i > $o] :
( ! [X1208: $i] :
( ( in @ X1208 @ X1200 )
=> ! [X1210: $i] :
( ( in @ X1210 @ X1202 )
=> ( ( in @ ( kpair @ X1208 @ X1210 ) @ X1204 )
=> ( X1206 @ ( kpair @ X1208 @ X1210 ) ) ) ) )
=> ! [X1212: $i] :
( ( in @ X1212 @ X1204 )
=> ( X1206 @ X1212 ) ) ) )
=> ( ! [X1214: $i,X1216: $i > $o] :
( ( ex1 @ X1214
@ ^ [V_41: $i] : ( X1216 @ V_41 ) )
=> ! [X1218: $i] :
( ( in @ X1218 @ X1214 )
=> ! [X1220: $i] :
( ( in @ X1220 @ X1214 )
=> ( ( X1216 @ X1218 )
=> ( ( X1216 @ X1220 )
=> ( X1218 = X1220 ) ) ) ) ) )
=> ( ! [X1222: $i,X1224: $i,X1226: $i] :
( ( func @ X1222 @ X1224 @ X1226 )
=> ! [X1228: $i] :
( ( in @ X1228 @ X1222 )
=> ( in @ ( kpair @ X1228 @ ( ap @ X1222 @ X1224 @ X1226 @ X1228 ) ) @ X1226 ) ) )
=> ( ! [X1230: $i,X1232: $i,X1234: $i] :
( ( in @ X1234 @ ( funcSet @ X1230 @ X1232 ) )
=> ! [X1236: $i] :
( ( in @ X1236 @ X1230 )
=> ( in @ ( kpair @ X1236 @ ( ap @ X1230 @ X1232 @ X1234 @ X1236 ) ) @ X1234 ) ) )
=> ( ! [X1238: $i,X1240: $i,X1242: $i] :
( ( func @ X1238 @ X1240 @ X1242 )
=> ! [X1244: $i] :
( ( in @ X1244 @ X1238 )
=> ! [X1246: $i] :
( ( in @ X1246 @ X1240 )
=> ( ( in @ ( kpair @ X1244 @ X1246 ) @ X1242 )
=> ( ( ap @ X1238 @ X1240 @ X1242 @ X1244 )
= X1246 ) ) ) ) )
=> ( ! [X1248: $i,X1250: $i,X1252: $i] :
( ( func @ X1248 @ X1250 @ X1252 )
=> ! [X1254: $i] :
( ( func @ X1248 @ X1250 @ X1254 )
=> ( ! [X1256: $i] :
( ( in @ X1256 @ X1248 )
=> ( ( ap @ X1248 @ X1250 @ X1252 @ X1256 )
= ( ap @ X1248 @ X1250 @ X1254 @ X1256 ) ) )
=> ! [X1258: $i] :
( ( in @ X1258 @ X1248 )
=> ! [X1260: $i] :
( ( in @ X1260 @ X1250 )
=> ( ( in @ ( kpair @ X1258 @ X1260 ) @ X1254 )
=> ( in @ ( kpair @ X1258 @ X1260 ) @ X1252 ) ) ) ) ) ) )
=> ( ! [X1262: $i,X1264: $i,X1266: $i] :
( ( in @ X1266 @ ( funcSet @ X1262 @ X1264 ) )
=> ! [X1268: $i] :
( ( in @ X1268 @ X1262 )
=> ! [X1270: $i] :
( ( in @ X1270 @ X1264 )
=> ( ( in @ ( kpair @ X1268 @ X1270 ) @ X1266 )
=> ( ( ap @ X1262 @ X1264 @ X1266 @ X1268 )
= X1270 ) ) ) ) )
=> ( ! [X1272: $i,X1274: $i,X1276: $i] :
( ( subset @ X1276 @ ( cartprod @ X1272 @ X1274 ) )
=> ! [X1278: $i] :
( ( subset @ X1278 @ ( cartprod @ X1272 @ X1274 ) )
=> ( ! [X1280: $i] :
( ( in @ X1280 @ X1272 )
=> ! [X1282: $i] :
( ( in @ X1282 @ X1274 )
=> ( ( in @ ( kpair @ X1280 @ X1282 ) @ X1276 )
=> ( in @ ( kpair @ X1280 @ X1282 ) @ X1278 ) ) ) )
=> ( subset @ X1276 @ X1278 ) ) ) )
=> ( ! [X1284: $i,X1286: $i,X1288: $i] :
( ( subset @ X1288 @ ( cartprod @ X1284 @ X1286 ) )
=> ! [X1290: $i] :
( ( subset @ X1290 @ ( cartprod @ X1284 @ X1286 ) )
=> ( ! [X1292: $i] :
( ( in @ X1292 @ X1284 )
=> ! [X1294: $i] :
( ( in @ X1294 @ X1286 )
=> ( ( in @ ( kpair @ X1292 @ X1294 ) @ X1288 )
=> ( in @ ( kpair @ X1292 @ X1294 ) @ X1290 ) ) ) )
=> ( ! [X1296: $i] :
( ( in @ X1296 @ X1284 )
=> ! [X1298: $i] :
( ( in @ X1298 @ X1286 )
=> ( ( in @ ( kpair @ X1296 @ X1298 ) @ X1290 )
=> ( in @ ( kpair @ X1296 @ X1298 ) @ X1288 ) ) ) )
=> ( X1288 = X1290 ) ) ) ) )
=> ( ! [X1300: $i,X1302: $i,X1304: $i] :
( ( func @ X1300 @ X1302 @ X1304 )
=> ! [X1306: $i] :
( ( func @ X1300 @ X1302 @ X1306 )
=> ( ! [X1308: $i] :
( ( in @ X1308 @ X1300 )
=> ( ( ap @ X1300 @ X1302 @ X1304 @ X1308 )
= ( ap @ X1300 @ X1302 @ X1306 @ X1308 ) ) )
=> ( X1304 = X1306 ) ) ) )
=> ( ! [X1310: $i,X1312: $i,X1314: $i] :
( ( in @ X1314 @ ( funcSet @ X1310 @ X1312 ) )
=> ! [X1316: $i] :
( ( in @ X1316 @ ( funcSet @ X1310 @ X1312 ) )
=> ( ! [X1318: $i] :
( ( in @ X1318 @ X1310 )
=> ( ( ap @ X1310 @ X1312 @ X1314 @ X1318 )
= ( ap @ X1310 @ X1312 @ X1316 @ X1318 ) ) )
=> ( X1314 = X1316 ) ) ) )
=> ( ! [X1320: $i,X1322: $i,X1324: $i] :
( ( in @ X1324 @ ( funcSet @ X1320 @ X1322 ) )
=> ! [X1326: $i] :
( ( in @ X1326 @ X1320 )
=> ( ( ap @ X1320 @ X1322 @ X1324 @ X1326 )
= ( ap @ X1320 @ X1322 @ X1324 @ X1326 ) ) ) )
=> ( ! [X1328: $i,X1330: $i,X1332: $i] :
( ( func @ X1328 @ X1330 @ X1332 )
=> ! [X1334: $i] :
( ( in @ X1334 @ X1328 )
=> ( ( ap @ X1328 @ X1330 @ X1332 @ X1334 )
= ( ap @ X1328 @ X1330 @ X1332 @ X1334 ) ) ) )
=> ( ! [X1336: $i,X1338: $i,X1340: $i > $i] :
( ! [X1342: $i] :
( ( in @ X1342 @ X1336 )
=> ( in @ ( X1340 @ X1342 ) @ X1338 ) )
=> ! [X1344: $i] :
( ( in @ X1344 @ X1336 )
=> ( ( ap @ X1336 @ X1338
@ ( lam @ X1336 @ X1338
@ ^ [V_42: $i] : ( X1340 @ V_42 ) )
@ X1344 )
= ( X1340 @ X1344 ) ) ) )
=> ( ! [X1346: $i,X1348: $i,X1350: $i] :
( ( func @ X1346 @ X1348 @ X1350 )
=> ( ( lam @ X1346 @ X1348
@ ^ [V_43: $i] : ( ap @ X1346 @ X1348 @ X1350 @ V_43 ) )
= X1350 ) )
=> ( ! [X1352: $i,X1354: $i,X1356: $i > $i] :
( ! [X1358: $i] :
( ( in @ X1358 @ X1352 )
=> ( in @ ( X1356 @ X1358 ) @ X1354 ) )
=> ( ( lam @ X1352 @ X1354
@ ^ [V_44: $i] : ( X1356 @ V_44 ) )
= ( lam @ X1352 @ X1354
@ ^ [V_45: $i] : ( X1356 @ V_45 ) ) ) )
=> ( ! [X1360: $i,X1362: $i,X1364: $i > $i] :
( ! [X1366: $i] :
( ( in @ X1366 @ X1360 )
=> ( in @ ( X1364 @ X1366 ) @ X1362 ) )
=> ! [X1368: $i] :
( ( in @ X1368 @ X1360 )
=> ( ( ap @ X1360 @ X1362
@ ( lam @ X1360 @ X1362
@ ^ [V_46: $i] : ( X1364 @ V_46 ) )
@ X1368 )
= ( X1364 @ X1368 ) ) ) )
=> ( ! [X1370: $i,X1372: $i,X1374: $i] :
( ( in @ X1374 @ ( funcSet @ X1370 @ X1372 ) )
=> ( ( lam @ X1370 @ X1372
@ ^ [V_47: $i] : ( ap @ X1370 @ X1372 @ X1374 @ V_47 ) )
= X1374 ) )
=> ( ! [X1376: $i,X1378: $o,X1380: $i] :
( ( in @ X1380 @ X1376 )
=> ! [X1382: $i] :
( ( in @ X1382 @ X1376 )
=> ( ~ X1378
=> ( in @ X1382
@ ( dsetconstr @ X1376
@ ^ [V_48: $i] :
( ( ( V_48 = X1382 )
& ~ X1378 )
| ( ( V_48 = X1380 )
& X1378 ) ) ) ) ) ) )
=> ( ! [X1384: $i,X1386: $o,X1388: $i] :
( ( in @ X1388 @ X1384 )
=> ! [X1390: $i] :
( ( in @ X1390 @ X1384 )
=> ( ~ X1386
=> ( ( dsetconstr @ X1384
@ ^ [V_49: $i] :
( ( ( V_49 = X1390 )
& ~ X1386 )
| ( ( V_49 = X1388 )
& X1386 ) ) )
= ( setadjoin @ X1390 @ emptyset ) ) ) ) )
=> ( ! [X1392: $i,X1394: $o,X1396: $i] :
( ( in @ X1396 @ X1392 )
=> ! [X1398: $i] :
( ( in @ X1398 @ X1392 )
=> ( X1394
=> ( in @ X1396
@ ( dsetconstr @ X1392
@ ^ [V_50: $i] :
( ( ( V_50 = X1398 )
& ~ X1394 )
| ( ( V_50 = X1396 )
& X1394 ) ) ) ) ) ) )
=> ( ! [X1400: $i,X1402: $o,X1404: $i] :
( ( in @ X1404 @ X1400 )
=> ! [X1406: $i] :
( ( in @ X1406 @ X1400 )
=> ( X1402
=> ( ( dsetconstr @ X1400
@ ^ [V_51: $i] :
( ( ( V_51 = X1406 )
& ~ X1402 )
| ( ( V_51 = X1404 )
& X1402 ) ) )
= ( setadjoin @ X1404 @ emptyset ) ) ) ) )
=> ( ! [X1408: $i,X1410: $o,X1412: $i] :
( ( in @ X1412 @ X1408 )
=> ! [X1414: $i] :
( ( in @ X1414 @ X1408 )
=> ( singleton
@ ( dsetconstr @ X1408
@ ^ [V_52: $i] :
( ( ( V_52 = X1414 )
& ~ X1410 )
| ( ( V_52 = X1412 )
& X1410 ) ) ) ) ) )
=> ( ! [X1416: $i,X1418: $o,X1420: $i] :
( ( in @ X1420 @ X1416 )
=> ! [X1422: $i] :
( ( in @ X1422 @ X1416 )
=> ( in @ ( if @ X1416 @ X1418 @ X1420 @ X1422 ) @ X1416 ) ) )
=> ( ! [X1424: $i] :
( ( singleton @ X1424 )
=> ! [X1426: $i] :
( ( in @ X1426 @ X1424 )
=> ( ( setunion @ X1424 )
= X1426 ) ) )
=> ( ! [X1428: $i,X1430: $o,X1432: $i] :
( ( in @ X1432 @ X1428 )
=> ! [X1434: $i] :
( ( in @ X1434 @ X1428 )
=> ( X1430
=> ( ( if @ X1428 @ X1430 @ X1432 @ X1434 )
= X1432 ) ) ) )
=> ( ! [X1436: $i,X1438: $o,X1440: $i] :
( ( in @ X1440 @ X1436 )
=> ! [X1442: $i] :
( ( in @ X1442 @ X1436 )
=> ( ~ X1438
=> ( ( if @ X1436 @ X1438 @ X1440 @ X1442 )
= X1442 ) ) ) )
=> ( ! [X1444: $i,X1446: $o,X1448: $i] :
( ( in @ X1448 @ X1444 )
=> ! [X1450: $i] :
( ( in @ X1450 @ X1444 )
=> ( in @ ( if @ X1444 @ X1446 @ X1448 @ X1450 ) @ ( setadjoin @ X1448 @ ( setadjoin @ X1450 @ emptyset ) ) ) ) )
=> ( ! [X1452: $i,X1454: $i] :
( ( in @ X1454 @ ( powerset @ X1452 ) )
=> ! [X1456: $i] :
( ( in @ X1456 @ ( powerset @ X1452 ) )
=> ( in @ ( binintersect @ X1454 @ X1456 ) @ ( powerset @ X1452 ) ) ) )
=> ( ! [X1458: $i,X1460: $i] :
( ( in @ X1460 @ ( powerset @ X1458 ) )
=> ! [X1462: $i] :
( ( in @ X1462 @ ( powerset @ X1458 ) )
=> ( in @ ( binunion @ X1460 @ X1462 ) @ ( powerset @ X1458 ) ) ) )
=> ( ! [X1464: $i,X1466: $i] :
( ( in @ X1466 @ ( powerset @ X1464 ) )
=> ( in @ ( powerset @ X1466 ) @ ( powerset @ ( powerset @ X1464 ) ) ) )
=> ( ! [X1468: $i,X1470: $i] :
( ( in @ X1470 @ ( powerset @ X1468 ) )
=> ! [X1472: $i] :
( ( in @ X1472 @ ( powerset @ X1468 ) )
=> ( in @ ( setminus @ X1470 @ X1472 ) @ ( powerset @ X1468 ) ) ) )
=> ( ! [X1474: $i,X1476: $i] :
( ( in @ X1476 @ ( powerset @ X1474 ) )
=> ( in @ ( setminus @ X1474 @ X1476 ) @ ( powerset @ X1474 ) ) )
=> ( ! [X1478: $i,X1480: $i] :
( ( in @ X1480 @ ( powerset @ X1478 ) )
=> ! [X1482: $i] :
( ( in @ X1482 @ ( powerset @ X1478 ) )
=> ( ! [X1484: $i] :
( ( in @ X1484 @ X1478 )
=> ( ( in @ X1484 @ X1480 )
=> ( in @ X1484 @ X1482 ) ) )
=> ( ! [X1486: $i] :
( ( in @ X1486 @ X1478 )
=> ( ( in @ X1486 @ X1482 )
=> ( in @ X1486 @ X1480 ) ) )
=> ( X1480 = X1482 ) ) ) ) )
=> ( ! [X1488: $i,X1490: $i] :
( ( in @ X1490 @ ( powerset @ X1488 ) )
=> ! [X1492: $i] :
( ( in @ X1492 @ ( powerset @ X1488 ) )
=> ( ! [X1494: $i] :
( ( in @ X1494 @ X1488 )
=> ( ( in @ X1494 @ X1490 )
=> ( in @ X1494 @ X1492 ) ) )
=> ( subset @ X1490 @ X1492 ) ) ) )
=> ( ! [X1496: $i,X1498: $i] :
( ( in @ X1498 @ ( powerset @ X1496 ) )
=> ! [X1500: $i] :
( ( in @ X1500 @ ( powerset @ X1496 ) )
=> ( ! [X1502: $i] :
( ( in @ X1502 @ X1496 )
=> ( ( in @ X1502 @ X1498 )
=> ( in @ X1502 @ X1500 ) ) )
=> ( in @ X1498 @ ( powerset @ X1500 ) ) ) ) )
=> ( ! [X1504: $i,X1506: $i] :
( ( in @ X1506 @ ( powerset @ X1504 ) )
=> ! [X1508: $i] :
( ( in @ X1508 @ ( powerset @ X1504 ) )
=> ! [X1510: $i] :
( ( in @ X1510 @ X1504 )
=> ( ( in @ X1506 @ ( powerset @ X1508 ) )
=> ( ( in @ X1510 @ X1506 )
=> ( in @ X1510 @ X1508 ) ) ) ) ) )
=> ( ! [X1512: $i,X1514: $i] :
( ( in @ X1514 @ ( powerset @ X1512 ) )
=> ! [X1516: $i] :
( ( in @ X1516 @ X1512 )
=> ( ( in @ X1516 @ X1514 )
=> ~ ( in @ X1516 @ ( setminus @ X1512 @ X1514 ) ) ) ) )
=> ( ! [X1518: $i,X1520: $i] :
( ( in @ X1520 @ ( powerset @ X1518 ) )
=> ! [X1522: $i] :
( ( in @ X1522 @ X1518 )
=> ( ~ ( in @ X1522 @ ( setminus @ X1518 @ X1520 ) )
=> ( in @ X1522 @ X1520 ) ) ) )
=> ( ! [X1524: $i,X1526: $i] :
( ( in @ X1526 @ ( powerset @ X1524 ) )
=> ! [X1528: $i] :
( ( in @ X1528 @ ( powerset @ X1524 ) )
=> ! [X1530: $i] :
( ( in @ X1530 @ X1524 )
=> ( ~ ( in @ X1530 @ X1526 )
=> ~ ( in @ X1530 @ ( binintersect @ X1526 @ X1528 ) ) ) ) ) )
=> ( ! [X1532: $i,X1534: $i] :
( ( in @ X1534 @ ( powerset @ X1532 ) )
=> ! [X1536: $i] :
( ( in @ X1536 @ ( powerset @ X1532 ) )
=> ! [X1538: $i] :
( ( in @ X1538 @ X1532 )
=> ( ~ ( in @ X1538 @ X1536 )
=> ~ ( in @ X1538 @ ( binintersect @ X1534 @ X1536 ) ) ) ) ) )
=> ( ! [X1540: $i,X1542: $i] :
( ( in @ X1542 @ ( powerset @ X1540 ) )
=> ! [X1544: $i] :
( ( in @ X1544 @ ( powerset @ X1540 ) )
=> ! [X1546: $i] :
( ( in @ X1546 @ X1540 )
=> ( ( subset @ X1542 @ ( setminus @ X1540 @ X1544 ) )
=> ( ( in @ X1546 @ X1544 )
=> ~ ( in @ X1546 @ X1542 ) ) ) ) ) )
=> ( ! [X1548: $i,X1550: $i] :
( ( in @ X1550 @ ( powerset @ X1548 ) )
=> ! [X1552: $i] :
( ( in @ X1552 @ ( powerset @ X1548 ) )
=> ! [X1554: $i] :
( ( in @ X1554 @ X1548 )
=> ( ( subset @ X1550 @ X1552 )
=> ( ~ ( in @ X1554 @ X1552 )
=> ~ ( in @ X1554 @ X1550 ) ) ) ) ) )
=> ( ! [X1556: $i,X1558: $i] :
( ( in @ X1558 @ ( powerset @ X1556 ) )
=> ! [X1560: $i] :
( ( in @ X1560 @ ( powerset @ X1556 ) )
=> ( ( subset @ X1558 @ X1560 )
=> ( subset @ ( setminus @ X1556 @ X1560 ) @ ( setminus @ X1556 @ X1558 ) ) ) ) )
=> ( ! [X1562: $i,X1564: $i] :
( ( in @ X1564 @ ( powerset @ X1562 ) )
=> ! [X1566: $i] :
( ( in @ X1566 @ ( powerset @ X1562 ) )
=> ( ( subset @ ( setminus @ X1562 @ X1566 ) @ ( setminus @ X1562 @ X1564 ) )
=> ( subset @ X1564 @ X1566 ) ) ) )
=> ( ! [X1568: $i,X1570: $i] :
( ( in @ X1570 @ ( powerset @ X1568 ) )
=> ! [X1572: $i] :
( ( in @ X1572 @ X1568 )
=> ( ( in @ X1572 @ X1570 )
=> ( in @ X1572 @ ( setminus @ X1568 @ ( setminus @ X1568 @ X1570 ) ) ) ) ) )
=> ( ! [X1574: $i,X1576: $i] :
( ( in @ X1576 @ ( powerset @ X1574 ) )
=> ! [X1578: $i] :
( ( in @ X1578 @ X1574 )
=> ( ( in @ X1578 @ ( setminus @ X1574 @ ( setminus @ X1574 @ X1576 ) ) )
=> ( in @ X1578 @ X1576 ) ) ) )
=> ( ! [X1580: $i,X1582: $i] :
( ( in @ X1582 @ ( powerset @ X1580 ) )
=> ( subset @ X1582 @ ( setminus @ X1580 @ ( setminus @ X1580 @ X1582 ) ) ) )
=> ( ! [X1584: $i,X1586: $i] :
( ( in @ X1586 @ ( powerset @ X1584 ) )
=> ( subset @ ( setminus @ X1584 @ ( setminus @ X1584 @ X1586 ) ) @ X1586 ) )
=> ( ! [X1588: $i,X1590: $i] :
( ( in @ X1590 @ ( powerset @ X1588 ) )
=> ( X1590
= ( setminus @ X1588 @ ( setminus @ X1588 @ X1590 ) ) ) )
=> ( ! [X1592: $i,X1594: $i] :
( ( in @ X1594 @ ( powerset @ X1592 ) )
=> ! [X1596: $i] :
( ( in @ X1596 @ ( powerset @ X1592 ) )
=> ! [X1598: $i] :
( ( in @ X1598 @ X1592 )
=> ( ( in @ X1598 @ ( setminus @ X1592 @ X1594 ) )
=> ~ ( in @ X1598 @ ( binintersect @ X1594 @ X1596 ) ) ) ) ) )
=> ( ! [X1600: $i,X1602: $i] :
( ( in @ X1602 @ ( powerset @ X1600 ) )
=> ! [X1604: $i] :
( ( in @ X1604 @ ( powerset @ X1600 ) )
=> ! [X1606: $i] :
( ( in @ X1606 @ X1600 )
=> ( ( in @ X1606 @ ( setminus @ X1600 @ X1602 ) )
=> ( in @ X1606 @ ( setminus @ X1600 @ ( binintersect @ X1602 @ X1604 ) ) ) ) ) ) )
=> ( ! [X1608: $i,X1610: $i] :
( ( in @ X1610 @ ( powerset @ X1608 ) )
=> ! [X1612: $i] :
( ( in @ X1612 @ ( powerset @ X1608 ) )
=> ( subset @ ( setminus @ X1608 @ X1610 ) @ ( setminus @ X1608 @ ( binintersect @ X1610 @ X1612 ) ) ) ) )
=> ( ! [X1614: $i,X1616: $i] :
( ( in @ X1616 @ ( powerset @ X1614 ) )
=> ! [X1618: $i] :
( ( in @ X1618 @ ( powerset @ X1614 ) )
=> ( in @ ( setminus @ X1614 @ X1616 ) @ ( powerset @ ( setminus @ X1614 @ ( binintersect @ X1616 @ X1618 ) ) ) ) ) )
=> ( ! [X1620: $i,X1622: $i] :
( ( in @ X1622 @ ( powerset @ X1620 ) )
=> ! [X1624: $i] :
( ( in @ X1624 @ ( powerset @ X1620 ) )
=> ( ( subset @ X1622 @ ( setminus @ X1620 @ X1624 ) )
=> ! [X1626: $i] :
( ( in @ X1626 @ X1620 )
=> ( ( in @ X1626 @ X1624 )
=> ( in @ X1626 @ ( setminus @ X1620 @ X1622 ) ) ) ) ) ) )
=> ( ! [X1628: $i,X1630: $i] :
( ( in @ X1630 @ ( powerset @ X1628 ) )
=> ! [X1632: $i] :
( ( in @ X1632 @ ( powerset @ X1628 ) )
=> ( ( subset @ X1630 @ ( setminus @ X1628 @ X1632 ) )
=> ( subset @ X1632 @ ( setminus @ X1628 @ X1630 ) ) ) ) )
=> ( ! [X1634: $i,X1636: $i] :
( ( in @ X1636 @ ( powerset @ X1634 ) )
=> ! [X1638: $i] :
( ( in @ X1638 @ ( powerset @ X1634 ) )
=> ! [X1640: $i] :
( ( in @ X1640 @ X1634 )
=> ( ~ ( in @ X1640 @ ( binunion @ X1636 @ X1638 ) )
=> ~ ( in @ X1640 @ X1636 ) ) ) ) )
=> ( ! [X1642: $i,X1644: $i] :
( ( in @ X1644 @ ( powerset @ X1642 ) )
=> ! [X1646: $i] :
( ( in @ X1646 @ ( powerset @ X1642 ) )
=> ! [X1648: $i] :
( ( in @ X1648 @ X1642 )
=> ( ~ ( in @ X1648 @ ( binunion @ X1644 @ X1646 ) )
=> ~ ( in @ X1648 @ X1646 ) ) ) ) )
=> ( ! [X1650: $i,X1652: $i] :
( ( in @ X1652 @ ( powerset @ X1650 ) )
=> ! [X1654: $i] :
( ( in @ X1654 @ ( powerset @ X1650 ) )
=> ! [X1656: $i] :
( ( in @ X1656 @ ( powerset @ X1650 ) )
=> ! [X1658: $i] :
( ( in @ X1658 @ X1650 )
=> ( ( in @ X1658 @ ( binintersect @ X1652 @ X1654 ) )
=> ( in @ X1658 @ ( binunion @ X1652 @ X1656 ) ) ) ) ) ) )
=> ( ! [X1660: $i,X1662: $i] :
( ( in @ X1662 @ ( powerset @ X1660 ) )
=> ! [X1664: $i] :
( ( in @ X1664 @ ( powerset @ X1660 ) )
=> ! [X1666: $i] :
( ( in @ X1666 @ ( powerset @ X1660 ) )
=> ! [X1668: $i] :
( ( in @ X1668 @ X1660 )
=> ( ( in @ X1668 @ ( binintersect @ X1662 @ X1664 ) )
=> ( in @ X1668 @ ( binunion @ X1664 @ X1666 ) ) ) ) ) ) )
=> ( ! [X1670: $i,X1672: $i] :
( ( in @ X1672 @ ( powerset @ X1670 ) )
=> ! [X1674: $i] :
( ( in @ X1674 @ ( powerset @ X1670 ) )
=> ! [X1676: $i] :
( ( in @ X1676 @ ( powerset @ X1670 ) )
=> ! [X1678: $i] :
( ( in @ X1678 @ X1670 )
=> ( ( in @ X1678 @ ( binintersect @ X1672 @ X1674 ) )
=> ( in @ X1678 @ ( binintersect @ ( binunion @ X1672 @ X1676 ) @ ( binunion @ X1674 @ X1676 ) ) ) ) ) ) ) )
=> ( ! [X1680: $i,X1682: $i] :
( ( in @ X1682 @ ( powerset @ X1680 ) )
=> ! [X1684: $i] :
( ( in @ X1684 @ ( powerset @ X1680 ) )
=> ! [X1686: $i] :
( ( in @ X1686 @ ( powerset @ X1680 ) )
=> ( in @ ( binintersect @ X1682 @ X1684 ) @ ( powerset @ ( binintersect @ ( binunion @ X1682 @ X1686 ) @ ( binunion @ X1684 @ X1686 ) ) ) ) ) ) )
=> ( ! [X1688: $i,X1690: $i] :
( ( in @ X1690 @ ( powerset @ X1688 ) )
=> ! [X1692: $i] :
( ( in @ X1692 @ ( powerset @ X1688 ) )
=> ! [X1694: $i] :
( ( in @ X1694 @ X1688 )
=> ( ( in @ X1694 @ ( setminus @ X1688 @ ( binunion @ X1690 @ X1692 ) ) )
=> ~ ( in @ X1694 @ X1690 ) ) ) ) )
=> ( ! [X1696: $i,X1698: $i] :
( ( in @ X1698 @ ( powerset @ X1696 ) )
=> ! [X1700: $i] :
( ( in @ X1700 @ ( powerset @ X1696 ) )
=> ! [X1702: $i] :
( ( in @ X1702 @ X1696 )
=> ( ( in @ X1702 @ ( setminus @ X1696 @ ( binunion @ X1698 @ X1700 ) ) )
=> ( in @ X1702 @ ( setminus @ X1696 @ X1698 ) ) ) ) ) )
=> ( ! [X1704: $i,X1706: $i] :
( ( in @ X1706 @ ( powerset @ X1704 ) )
=> ! [X1708: $i] :
( ( in @ X1708 @ ( powerset @ X1704 ) )
=> ! [X1710: $o,X1712: $i] :
( ( in @ X1712 @ X1704 )
=> ( ( in @ X1712 @ ( binunion @ X1706 @ X1708 ) )
=> ( ( ( in @ X1712 @ X1706 )
=> X1710 )
=> ( ( ( in @ X1712 @ X1708 )
=> X1710 )
=> X1710 ) ) ) ) ) )
=> ( ! [X1714: $i,X1716: $i] :
( ( in @ X1716 @ ( powerset @ X1714 ) )
=> ! [X1718: $i] :
( ( in @ X1718 @ ( powerset @ X1714 ) )
=> ! [X1720: $i] :
( ( in @ X1720 @ X1714 )
=> ( ~ ( in @ X1720 @ X1716 )
=> ( ~ ( in @ X1720 @ X1718 )
=> ~ ( in @ X1720 @ ( binunion @ X1716 @ X1718 ) ) ) ) ) ) )
=> ( ! [X1722: $i,X1724: $i] :
( ( in @ X1724 @ ( powerset @ X1722 ) )
=> ! [X1726: $i] :
( ( in @ X1726 @ ( powerset @ X1722 ) )
=> ! [X1728: $i] :
( ( in @ X1728 @ X1722 )
=> ( ( in @ X1728 @ ( setminus @ X1722 @ ( binunion @ X1724 @ X1726 ) ) )
=> ( in @ X1728 @ ( setminus @ X1722 @ X1724 ) ) ) ) ) )
=> ( ! [X1730: $i,X1732: $i] :
( ( in @ X1732 @ ( powerset @ X1730 ) )
=> ! [X1734: $i] :
( ( in @ X1734 @ ( powerset @ X1730 ) )
=> ( in @ ( setminus @ X1730 @ ( binunion @ X1732 @ X1734 ) ) @ ( powerset @ ( setminus @ X1730 @ X1732 ) ) ) ) )
=> ( ! [X1736: $i,X1738: $i] :
( ( in @ X1738 @ ( powerset @ X1736 ) )
=> ! [X1740: $i] :
( ( in @ X1740 @ ( powerset @ X1736 ) )
=> ! [X1742: $i] :
( ( in @ X1742 @ X1736 )
=> ( ( in @ X1742 @ ( setminus @ X1736 @ ( binunion @ X1738 @ X1740 ) ) )
=> ( in @ X1742 @ ( setminus @ X1736 @ X1740 ) ) ) ) ) )
=> ( ! [X1744: $i,X1746: $i] :
( ( in @ X1746 @ ( powerset @ X1744 ) )
=> ! [X1748: $i] :
( ( in @ X1748 @ ( powerset @ X1744 ) )
=> ! [X1750: $i] :
( ( in @ X1750 @ X1744 )
=> ( ( in @ X1750 @ ( setminus @ X1744 @ ( binintersect @ X1746 @ X1748 ) ) )
=> ( in @ X1750 @ ( binunion @ ( setminus @ X1744 @ X1746 ) @ ( setminus @ X1744 @ X1748 ) ) ) ) ) ) )
=> ( ! [X1752: $i,X1754: $i] :
( ( in @ X1754 @ ( powerset @ X1752 ) )
=> ! [X1756: $i] :
( ( in @ X1756 @ ( powerset @ X1752 ) )
=> ! [X1758: $i] :
( ( in @ X1758 @ X1752 )
=> ( ( in @ X1758 @ ( binunion @ ( setminus @ X1752 @ X1754 ) @ ( setminus @ X1752 @ X1756 ) ) )
=> ( in @ X1758 @ ( setminus @ X1752 @ ( binintersect @ X1754 @ X1756 ) ) ) ) ) ) )
=> ( ! [X1760: $i,X1762: $i] :
( ( in @ X1762 @ ( powerset @ X1760 ) )
=> ! [X1764: $i] :
( ( in @ X1764 @ ( powerset @ X1760 ) )
=> ( ( setminus @ X1760 @ ( binintersect @ X1762 @ X1764 ) )
= ( binunion @ ( setminus @ X1760 @ X1762 ) @ ( setminus @ X1760 @ X1764 ) ) ) ) )
=> ( ! [X1766: $i,X1768: $i] :
( ( in @ X1768 @ ( powerset @ X1766 ) )
=> ! [X1770: $i] :
( ( in @ X1770 @ ( powerset @ X1766 ) )
=> ! [X1772: $i] :
( ( in @ X1772 @ X1766 )
=> ( ( in @ X1772 @ ( setminus @ X1766 @ ( binunion @ X1768 @ X1770 ) ) )
=> ( in @ X1772 @ ( binintersect @ ( setminus @ X1766 @ X1768 ) @ ( setminus @ X1766 @ X1770 ) ) ) ) ) ) )
=> ( ! [X1774: $i,X1776: $i] :
( ( in @ X1776 @ ( powerset @ X1774 ) )
=> ! [X1778: $i] :
( ( in @ X1778 @ ( powerset @ X1774 ) )
=> ! [X1780: $i] :
( ( in @ X1780 @ X1774 )
=> ( ( in @ X1780 @ ( setminus @ X1774 @ X1776 ) )
=> ( ( in @ X1780 @ ( setminus @ X1774 @ X1778 ) )
=> ( in @ X1780 @ ( setminus @ X1774 @ ( binunion @ X1776 @ X1778 ) ) ) ) ) ) ) )
=> ( ! [X1782: $i,X1784: $i] :
( ( in @ X1784 @ ( powerset @ X1782 ) )
=> ! [X1786: $i] :
( ( in @ X1786 @ ( powerset @ X1782 ) )
=> ! [X1788: $i] :
( ( in @ X1788 @ X1782 )
=> ( ( in @ X1788 @ ( binintersect @ ( setminus @ X1782 @ X1784 ) @ ( setminus @ X1782 @ X1786 ) ) )
=> ( in @ X1788 @ ( setminus @ X1782 @ ( binunion @ X1784 @ X1786 ) ) ) ) ) ) )
=> ( ! [X1790: $i,X1792: $i] :
( ( in @ X1792 @ ( powerset @ X1790 ) )
=> ! [X1794: $i] :
( ( in @ X1794 @ ( powerset @ X1790 ) )
=> ( ( setminus @ X1790 @ ( binunion @ X1792 @ X1794 ) )
= ( binintersect @ ( setminus @ X1790 @ X1792 ) @ ( setminus @ X1790 @ X1794 ) ) ) ) )
=> ( ! [X1796: $i,X1798: $i] :
( ( in @ X1798 @ ( powerset @ X1796 ) )
=> ! [X1800: $i] :
( ( in @ X1800 @ ( powerset @ X1796 ) )
=> ! [X1802: $i] :
( ( in @ X1802 @ ( binintersect @ X1798 @ X1800 ) )
=> ( in @ X1802 @ X1796 ) ) ) )
=> ( ! [X1804: $i,X1806: $i] :
( ( in @ X1806 @ ( powerset @ X1804 ) )
=> ! [X1808: $i] :
( ( in @ X1808 @ ( powerset @ X1804 ) )
=> ! [X1810: $i] :
( ( in @ X1810 @ ( powerset @ X1804 ) )
=> ( ( subset @ X1806 @ X1810 )
=> ( subset @ ( binintersect @ X1806 @ X1808 ) @ X1810 ) ) ) ) )
=> ( ! [X1812: $i,X1814: $i] :
( ( in @ X1814 @ ( powerset @ X1812 ) )
=> ! [X1816: $i] :
( ( in @ X1816 @ ( powerset @ X1812 ) )
=> ! [X1818: $i] :
( ( in @ X1818 @ ( powerset @ X1812 ) )
=> ( ( subset @ X1816 @ X1818 )
=> ( subset @ ( binintersect @ X1814 @ X1816 ) @ X1818 ) ) ) ) )
=> ( ! [X1820: $i,X1822: $i] :
( ( in @ X1822 @ ( powerset @ X1820 ) )
=> ! [X1824: $i] :
( ( in @ X1824 @ ( powerset @ X1820 ) )
=> ! [X1826: $i] :
( ( in @ X1826 @ ( powerset @ X1820 ) )
=> ( ( subset @ X1822 @ X1824 )
=> ( ( subset @ X1822 @ X1826 )
=> ( subset @ X1822 @ ( binintersect @ X1824 @ X1826 ) ) ) ) ) ) )
=> ( ! [X1828: $i,X1830: $i] :
( ( in @ X1830 @ ( powerset @ X1828 ) )
=> ! [X1832: $i] :
( ( in @ X1832 @ ( powerset @ X1828 ) )
=> ! [X1834: $i] :
( ( in @ X1834 @ ( powerset @ X1828 ) )
=> ! [X1836: $i] :
( ( in @ X1836 @ ( powerset @ X1828 ) )
=> ( ( subset @ X1830 @ X1834 )
=> ( ( subset @ X1832 @ X1836 )
=> ( subset @ ( binintersect @ X1830 @ X1832 ) @ ( binintersect @ X1834 @ X1836 ) ) ) ) ) ) ) )
=> ( ! [X1838: $i,X1840: $i] :
( ( in @ X1840 @ ( powerset @ X1838 ) )
=> ! [X1842: $i] :
( ( in @ X1842 @ ( powerset @ X1838 ) )
=> ( in @ ( setminus @ X1838 @ X1840 ) @ ( powerset @ ( setminus @ X1838 @ ( binintersect @ X1840 @ X1842 ) ) ) ) ) )
=> ( ! [X1844: $i,X1846: $i] :
( ( in @ X1846 @ ( powerset @ X1844 ) )
=> ! [X1848: $i] :
( ( in @ X1848 @ ( powerset @ X1844 ) )
=> ! [X1850: $i] :
( ( in @ X1850 @ ( powerset @ X1844 ) )
=> ! [X1852: $i] :
( ( in @ X1852 @ ( powerset @ X1844 ) )
=> ( ( setminus @ X1844 @ ( binintersect @ ( binunion @ X1846 @ X1848 ) @ ( binunion @ X1850 @ X1852 ) ) )
= ( binunion @ ( binintersect @ ( setminus @ X1844 @ X1846 ) @ ( setminus @ X1844 @ X1848 ) ) @ ( binintersect @ ( setminus @ X1844 @ X1850 ) @ ( setminus @ X1844 @ X1852 ) ) ) ) ) ) ) )
=> ( ! [X1854: $i,X1856: $i] :
( ( in @ X1856 @ ( powerset @ X1854 ) )
=> ! [X1858: $i] :
( ( in @ X1858 @ ( powerset @ X1854 ) )
=> ! [X1860: $i] :
( ( in @ X1860 @ ( powerset @ X1854 ) )
=> ( in @ ( binintersect @ X1856 @ X1858 ) @ ( powerset @ ( binintersect @ ( binunion @ X1856 @ X1860 ) @ ( binunion @ X1858 @ X1860 ) ) ) ) ) ) )
=> ( ! [X1862: $i,X1864: $i] :
( ( in @ X1864 @ ( powerset @ X1862 ) )
=> ! [X1866: $i] :
( ( in @ X1866 @ ( powerset @ X1862 ) )
=> ( ( subset @ X1864 @ ( setminus @ X1862 @ X1866 ) )
=> ( subset @ X1866 @ ( setminus @ X1862 @ X1864 ) ) ) ) )
=> ( ! [X1868: $i,X1870: $i] :
( ( in @ X1870 @ ( powerset @ X1868 ) )
=> ! [X1872: $i] :
( ( in @ X1872 @ ( powerset @ X1868 ) )
=> ( in @ ( setminus @ X1868 @ ( binunion @ X1870 @ X1872 ) ) @ ( powerset @ ( setminus @ X1868 @ X1870 ) ) ) ) )
=> ( ! [X1874: $i,X1876: $i] :
( ( subset @ X1876 @ ( cartprod @ X1874 @ X1874 ) )
=> ! [X1878: $i > $o] :
( ! [X1880: $i] :
( ( in @ X1880 @ X1874 )
=> ! [X1882: $i] :
( ( in @ X1882 @ X1874 )
=> ( ( in @ ( kpair @ X1880 @ X1882 ) @ X1876 )
=> ( X1878 @ ( kpair @ X1880 @ X1882 ) ) ) ) )
=> ! [X1884: $i] :
( ( in @ X1884 @ X1876 )
=> ( X1878 @ X1884 ) ) ) )
=> ( ! [X1886: $i,X1888: $i] :
( ( in @ X1888 @ ( breln1Set @ X1886 ) )
=> ( subset @ X1888 @ ( cartprod @ X1886 @ X1886 ) ) )
=> ( ! [X1890: $i,X1892: $i,X1894: $i > $i > $o] :
( ! [X1896: $i] :
( ( in @ X1896 @ X1890 )
=> ? [X1898: $i] :
( ( in @ X1898 @ X1892 )
& ( X1894 @ X1896 @ X1898 ) ) )
=> ! [X1900: $i] :
( ( in @ X1900 @ ( breln1Set @ X1892 ) )
=> ( ( reflwellordering @ X1892 @ X1900 )
=> ! [X1902: $i] :
( ( in @ X1902 @ X1890 )
=> ( singleton
@ ( dsetconstr @ X1892
@ ^ [V_53: $i] :
( ! [X1904: $i] :
( ( in @ X1904 @ X1892 )
=> ( ( X1894 @ X1902 @ X1904 )
=> ( in @ ( kpair @ V_53 @ X1904 ) @ X1900 ) ) )
& ( X1894 @ X1902 @ V_53 ) ) ) ) ) ) ) )
=> ( ! [X1906: $i,X1908: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1906 @ X1906
@ ^ [V_54: $i,V_55: $i] : ( X1908 @ V_54 @ V_55 ) )
@ ( cartprod @ X1906 @ X1906 ) )
=> ( ! [X1910: $i,X1912: $i] :
( ( subset @ X1912 @ ( cartprod @ X1910 @ X1910 ) )
=> ! [X1914: $i > $o] :
( ! [X1916: $i] :
( ( in @ X1916 @ X1910 )
=> ! [X1918: $i] :
( ( in @ X1918 @ X1910 )
=> ( ( in @ ( kpair @ X1916 @ X1918 ) @ X1912 )
=> ( X1914 @ ( kpair @ X1916 @ X1918 ) ) ) ) )
=> ! [X1920: $i] :
( ( in @ X1920 @ X1912 )
=> ( X1914 @ X1920 ) ) ) )
=> ( ! [X1922: $i,X1924: $i] :
( ( subset @ X1924 @ ( cartprod @ X1922 @ X1922 ) )
=> ! [X1926: $i] :
( ( subset @ X1926 @ ( cartprod @ X1922 @ X1922 ) )
=> ( ! [X1928: $i] :
( ( in @ X1928 @ X1922 )
=> ! [X1930: $i] :
( ( in @ X1930 @ X1922 )
=> ( ( in @ ( kpair @ X1928 @ X1930 ) @ X1924 )
=> ( in @ ( kpair @ X1928 @ X1930 ) @ X1926 ) ) ) )
=> ( subset @ X1924 @ X1926 ) ) ) )
=> ( ! [X1932: $i,X1934: $i] :
( ( subset @ X1934 @ ( cartprod @ X1932 @ X1932 ) )
=> ! [X1936: $i] :
( ( subset @ X1936 @ ( cartprod @ X1932 @ X1932 ) )
=> ( ! [X1938: $i] :
( ( in @ X1938 @ X1932 )
=> ! [X1940: $i] :
( ( in @ X1940 @ X1932 )
=> ( ( in @ ( kpair @ X1938 @ X1940 ) @ X1934 )
=> ( in @ ( kpair @ X1938 @ X1940 ) @ X1936 ) ) ) )
=> ( ! [X1942: $i] :
( ( in @ X1942 @ X1932 )
=> ! [X1944: $i] :
( ( in @ X1944 @ X1932 )
=> ( ( in @ ( kpair @ X1942 @ X1944 ) @ X1936 )
=> ( in @ ( kpair @ X1942 @ X1944 ) @ X1934 ) ) ) )
=> ( X1934 = X1936 ) ) ) ) )
=> ( ! [X1946: $i,X1948: $i] :
( ( subset @ X1948 @ ( cartprod @ X1946 @ X1946 ) )
=> ( subset
@ ( dpsetconstr @ X1946 @ X1946
@ ^ [V_56: $i,V_57: $i] : ( in @ ( kpair @ V_57 @ V_56 ) @ X1948 ) )
@ ( cartprod @ X1946 @ X1946 ) ) )
=> ( ! [X1950: $i,X1952: $i] :
( ( subset @ X1952 @ ( cartprod @ X1950 @ X1950 ) )
=> ! [X1954: $i] :
( ( in @ X1954 @ X1950 )
=> ! [X1956: $i] :
( ( in @ X1956 @ X1950 )
=> ( ( in @ ( kpair @ X1954 @ X1956 ) @ X1952 )
=> ( in @ ( kpair @ X1956 @ X1954 )
@ ( dpsetconstr @ X1950 @ X1950
@ ^ [V_58: $i,V_59: $i] : ( in @ ( kpair @ V_59 @ V_58 ) @ X1952 ) ) ) ) ) ) )
=> ! [X1958: $i,X1960: $i] :
( ( subset @ X1960 @ ( cartprod @ X1958 @ X1958 ) )
=> ! [X1962: $i] :
( ( in @ X1962 @ X1958 )
=> ! [X1964: $i] :
( ( in @ X1964 @ X1958 )
=> ( ( in @ ( kpair @ X1964 @ X1962 )
@ ( dpsetconstr @ X1958 @ X1958
@ ^ [V_60: $i,V_61: $i] : ( in @ ( kpair @ V_61 @ V_60 ) @ X1960 ) ) )
=> ( in @ ( kpair @ X1962 @ X1964 ) @ X1960 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(zf_stmt_1,type,
zip_tseitin_5: $i > ( $i > $o ) > $o ).
thf(zf_stmt_2,axiom,
! [X422: $i,X414: $i > $o] :
( ( zip_tseitin_5 @ X422 @ X414 )
<=> ( ! [X424: $i] : ( zip_tseitin_4 @ X424 @ X422 @ X414 )
& ( X414 @ X422 ) ) ) ).
thf(zf_stmt_3,type,
zip_tseitin_4: $i > $i > ( $i > $o ) > $o ).
thf(zf_stmt_4,axiom,
! [X424: $i,X422: $i,X414: $i > $o] :
( ( zip_tseitin_4 @ X424 @ X422 @ X414 )
<=> ( ( X414 @ X424 )
=> ( X422 = X424 ) ) ) ).
thf(zf_stmt_5,type,
zip_tseitin_3: $i > $i > $i > ( $i > $i > $o ) > $o ).
thf(zf_stmt_6,axiom,
! [X52: $i,X50: $i,X40: $i,X38: $i > $i > $o] :
( ( zip_tseitin_3 @ X52 @ X50 @ X40 @ X38 )
<=> ( ( X38 @ X52 @ X50 )
& ( in @ X52 @ X40 ) ) ) ).
thf(zf_stmt_7,type,
zip_tseitin_2: $i > $i > $i > $o ).
thf(zf_stmt_8,axiom,
! [X28: $i,X26: $i,X24: $i] :
( ( zip_tseitin_2 @ X28 @ X26 @ X24 )
<=> ( ( in @ X28 @ X24 )
& ( in @ X26 @ X28 ) ) ) ).
thf(zf_stmt_9,type,
zip_tseitin_1: $i > $i > $i > $o ).
thf(zf_stmt_10,axiom,
! [X22: $i,X20: $i,X18: $i] :
( ( zip_tseitin_1 @ X22 @ X20 @ X18 )
<=> ( ( in @ X22 @ X20 )
=> ( in @ X22 @ X18 ) ) ) ).
thf(zf_stmt_11,type,
zip_tseitin_0: $i > $i > $i > $o ).
thf(zf_stmt_12,axiom,
! [X16: $i,X14: $i,X12: $i] :
( ( zip_tseitin_0 @ X16 @ X14 @ X12 )
<=> ( ( X16 = X12 )
| ( in @ X16 @ X14 ) ) ) ).
thf(zf_stmt_13,conjecture,
( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
<=> ( in @ X8 @ X6 ) )
=> ( X4 = X6 ) )
=> ( ! [X10: $i] :
~ ( in @ X10 @ emptyset )
=> ( ! [X12: $i,X14: $i,X16: $i] :
( ( in @ X16 @ ( setadjoin @ X12 @ X14 ) )
<=> ( zip_tseitin_0 @ X16 @ X14 @ X12 ) )
=> ( ! [X18: $i,X20: $i] :
( ( in @ X20 @ ( powerset @ X18 ) )
<=> ! [X22: $i] : ( zip_tseitin_1 @ X22 @ X20 @ X18 ) )
=> ( ! [X24: $i,X26: $i] :
( ( in @ X26 @ ( setunion @ X24 ) )
<=> ? [X28: $i] : ( zip_tseitin_2 @ X28 @ X26 @ X24 ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X30: $i] :
( ( in @ X30 @ omega )
=> ( in @ ( setadjoin @ X30 @ X30 ) @ omega ) )
=> ( ! [X32: $i] :
( ( ( in @ emptyset @ X32 )
& ! [X34: $i] :
( ( ( in @ X34 @ omega )
& ( in @ X34 @ X32 ) )
=> ( in @ ( setadjoin @ X34 @ X34 ) @ X32 ) ) )
=> ! [X36: $i] :
( ( in @ X36 @ omega )
=> ( in @ X36 @ X32 ) ) )
=> ( ! [X38: $i > $i > $o,X40: $i] :
( ! [X42: $i] :
( ( in @ X42 @ X40 )
=> ? [X44: $i] :
( ! [X46: $i] :
( ( X38 @ X42 @ X46 )
=> ( X44 = X46 ) )
& ( X38 @ X42 @ X44 ) ) )
=> ? [X48: $i] :
! [X50: $i] :
( ( in @ X50 @ X48 )
<=> ? [X52: $i] : ( zip_tseitin_3 @ X52 @ X50 @ X40 @ X38 ) ) )
=> ( ! [X54: $i] :
( ? [X56: $i] : ( in @ X56 @ X54 )
=> ? [X58: $i] :
( ~ ? [X60: $i] :
( ( in @ X60 @ X54 )
& ( in @ X60 @ X58 ) )
& ( in @ X58 @ X54 ) ) )
=> ( ! [X62: $i] :
? [X64: $i] :
( ! [X66: $i] :
( ( ! [X68: $i] :
( ( in @ X68 @ X66 )
=> ( in @ X68 @ X62 ) )
& ? [X70: $i] : ( in @ X70 @ X66 ) )
=> ? [X72: $i,X74: $i] :
( ! [X76: $i] :
( ( in @ X76 @ X64 )
=> ( ! [X78: $i] :
( ( in @ X78 @ X76 )
=> ( in @ X78 @ X72 ) )
| ( in @ X74 @ X76 ) ) )
& ~ ? [X80: $i] :
( ( in @ X80 @ X66 )
& ( in @ X80 @ X72 ) )
& ( in @ X74 @ X66 )
& ( in @ X72 @ X64 ) ) )
& ! [X82: $i,X84: $i] :
( ( ( in @ X82 @ X64 )
& ( in @ X84 @ X64 ) )
=> ( ! [X86: $i] :
( ( in @ X86 @ X82 )
=> ( in @ X86 @ X84 ) )
| ! [X88: $i] :
( ( in @ X88 @ X84 )
=> ( in @ X88 @ X82 ) ) ) )
& ! [X90: $i,X92: $i] :
( ( ( in @ X90 @ X62 )
& ( in @ X92 @ X62 ) )
=> ( ! [X94: $i] :
( ( in @ X94 @ X64 )
=> ( ( in @ X90 @ X94 )
<=> ( in @ X92 @ X94 ) ) )
=> ( X90 = X92 ) ) )
& ! [X96: $i] :
( ( in @ X96 @ X64 )
=> ! [X98: $i] :
( ( in @ X98 @ X96 )
=> ( in @ X98 @ X62 ) ) ) )
=> ( ! [X100: $i > $o] :
( ? [X102: $i] :
( ! [X104: $i] :
( ( X100 @ X104 )
=> ( X102 = X104 ) )
& ( X100 @ X102 ) )
=> ( X100
@ ( descr
@ ^ [V_1: $i] : ( X100 @ V_1 ) ) ) )
=> ( ! [X106: $i,X108: $i > $o,X110: $i] :
( ( in @ X110 @ X106 )
=> ( ( X108 @ X110 )
=> ( in @ X110
@ ( dsetconstr @ X106
@ ^ [V_2: $i] : ( X108 @ V_2 ) ) ) ) )
=> ( ! [X112: $i,X114: $i > $o,X116: $i] :
( ( in @ X116
@ ( dsetconstr @ X112
@ ^ [V_3: $i] : ( X114 @ V_3 ) ) )
=> ( in @ X116 @ X112 ) )
=> ( ! [X118: $i,X120: $i > $o,X122: $i] :
( ( in @ X122
@ ( dsetconstr @ X118
@ ^ [V_4: $i] : ( X120 @ V_4 ) ) )
=> ( X120 @ X122 ) )
=> ( ! [X124: $i > $o] :
( ? [X126: $i] :
( ! [X128: $i] :
( ( X124 @ X128 )
=> ( X126 = X128 ) )
& ( X124 @ X126 ) )
=> ? [X130: $i] :
( ! [X132: $i] :
( ( X124 @ X132 )
=> ( X130 = X132 ) )
& ( X124 @ X130 ) ) )
=> ( ! [X134: $o,X136: $i] :
( ( in @ X136 @ ( prop2set @ X134 ) )
=> X134 )
=> ( ! [X138: $i] :
( ( in @ X138 @ emptyset )
=> ! [X140: $o] : X140 )
=> ( ! [X142: $i] :
~ ( in @ X142 @ emptyset )
=> ( ! [X144: $i] :
~ ( in @ X144 @ emptyset )
=> ( ! [X146: $i > $o] :
( ? [X148: $i] :
( ! [X150: $i] :
( ( X146 @ X150 )
=> ( X148 = X150 ) )
& ( X146 @ X148 ) )
=> ? [X152: $i] : ( X146 @ X152 ) )
=> ( ! [X154: $i,X156: $i] :
( ! [X158: $i] :
( ( in @ X158 @ X154 )
=> ( in @ X158 @ X156 ) )
=> ( ! [X160: $i] :
( ( in @ X160 @ X156 )
=> ( in @ X160 @ X154 ) )
=> ( X154 = X156 ) ) )
=> ( ! [X162: $i] :
( ! [X164: $i] :
~ ( in @ X164 @ X162 )
=> ( X162 = emptyset ) )
=> ( ! [X166: $i] :
( ! [X168: $i] :
~ ( in @ X168 @ X166 )
=> ( X166 = emptyset ) )
=> ( ! [X170: $i,X172: $i > $o,X174: $i] :
( ( in @ X174 @ X170 )
=> ( ( in @ X174
@ ( dsetconstr @ X170
@ ^ [V_5: $i] : ( X172 @ V_5 ) ) )
<=> ( X172 @ X174 ) ) )
=> ( ! [X176: $i] :
( ( X176 != emptyset )
=> ? [X178: $i] : ( in @ X178 @ X176 ) )
=> ( ! [X180: $i,X182: $i > $o,X184: $i] :
( ( in @ X184 @ X180 )
=> ( ( X182 @ X184 )
=> ( ( dsetconstr @ X180
@ ^ [V_6: $i] : ( X182 @ V_6 ) )
!= emptyset ) ) )
=> ( ! [X186: $i] :
( ? [X188: $i] : ( in @ X188 @ X186 )
=> ( X186 != emptyset ) )
=> ( ! [X190: $i,X192: $i] : ( in @ X190 @ ( setadjoin @ X190 @ X192 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X194: $i,X196: $i,X198: $i] :
( ( in @ X198 @ X196 )
=> ( in @ X198 @ ( setadjoin @ X194 @ X196 ) ) )
=> ( ! [X200: $i,X202: $i,X204: $i] :
( ( in @ X204 @ ( setadjoin @ X200 @ X202 ) )
=> ! [X206: $o] :
( ( ( X204 = X200 )
=> X206 )
=> ( ( ( in @ X204 @ X202 )
=> X206 )
=> X206 ) ) )
=> ( ! [X208: $i,X210: $i,X212: $i] :
( ( in @ X212 @ ( setadjoin @ X208 @ X210 ) )
=> ( ( X212 = X208 )
| ( in @ X212 @ X210 ) ) )
=> ( ! [X214: $i] :
( ( dsetconstr @ X214
@ ^ [V_7: $i] : $true )
= X214 )
=> ( ! [X216: $i,X218: $i] :
( ! [X220: $i] :
( ( in @ X220 @ X218 )
=> ( in @ X220 @ X216 ) )
=> ( in @ X218 @ ( powerset @ X216 ) ) )
=> ( ! [X222: $i] : ( in @ emptyset @ ( powerset @ X222 ) )
=> ( ! [X224: $i] : ( in @ emptyset @ ( powerset @ X224 ) )
=> ( ! [X226: $i,X228: $i,X230: $i] :
( ( in @ X228 @ ( powerset @ X226 ) )
=> ( ( in @ X230 @ X228 )
=> ( in @ X230 @ X226 ) ) )
=> ( ! [X232: $i,X234: $i,X236: $i] :
( ( in @ X234 @ X236 )
=> ( ( in @ X236 @ X232 )
=> ( in @ X234 @ ( setunion @ X232 ) ) ) )
=> ( ! [X238: $i,X240: $i] :
( ( in @ X240 @ ( setunion @ X238 ) )
=> ! [X242: $o] :
( ! [X244: $i] :
( ( in @ X240 @ X244 )
=> ( ( in @ X244 @ X238 )
=> X242 ) )
=> X242 ) )
=> ( ! [X246: $i,X248: $i] :
( ( in @ X248 @ X246 )
=> ( in @ X248 @ ( powerset @ ( setunion @ X246 ) ) ) )
=> ( ! [X250: $i > $o] :
( ? [X252: $i] :
( ! [X254: $i] :
( ( X250 @ X254 )
=> ( X252 = X254 ) )
& ( X250 @ X252 ) )
=> ? [X256: $i] :
! [X258: $i] :
( ( X250 @ X258 )
<=> ( X258 = X256 ) ) )
=> ( ! [X260: $i] :
( ( X260 != emptyset )
=> ? [X262: $i] : ( in @ X262 @ X260 ) )
=> ( ! [X264: $i,X266: $i] :
( ( in @ X264 @ ( setadjoin @ X266 @ emptyset ) )
=> ( X264 = X266 ) )
=> ( ! [X268: $i,X270: $i] :
( ( X268 != X270 )
=> ~ ( in @ X270 @ ( setadjoin @ X268 @ emptyset ) ) )
=> ( ! [X272: $i,X274: $i] :
( ( X272 = X274 )
=> ( in @ X272 @ ( setadjoin @ X274 @ emptyset ) ) )
=> ( ! [X276: $i,X278: $i] :
( ( in @ X276 @ ( setadjoin @ X278 @ emptyset ) )
=> ( in @ X278 @ ( setadjoin @ X276 @ emptyset ) ) )
=> ( ! [X280: $i,X282: $i,X284: $i] :
( ( in @ X284 @ ( setadjoin @ X280 @ ( setadjoin @ X282 @ emptyset ) ) )
=> ( ( X284 = X280 )
| ( X284 = X282 ) ) )
=> ( ! [X286: $i,X288: $i] : ( in @ X286 @ ( setadjoin @ X286 @ ( setadjoin @ X288 @ emptyset ) ) )
=> ( ! [X290: $i,X292: $i] : ( in @ X292 @ ( setadjoin @ X290 @ ( setadjoin @ X292 @ emptyset ) ) )
=> ( ! [X294: $i,X296: $i > $o] :
( ? [X298: $i] :
( ( X296 @ X298 )
& ( in @ X298 @ X294 ) )
=> ( ( dsetconstr @ X294
@ ^ [V_8: $i] : ( X296 @ V_8 ) )
!= emptyset ) )
=> ( ! [X300: $i > $o,X302: $i] :
( ( in @ X302 @ emptyset )
=> ( X300 @ X302 ) )
=> ( ! [X304: $i,X306: $i > $o] :
( ~ ! [X308: $i] :
( ( in @ X308 @ X304 )
=> ( X306 @ X308 ) )
=> ? [X310: $i] :
( ~ ( X306 @ X310 )
& ( in @ X310 @ X304 ) ) )
=> ( ! [X312: $i,X314: $i > $o] :
( ! [X316: $i] :
( ( in @ X316 @ X312 )
=> ~ ( X314 @ X316 ) )
=> ~ ? [X318: $i] :
( ( X314 @ X318 )
& ( in @ X318 @ X312 ) ) )
=> ( ! [X320: $i,X322: $i > $o] :
( ~ ? [X324: $i] :
( ( X322 @ X324 )
& ( in @ X324 @ X320 ) )
=> ! [X326: $i] :
( ( in @ X326 @ X320 )
=> ~ ( X322 @ X326 ) ) )
=> ( ! [X328: $i,X330: $i > $o] :
( ? [X332: $i] :
( ~ ( X330 @ X332 )
& ( in @ X332 @ X328 ) )
=> ~ ! [X334: $i] :
( ( in @ X334 @ X328 )
=> ( X330 @ X334 ) ) )
=> ( ! [X336: $o] :
( X336
=> ( in @ emptyset @ ( prop2set @ X336 ) ) )
=> ( ! [X338: $o] :
( X338
=> ( set2prop @ ( prop2set @ X338 ) ) )
=> ( ! [X340: $i,X342: $i > $o] :
( ~ ? [X344: $i] :
( ( X342 @ X344 )
& ( in @ X344 @ X340 ) )
=> ! [X346: $i] :
( ( in @ X346 @ X340 )
=> ~ ( X342 @ X346 ) ) )
=> ( ! [X348: $i,X350: $i > $o] :
( ~ ! [X352: $i] :
( ( in @ X352 @ X348 )
=> ( X350 @ X352 ) )
=> ? [X354: $i] :
( ~ ( X350 @ X354 )
& ( in @ X354 @ X348 ) ) )
=> ( ! [X356: $i > $o] :
( ? [X358: $i] :
( ! [X360: $i] :
( ( X356 @ X360 )
=> ( X358 = X360 ) )
& ( X356 @ X358 ) )
=> ? [X362: $i] :
( ! [X364: $i] :
( ( X356 @ X364 )
=> ( X362 = X364 ) )
& ( X356 @ X362 ) ) )
=> ( ! [X366: $i > $o] :
( ? [X368: $i] : ( X366 @ X368 )
=> ( ! [X370: $i,X372: $i] :
( ( X366 @ X370 )
=> ( ( X366 @ X372 )
=> ( X370 = X372 ) ) )
=> ? [X374: $i] :
( ! [X376: $i] :
( ( X366 @ X376 )
=> ( X374 = X376 ) )
& ( X366 @ X374 ) ) ) )
=> ( ! [X378: $i > $o] :
( ? [X380: $i] :
! [X382: $i] :
( ( X378 @ X382 )
<=> ( X382 = X380 ) )
=> ? [X384: $i] :
( ! [X386: $i] :
( ( X378 @ X386 )
=> ( X384 = X386 ) )
& ( X378 @ X384 ) ) )
=> ( ! [X388: $i,X390: $i] :
( ( X388 = X390 )
=> ! [X392: $i,X394: $i] :
( ( X392 = X394 )
=> ( ( in @ X392 @ X388 )
=> ( in @ X394 @ X390 ) ) ) )
=> ( ! [X396: $i,X398: $i] :
( ( X396 = X398 )
=> ! [X400: $i,X402: $i] :
( ( X400 = X402 )
=> ( ( in @ X400 @ X396 )
<=> ( in @ X402 @ X398 ) ) ) )
=> ( ! [X404: $i > $o] :
( ? [X406: $i] :
( ! [X408: $i] :
( ( X404 @ X408 )
=> ( X406 = X408 ) )
& ( X404 @ X406 ) )
=> ! [X410: $i,X412: $i] :
( ( X404 @ X410 )
=> ( ( X404 @ X412 )
=> ( X410 = X412 ) ) ) )
=> ( ! [X414: $i > $o,X416: $i > $o] :
( ! [X418: $i,X420: $i] :
( ( X418 = X420 )
=> ( ( X414 @ X418 )
<=> ( X416 @ X420 ) ) )
=> ( ? [X422: $i] : ( zip_tseitin_5 @ X422 @ X414 )
<=> ? [X426: $i] : ( zip_tseitin_5 @ X426 @ X416 ) ) )
=> ( ( emptyset = emptyset )
=> ( ! [X430: $i,X432: $i] :
( ( X430 = X432 )
=> ! [X434: $i,X436: $i] :
( ( X434 = X436 )
=> ( ( setadjoin @ X430 @ X434 )
= ( setadjoin @ X432 @ X436 ) ) ) )
=> ( ! [X438: $i,X440: $i] :
( ( X438 = X440 )
=> ( ( powerset @ X438 )
= ( powerset @ X440 ) ) )
=> ( ! [X442: $i,X444: $i] :
( ( X442 = X444 )
=> ( ( setunion @ X442 )
= ( setunion @ X444 ) ) )
=> ( ( omega = omega )
=> ( ! [X446: $i > $o] :
( ? [X448: $i] :
( ! [X450: $i] :
( ( X446 @ X450 )
=> ( X448 = X450 ) )
& ( X446 @ X448 ) )
=> ! [X452: $i,X454: $i] :
( ( X446 @ X452 )
=> ( ( X446 @ X454 )
=> ( X452 = X454 ) ) ) )
=> ( ! [X456: $i > $o,X458: $i > $o] :
( ! [X460: $i,X462: $i] :
( ( X460 = X462 )
=> ( ( X456 @ X460 )
<=> ( X458 @ X462 ) ) )
=> ( ? [X464: $i] :
( ! [X466: $i] :
( ( X456 @ X466 )
=> ( X464 = X466 ) )
& ( X456 @ X464 ) )
=> ( ? [X468: $i] :
( ! [X470: $i] :
( ( X458 @ X470 )
=> ( X468 = X470 ) )
& ( X458 @ X468 ) )
=> ( ( descr
@ ^ [V_9: $i] : ( X456 @ V_9 ) )
= ( descr
@ ^ [V_10: $i] : ( X458 @ V_10 ) ) ) ) ) )
=> ( ! [X472: $i,X474: $i] :
( ( X472 = X474 )
=> ! [X476: $i > $o,X478: $i > $o] :
( ! [X480: $i] :
( ( in @ X480 @ X472 )
=> ! [X482: $i] :
( ( in @ X482 @ X474 )
=> ( ( X480 = X482 )
=> ( ( X476 @ X480 )
<=> ( X478 @ X482 ) ) ) ) )
=> ( ( dsetconstr @ X472
@ ^ [V_11: $i] : ( X476 @ V_11 ) )
= ( dsetconstr @ X474
@ ^ [V_12: $i] : ( X478 @ V_12 ) ) ) ) )
=> ( ! [X484: $i,X486: $i] :
( ! [X488: $i] :
( ( in @ X488 @ X484 )
=> ( in @ X488 @ X486 ) )
=> ( subset @ X484 @ X486 ) )
=> ( ! [X490: $i,X492: $i] :
( ( X490 = X492 )
=> ( subset @ X492 @ X490 ) )
=> ( ! [X494: $i,X496: $i] :
( ( X494 = X496 )
=> ( subset @ X494 @ X496 ) )
=> ( ! [X498: $i,X500: $i] :
( ! [X502: $i] :
( ( in @ X502 @ X498 )
=> ( in @ X502 @ X500 ) )
=> ( subset @ X498 @ X500 ) )
=> ( ! [X504: $i] : ( subset @ emptyset @ X504 )
=> ( ! [X506: $i,X508: $i,X510: $i] :
( ( subset @ X506 @ X508 )
=> ( ( in @ X510 @ X506 )
=> ( in @ X510 @ X508 ) ) )
=> ( ! [X512: $i,X514: $i,X516: $i] :
( ( subset @ X512 @ X514 )
=> ( ~ ( in @ X516 @ X514 )
=> ~ ( in @ X516 @ X512 ) ) )
=> ( ! [X518: $i,X520: $i,X522: $i] :
( ( in @ X522 @ X518 )
=> ( ~ ( in @ X522 @ X520 )
=> ~ ( subset @ X518 @ X520 ) ) )
=> ( ! [X524: $i,X526: $i] :
( ~ ( subset @ X524 @ X526 )
=> ( X524 != X526 ) )
=> ( ! [X528: $i,X530: $i,X532: $i] :
( ( in @ X532 @ X528 )
=> ( ~ ( in @ X532 @ X530 )
=> ( X528 != X530 ) ) )
=> ( ! [X534: $i] : ( subset @ X534 @ X534 )
=> ( ! [X536: $i,X538: $i,X540: $i] :
( ( subset @ X536 @ X538 )
=> ( ( subset @ X538 @ X540 )
=> ( subset @ X536 @ X540 ) ) )
=> ( ! [X542: $i,X544: $i] : ( subset @ X544 @ ( setadjoin @ X542 @ X544 ) )
=> ( ! [X546: $i,X548: $i,X550: $i] :
( ( subset @ X546 @ X550 )
=> ( subset @ X546 @ ( setadjoin @ X548 @ X550 ) ) )
=> ( ! [X552: $i,X554: $i] :
( ( subset @ X552 @ X554 )
=> ( in @ X552 @ ( powerset @ X554 ) ) )
=> ( ! [X556: $i,X558: $i] :
( ( subset @ X556 @ X558 )
=> ( ( subset @ X558 @ X556 )
=> ( X556 = X558 ) ) )
=> ( ! [X560: $i] :
( ( subset @ X560 @ emptyset )
=> ( X560 = emptyset ) )
=> ( ! [X562: $i,X564: $i] :
( ( subset @ X564 @ X562 )
=> ( in @ X564 @ ( powerset @ X562 ) ) )
=> ( ! [X566: $i,X568: $i] :
( ( in @ X568 @ ( powerset @ X566 ) )
=> ( subset @ X568 @ X566 ) )
=> ( ! [X570: $i] : ( in @ X570 @ ( powerset @ X570 ) )
=> ( ! [X572: $i,X574: $i] :
( ( subset @ X572 @ X574 )
=> ( subset @ ( powerset @ X572 ) @ ( powerset @ X574 ) ) )
=> ( ! [X576: $i,X578: $i > $o] :
( in
@ ( dsetconstr @ X576
@ ^ [V_13: $i] : ( X578 @ V_13 ) )
@ ( powerset @ X576 ) )
=> ( ! [X580: $i,X582: $i > $o] :
( subset
@ ( dsetconstr @ X580
@ ^ [V_14: $i] : ( X582 @ V_14 ) )
@ X580 )
=> ( ! [X584: $i,X586: $i,X588: $i] :
( ( in @ X588 @ X584 )
=> ( in @ X588 @ ( binunion @ X584 @ X586 ) ) )
=> ( ! [X590: $i,X592: $i] : ( in @ X592 @ ( setadjoin @ X590 @ ( setadjoin @ X592 @ emptyset ) ) )
=> ( ! [X594: $i,X596: $i,X598: $i] :
( ( in @ X598 @ X596 )
=> ( in @ X598 @ ( binunion @ X594 @ X596 ) ) )
=> ( ! [X600: $i,X602: $i,X604: $i,X606: $o] :
( ( in @ X604 @ ( binunion @ X600 @ X602 ) )
=> ( ( ( in @ X604 @ X600 )
=> X606 )
=> ( ( ( in @ X604 @ X602 )
=> X606 )
=> X606 ) ) )
=> ( ! [X608: $i,X610: $i,X612: $i] :
( ( in @ X612 @ ( binunion @ X608 @ X610 ) )
=> ( ( in @ X612 @ X608 )
| ( in @ X612 @ X610 ) ) )
=> ( ! [X614: $i,X616: $i] : ( subset @ X614 @ ( binunion @ X614 @ X616 ) )
=> ( ! [X618: $i,X620: $i] : ( subset @ X620 @ ( binunion @ X618 @ X620 ) )
=> ( ! [X622: $i,X624: $i,X626: $i] :
( ( in @ X626 @ X622 )
=> ( ( in @ X626 @ X624 )
=> ( in @ X626 @ ( binintersect @ X622 @ X624 ) ) ) )
=> ( ! [X628: $i,X630: $i,X632: $i] :
( ( subset @ X632 @ X628 )
=> ( ( subset @ X632 @ X630 )
=> ( subset @ X632 @ ( binintersect @ X628 @ X630 ) ) ) )
=> ( ! [X634: $i,X636: $i,X638: $i] :
( ( in @ X638 @ ( binintersect @ X634 @ X636 ) )
=> ( in @ X638 @ X634 ) )
=> ( ! [X640: $i,X642: $i] : ( subset @ ( binintersect @ X640 @ X642 ) @ X640 )
=> ( ! [X644: $i,X646: $i] :
( ( subset @ X644 @ X646 )
=> ( ( binintersect @ X644 @ X646 )
= X644 ) )
=> ( ! [X648: $i,X650: $i] :
( ( ( binintersect @ X648 @ X650 )
= X650 )
=> ( subset @ X650 @ X648 ) )
=> ( ! [X652: $i,X654: $i,X656: $i] :
( ( in @ X656 @ ( binintersect @ X652 @ X654 ) )
=> ( in @ X656 @ X654 ) )
=> ( ! [X658: $i,X660: $i] :
( ~ ? [X662: $i] :
( ( in @ X662 @ X660 )
& ( in @ X662 @ X658 ) )
=> ( ( binintersect @ X658 @ X660 )
= emptyset ) )
=> ( ! [X664: $i,X666: $i] : ( subset @ ( binintersect @ X664 @ X666 ) @ X666 )
=> ( ! [X668: $i,X670: $i] :
( ( subset @ X670 @ X668 )
=> ( ( binintersect @ X668 @ X670 )
= X670 ) )
=> ( ! [X672: $i,X674: $i] :
( ( ( binintersect @ X672 @ X674 )
= X672 )
=> ( subset @ X672 @ X674 ) )
=> ( ! [X676: $i,X678: $i,X680: $i] :
( ( binintersect @ X676 @ ( binunion @ X678 @ X680 ) )
= ( binunion @ ( binintersect @ X676 @ X678 ) @ ( binintersect @ X676 @ X680 ) ) )
=> ( ! [X682: $i,X684: $i,X686: $i] :
( ( in @ X686 @ X682 )
=> ( ~ ( in @ X686 @ X684 )
=> ( in @ X686 @ ( setminus @ X682 @ X684 ) ) ) )
=> ( ! [X688: $i,X690: $i,X692: $i] :
( ( in @ X692 @ ( setminus @ X688 @ X690 ) )
=> ( in @ X692 @ X688 ) )
=> ( ! [X694: $i,X696: $i,X698: $i] :
( ( in @ X698 @ ( setminus @ X694 @ X696 ) )
=> ~ ( in @ X698 @ X696 ) )
=> ( ! [X700: $i,X702: $i] :
( ( subset @ X700 @ X702 )
=> ( ( setminus @ X700 @ X702 )
= emptyset ) )
=> ( ! [X704: $i,X706: $i,X708: $i] :
( ~ ( in @ X708 @ ( setminus @ X704 @ X706 ) )
=> ( ( in @ X708 @ X704 )
=> ( in @ X708 @ X706 ) ) )
=> ( ! [X710: $i,X712: $i,X714: $i] :
( ~ ( in @ X714 @ ( setminus @ X710 @ X712 ) )
=> ( ~ ( in @ X714 @ X712 )
=> ~ ( in @ X714 @ X710 ) ) )
=> ( ! [X716: $i,X718: $i,X720: $i] :
( ~ ( in @ X720 @ X716 )
=> ~ ( in @ X720 @ ( setminus @ X716 @ X718 ) ) )
=> ( ! [X722: $i,X724: $i,X726: $i] :
( ( in @ X726 @ X724 )
=> ~ ( in @ X726 @ ( setminus @ X722 @ X724 ) ) )
=> ( ! [X728: $i,X730: $i] : ( subset @ ( setminus @ X728 @ X730 ) @ X728 )
=> ( ! [X732: $i,X734: $i] :
( ( ( setminus @ X732 @ X734 )
= emptyset )
=> ( subset @ X732 @ X734 ) )
=> ( ! [X736: $i,X738: $i,X740: $i] :
( ( in @ X740 @ ( symdiff @ X736 @ X738 ) )
=> ! [X742: $o] :
( ( ( in @ X740 @ X736 )
=> ( ~ ( in @ X740 @ X738 )
=> X742 ) )
=> ( ( ~ ( in @ X740 @ X736 )
=> ( ( in @ X740 @ X738 )
=> X742 ) )
=> X742 ) ) )
=> ( ! [X744: $i,X746: $i,X748: $i] :
( ( in @ X748 @ X744 )
=> ( ~ ( in @ X748 @ X746 )
=> ( in @ X748 @ ( symdiff @ X744 @ X746 ) ) ) )
=> ( ! [X750: $i,X752: $i,X754: $i] :
( ~ ( in @ X754 @ X750 )
=> ( ( in @ X754 @ X752 )
=> ( in @ X754 @ ( symdiff @ X750 @ X752 ) ) ) )
=> ( ! [X756: $i,X758: $i,X760: $i] :
( ( in @ X760 @ X756 )
=> ( ( in @ X760 @ X758 )
=> ~ ( in @ X760 @ ( symdiff @ X756 @ X758 ) ) ) )
=> ( ! [X762: $i,X764: $i,X766: $i] :
( ~ ( in @ X766 @ X762 )
=> ( ~ ( in @ X766 @ X764 )
=> ~ ( in @ X766 @ ( symdiff @ X762 @ X764 ) ) ) )
=> ( ! [X768: $i,X770: $i] : ( in @ X770 @ ( setadjoin @ X768 @ ( setadjoin @ X770 @ emptyset ) ) )
=> ( ! [X772: $i,X774: $i] : ( in @ X772 @ ( setunion @ ( setadjoin @ ( setadjoin @ X772 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X772 @ ( setadjoin @ X774 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X776: $i,X778: $i] : ( in @ X778 @ ( setunion @ ( setadjoin @ ( setadjoin @ X776 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X776 @ ( setadjoin @ X778 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X780: $i,X782: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X780 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X780 @ ( setadjoin @ X782 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X784: $i,X786: $i] : ( iskpair @ ( kpair @ X784 @ X786 ) )
=> ( ! [X788: $i,X790: $i] :
( ( in @ X790 @ X788 )
=> ( subset @ ( setadjoin @ X790 @ emptyset ) @ X788 ) )
=> ( ! [X792: $i,X794: $i] :
( ( in @ X794 @ X792 )
=> ( in @ ( setadjoin @ X794 @ emptyset ) @ ( powerset @ X792 ) ) )
=> ( ! [X796: $i,X798: $i,X800: $i] :
( ( in @ X800 @ X796 )
=> ( in @ ( setadjoin @ X800 @ emptyset ) @ ( powerset @ ( binunion @ X796 @ X798 ) ) ) )
=> ( ! [X802: $i,X804: $i,X806: $i] :
( ( in @ X806 @ ( setadjoin @ X802 @ ( setadjoin @ X804 @ emptyset ) ) )
=> ( ( X806 = X802 )
| ( X806 = X804 ) ) )
=> ( ! [X808: $i,X810: $i,X812: $i] :
( ( in @ X812 @ X808 )
=> ! [X814: $i] :
( ( in @ X814 @ X810 )
=> ( subset @ ( setadjoin @ X812 @ ( setadjoin @ X814 @ emptyset ) ) @ ( binunion @ X808 @ X810 ) ) ) )
=> ( ! [X816: $i,X818: $i,X820: $i] :
( ( in @ X820 @ X816 )
=> ! [X822: $i] :
( ( in @ X822 @ X818 )
=> ( in @ ( setadjoin @ X820 @ ( setadjoin @ X822 @ emptyset ) ) @ ( powerset @ ( binunion @ X816 @ X818 ) ) ) ) )
=> ( ! [X824: $i,X826: $i,X828: $i] :
( ( in @ X828 @ X824 )
=> ! [X830: $i] :
( ( in @ X830 @ X826 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X828 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X828 @ ( setadjoin @ X830 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X824 @ X826 ) ) ) ) )
=> ( ! [X832: $i,X834: $i,X836: $i] :
( ( in @ X836 @ X832 )
=> ! [X838: $i] :
( ( in @ X838 @ X834 )
=> ( in @ ( setadjoin @ ( setadjoin @ X836 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X836 @ ( setadjoin @ X838 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X832 @ X834 ) ) ) ) ) )
=> ( ! [X840: $i,X842: $i,X844: $i] :
( ( in @ X844 @ X840 )
=> ! [X846: $i] :
( ( in @ X846 @ X842 )
=> ( in @ ( kpair @ X844 @ X846 ) @ ( powerset @ ( powerset @ ( binunion @ X840 @ X842 ) ) ) ) ) )
=> ( ! [X848: $i,X850: $i,X852: $i] :
( ( in @ X852 @ X848 )
=> ! [X854: $i] :
( ( in @ X854 @ X850 )
=> ( in @ ( kpair @ X852 @ X854 ) @ ( cartprod @ X848 @ X850 ) ) ) )
=> ( ! [X856: $i,X858: $i,X860: $i] :
( ( in @ X860 @ ( cartprod @ X856 @ X858 ) )
=> ? [X862: $i] :
( ? [X864: $i] :
( ( X860
= ( kpair @ X862 @ X864 ) )
& ( in @ X864 @ X858 ) )
& ( in @ X862 @ X856 ) ) )
=> ( ! [X866: $i,X868: $i,X870: $i] :
( ( in @ X870 @ ( cartprod @ X866 @ X868 ) )
=> ( iskpair @ X870 ) )
=> ( ! [X872: $i,X874: $i] :
( ( in @ X874 @ ( setunion @ X872 ) )
=> ? [X876: $i] :
( ( in @ X874 @ X876 )
& ( in @ X876 @ X872 ) ) )
=> ( ! [X878: $i] : ( subset @ ( setunion @ ( setadjoin @ X878 @ emptyset ) ) @ X878 )
=> ( ! [X880: $i] : ( subset @ X880 @ ( setunion @ ( setadjoin @ X880 @ emptyset ) ) )
=> ( ! [X882: $i] :
( ( setunion @ ( setadjoin @ X882 @ emptyset ) )
= X882 )
=> ( ! [X884: $i,X886: $i > $o] :
( ! [X888: $i] :
( ( in @ X888 @ X884 )
=> ! [X890: $i] :
( ( in @ X890 @ X884 )
=> ( ( X886 @ X888 )
=> ( ( X886 @ X890 )
=> ( X888 = X890 ) ) ) ) )
=> ( ? [X892: $i] :
( ( X886 @ X892 )
& ( in @ X892 @ X884 ) )
=> ( singleton
@ ( dsetconstr @ X884
@ ^ [V_15: $i] : ( X886 @ V_15 ) ) ) ) )
=> ( ! [X894: $i,X896: $i > $o] :
( ( ex1 @ X894
@ ^ [V_16: $i] : ( X896 @ V_16 ) )
=> ? [X898: $i] :
( ( X896 @ X898 )
& ( in @ X898 @ X894 ) ) )
=> ( ! [X900: $i,X902: $i > $o,X904: $i] :
( ( in @ X904 @ X900 )
=> ( ( X902 @ X904 )
=> ( ! [X906: $i] :
( ( in @ X906 @ X900 )
=> ( ( X902 @ X906 )
=> ( X906 = X904 ) ) )
=> ( ex1 @ X900
@ ^ [V_17: $i] : ( X902 @ V_17 ) ) ) ) )
=> ( ! [X908: $i,X910: $i > $o] :
( ! [X912: $i] :
( ( in @ X912 @ X908 )
=> ! [X914: $i] :
( ( in @ X914 @ X908 )
=> ( ( X910 @ X912 )
=> ( ( X910 @ X914 )
=> ( X912 = X914 ) ) ) ) )
=> ( ? [X916: $i] :
( ( X910 @ X916 )
& ( in @ X916 @ X908 ) )
=> ( ex1 @ X908
@ ^ [V_18: $i] : ( X910 @ V_18 ) ) ) )
=> ( ! [X918: $i,X920: $i] :
( ( ( setadjoin @ X918 @ emptyset )
= ( setadjoin @ X920 @ emptyset ) )
=> ( X918 = X920 ) )
=> ( ! [X922: $i,X924: $i,X926: $i] :
( ( in @ ( setadjoin @ X926 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X922 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X922 @ ( setadjoin @ X924 @ emptyset ) ) @ emptyset ) ) )
=> ( X922 = X926 ) )
=> ( ! [X928: $i] :
( ( iskpair @ X928 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X928 )
@ ^ [V_19: $i] : ( in @ ( setadjoin @ V_19 @ emptyset ) @ X928 ) ) ) )
=> ( ! [X930: $i] :
( ( singleton @ X930 )
=> ( in @ ( setunion @ X930 ) @ X930 ) )
=> ( ! [X932: $i,X934: $i] :
( ( kfst @ ( kpair @ X932 @ X934 ) )
= X932 )
=> ( ! [X936: $i,X938: $i,X940: $i] :
( ( in @ X940 @ ( cartprod @ X936 @ X938 ) )
=> ( in @ ( kfst @ X940 ) @ X936 ) )
=> ( ! [X942: $i,X944: $i,X946: $i,X948: $i] :
( ( ( setadjoin @ ( setadjoin @ X942 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X942 @ ( setadjoin @ X944 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X946 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X946 @ ( setadjoin @ X948 @ emptyset ) ) @ emptyset ) ) )
=> ( X942 = X946 ) )
=> ( ! [X950: $i,X952: $i,X954: $i,X956: $i] :
( ( ( kpair @ X950 @ X952 )
= ( kpair @ X954 @ X956 ) )
=> ( X950 = X954 ) )
=> ( ! [X958: $i,X960: $i] :
( ( X958 = X960 )
=> ( ( setadjoin @ X958 @ ( setadjoin @ X960 @ emptyset ) )
= ( setadjoin @ X958 @ emptyset ) ) )
=> ( ! [X962: $i,X964: $i] :
( ( X962 = X964 )
=> ( ( setadjoin @ ( setadjoin @ X962 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X962 @ ( setadjoin @ X964 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X962 @ emptyset ) @ emptyset ) ) )
=> ( ! [X966: $i,X968: $i,X970: $i,X972: $i] :
( ( ( setadjoin @ ( setadjoin @ X966 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X966 @ ( setadjoin @ X968 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X970 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X970 @ ( setadjoin @ X972 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X970 = X972 )
=> ( X968 = X972 ) ) )
=> ( ! [X974: $i,X976: $i,X978: $i] :
( ( ( setadjoin @ X974 @ ( setadjoin @ X976 @ emptyset ) )
= ( setadjoin @ X978 @ emptyset ) )
=> ( X974 = X976 ) )
=> ( ! [X980: $i,X982: $i,X984: $i,X986: $i] :
( ( ( setadjoin @ ( setadjoin @ X980 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X980 @ ( setadjoin @ X982 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X984 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X984 @ ( setadjoin @ X986 @ emptyset ) ) @ emptyset ) ) )
=> ( X982 = X986 ) )
=> ( ! [X988: $i,X990: $i,X992: $i,X994: $i] :
( ( ( kpair @ X988 @ X990 )
= ( kpair @ X992 @ X994 ) )
=> ( X990 = X994 ) )
=> ( ! [X996: $i] :
( ( iskpair @ X996 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X996 )
@ ^ [V_20: $i] :
( X996
= ( kpair @ ( kfst @ X996 ) @ V_20 ) ) ) ) )
=> ( ! [X998: $i,X1000: $i] :
( ( ksnd @ ( kpair @ X998 @ X1000 ) )
= X1000 )
=> ( ! [X1002: $i] :
( ( iskpair @ X1002 )
=> ( ( kpair @ ( kfst @ X1002 ) @ ( ksnd @ X1002 ) )
= X1002 ) )
=> ( ! [X1004: $i,X1006: $i,X1008: $i] :
( ( in @ X1008 @ ( cartprod @ X1004 @ X1006 ) )
=> ( in @ ( ksnd @ X1008 ) @ X1006 ) )
=> ( ! [X1010: $i,X1012: $i,X1014: $i,X1016: $i] :
( ( in @ ( kpair @ X1014 @ X1016 ) @ ( cartprod @ X1010 @ X1012 ) )
=> ( in @ X1014 @ X1010 ) )
=> ( ! [X1018: $i,X1020: $i,X1022: $i,X1024: $i] :
( ( in @ ( kpair @ X1022 @ X1024 ) @ ( cartprod @ X1018 @ X1020 ) )
=> ( in @ X1024 @ X1020 ) )
=> ( ! [X1026: $i,X1028: $i,X1030: $i] :
( ( in @ X1030 @ X1026 )
=> ! [X1032: $i] :
( ( in @ X1032 @ X1028 )
=> ( ( kpair @ X1030 @ X1032 )
= ( kpair @ X1030 @ X1032 ) ) ) )
=> ( ! [X1034: $i,X1036: $i,X1038: $i] :
( ( in @ X1038 @ X1034 )
=> ! [X1040: $i] :
( ( in @ X1040 @ X1036 )
=> ( ( kfst @ ( kpair @ X1038 @ X1040 ) )
= X1038 ) ) )
=> ( ! [X1042: $i,X1044: $i,X1046: $i] :
( ( in @ X1046 @ X1042 )
=> ! [X1048: $i] :
( ( in @ X1048 @ X1044 )
=> ( ( ksnd @ ( kpair @ X1046 @ X1048 ) )
= X1048 ) ) )
=> ( ! [X1050: $i,X1052: $i,X1054: $i] :
( ( in @ X1054 @ ( cartprod @ X1050 @ X1052 ) )
=> ( ( kpair @ ( kfst @ X1054 ) @ ( ksnd @ X1054 ) )
= X1054 ) )
=> ( ! [X1056: $i,X1058: $i,X1060: $i > $i > $o,X1062: $i] :
( ( in @ X1062 @ X1056 )
=> ! [X1064: $i] :
( ( in @ X1064 @ X1058 )
=> ( ( X1060 @ X1062 @ X1064 )
=> ( in @ ( kpair @ X1062 @ X1064 )
@ ( dpsetconstr @ X1056 @ X1058
@ ^ [V_21: $i,V_22: $i] : ( X1060 @ V_21 @ V_22 ) ) ) ) ) )
=> ( ! [X1066: $i,X1068: $i,X1070: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1066 @ X1068
@ ^ [V_23: $i,V_24: $i] : ( X1070 @ V_23 @ V_24 ) )
@ ( cartprod @ X1066 @ X1068 ) )
=> ( ! [X1072: $i,X1074: $i,X1076: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1072 @ X1074
@ ^ [V_25: $i,V_26: $i] : ( X1076 @ V_25 @ V_26 ) )
@ ( cartprod @ X1072 @ X1074 ) )
=> ( ! [X1078: $i,X1080: $i,X1082: $i > $i > $o,X1084: $i] :
( ( in @ X1084 @ X1078 )
=> ! [X1086: $i] :
( ( in @ X1086 @ X1080 )
=> ( ( in @ ( kpair @ X1084 @ X1086 )
@ ( dpsetconstr @ X1078 @ X1080
@ ^ [V_27: $i,V_28: $i] : ( X1082 @ V_27 @ V_28 ) ) )
=> ( X1082 @ X1084 @ X1086 ) ) ) )
=> ( ! [X1088: $i,X1090: $i,X1092: $i > $i > $o,X1094: $i,X1096: $i] :
( ( in @ ( kpair @ X1094 @ X1096 )
@ ( dpsetconstr @ X1088 @ X1090
@ ^ [V_29: $i,V_30: $i] : ( X1092 @ V_29 @ V_30 ) ) )
=> ( in @ X1094 @ X1088 ) )
=> ( ! [X1098: $i,X1100: $i,X1102: $i > $i > $o,X1104: $i,X1106: $i] :
( ( in @ ( kpair @ X1104 @ X1106 )
@ ( dpsetconstr @ X1098 @ X1100
@ ^ [V_31: $i,V_32: $i] : ( X1102 @ V_31 @ V_32 ) ) )
=> ( in @ X1106 @ X1100 ) )
=> ( ! [X1108: $i,X1110: $i,X1112: $i > $i > $o,X1114: $i,X1116: $i] :
( ( in @ ( kpair @ X1114 @ X1116 )
@ ( dpsetconstr @ X1108 @ X1110
@ ^ [V_33: $i,V_34: $i] : ( X1112 @ V_33 @ V_34 ) ) )
=> ( X1112 @ X1114 @ X1116 ) )
=> ( ! [X1118: $i,X1120: $i,X1122: $i] :
( ( func @ X1118 @ X1120 @ X1122 )
=> ! [X1124: $i] :
( ( in @ X1124 @ X1118 )
=> ( singleton
@ ( dsetconstr @ X1120
@ ^ [V_35: $i] : ( in @ ( kpair @ X1124 @ V_35 ) @ X1122 ) ) ) ) )
=> ( ! [X1126: $i,X1128: $i,X1130: $i] :
( ( func @ X1126 @ X1128 @ X1130 )
=> ! [X1132: $i] :
( ( in @ X1132 @ X1126 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X1128
@ ^ [V_36: $i] : ( in @ ( kpair @ X1132 @ V_36 ) @ X1130 ) ) )
@ X1128 ) ) )
=> ( ! [X1134: $i,X1136: $i,X1138: $i] :
( ( func @ X1134 @ X1136 @ X1138 )
=> ! [X1140: $i] :
( ( in @ X1140 @ X1134 )
=> ( in @ ( ap @ X1134 @ X1136 @ X1138 @ X1140 ) @ X1136 ) ) )
=> ( ! [X1142: $i,X1144: $i,X1146: $i] :
( ( in @ X1146 @ ( funcSet @ X1142 @ X1144 ) )
=> ( func @ X1142 @ X1144 @ X1146 ) )
=> ( ! [X1148: $i,X1150: $i,X1152: $i] :
( ( in @ X1152 @ ( funcSet @ X1148 @ X1150 ) )
=> ! [X1154: $i] :
( ( in @ X1154 @ X1148 )
=> ( in @ ( ap @ X1148 @ X1150 @ X1152 @ X1154 ) @ X1150 ) ) )
=> ( ! [X1156: $i,X1158: $i,X1160: $i] :
( ( func @ X1156 @ X1158 @ X1160 )
=> ( in @ X1160 @ ( funcSet @ X1156 @ X1158 ) ) )
=> ( ! [X1162: $i,X1164: $i,X1166: $i > $i] :
( ! [X1168: $i] :
( ( in @ X1168 @ X1162 )
=> ( in @ ( X1166 @ X1168 ) @ X1164 ) )
=> ( func @ X1162 @ X1164
@ ( dpsetconstr @ X1162 @ X1164
@ ^ [V_37: $i,V_38: $i] :
( ( X1166 @ V_37 )
= V_38 ) ) ) )
=> ( ! [X1170: $i,X1172: $i,X1174: $i > $i] :
( ! [X1176: $i] :
( ( in @ X1176 @ X1170 )
=> ( in @ ( X1174 @ X1176 ) @ X1172 ) )
=> ( func @ X1170 @ X1172
@ ( lam @ X1170 @ X1172
@ ^ [V_39: $i] : ( X1174 @ V_39 ) ) ) )
=> ( ! [X1178: $i,X1180: $i,X1182: $i > $i] :
( ! [X1184: $i] :
( ( in @ X1184 @ X1178 )
=> ( in @ ( X1182 @ X1184 ) @ X1180 ) )
=> ( in
@ ( lam @ X1178 @ X1180
@ ^ [V_40: $i] : ( X1182 @ V_40 ) )
@ ( funcSet @ X1178 @ X1180 ) ) )
=> ( ! [X1186: $i,X1188: $i,X1190: $i] :
( ( subset @ X1190 @ ( cartprod @ X1186 @ X1188 ) )
=> ! [X1192: $i > $o] :
( ! [X1194: $i] :
( ( in @ X1194 @ X1186 )
=> ! [X1196: $i] :
( ( in @ X1196 @ X1188 )
=> ( ( in @ ( kpair @ X1194 @ X1196 ) @ X1190 )
=> ( X1192 @ ( kpair @ X1194 @ X1196 ) ) ) ) )
=> ! [X1198: $i] :
( ( in @ X1198 @ X1190 )
=> ( X1192 @ X1198 ) ) ) )
=> ( ! [X1200: $i,X1202: $i,X1204: $i] :
( ( subset @ X1204 @ ( cartprod @ X1200 @ X1202 ) )
=> ! [X1206: $i > $o] :
( ! [X1208: $i] :
( ( in @ X1208 @ X1200 )
=> ! [X1210: $i] :
( ( in @ X1210 @ X1202 )
=> ( ( in @ ( kpair @ X1208 @ X1210 ) @ X1204 )
=> ( X1206 @ ( kpair @ X1208 @ X1210 ) ) ) ) )
=> ! [X1212: $i] :
( ( in @ X1212 @ X1204 )
=> ( X1206 @ X1212 ) ) ) )
=> ( ! [X1214: $i,X1216: $i > $o] :
( ( ex1 @ X1214
@ ^ [V_41: $i] : ( X1216 @ V_41 ) )
=> ! [X1218: $i] :
( ( in @ X1218 @ X1214 )
=> ! [X1220: $i] :
( ( in @ X1220 @ X1214 )
=> ( ( X1216 @ X1218 )
=> ( ( X1216 @ X1220 )
=> ( X1218 = X1220 ) ) ) ) ) )
=> ( ! [X1222: $i,X1224: $i,X1226: $i] :
( ( func @ X1222 @ X1224 @ X1226 )
=> ! [X1228: $i] :
( ( in @ X1228 @ X1222 )
=> ( in @ ( kpair @ X1228 @ ( ap @ X1222 @ X1224 @ X1226 @ X1228 ) ) @ X1226 ) ) )
=> ( ! [X1230: $i,X1232: $i,X1234: $i] :
( ( in @ X1234 @ ( funcSet @ X1230 @ X1232 ) )
=> ! [X1236: $i] :
( ( in @ X1236 @ X1230 )
=> ( in @ ( kpair @ X1236 @ ( ap @ X1230 @ X1232 @ X1234 @ X1236 ) ) @ X1234 ) ) )
=> ( ! [X1238: $i,X1240: $i,X1242: $i] :
( ( func @ X1238 @ X1240 @ X1242 )
=> ! [X1244: $i] :
( ( in @ X1244 @ X1238 )
=> ! [X1246: $i] :
( ( in @ X1246 @ X1240 )
=> ( ( in @ ( kpair @ X1244 @ X1246 ) @ X1242 )
=> ( ( ap @ X1238 @ X1240 @ X1242 @ X1244 )
= X1246 ) ) ) ) )
=> ( ! [X1248: $i,X1250: $i,X1252: $i] :
( ( func @ X1248 @ X1250 @ X1252 )
=> ! [X1254: $i] :
( ( func @ X1248 @ X1250 @ X1254 )
=> ( ! [X1256: $i] :
( ( in @ X1256 @ X1248 )
=> ( ( ap @ X1248 @ X1250 @ X1252 @ X1256 )
= ( ap @ X1248 @ X1250 @ X1254 @ X1256 ) ) )
=> ! [X1258: $i] :
( ( in @ X1258 @ X1248 )
=> ! [X1260: $i] :
( ( in @ X1260 @ X1250 )
=> ( ( in @ ( kpair @ X1258 @ X1260 ) @ X1254 )
=> ( in @ ( kpair @ X1258 @ X1260 ) @ X1252 ) ) ) ) ) ) )
=> ( ! [X1262: $i,X1264: $i,X1266: $i] :
( ( in @ X1266 @ ( funcSet @ X1262 @ X1264 ) )
=> ! [X1268: $i] :
( ( in @ X1268 @ X1262 )
=> ! [X1270: $i] :
( ( in @ X1270 @ X1264 )
=> ( ( in @ ( kpair @ X1268 @ X1270 ) @ X1266 )
=> ( ( ap @ X1262 @ X1264 @ X1266 @ X1268 )
= X1270 ) ) ) ) )
=> ( ! [X1272: $i,X1274: $i,X1276: $i] :
( ( subset @ X1276 @ ( cartprod @ X1272 @ X1274 ) )
=> ! [X1278: $i] :
( ( subset @ X1278 @ ( cartprod @ X1272 @ X1274 ) )
=> ( ! [X1280: $i] :
( ( in @ X1280 @ X1272 )
=> ! [X1282: $i] :
( ( in @ X1282 @ X1274 )
=> ( ( in @ ( kpair @ X1280 @ X1282 ) @ X1276 )
=> ( in @ ( kpair @ X1280 @ X1282 ) @ X1278 ) ) ) )
=> ( subset @ X1276 @ X1278 ) ) ) )
=> ( ! [X1284: $i,X1286: $i,X1288: $i] :
( ( subset @ X1288 @ ( cartprod @ X1284 @ X1286 ) )
=> ! [X1290: $i] :
( ( subset @ X1290 @ ( cartprod @ X1284 @ X1286 ) )
=> ( ! [X1292: $i] :
( ( in @ X1292 @ X1284 )
=> ! [X1294: $i] :
( ( in @ X1294 @ X1286 )
=> ( ( in @ ( kpair @ X1292 @ X1294 ) @ X1288 )
=> ( in @ ( kpair @ X1292 @ X1294 ) @ X1290 ) ) ) )
=> ( ! [X1296: $i] :
( ( in @ X1296 @ X1284 )
=> ! [X1298: $i] :
( ( in @ X1298 @ X1286 )
=> ( ( in @ ( kpair @ X1296 @ X1298 ) @ X1290 )
=> ( in @ ( kpair @ X1296 @ X1298 ) @ X1288 ) ) ) )
=> ( X1288 = X1290 ) ) ) ) )
=> ( ! [X1300: $i,X1302: $i,X1304: $i] :
( ( func @ X1300 @ X1302 @ X1304 )
=> ! [X1306: $i] :
( ( func @ X1300 @ X1302 @ X1306 )
=> ( ! [X1308: $i] :
( ( in @ X1308 @ X1300 )
=> ( ( ap @ X1300 @ X1302 @ X1304 @ X1308 )
= ( ap @ X1300 @ X1302 @ X1306 @ X1308 ) ) )
=> ( X1304 = X1306 ) ) ) )
=> ( ! [X1310: $i,X1312: $i,X1314: $i] :
( ( in @ X1314 @ ( funcSet @ X1310 @ X1312 ) )
=> ! [X1316: $i] :
( ( in @ X1316 @ ( funcSet @ X1310 @ X1312 ) )
=> ( ! [X1318: $i] :
( ( in @ X1318 @ X1310 )
=> ( ( ap @ X1310 @ X1312 @ X1314 @ X1318 )
= ( ap @ X1310 @ X1312 @ X1316 @ X1318 ) ) )
=> ( X1314 = X1316 ) ) ) )
=> ( ! [X1320: $i,X1322: $i,X1324: $i] :
( ( in @ X1324 @ ( funcSet @ X1320 @ X1322 ) )
=> ! [X1326: $i] :
( ( in @ X1326 @ X1320 )
=> ( ( ap @ X1320 @ X1322 @ X1324 @ X1326 )
= ( ap @ X1320 @ X1322 @ X1324 @ X1326 ) ) ) )
=> ( ! [X1328: $i,X1330: $i,X1332: $i] :
( ( func @ X1328 @ X1330 @ X1332 )
=> ! [X1334: $i] :
( ( in @ X1334 @ X1328 )
=> ( ( ap @ X1328 @ X1330 @ X1332 @ X1334 )
= ( ap @ X1328 @ X1330 @ X1332 @ X1334 ) ) ) )
=> ( ! [X1336: $i,X1338: $i,X1340: $i > $i] :
( ! [X1342: $i] :
( ( in @ X1342 @ X1336 )
=> ( in @ ( X1340 @ X1342 ) @ X1338 ) )
=> ! [X1344: $i] :
( ( in @ X1344 @ X1336 )
=> ( ( ap @ X1336 @ X1338
@ ( lam @ X1336 @ X1338
@ ^ [V_42: $i] : ( X1340 @ V_42 ) )
@ X1344 )
= ( X1340 @ X1344 ) ) ) )
=> ( ! [X1346: $i,X1348: $i,X1350: $i] :
( ( func @ X1346 @ X1348 @ X1350 )
=> ( ( lam @ X1346 @ X1348
@ ^ [V_43: $i] : ( ap @ X1346 @ X1348 @ X1350 @ V_43 ) )
= X1350 ) )
=> ( ! [X1352: $i,X1354: $i,X1356: $i > $i] :
( ! [X1358: $i] :
( ( in @ X1358 @ X1352 )
=> ( in @ ( X1356 @ X1358 ) @ X1354 ) )
=> ( ( lam @ X1352 @ X1354
@ ^ [V_44: $i] : ( X1356 @ V_44 ) )
= ( lam @ X1352 @ X1354
@ ^ [V_45: $i] : ( X1356 @ V_45 ) ) ) )
=> ( ! [X1360: $i,X1362: $i,X1364: $i > $i] :
( ! [X1366: $i] :
( ( in @ X1366 @ X1360 )
=> ( in @ ( X1364 @ X1366 ) @ X1362 ) )
=> ! [X1368: $i] :
( ( in @ X1368 @ X1360 )
=> ( ( ap @ X1360 @ X1362
@ ( lam @ X1360 @ X1362
@ ^ [V_46: $i] : ( X1364 @ V_46 ) )
@ X1368 )
= ( X1364 @ X1368 ) ) ) )
=> ( ! [X1370: $i,X1372: $i,X1374: $i] :
( ( in @ X1374 @ ( funcSet @ X1370 @ X1372 ) )
=> ( ( lam @ X1370 @ X1372
@ ^ [V_47: $i] : ( ap @ X1370 @ X1372 @ X1374 @ V_47 ) )
= X1374 ) )
=> ( ! [X1376: $i,X1378: $o,X1380: $i] :
( ( in @ X1380 @ X1376 )
=> ! [X1382: $i] :
( ( in @ X1382 @ X1376 )
=> ( ~ X1378
=> ( in @ X1382
@ ( dsetconstr @ X1376
@ ^ [V_48: $i] :
( ( ( V_48 = X1382 )
& ~ X1378 )
| ( ( V_48 = X1380 )
& X1378 ) ) ) ) ) ) )
=> ( ! [X1384: $i,X1386: $o,X1388: $i] :
( ( in @ X1388 @ X1384 )
=> ! [X1390: $i] :
( ( in @ X1390 @ X1384 )
=> ( ~ X1386
=> ( ( dsetconstr @ X1384
@ ^ [V_49: $i] :
( ( ( V_49 = X1390 )
& ~ X1386 )
| ( ( V_49 = X1388 )
& X1386 ) ) )
= ( setadjoin @ X1390 @ emptyset ) ) ) ) )
=> ( ! [X1392: $i,X1394: $o,X1396: $i] :
( ( in @ X1396 @ X1392 )
=> ! [X1398: $i] :
( ( in @ X1398 @ X1392 )
=> ( X1394
=> ( in @ X1396
@ ( dsetconstr @ X1392
@ ^ [V_50: $i] :
( ( ( V_50 = X1398 )
& ~ X1394 )
| ( ( V_50 = X1396 )
& X1394 ) ) ) ) ) ) )
=> ( ! [X1400: $i,X1402: $o,X1404: $i] :
( ( in @ X1404 @ X1400 )
=> ! [X1406: $i] :
( ( in @ X1406 @ X1400 )
=> ( X1402
=> ( ( dsetconstr @ X1400
@ ^ [V_51: $i] :
( ( ( V_51 = X1406 )
& ~ X1402 )
| ( ( V_51 = X1404 )
& X1402 ) ) )
= ( setadjoin @ X1404 @ emptyset ) ) ) ) )
=> ( ! [X1408: $i,X1410: $o,X1412: $i] :
( ( in @ X1412 @ X1408 )
=> ! [X1414: $i] :
( ( in @ X1414 @ X1408 )
=> ( singleton
@ ( dsetconstr @ X1408
@ ^ [V_52: $i] :
( ( ( V_52 = X1414 )
& ~ X1410 )
| ( ( V_52 = X1412 )
& X1410 ) ) ) ) ) )
=> ( ! [X1416: $i,X1418: $o,X1420: $i] :
( ( in @ X1420 @ X1416 )
=> ! [X1422: $i] :
( ( in @ X1422 @ X1416 )
=> ( in @ ( if @ X1416 @ X1418 @ X1420 @ X1422 ) @ X1416 ) ) )
=> ( ! [X1424: $i] :
( ( singleton @ X1424 )
=> ! [X1426: $i] :
( ( in @ X1426 @ X1424 )
=> ( ( setunion @ X1424 )
= X1426 ) ) )
=> ( ! [X1428: $i,X1430: $o,X1432: $i] :
( ( in @ X1432 @ X1428 )
=> ! [X1434: $i] :
( ( in @ X1434 @ X1428 )
=> ( X1430
=> ( ( if @ X1428 @ X1430 @ X1432 @ X1434 )
= X1432 ) ) ) )
=> ( ! [X1436: $i,X1438: $o,X1440: $i] :
( ( in @ X1440 @ X1436 )
=> ! [X1442: $i] :
( ( in @ X1442 @ X1436 )
=> ( ~ X1438
=> ( ( if @ X1436 @ X1438 @ X1440 @ X1442 )
= X1442 ) ) ) )
=> ( ! [X1444: $i,X1446: $o,X1448: $i] :
( ( in @ X1448 @ X1444 )
=> ! [X1450: $i] :
( ( in @ X1450 @ X1444 )
=> ( in @ ( if @ X1444 @ X1446 @ X1448 @ X1450 ) @ ( setadjoin @ X1448 @ ( setadjoin @ X1450 @ emptyset ) ) ) ) )
=> ( ! [X1452: $i,X1454: $i] :
( ( in @ X1454 @ ( powerset @ X1452 ) )
=> ! [X1456: $i] :
( ( in @ X1456 @ ( powerset @ X1452 ) )
=> ( in @ ( binintersect @ X1454 @ X1456 ) @ ( powerset @ X1452 ) ) ) )
=> ( ! [X1458: $i,X1460: $i] :
( ( in @ X1460 @ ( powerset @ X1458 ) )
=> ! [X1462: $i] :
( ( in @ X1462 @ ( powerset @ X1458 ) )
=> ( in @ ( binunion @ X1460 @ X1462 ) @ ( powerset @ X1458 ) ) ) )
=> ( ! [X1464: $i,X1466: $i] :
( ( in @ X1466 @ ( powerset @ X1464 ) )
=> ( in @ ( powerset @ X1466 ) @ ( powerset @ ( powerset @ X1464 ) ) ) )
=> ( ! [X1468: $i,X1470: $i] :
( ( in @ X1470 @ ( powerset @ X1468 ) )
=> ! [X1472: $i] :
( ( in @ X1472 @ ( powerset @ X1468 ) )
=> ( in @ ( setminus @ X1470 @ X1472 ) @ ( powerset @ X1468 ) ) ) )
=> ( ! [X1474: $i,X1476: $i] :
( ( in @ X1476 @ ( powerset @ X1474 ) )
=> ( in @ ( setminus @ X1474 @ X1476 ) @ ( powerset @ X1474 ) ) )
=> ( ! [X1478: $i,X1480: $i] :
( ( in @ X1480 @ ( powerset @ X1478 ) )
=> ! [X1482: $i] :
( ( in @ X1482 @ ( powerset @ X1478 ) )
=> ( ! [X1484: $i] :
( ( in @ X1484 @ X1478 )
=> ( ( in @ X1484 @ X1480 )
=> ( in @ X1484 @ X1482 ) ) )
=> ( ! [X1486: $i] :
( ( in @ X1486 @ X1478 )
=> ( ( in @ X1486 @ X1482 )
=> ( in @ X1486 @ X1480 ) ) )
=> ( X1480 = X1482 ) ) ) ) )
=> ( ! [X1488: $i,X1490: $i] :
( ( in @ X1490 @ ( powerset @ X1488 ) )
=> ! [X1492: $i] :
( ( in @ X1492 @ ( powerset @ X1488 ) )
=> ( ! [X1494: $i] :
( ( in @ X1494 @ X1488 )
=> ( ( in @ X1494 @ X1490 )
=> ( in @ X1494 @ X1492 ) ) )
=> ( subset @ X1490 @ X1492 ) ) ) )
=> ( ! [X1496: $i,X1498: $i] :
( ( in @ X1498 @ ( powerset @ X1496 ) )
=> ! [X1500: $i] :
( ( in @ X1500 @ ( powerset @ X1496 ) )
=> ( ! [X1502: $i] :
( ( in @ X1502 @ X1496 )
=> ( ( in @ X1502 @ X1498 )
=> ( in @ X1502 @ X1500 ) ) )
=> ( in @ X1498 @ ( powerset @ X1500 ) ) ) ) )
=> ( ! [X1504: $i,X1506: $i] :
( ( in @ X1506 @ ( powerset @ X1504 ) )
=> ! [X1508: $i] :
( ( in @ X1508 @ ( powerset @ X1504 ) )
=> ! [X1510: $i] :
( ( in @ X1510 @ X1504 )
=> ( ( in @ X1506 @ ( powerset @ X1508 ) )
=> ( ( in @ X1510 @ X1506 )
=> ( in @ X1510 @ X1508 ) ) ) ) ) )
=> ( ! [X1512: $i,X1514: $i] :
( ( in @ X1514 @ ( powerset @ X1512 ) )
=> ! [X1516: $i] :
( ( in @ X1516 @ X1512 )
=> ( ( in @ X1516 @ X1514 )
=> ~ ( in @ X1516 @ ( setminus @ X1512 @ X1514 ) ) ) ) )
=> ( ! [X1518: $i,X1520: $i] :
( ( in @ X1520 @ ( powerset @ X1518 ) )
=> ! [X1522: $i] :
( ( in @ X1522 @ X1518 )
=> ( ~ ( in @ X1522 @ ( setminus @ X1518 @ X1520 ) )
=> ( in @ X1522 @ X1520 ) ) ) )
=> ( ! [X1524: $i,X1526: $i] :
( ( in @ X1526 @ ( powerset @ X1524 ) )
=> ! [X1528: $i] :
( ( in @ X1528 @ ( powerset @ X1524 ) )
=> ! [X1530: $i] :
( ( in @ X1530 @ X1524 )
=> ( ~ ( in @ X1530 @ X1526 )
=> ~ ( in @ X1530 @ ( binintersect @ X1526 @ X1528 ) ) ) ) ) )
=> ( ! [X1532: $i,X1534: $i] :
( ( in @ X1534 @ ( powerset @ X1532 ) )
=> ! [X1536: $i] :
( ( in @ X1536 @ ( powerset @ X1532 ) )
=> ! [X1538: $i] :
( ( in @ X1538 @ X1532 )
=> ( ~ ( in @ X1538 @ X1536 )
=> ~ ( in @ X1538 @ ( binintersect @ X1534 @ X1536 ) ) ) ) ) )
=> ( ! [X1540: $i,X1542: $i] :
( ( in @ X1542 @ ( powerset @ X1540 ) )
=> ! [X1544: $i] :
( ( in @ X1544 @ ( powerset @ X1540 ) )
=> ! [X1546: $i] :
( ( in @ X1546 @ X1540 )
=> ( ( subset @ X1542 @ ( setminus @ X1540 @ X1544 ) )
=> ( ( in @ X1546 @ X1544 )
=> ~ ( in @ X1546 @ X1542 ) ) ) ) ) )
=> ( ! [X1548: $i,X1550: $i] :
( ( in @ X1550 @ ( powerset @ X1548 ) )
=> ! [X1552: $i] :
( ( in @ X1552 @ ( powerset @ X1548 ) )
=> ! [X1554: $i] :
( ( in @ X1554 @ X1548 )
=> ( ( subset @ X1550 @ X1552 )
=> ( ~ ( in @ X1554 @ X1552 )
=> ~ ( in @ X1554 @ X1550 ) ) ) ) ) )
=> ( ! [X1556: $i,X1558: $i] :
( ( in @ X1558 @ ( powerset @ X1556 ) )
=> ! [X1560: $i] :
( ( in @ X1560 @ ( powerset @ X1556 ) )
=> ( ( subset @ X1558 @ X1560 )
=> ( subset @ ( setminus @ X1556 @ X1560 ) @ ( setminus @ X1556 @ X1558 ) ) ) ) )
=> ( ! [X1562: $i,X1564: $i] :
( ( in @ X1564 @ ( powerset @ X1562 ) )
=> ! [X1566: $i] :
( ( in @ X1566 @ ( powerset @ X1562 ) )
=> ( ( subset @ ( setminus @ X1562 @ X1566 ) @ ( setminus @ X1562 @ X1564 ) )
=> ( subset @ X1564 @ X1566 ) ) ) )
=> ( ! [X1568: $i,X1570: $i] :
( ( in @ X1570 @ ( powerset @ X1568 ) )
=> ! [X1572: $i] :
( ( in @ X1572 @ X1568 )
=> ( ( in @ X1572 @ X1570 )
=> ( in @ X1572 @ ( setminus @ X1568 @ ( setminus @ X1568 @ X1570 ) ) ) ) ) )
=> ( ! [X1574: $i,X1576: $i] :
( ( in @ X1576 @ ( powerset @ X1574 ) )
=> ! [X1578: $i] :
( ( in @ X1578 @ X1574 )
=> ( ( in @ X1578 @ ( setminus @ X1574 @ ( setminus @ X1574 @ X1576 ) ) )
=> ( in @ X1578 @ X1576 ) ) ) )
=> ( ! [X1580: $i,X1582: $i] :
( ( in @ X1582 @ ( powerset @ X1580 ) )
=> ( subset @ X1582 @ ( setminus @ X1580 @ ( setminus @ X1580 @ X1582 ) ) ) )
=> ( ! [X1584: $i,X1586: $i] :
( ( in @ X1586 @ ( powerset @ X1584 ) )
=> ( subset @ ( setminus @ X1584 @ ( setminus @ X1584 @ X1586 ) ) @ X1586 ) )
=> ( ! [X1588: $i,X1590: $i] :
( ( in @ X1590 @ ( powerset @ X1588 ) )
=> ( X1590
= ( setminus @ X1588 @ ( setminus @ X1588 @ X1590 ) ) ) )
=> ( ! [X1592: $i,X1594: $i] :
( ( in @ X1594 @ ( powerset @ X1592 ) )
=> ! [X1596: $i] :
( ( in @ X1596 @ ( powerset @ X1592 ) )
=> ! [X1598: $i] :
( ( in @ X1598 @ X1592 )
=> ( ( in @ X1598 @ ( setminus @ X1592 @ X1594 ) )
=> ~ ( in @ X1598 @ ( binintersect @ X1594 @ X1596 ) ) ) ) ) )
=> ( ! [X1600: $i,X1602: $i] :
( ( in @ X1602 @ ( powerset @ X1600 ) )
=> ! [X1604: $i] :
( ( in @ X1604 @ ( powerset @ X1600 ) )
=> ! [X1606: $i] :
( ( in @ X1606 @ X1600 )
=> ( ( in @ X1606 @ ( setminus @ X1600 @ X1602 ) )
=> ( in @ X1606 @ ( setminus @ X1600 @ ( binintersect @ X1602 @ X1604 ) ) ) ) ) ) )
=> ( ! [X1608: $i,X1610: $i] :
( ( in @ X1610 @ ( powerset @ X1608 ) )
=> ! [X1612: $i] :
( ( in @ X1612 @ ( powerset @ X1608 ) )
=> ( subset @ ( setminus @ X1608 @ X1610 ) @ ( setminus @ X1608 @ ( binintersect @ X1610 @ X1612 ) ) ) ) )
=> ( ! [X1614: $i,X1616: $i] :
( ( in @ X1616 @ ( powerset @ X1614 ) )
=> ! [X1618: $i] :
( ( in @ X1618 @ ( powerset @ X1614 ) )
=> ( in @ ( setminus @ X1614 @ X1616 ) @ ( powerset @ ( setminus @ X1614 @ ( binintersect @ X1616 @ X1618 ) ) ) ) ) )
=> ( ! [X1620: $i,X1622: $i] :
( ( in @ X1622 @ ( powerset @ X1620 ) )
=> ! [X1624: $i] :
( ( in @ X1624 @ ( powerset @ X1620 ) )
=> ( ( subset @ X1622 @ ( setminus @ X1620 @ X1624 ) )
=> ! [X1626: $i] :
( ( in @ X1626 @ X1620 )
=> ( ( in @ X1626 @ X1624 )
=> ( in @ X1626 @ ( setminus @ X1620 @ X1622 ) ) ) ) ) ) )
=> ( ! [X1628: $i,X1630: $i] :
( ( in @ X1630 @ ( powerset @ X1628 ) )
=> ! [X1632: $i] :
( ( in @ X1632 @ ( powerset @ X1628 ) )
=> ( ( subset @ X1630 @ ( setminus @ X1628 @ X1632 ) )
=> ( subset @ X1632 @ ( setminus @ X1628 @ X1630 ) ) ) ) )
=> ( ! [X1634: $i,X1636: $i] :
( ( in @ X1636 @ ( powerset @ X1634 ) )
=> ! [X1638: $i] :
( ( in @ X1638 @ ( powerset @ X1634 ) )
=> ! [X1640: $i] :
( ( in @ X1640 @ X1634 )
=> ( ~ ( in @ X1640 @ ( binunion @ X1636 @ X1638 ) )
=> ~ ( in @ X1640 @ X1636 ) ) ) ) )
=> ( ! [X1642: $i,X1644: $i] :
( ( in @ X1644 @ ( powerset @ X1642 ) )
=> ! [X1646: $i] :
( ( in @ X1646 @ ( powerset @ X1642 ) )
=> ! [X1648: $i] :
( ( in @ X1648 @ X1642 )
=> ( ~ ( in @ X1648 @ ( binunion @ X1644 @ X1646 ) )
=> ~ ( in @ X1648 @ X1646 ) ) ) ) )
=> ( ! [X1650: $i,X1652: $i] :
( ( in @ X1652 @ ( powerset @ X1650 ) )
=> ! [X1654: $i] :
( ( in @ X1654 @ ( powerset @ X1650 ) )
=> ! [X1656: $i] :
( ( in @ X1656 @ ( powerset @ X1650 ) )
=> ! [X1658: $i] :
( ( in @ X1658 @ X1650 )
=> ( ( in @ X1658 @ ( binintersect @ X1652 @ X1654 ) )
=> ( in @ X1658 @ ( binunion @ X1652 @ X1656 ) ) ) ) ) ) )
=> ( ! [X1660: $i,X1662: $i] :
( ( in @ X1662 @ ( powerset @ X1660 ) )
=> ! [X1664: $i] :
( ( in @ X1664 @ ( powerset @ X1660 ) )
=> ! [X1666: $i] :
( ( in @ X1666 @ ( powerset @ X1660 ) )
=> ! [X1668: $i] :
( ( in @ X1668 @ X1660 )
=> ( ( in @ X1668 @ ( binintersect @ X1662 @ X1664 ) )
=> ( in @ X1668 @ ( binunion @ X1664 @ X1666 ) ) ) ) ) ) )
=> ( ! [X1670: $i,X1672: $i] :
( ( in @ X1672 @ ( powerset @ X1670 ) )
=> ! [X1674: $i] :
( ( in @ X1674 @ ( powerset @ X1670 ) )
=> ! [X1676: $i] :
( ( in @ X1676 @ ( powerset @ X1670 ) )
=> ! [X1678: $i] :
( ( in @ X1678 @ X1670 )
=> ( ( in @ X1678 @ ( binintersect @ X1672 @ X1674 ) )
=> ( in @ X1678 @ ( binintersect @ ( binunion @ X1672 @ X1676 ) @ ( binunion @ X1674 @ X1676 ) ) ) ) ) ) ) )
=> ( ! [X1680: $i,X1682: $i] :
( ( in @ X1682 @ ( powerset @ X1680 ) )
=> ! [X1684: $i] :
( ( in @ X1684 @ ( powerset @ X1680 ) )
=> ! [X1686: $i] :
( ( in @ X1686 @ ( powerset @ X1680 ) )
=> ( in @ ( binintersect @ X1682 @ X1684 ) @ ( powerset @ ( binintersect @ ( binunion @ X1682 @ X1686 ) @ ( binunion @ X1684 @ X1686 ) ) ) ) ) ) )
=> ( ! [X1688: $i,X1690: $i] :
( ( in @ X1690 @ ( powerset @ X1688 ) )
=> ! [X1692: $i] :
( ( in @ X1692 @ ( powerset @ X1688 ) )
=> ! [X1694: $i] :
( ( in @ X1694 @ X1688 )
=> ( ( in @ X1694 @ ( setminus @ X1688 @ ( binunion @ X1690 @ X1692 ) ) )
=> ~ ( in @ X1694 @ X1690 ) ) ) ) )
=> ( ! [X1696: $i,X1698: $i] :
( ( in @ X1698 @ ( powerset @ X1696 ) )
=> ! [X1700: $i] :
( ( in @ X1700 @ ( powerset @ X1696 ) )
=> ! [X1702: $i] :
( ( in @ X1702 @ X1696 )
=> ( ( in @ X1702 @ ( setminus @ X1696 @ ( binunion @ X1698 @ X1700 ) ) )
=> ( in @ X1702 @ ( setminus @ X1696 @ X1698 ) ) ) ) ) )
=> ( ! [X1704: $i,X1706: $i] :
( ( in @ X1706 @ ( powerset @ X1704 ) )
=> ! [X1708: $i] :
( ( in @ X1708 @ ( powerset @ X1704 ) )
=> ! [X1710: $o,X1712: $i] :
( ( in @ X1712 @ X1704 )
=> ( ( in @ X1712 @ ( binunion @ X1706 @ X1708 ) )
=> ( ( ( in @ X1712 @ X1706 )
=> X1710 )
=> ( ( ( in @ X1712 @ X1708 )
=> X1710 )
=> X1710 ) ) ) ) ) )
=> ( ! [X1714: $i,X1716: $i] :
( ( in @ X1716 @ ( powerset @ X1714 ) )
=> ! [X1718: $i] :
( ( in @ X1718 @ ( powerset @ X1714 ) )
=> ! [X1720: $i] :
( ( in @ X1720 @ X1714 )
=> ( ~ ( in @ X1720 @ X1716 )
=> ( ~ ( in @ X1720 @ X1718 )
=> ~ ( in @ X1720 @ ( binunion @ X1716 @ X1718 ) ) ) ) ) ) )
=> ( ! [X1722: $i,X1724: $i] :
( ( in @ X1724 @ ( powerset @ X1722 ) )
=> ! [X1726: $i] :
( ( in @ X1726 @ ( powerset @ X1722 ) )
=> ! [X1728: $i] :
( ( in @ X1728 @ X1722 )
=> ( ( in @ X1728 @ ( setminus @ X1722 @ ( binunion @ X1724 @ X1726 ) ) )
=> ( in @ X1728 @ ( setminus @ X1722 @ X1724 ) ) ) ) ) )
=> ( ! [X1730: $i,X1732: $i] :
( ( in @ X1732 @ ( powerset @ X1730 ) )
=> ! [X1734: $i] :
( ( in @ X1734 @ ( powerset @ X1730 ) )
=> ( in @ ( setminus @ X1730 @ ( binunion @ X1732 @ X1734 ) ) @ ( powerset @ ( setminus @ X1730 @ X1732 ) ) ) ) )
=> ( ! [X1736: $i,X1738: $i] :
( ( in @ X1738 @ ( powerset @ X1736 ) )
=> ! [X1740: $i] :
( ( in @ X1740 @ ( powerset @ X1736 ) )
=> ! [X1742: $i] :
( ( in @ X1742 @ X1736 )
=> ( ( in @ X1742 @ ( setminus @ X1736 @ ( binunion @ X1738 @ X1740 ) ) )
=> ( in @ X1742 @ ( setminus @ X1736 @ X1740 ) ) ) ) ) )
=> ( ! [X1744: $i,X1746: $i] :
( ( in @ X1746 @ ( powerset @ X1744 ) )
=> ! [X1748: $i] :
( ( in @ X1748 @ ( powerset @ X1744 ) )
=> ! [X1750: $i] :
( ( in @ X1750 @ X1744 )
=> ( ( in @ X1750 @ ( setminus @ X1744 @ ( binintersect @ X1746 @ X1748 ) ) )
=> ( in @ X1750 @ ( binunion @ ( setminus @ X1744 @ X1746 ) @ ( setminus @ X1744 @ X1748 ) ) ) ) ) ) )
=> ( ! [X1752: $i,X1754: $i] :
( ( in @ X1754 @ ( powerset @ X1752 ) )
=> ! [X1756: $i] :
( ( in @ X1756 @ ( powerset @ X1752 ) )
=> ! [X1758: $i] :
( ( in @ X1758 @ X1752 )
=> ( ( in @ X1758 @ ( binunion @ ( setminus @ X1752 @ X1754 ) @ ( setminus @ X1752 @ X1756 ) ) )
=> ( in @ X1758 @ ( setminus @ X1752 @ ( binintersect @ X1754 @ X1756 ) ) ) ) ) ) )
=> ( ! [X1760: $i,X1762: $i] :
( ( in @ X1762 @ ( powerset @ X1760 ) )
=> ! [X1764: $i] :
( ( in @ X1764 @ ( powerset @ X1760 ) )
=> ( ( setminus @ X1760 @ ( binintersect @ X1762 @ X1764 ) )
= ( binunion @ ( setminus @ X1760 @ X1762 ) @ ( setminus @ X1760 @ X1764 ) ) ) ) )
=> ( ! [X1766: $i,X1768: $i] :
( ( in @ X1768 @ ( powerset @ X1766 ) )
=> ! [X1770: $i] :
( ( in @ X1770 @ ( powerset @ X1766 ) )
=> ! [X1772: $i] :
( ( in @ X1772 @ X1766 )
=> ( ( in @ X1772 @ ( setminus @ X1766 @ ( binunion @ X1768 @ X1770 ) ) )
=> ( in @ X1772 @ ( binintersect @ ( setminus @ X1766 @ X1768 ) @ ( setminus @ X1766 @ X1770 ) ) ) ) ) ) )
=> ( ! [X1774: $i,X1776: $i] :
( ( in @ X1776 @ ( powerset @ X1774 ) )
=> ! [X1778: $i] :
( ( in @ X1778 @ ( powerset @ X1774 ) )
=> ! [X1780: $i] :
( ( in @ X1780 @ X1774 )
=> ( ( in @ X1780 @ ( setminus @ X1774 @ X1776 ) )
=> ( ( in @ X1780 @ ( setminus @ X1774 @ X1778 ) )
=> ( in @ X1780 @ ( setminus @ X1774 @ ( binunion @ X1776 @ X1778 ) ) ) ) ) ) ) )
=> ( ! [X1782: $i,X1784: $i] :
( ( in @ X1784 @ ( powerset @ X1782 ) )
=> ! [X1786: $i] :
( ( in @ X1786 @ ( powerset @ X1782 ) )
=> ! [X1788: $i] :
( ( in @ X1788 @ X1782 )
=> ( ( in @ X1788 @ ( binintersect @ ( setminus @ X1782 @ X1784 ) @ ( setminus @ X1782 @ X1786 ) ) )
=> ( in @ X1788 @ ( setminus @ X1782 @ ( binunion @ X1784 @ X1786 ) ) ) ) ) ) )
=> ( ! [X1790: $i,X1792: $i] :
( ( in @ X1792 @ ( powerset @ X1790 ) )
=> ! [X1794: $i] :
( ( in @ X1794 @ ( powerset @ X1790 ) )
=> ( ( setminus @ X1790 @ ( binunion @ X1792 @ X1794 ) )
= ( binintersect @ ( setminus @ X1790 @ X1792 ) @ ( setminus @ X1790 @ X1794 ) ) ) ) )
=> ( ! [X1796: $i,X1798: $i] :
( ( in @ X1798 @ ( powerset @ X1796 ) )
=> ! [X1800: $i] :
( ( in @ X1800 @ ( powerset @ X1796 ) )
=> ! [X1802: $i] :
( ( in @ X1802 @ ( binintersect @ X1798 @ X1800 ) )
=> ( in @ X1802 @ X1796 ) ) ) )
=> ( ! [X1804: $i,X1806: $i] :
( ( in @ X1806 @ ( powerset @ X1804 ) )
=> ! [X1808: $i] :
( ( in @ X1808 @ ( powerset @ X1804 ) )
=> ! [X1810: $i] :
( ( in @ X1810 @ ( powerset @ X1804 ) )
=> ( ( subset @ X1806 @ X1810 )
=> ( subset @ ( binintersect @ X1806 @ X1808 ) @ X1810 ) ) ) ) )
=> ( ! [X1812: $i,X1814: $i] :
( ( in @ X1814 @ ( powerset @ X1812 ) )
=> ! [X1816: $i] :
( ( in @ X1816 @ ( powerset @ X1812 ) )
=> ! [X1818: $i] :
( ( in @ X1818 @ ( powerset @ X1812 ) )
=> ( ( subset @ X1816 @ X1818 )
=> ( subset @ ( binintersect @ X1814 @ X1816 ) @ X1818 ) ) ) ) )
=> ( ! [X1820: $i,X1822: $i] :
( ( in @ X1822 @ ( powerset @ X1820 ) )
=> ! [X1824: $i] :
( ( in @ X1824 @ ( powerset @ X1820 ) )
=> ! [X1826: $i] :
( ( in @ X1826 @ ( powerset @ X1820 ) )
=> ( ( subset @ X1822 @ X1824 )
=> ( ( subset @ X1822 @ X1826 )
=> ( subset @ X1822 @ ( binintersect @ X1824 @ X1826 ) ) ) ) ) ) )
=> ( ! [X1828: $i,X1830: $i] :
( ( in @ X1830 @ ( powerset @ X1828 ) )
=> ! [X1832: $i] :
( ( in @ X1832 @ ( powerset @ X1828 ) )
=> ! [X1834: $i] :
( ( in @ X1834 @ ( powerset @ X1828 ) )
=> ! [X1836: $i] :
( ( in @ X1836 @ ( powerset @ X1828 ) )
=> ( ( subset @ X1830 @ X1834 )
=> ( ( subset @ X1832 @ X1836 )
=> ( subset @ ( binintersect @ X1830 @ X1832 ) @ ( binintersect @ X1834 @ X1836 ) ) ) ) ) ) ) )
=> ( ! [X1838: $i,X1840: $i] :
( ( in @ X1840 @ ( powerset @ X1838 ) )
=> ! [X1842: $i] :
( ( in @ X1842 @ ( powerset @ X1838 ) )
=> ( in @ ( setminus @ X1838 @ X1840 ) @ ( powerset @ ( setminus @ X1838 @ ( binintersect @ X1840 @ X1842 ) ) ) ) ) )
=> ( ! [X1844: $i,X1846: $i] :
( ( in @ X1846 @ ( powerset @ X1844 ) )
=> ! [X1848: $i] :
( ( in @ X1848 @ ( powerset @ X1844 ) )
=> ! [X1850: $i] :
( ( in @ X1850 @ ( powerset @ X1844 ) )
=> ! [X1852: $i] :
( ( in @ X1852 @ ( powerset @ X1844 ) )
=> ( ( setminus @ X1844 @ ( binintersect @ ( binunion @ X1846 @ X1848 ) @ ( binunion @ X1850 @ X1852 ) ) )
= ( binunion @ ( binintersect @ ( setminus @ X1844 @ X1846 ) @ ( setminus @ X1844 @ X1848 ) ) @ ( binintersect @ ( setminus @ X1844 @ X1850 ) @ ( setminus @ X1844 @ X1852 ) ) ) ) ) ) ) )
=> ( ! [X1854: $i,X1856: $i] :
( ( in @ X1856 @ ( powerset @ X1854 ) )
=> ! [X1858: $i] :
( ( in @ X1858 @ ( powerset @ X1854 ) )
=> ! [X1860: $i] :
( ( in @ X1860 @ ( powerset @ X1854 ) )
=> ( in @ ( binintersect @ X1856 @ X1858 ) @ ( powerset @ ( binintersect @ ( binunion @ X1856 @ X1860 ) @ ( binunion @ X1858 @ X1860 ) ) ) ) ) ) )
=> ( ! [X1862: $i,X1864: $i] :
( ( in @ X1864 @ ( powerset @ X1862 ) )
=> ! [X1866: $i] :
( ( in @ X1866 @ ( powerset @ X1862 ) )
=> ( ( subset @ X1864 @ ( setminus @ X1862 @ X1866 ) )
=> ( subset @ X1866 @ ( setminus @ X1862 @ X1864 ) ) ) ) )
=> ( ! [X1868: $i,X1870: $i] :
( ( in @ X1870 @ ( powerset @ X1868 ) )
=> ! [X1872: $i] :
( ( in @ X1872 @ ( powerset @ X1868 ) )
=> ( in @ ( setminus @ X1868 @ ( binunion @ X1870 @ X1872 ) ) @ ( powerset @ ( setminus @ X1868 @ X1870 ) ) ) ) )
=> ( ! [X1874: $i,X1876: $i] :
( ( subset @ X1876 @ ( cartprod @ X1874 @ X1874 ) )
=> ! [X1878: $i > $o] :
( ! [X1880: $i] :
( ( in @ X1880 @ X1874 )
=> ! [X1882: $i] :
( ( in @ X1882 @ X1874 )
=> ( ( in @ ( kpair @ X1880 @ X1882 ) @ X1876 )
=> ( X1878 @ ( kpair @ X1880 @ X1882 ) ) ) ) )
=> ! [X1884: $i] :
( ( in @ X1884 @ X1876 )
=> ( X1878 @ X1884 ) ) ) )
=> ( ! [X1886: $i,X1888: $i] :
( ( in @ X1888 @ ( breln1Set @ X1886 ) )
=> ( subset @ X1888 @ ( cartprod @ X1886 @ X1886 ) ) )
=> ( ! [X1890: $i,X1892: $i,X1894: $i > $i > $o] :
( ! [X1896: $i] :
( ( in @ X1896 @ X1890 )
=> ? [X1898: $i] :
( ( X1894 @ X1896 @ X1898 )
& ( in @ X1898 @ X1892 ) ) )
=> ! [X1900: $i] :
( ( in @ X1900 @ ( breln1Set @ X1892 ) )
=> ( ( reflwellordering @ X1892 @ X1900 )
=> ! [X1902: $i] :
( ( in @ X1902 @ X1890 )
=> ( singleton
@ ( dsetconstr @ X1892
@ ^ [V_53: $i] :
( ! [X1904: $i] :
( ( in @ X1904 @ X1892 )
=> ( ( X1894 @ X1902 @ X1904 )
=> ( in @ ( kpair @ V_53 @ X1904 ) @ X1900 ) ) )
& ( X1894 @ X1902 @ V_53 ) ) ) ) ) ) ) )
=> ( ! [X1906: $i,X1908: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1906 @ X1906
@ ^ [V_54: $i,V_55: $i] : ( X1908 @ V_54 @ V_55 ) )
@ ( cartprod @ X1906 @ X1906 ) )
=> ( ! [X1910: $i,X1912: $i] :
( ( subset @ X1912 @ ( cartprod @ X1910 @ X1910 ) )
=> ! [X1914: $i > $o] :
( ! [X1916: $i] :
( ( in @ X1916 @ X1910 )
=> ! [X1918: $i] :
( ( in @ X1918 @ X1910 )
=> ( ( in @ ( kpair @ X1916 @ X1918 ) @ X1912 )
=> ( X1914 @ ( kpair @ X1916 @ X1918 ) ) ) ) )
=> ! [X1920: $i] :
( ( in @ X1920 @ X1912 )
=> ( X1914 @ X1920 ) ) ) )
=> ( ! [X1922: $i,X1924: $i] :
( ( subset @ X1924 @ ( cartprod @ X1922 @ X1922 ) )
=> ! [X1926: $i] :
( ( subset @ X1926 @ ( cartprod @ X1922 @ X1922 ) )
=> ( ! [X1928: $i] :
( ( in @ X1928 @ X1922 )
=> ! [X1930: $i] :
( ( in @ X1930 @ X1922 )
=> ( ( in @ ( kpair @ X1928 @ X1930 ) @ X1924 )
=> ( in @ ( kpair @ X1928 @ X1930 ) @ X1926 ) ) ) )
=> ( subset @ X1924 @ X1926 ) ) ) )
=> ( ! [X1932: $i,X1934: $i] :
( ( subset @ X1934 @ ( cartprod @ X1932 @ X1932 ) )
=> ! [X1936: $i] :
( ( subset @ X1936 @ ( cartprod @ X1932 @ X1932 ) )
=> ( ! [X1938: $i] :
( ( in @ X1938 @ X1932 )
=> ! [X1940: $i] :
( ( in @ X1940 @ X1932 )
=> ( ( in @ ( kpair @ X1938 @ X1940 ) @ X1934 )
=> ( in @ ( kpair @ X1938 @ X1940 ) @ X1936 ) ) ) )
=> ( ! [X1942: $i] :
( ( in @ X1942 @ X1932 )
=> ! [X1944: $i] :
( ( in @ X1944 @ X1932 )
=> ( ( in @ ( kpair @ X1942 @ X1944 ) @ X1936 )
=> ( in @ ( kpair @ X1942 @ X1944 ) @ X1934 ) ) ) )
=> ( X1934 = X1936 ) ) ) ) )
=> ( ! [X1946: $i,X1948: $i] :
( ( subset @ X1948 @ ( cartprod @ X1946 @ X1946 ) )
=> ( subset
@ ( dpsetconstr @ X1946 @ X1946
@ ^ [V_56: $i,V_57: $i] : ( in @ ( kpair @ V_57 @ V_56 ) @ X1948 ) )
@ ( cartprod @ X1946 @ X1946 ) ) )
=> ( ! [X1950: $i,X1952: $i] :
( ( subset @ X1952 @ ( cartprod @ X1950 @ X1950 ) )
=> ! [X1954: $i] :
( ( in @ X1954 @ X1950 )
=> ! [X1956: $i] :
( ( in @ X1956 @ X1950 )
=> ( ( in @ ( kpair @ X1954 @ X1956 ) @ X1952 )
=> ( in @ ( kpair @ X1956 @ X1954 )
@ ( dpsetconstr @ X1950 @ X1950
@ ^ [V_58: $i,V_59: $i] : ( in @ ( kpair @ V_59 @ V_58 ) @ X1952 ) ) ) ) ) ) )
=> ! [X1958: $i,X1960: $i] :
( ( subset @ X1960 @ ( cartprod @ X1958 @ X1958 ) )
=> ! [X1962: $i] :
( ( in @ X1962 @ X1958 )
=> ! [X1964: $i] :
( ( in @ X1964 @ X1958 )
=> ( ( in @ ( kpair @ X1964 @ X1962 )
@ ( dpsetconstr @ X1958 @ X1958
@ ^ [V_60: $i,V_61: $i] : ( in @ ( kpair @ V_61 @ V_60 ) @ X1960 ) ) )
=> ( in @ ( kpair @ X1962 @ X1964 ) @ X1960 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(zf_stmt_14,negated_conjecture,
~ ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
<=> ( in @ X8 @ X6 ) )
=> ( X4 = X6 ) )
=> ( ! [X10: $i] :
~ ( in @ X10 @ emptyset )
=> ( ! [X12: $i,X14: $i,X16: $i] :
( ( in @ X16 @ ( setadjoin @ X12 @ X14 ) )
<=> ( zip_tseitin_0 @ X16 @ X14 @ X12 ) )
=> ( ! [X18: $i,X20: $i] :
( ( in @ X20 @ ( powerset @ X18 ) )
<=> ! [X22: $i] : ( zip_tseitin_1 @ X22 @ X20 @ X18 ) )
=> ( ! [X24: $i,X26: $i] :
( ( in @ X26 @ ( setunion @ X24 ) )
<=> ? [X28: $i] : ( zip_tseitin_2 @ X28 @ X26 @ X24 ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X30: $i] :
( ( in @ X30 @ omega )
=> ( in @ ( setadjoin @ X30 @ X30 ) @ omega ) )
=> ( ! [X32: $i] :
( ( ( in @ emptyset @ X32 )
& ! [X34: $i] :
( ( ( in @ X34 @ omega )
& ( in @ X34 @ X32 ) )
=> ( in @ ( setadjoin @ X34 @ X34 ) @ X32 ) ) )
=> ! [X36: $i] :
( ( in @ X36 @ omega )
=> ( in @ X36 @ X32 ) ) )
=> ( ! [X38: $i > $i > $o,X40: $i] :
( ! [X42: $i] :
( ( in @ X42 @ X40 )
=> ? [X44: $i] :
( ! [X46: $i] :
( ( X38 @ X42 @ X46 )
=> ( X44 = X46 ) )
& ( X38 @ X42 @ X44 ) ) )
=> ? [X48: $i] :
! [X50: $i] :
( ( in @ X50 @ X48 )
<=> ? [X52: $i] : ( zip_tseitin_3 @ X52 @ X50 @ X40 @ X38 ) ) )
=> ( ! [X54: $i] :
( ? [X56: $i] : ( in @ X56 @ X54 )
=> ? [X58: $i] :
( ~ ? [X60: $i] :
( ( in @ X60 @ X54 )
& ( in @ X60 @ X58 ) )
& ( in @ X58 @ X54 ) ) )
=> ( ! [X62: $i] :
? [X64: $i] :
( ! [X66: $i] :
( ( ! [X68: $i] :
( ( in @ X68 @ X66 )
=> ( in @ X68 @ X62 ) )
& ? [X70: $i] : ( in @ X70 @ X66 ) )
=> ? [X72: $i,X74: $i] :
( ! [X76: $i] :
( ( in @ X76 @ X64 )
=> ( ! [X78: $i] :
( ( in @ X78 @ X76 )
=> ( in @ X78 @ X72 ) )
| ( in @ X74 @ X76 ) ) )
& ~ ? [X80: $i] :
( ( in @ X80 @ X66 )
& ( in @ X80 @ X72 ) )
& ( in @ X74 @ X66 )
& ( in @ X72 @ X64 ) ) )
& ! [X82: $i,X84: $i] :
( ( ( in @ X82 @ X64 )
& ( in @ X84 @ X64 ) )
=> ( ! [X86: $i] :
( ( in @ X86 @ X82 )
=> ( in @ X86 @ X84 ) )
| ! [X88: $i] :
( ( in @ X88 @ X84 )
=> ( in @ X88 @ X82 ) ) ) )
& ! [X90: $i,X92: $i] :
( ( ( in @ X90 @ X62 )
& ( in @ X92 @ X62 ) )
=> ( ! [X94: $i] :
( ( in @ X94 @ X64 )
=> ( ( in @ X90 @ X94 )
<=> ( in @ X92 @ X94 ) ) )
=> ( X90 = X92 ) ) )
& ! [X96: $i] :
( ( in @ X96 @ X64 )
=> ! [X98: $i] :
( ( in @ X98 @ X96 )
=> ( in @ X98 @ X62 ) ) ) )
=> ( ! [X100: $i > $o] :
( ? [X102: $i] :
( ! [X104: $i] :
( ( X100 @ X104 )
=> ( X102 = X104 ) )
& ( X100 @ X102 ) )
=> ( X100
@ ( descr
@ ^ [V_1: $i] : ( X100 @ V_1 ) ) ) )
=> ( ! [X106: $i,X108: $i > $o,X110: $i] :
( ( in @ X110 @ X106 )
=> ( ( X108 @ X110 )
=> ( in @ X110
@ ( dsetconstr @ X106
@ ^ [V_2: $i] : ( X108 @ V_2 ) ) ) ) )
=> ( ! [X112: $i,X114: $i > $o,X116: $i] :
( ( in @ X116
@ ( dsetconstr @ X112
@ ^ [V_3: $i] : ( X114 @ V_3 ) ) )
=> ( in @ X116 @ X112 ) )
=> ( ! [X118: $i,X120: $i > $o,X122: $i] :
( ( in @ X122
@ ( dsetconstr @ X118
@ ^ [V_4: $i] : ( X120 @ V_4 ) ) )
=> ( X120 @ X122 ) )
=> ( ! [X124: $i > $o] :
( ? [X126: $i] :
( ! [X128: $i] :
( ( X124 @ X128 )
=> ( X126 = X128 ) )
& ( X124 @ X126 ) )
=> ? [X130: $i] :
( ! [X132: $i] :
( ( X124 @ X132 )
=> ( X130 = X132 ) )
& ( X124 @ X130 ) ) )
=> ( ! [X134: $o,X136: $i] :
( ( in @ X136 @ ( prop2set @ X134 ) )
=> X134 )
=> ( ! [X138: $i] :
( ( in @ X138 @ emptyset )
=> ! [X140: $o] : X140 )
=> ( ! [X142: $i] :
~ ( in @ X142 @ emptyset )
=> ( ! [X144: $i] :
~ ( in @ X144 @ emptyset )
=> ( ! [X146: $i > $o] :
( ? [X148: $i] :
( ! [X150: $i] :
( ( X146 @ X150 )
=> ( X148 = X150 ) )
& ( X146 @ X148 ) )
=> ? [X152: $i] : ( X146 @ X152 ) )
=> ( ! [X154: $i,X156: $i] :
( ! [X158: $i] :
( ( in @ X158 @ X154 )
=> ( in @ X158 @ X156 ) )
=> ( ! [X160: $i] :
( ( in @ X160 @ X156 )
=> ( in @ X160 @ X154 ) )
=> ( X154 = X156 ) ) )
=> ( ! [X162: $i] :
( ! [X164: $i] :
~ ( in @ X164 @ X162 )
=> ( X162 = emptyset ) )
=> ( ! [X166: $i] :
( ! [X168: $i] :
~ ( in @ X168 @ X166 )
=> ( X166 = emptyset ) )
=> ( ! [X170: $i,X172: $i > $o,X174: $i] :
( ( in @ X174 @ X170 )
=> ( ( in @ X174
@ ( dsetconstr @ X170
@ ^ [V_5: $i] : ( X172 @ V_5 ) ) )
<=> ( X172 @ X174 ) ) )
=> ( ! [X176: $i] :
( ( X176 != emptyset )
=> ? [X178: $i] : ( in @ X178 @ X176 ) )
=> ( ! [X180: $i,X182: $i > $o,X184: $i] :
( ( in @ X184 @ X180 )
=> ( ( X182 @ X184 )
=> ( ( dsetconstr @ X180
@ ^ [V_6: $i] : ( X182 @ V_6 ) )
!= emptyset ) ) )
=> ( ! [X186: $i] :
( ? [X188: $i] : ( in @ X188 @ X186 )
=> ( X186 != emptyset ) )
=> ( ! [X190: $i,X192: $i] : ( in @ X190 @ ( setadjoin @ X190 @ X192 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X194: $i,X196: $i,X198: $i] :
( ( in @ X198 @ X196 )
=> ( in @ X198 @ ( setadjoin @ X194 @ X196 ) ) )
=> ( ! [X200: $i,X202: $i,X204: $i] :
( ( in @ X204 @ ( setadjoin @ X200 @ X202 ) )
=> ! [X206: $o] :
( ( ( X204 = X200 )
=> X206 )
=> ( ( ( in @ X204 @ X202 )
=> X206 )
=> X206 ) ) )
=> ( ! [X208: $i,X210: $i,X212: $i] :
( ( in @ X212 @ ( setadjoin @ X208 @ X210 ) )
=> ( ( X212 = X208 )
| ( in @ X212 @ X210 ) ) )
=> ( ! [X214: $i] :
( ( dsetconstr @ X214
@ ^ [V_7: $i] : $true )
= X214 )
=> ( ! [X216: $i,X218: $i] :
( ! [X220: $i] :
( ( in @ X220 @ X218 )
=> ( in @ X220 @ X216 ) )
=> ( in @ X218 @ ( powerset @ X216 ) ) )
=> ( ! [X222: $i] : ( in @ emptyset @ ( powerset @ X222 ) )
=> ( ! [X224: $i] : ( in @ emptyset @ ( powerset @ X224 ) )
=> ( ! [X226: $i,X228: $i,X230: $i] :
( ( in @ X228 @ ( powerset @ X226 ) )
=> ( ( in @ X230 @ X228 )
=> ( in @ X230 @ X226 ) ) )
=> ( ! [X232: $i,X234: $i,X236: $i] :
( ( in @ X234 @ X236 )
=> ( ( in @ X236 @ X232 )
=> ( in @ X234 @ ( setunion @ X232 ) ) ) )
=> ( ! [X238: $i,X240: $i] :
( ( in @ X240 @ ( setunion @ X238 ) )
=> ! [X242: $o] :
( ! [X244: $i] :
( ( in @ X240 @ X244 )
=> ( ( in @ X244 @ X238 )
=> X242 ) )
=> X242 ) )
=> ( ! [X246: $i,X248: $i] :
( ( in @ X248 @ X246 )
=> ( in @ X248 @ ( powerset @ ( setunion @ X246 ) ) ) )
=> ( ! [X250: $i > $o] :
( ? [X252: $i] :
( ! [X254: $i] :
( ( X250 @ X254 )
=> ( X252 = X254 ) )
& ( X250 @ X252 ) )
=> ? [X256: $i] :
! [X258: $i] :
( ( X250 @ X258 )
<=> ( X258 = X256 ) ) )
=> ( ! [X260: $i] :
( ( X260 != emptyset )
=> ? [X262: $i] : ( in @ X262 @ X260 ) )
=> ( ! [X264: $i,X266: $i] :
( ( in @ X264 @ ( setadjoin @ X266 @ emptyset ) )
=> ( X264 = X266 ) )
=> ( ! [X268: $i,X270: $i] :
( ( X268 != X270 )
=> ~ ( in @ X270 @ ( setadjoin @ X268 @ emptyset ) ) )
=> ( ! [X272: $i,X274: $i] :
( ( X272 = X274 )
=> ( in @ X272 @ ( setadjoin @ X274 @ emptyset ) ) )
=> ( ! [X276: $i,X278: $i] :
( ( in @ X276 @ ( setadjoin @ X278 @ emptyset ) )
=> ( in @ X278 @ ( setadjoin @ X276 @ emptyset ) ) )
=> ( ! [X280: $i,X282: $i,X284: $i] :
( ( in @ X284 @ ( setadjoin @ X280 @ ( setadjoin @ X282 @ emptyset ) ) )
=> ( ( X284 = X280 )
| ( X284 = X282 ) ) )
=> ( ! [X286: $i,X288: $i] : ( in @ X286 @ ( setadjoin @ X286 @ ( setadjoin @ X288 @ emptyset ) ) )
=> ( ! [X290: $i,X292: $i] : ( in @ X292 @ ( setadjoin @ X290 @ ( setadjoin @ X292 @ emptyset ) ) )
=> ( ! [X294: $i,X296: $i > $o] :
( ? [X298: $i] :
( ( X296 @ X298 )
& ( in @ X298 @ X294 ) )
=> ( ( dsetconstr @ X294
@ ^ [V_8: $i] : ( X296 @ V_8 ) )
!= emptyset ) )
=> ( ! [X300: $i > $o,X302: $i] :
( ( in @ X302 @ emptyset )
=> ( X300 @ X302 ) )
=> ( ! [X304: $i,X306: $i > $o] :
( ~ ! [X308: $i] :
( ( in @ X308 @ X304 )
=> ( X306 @ X308 ) )
=> ? [X310: $i] :
( ~ ( X306 @ X310 )
& ( in @ X310 @ X304 ) ) )
=> ( ! [X312: $i,X314: $i > $o] :
( ! [X316: $i] :
( ( in @ X316 @ X312 )
=> ~ ( X314 @ X316 ) )
=> ~ ? [X318: $i] :
( ( X314 @ X318 )
& ( in @ X318 @ X312 ) ) )
=> ( ! [X320: $i,X322: $i > $o] :
( ~ ? [X324: $i] :
( ( X322 @ X324 )
& ( in @ X324 @ X320 ) )
=> ! [X326: $i] :
( ( in @ X326 @ X320 )
=> ~ ( X322 @ X326 ) ) )
=> ( ! [X328: $i,X330: $i > $o] :
( ? [X332: $i] :
( ~ ( X330 @ X332 )
& ( in @ X332 @ X328 ) )
=> ~ ! [X334: $i] :
( ( in @ X334 @ X328 )
=> ( X330 @ X334 ) ) )
=> ( ! [X336: $o] :
( X336
=> ( in @ emptyset @ ( prop2set @ X336 ) ) )
=> ( ! [X338: $o] :
( X338
=> ( set2prop @ ( prop2set @ X338 ) ) )
=> ( ! [X340: $i,X342: $i > $o] :
( ~ ? [X344: $i] :
( ( X342 @ X344 )
& ( in @ X344 @ X340 ) )
=> ! [X346: $i] :
( ( in @ X346 @ X340 )
=> ~ ( X342 @ X346 ) ) )
=> ( ! [X348: $i,X350: $i > $o] :
( ~ ! [X352: $i] :
( ( in @ X352 @ X348 )
=> ( X350 @ X352 ) )
=> ? [X354: $i] :
( ~ ( X350 @ X354 )
& ( in @ X354 @ X348 ) ) )
=> ( ! [X356: $i > $o] :
( ? [X358: $i] :
( ! [X360: $i] :
( ( X356 @ X360 )
=> ( X358 = X360 ) )
& ( X356 @ X358 ) )
=> ? [X362: $i] :
( ! [X364: $i] :
( ( X356 @ X364 )
=> ( X362 = X364 ) )
& ( X356 @ X362 ) ) )
=> ( ! [X366: $i > $o] :
( ? [X368: $i] : ( X366 @ X368 )
=> ( ! [X370: $i,X372: $i] :
( ( X366 @ X370 )
=> ( ( X366 @ X372 )
=> ( X370 = X372 ) ) )
=> ? [X374: $i] :
( ! [X376: $i] :
( ( X366 @ X376 )
=> ( X374 = X376 ) )
& ( X366 @ X374 ) ) ) )
=> ( ! [X378: $i > $o] :
( ? [X380: $i] :
! [X382: $i] :
( ( X378 @ X382 )
<=> ( X382 = X380 ) )
=> ? [X384: $i] :
( ! [X386: $i] :
( ( X378 @ X386 )
=> ( X384 = X386 ) )
& ( X378 @ X384 ) ) )
=> ( ! [X388: $i,X390: $i] :
( ( X388 = X390 )
=> ! [X392: $i,X394: $i] :
( ( X392 = X394 )
=> ( ( in @ X392 @ X388 )
=> ( in @ X394 @ X390 ) ) ) )
=> ( ! [X396: $i,X398: $i] :
( ( X396 = X398 )
=> ! [X400: $i,X402: $i] :
( ( X400 = X402 )
=> ( ( in @ X400 @ X396 )
<=> ( in @ X402 @ X398 ) ) ) )
=> ( ! [X404: $i > $o] :
( ? [X406: $i] :
( ! [X408: $i] :
( ( X404 @ X408 )
=> ( X406 = X408 ) )
& ( X404 @ X406 ) )
=> ! [X410: $i,X412: $i] :
( ( X404 @ X410 )
=> ( ( X404 @ X412 )
=> ( X410 = X412 ) ) ) )
=> ( ! [X414: $i > $o,X416: $i > $o] :
( ! [X418: $i,X420: $i] :
( ( X418 = X420 )
=> ( ( X414 @ X418 )
<=> ( X416 @ X420 ) ) )
=> ( ? [X422: $i] : ( zip_tseitin_5 @ X422 @ X414 )
<=> ? [X426: $i] : ( zip_tseitin_5 @ X426 @ X416 ) ) )
=> ( ( emptyset = emptyset )
=> ( ! [X430: $i,X432: $i] :
( ( X430 = X432 )
=> ! [X434: $i,X436: $i] :
( ( X434 = X436 )
=> ( ( setadjoin @ X430 @ X434 )
= ( setadjoin @ X432 @ X436 ) ) ) )
=> ( ! [X438: $i,X440: $i] :
( ( X438 = X440 )
=> ( ( powerset @ X438 )
= ( powerset @ X440 ) ) )
=> ( ! [X442: $i,X444: $i] :
( ( X442 = X444 )
=> ( ( setunion @ X442 )
= ( setunion @ X444 ) ) )
=> ( ( omega = omega )
=> ( ! [X446: $i > $o] :
( ? [X448: $i] :
( ! [X450: $i] :
( ( X446 @ X450 )
=> ( X448 = X450 ) )
& ( X446 @ X448 ) )
=> ! [X452: $i,X454: $i] :
( ( X446 @ X452 )
=> ( ( X446 @ X454 )
=> ( X452 = X454 ) ) ) )
=> ( ! [X456: $i > $o,X458: $i > $o] :
( ! [X460: $i,X462: $i] :
( ( X460 = X462 )
=> ( ( X456 @ X460 )
<=> ( X458 @ X462 ) ) )
=> ( ? [X464: $i] :
( ! [X466: $i] :
( ( X456 @ X466 )
=> ( X464 = X466 ) )
& ( X456 @ X464 ) )
=> ( ? [X468: $i] :
( ! [X470: $i] :
( ( X458 @ X470 )
=> ( X468 = X470 ) )
& ( X458 @ X468 ) )
=> ( ( descr
@ ^ [V_9: $i] : ( X456 @ V_9 ) )
= ( descr
@ ^ [V_10: $i] : ( X458 @ V_10 ) ) ) ) ) )
=> ( ! [X472: $i,X474: $i] :
( ( X472 = X474 )
=> ! [X476: $i > $o,X478: $i > $o] :
( ! [X480: $i] :
( ( in @ X480 @ X472 )
=> ! [X482: $i] :
( ( in @ X482 @ X474 )
=> ( ( X480 = X482 )
=> ( ( X476 @ X480 )
<=> ( X478 @ X482 ) ) ) ) )
=> ( ( dsetconstr @ X472
@ ^ [V_11: $i] : ( X476 @ V_11 ) )
= ( dsetconstr @ X474
@ ^ [V_12: $i] : ( X478 @ V_12 ) ) ) ) )
=> ( ! [X484: $i,X486: $i] :
( ! [X488: $i] :
( ( in @ X488 @ X484 )
=> ( in @ X488 @ X486 ) )
=> ( subset @ X484 @ X486 ) )
=> ( ! [X490: $i,X492: $i] :
( ( X490 = X492 )
=> ( subset @ X492 @ X490 ) )
=> ( ! [X494: $i,X496: $i] :
( ( X494 = X496 )
=> ( subset @ X494 @ X496 ) )
=> ( ! [X498: $i,X500: $i] :
( ! [X502: $i] :
( ( in @ X502 @ X498 )
=> ( in @ X502 @ X500 ) )
=> ( subset @ X498 @ X500 ) )
=> ( ! [X504: $i] : ( subset @ emptyset @ X504 )
=> ( ! [X506: $i,X508: $i,X510: $i] :
( ( subset @ X506 @ X508 )
=> ( ( in @ X510 @ X506 )
=> ( in @ X510 @ X508 ) ) )
=> ( ! [X512: $i,X514: $i,X516: $i] :
( ( subset @ X512 @ X514 )
=> ( ~ ( in @ X516 @ X514 )
=> ~ ( in @ X516 @ X512 ) ) )
=> ( ! [X518: $i,X520: $i,X522: $i] :
( ( in @ X522 @ X518 )
=> ( ~ ( in @ X522 @ X520 )
=> ~ ( subset @ X518 @ X520 ) ) )
=> ( ! [X524: $i,X526: $i] :
( ~ ( subset @ X524 @ X526 )
=> ( X524 != X526 ) )
=> ( ! [X528: $i,X530: $i,X532: $i] :
( ( in @ X532 @ X528 )
=> ( ~ ( in @ X532 @ X530 )
=> ( X528 != X530 ) ) )
=> ( ! [X534: $i] : ( subset @ X534 @ X534 )
=> ( ! [X536: $i,X538: $i,X540: $i] :
( ( subset @ X536 @ X538 )
=> ( ( subset @ X538 @ X540 )
=> ( subset @ X536 @ X540 ) ) )
=> ( ! [X542: $i,X544: $i] : ( subset @ X544 @ ( setadjoin @ X542 @ X544 ) )
=> ( ! [X546: $i,X548: $i,X550: $i] :
( ( subset @ X546 @ X550 )
=> ( subset @ X546 @ ( setadjoin @ X548 @ X550 ) ) )
=> ( ! [X552: $i,X554: $i] :
( ( subset @ X552 @ X554 )
=> ( in @ X552 @ ( powerset @ X554 ) ) )
=> ( ! [X556: $i,X558: $i] :
( ( subset @ X556 @ X558 )
=> ( ( subset @ X558 @ X556 )
=> ( X556 = X558 ) ) )
=> ( ! [X560: $i] :
( ( subset @ X560 @ emptyset )
=> ( X560 = emptyset ) )
=> ( ! [X562: $i,X564: $i] :
( ( subset @ X564 @ X562 )
=> ( in @ X564 @ ( powerset @ X562 ) ) )
=> ( ! [X566: $i,X568: $i] :
( ( in @ X568 @ ( powerset @ X566 ) )
=> ( subset @ X568 @ X566 ) )
=> ( ! [X570: $i] : ( in @ X570 @ ( powerset @ X570 ) )
=> ( ! [X572: $i,X574: $i] :
( ( subset @ X572 @ X574 )
=> ( subset @ ( powerset @ X572 ) @ ( powerset @ X574 ) ) )
=> ( ! [X576: $i,X578: $i > $o] :
( in
@ ( dsetconstr @ X576
@ ^ [V_13: $i] : ( X578 @ V_13 ) )
@ ( powerset @ X576 ) )
=> ( ! [X580: $i,X582: $i > $o] :
( subset
@ ( dsetconstr @ X580
@ ^ [V_14: $i] : ( X582 @ V_14 ) )
@ X580 )
=> ( ! [X584: $i,X586: $i,X588: $i] :
( ( in @ X588 @ X584 )
=> ( in @ X588 @ ( binunion @ X584 @ X586 ) ) )
=> ( ! [X590: $i,X592: $i] : ( in @ X592 @ ( setadjoin @ X590 @ ( setadjoin @ X592 @ emptyset ) ) )
=> ( ! [X594: $i,X596: $i,X598: $i] :
( ( in @ X598 @ X596 )
=> ( in @ X598 @ ( binunion @ X594 @ X596 ) ) )
=> ( ! [X600: $i,X602: $i,X604: $i,X606: $o] :
( ( in @ X604 @ ( binunion @ X600 @ X602 ) )
=> ( ( ( in @ X604 @ X600 )
=> X606 )
=> ( ( ( in @ X604 @ X602 )
=> X606 )
=> X606 ) ) )
=> ( ! [X608: $i,X610: $i,X612: $i] :
( ( in @ X612 @ ( binunion @ X608 @ X610 ) )
=> ( ( in @ X612 @ X608 )
| ( in @ X612 @ X610 ) ) )
=> ( ! [X614: $i,X616: $i] : ( subset @ X614 @ ( binunion @ X614 @ X616 ) )
=> ( ! [X618: $i,X620: $i] : ( subset @ X620 @ ( binunion @ X618 @ X620 ) )
=> ( ! [X622: $i,X624: $i,X626: $i] :
( ( in @ X626 @ X622 )
=> ( ( in @ X626 @ X624 )
=> ( in @ X626 @ ( binintersect @ X622 @ X624 ) ) ) )
=> ( ! [X628: $i,X630: $i,X632: $i] :
( ( subset @ X632 @ X628 )
=> ( ( subset @ X632 @ X630 )
=> ( subset @ X632 @ ( binintersect @ X628 @ X630 ) ) ) )
=> ( ! [X634: $i,X636: $i,X638: $i] :
( ( in @ X638 @ ( binintersect @ X634 @ X636 ) )
=> ( in @ X638 @ X634 ) )
=> ( ! [X640: $i,X642: $i] : ( subset @ ( binintersect @ X640 @ X642 ) @ X640 )
=> ( ! [X644: $i,X646: $i] :
( ( subset @ X644 @ X646 )
=> ( ( binintersect @ X644 @ X646 )
= X644 ) )
=> ( ! [X648: $i,X650: $i] :
( ( ( binintersect @ X648 @ X650 )
= X650 )
=> ( subset @ X650 @ X648 ) )
=> ( ! [X652: $i,X654: $i,X656: $i] :
( ( in @ X656 @ ( binintersect @ X652 @ X654 ) )
=> ( in @ X656 @ X654 ) )
=> ( ! [X658: $i,X660: $i] :
( ~ ? [X662: $i] :
( ( in @ X662 @ X660 )
& ( in @ X662 @ X658 ) )
=> ( ( binintersect @ X658 @ X660 )
= emptyset ) )
=> ( ! [X664: $i,X666: $i] : ( subset @ ( binintersect @ X664 @ X666 ) @ X666 )
=> ( ! [X668: $i,X670: $i] :
( ( subset @ X670 @ X668 )
=> ( ( binintersect @ X668 @ X670 )
= X670 ) )
=> ( ! [X672: $i,X674: $i] :
( ( ( binintersect @ X672 @ X674 )
= X672 )
=> ( subset @ X672 @ X674 ) )
=> ( ! [X676: $i,X678: $i,X680: $i] :
( ( binintersect @ X676 @ ( binunion @ X678 @ X680 ) )
= ( binunion @ ( binintersect @ X676 @ X678 ) @ ( binintersect @ X676 @ X680 ) ) )
=> ( ! [X682: $i,X684: $i,X686: $i] :
( ( in @ X686 @ X682 )
=> ( ~ ( in @ X686 @ X684 )
=> ( in @ X686 @ ( setminus @ X682 @ X684 ) ) ) )
=> ( ! [X688: $i,X690: $i,X692: $i] :
( ( in @ X692 @ ( setminus @ X688 @ X690 ) )
=> ( in @ X692 @ X688 ) )
=> ( ! [X694: $i,X696: $i,X698: $i] :
( ( in @ X698 @ ( setminus @ X694 @ X696 ) )
=> ~ ( in @ X698 @ X696 ) )
=> ( ! [X700: $i,X702: $i] :
( ( subset @ X700 @ X702 )
=> ( ( setminus @ X700 @ X702 )
= emptyset ) )
=> ( ! [X704: $i,X706: $i,X708: $i] :
( ~ ( in @ X708 @ ( setminus @ X704 @ X706 ) )
=> ( ( in @ X708 @ X704 )
=> ( in @ X708 @ X706 ) ) )
=> ( ! [X710: $i,X712: $i,X714: $i] :
( ~ ( in @ X714 @ ( setminus @ X710 @ X712 ) )
=> ( ~ ( in @ X714 @ X712 )
=> ~ ( in @ X714 @ X710 ) ) )
=> ( ! [X716: $i,X718: $i,X720: $i] :
( ~ ( in @ X720 @ X716 )
=> ~ ( in @ X720 @ ( setminus @ X716 @ X718 ) ) )
=> ( ! [X722: $i,X724: $i,X726: $i] :
( ( in @ X726 @ X724 )
=> ~ ( in @ X726 @ ( setminus @ X722 @ X724 ) ) )
=> ( ! [X728: $i,X730: $i] : ( subset @ ( setminus @ X728 @ X730 ) @ X728 )
=> ( ! [X732: $i,X734: $i] :
( ( ( setminus @ X732 @ X734 )
= emptyset )
=> ( subset @ X732 @ X734 ) )
=> ( ! [X736: $i,X738: $i,X740: $i] :
( ( in @ X740 @ ( symdiff @ X736 @ X738 ) )
=> ! [X742: $o] :
( ( ( in @ X740 @ X736 )
=> ( ~ ( in @ X740 @ X738 )
=> X742 ) )
=> ( ( ~ ( in @ X740 @ X736 )
=> ( ( in @ X740 @ X738 )
=> X742 ) )
=> X742 ) ) )
=> ( ! [X744: $i,X746: $i,X748: $i] :
( ( in @ X748 @ X744 )
=> ( ~ ( in @ X748 @ X746 )
=> ( in @ X748 @ ( symdiff @ X744 @ X746 ) ) ) )
=> ( ! [X750: $i,X752: $i,X754: $i] :
( ~ ( in @ X754 @ X750 )
=> ( ( in @ X754 @ X752 )
=> ( in @ X754 @ ( symdiff @ X750 @ X752 ) ) ) )
=> ( ! [X756: $i,X758: $i,X760: $i] :
( ( in @ X760 @ X756 )
=> ( ( in @ X760 @ X758 )
=> ~ ( in @ X760 @ ( symdiff @ X756 @ X758 ) ) ) )
=> ( ! [X762: $i,X764: $i,X766: $i] :
( ~ ( in @ X766 @ X762 )
=> ( ~ ( in @ X766 @ X764 )
=> ~ ( in @ X766 @ ( symdiff @ X762 @ X764 ) ) ) )
=> ( ! [X768: $i,X770: $i] : ( in @ X770 @ ( setadjoin @ X768 @ ( setadjoin @ X770 @ emptyset ) ) )
=> ( ! [X772: $i,X774: $i] : ( in @ X772 @ ( setunion @ ( setadjoin @ ( setadjoin @ X772 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X772 @ ( setadjoin @ X774 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X776: $i,X778: $i] : ( in @ X778 @ ( setunion @ ( setadjoin @ ( setadjoin @ X776 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X776 @ ( setadjoin @ X778 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X780: $i,X782: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X780 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X780 @ ( setadjoin @ X782 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X784: $i,X786: $i] : ( iskpair @ ( kpair @ X784 @ X786 ) )
=> ( ! [X788: $i,X790: $i] :
( ( in @ X790 @ X788 )
=> ( subset @ ( setadjoin @ X790 @ emptyset ) @ X788 ) )
=> ( ! [X792: $i,X794: $i] :
( ( in @ X794 @ X792 )
=> ( in @ ( setadjoin @ X794 @ emptyset ) @ ( powerset @ X792 ) ) )
=> ( ! [X796: $i,X798: $i,X800: $i] :
( ( in @ X800 @ X796 )
=> ( in @ ( setadjoin @ X800 @ emptyset ) @ ( powerset @ ( binunion @ X796 @ X798 ) ) ) )
=> ( ! [X802: $i,X804: $i,X806: $i] :
( ( in @ X806 @ ( setadjoin @ X802 @ ( setadjoin @ X804 @ emptyset ) ) )
=> ( ( X806 = X802 )
| ( X806 = X804 ) ) )
=> ( ! [X808: $i,X810: $i,X812: $i] :
( ( in @ X812 @ X808 )
=> ! [X814: $i] :
( ( in @ X814 @ X810 )
=> ( subset @ ( setadjoin @ X812 @ ( setadjoin @ X814 @ emptyset ) ) @ ( binunion @ X808 @ X810 ) ) ) )
=> ( ! [X816: $i,X818: $i,X820: $i] :
( ( in @ X820 @ X816 )
=> ! [X822: $i] :
( ( in @ X822 @ X818 )
=> ( in @ ( setadjoin @ X820 @ ( setadjoin @ X822 @ emptyset ) ) @ ( powerset @ ( binunion @ X816 @ X818 ) ) ) ) )
=> ( ! [X824: $i,X826: $i,X828: $i] :
( ( in @ X828 @ X824 )
=> ! [X830: $i] :
( ( in @ X830 @ X826 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X828 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X828 @ ( setadjoin @ X830 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X824 @ X826 ) ) ) ) )
=> ( ! [X832: $i,X834: $i,X836: $i] :
( ( in @ X836 @ X832 )
=> ! [X838: $i] :
( ( in @ X838 @ X834 )
=> ( in @ ( setadjoin @ ( setadjoin @ X836 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X836 @ ( setadjoin @ X838 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X832 @ X834 ) ) ) ) ) )
=> ( ! [X840: $i,X842: $i,X844: $i] :
( ( in @ X844 @ X840 )
=> ! [X846: $i] :
( ( in @ X846 @ X842 )
=> ( in @ ( kpair @ X844 @ X846 ) @ ( powerset @ ( powerset @ ( binunion @ X840 @ X842 ) ) ) ) ) )
=> ( ! [X848: $i,X850: $i,X852: $i] :
( ( in @ X852 @ X848 )
=> ! [X854: $i] :
( ( in @ X854 @ X850 )
=> ( in @ ( kpair @ X852 @ X854 ) @ ( cartprod @ X848 @ X850 ) ) ) )
=> ( ! [X856: $i,X858: $i,X860: $i] :
( ( in @ X860 @ ( cartprod @ X856 @ X858 ) )
=> ? [X862: $i] :
( ? [X864: $i] :
( ( X860
= ( kpair @ X862 @ X864 ) )
& ( in @ X864 @ X858 ) )
& ( in @ X862 @ X856 ) ) )
=> ( ! [X866: $i,X868: $i,X870: $i] :
( ( in @ X870 @ ( cartprod @ X866 @ X868 ) )
=> ( iskpair @ X870 ) )
=> ( ! [X872: $i,X874: $i] :
( ( in @ X874 @ ( setunion @ X872 ) )
=> ? [X876: $i] :
( ( in @ X874 @ X876 )
& ( in @ X876 @ X872 ) ) )
=> ( ! [X878: $i] : ( subset @ ( setunion @ ( setadjoin @ X878 @ emptyset ) ) @ X878 )
=> ( ! [X880: $i] : ( subset @ X880 @ ( setunion @ ( setadjoin @ X880 @ emptyset ) ) )
=> ( ! [X882: $i] :
( ( setunion @ ( setadjoin @ X882 @ emptyset ) )
= X882 )
=> ( ! [X884: $i,X886: $i > $o] :
( ! [X888: $i] :
( ( in @ X888 @ X884 )
=> ! [X890: $i] :
( ( in @ X890 @ X884 )
=> ( ( X886 @ X888 )
=> ( ( X886 @ X890 )
=> ( X888 = X890 ) ) ) ) )
=> ( ? [X892: $i] :
( ( X886 @ X892 )
& ( in @ X892 @ X884 ) )
=> ( singleton
@ ( dsetconstr @ X884
@ ^ [V_15: $i] : ( X886 @ V_15 ) ) ) ) )
=> ( ! [X894: $i,X896: $i > $o] :
( ( ex1 @ X894
@ ^ [V_16: $i] : ( X896 @ V_16 ) )
=> ? [X898: $i] :
( ( X896 @ X898 )
& ( in @ X898 @ X894 ) ) )
=> ( ! [X900: $i,X902: $i > $o,X904: $i] :
( ( in @ X904 @ X900 )
=> ( ( X902 @ X904 )
=> ( ! [X906: $i] :
( ( in @ X906 @ X900 )
=> ( ( X902 @ X906 )
=> ( X906 = X904 ) ) )
=> ( ex1 @ X900
@ ^ [V_17: $i] : ( X902 @ V_17 ) ) ) ) )
=> ( ! [X908: $i,X910: $i > $o] :
( ! [X912: $i] :
( ( in @ X912 @ X908 )
=> ! [X914: $i] :
( ( in @ X914 @ X908 )
=> ( ( X910 @ X912 )
=> ( ( X910 @ X914 )
=> ( X912 = X914 ) ) ) ) )
=> ( ? [X916: $i] :
( ( X910 @ X916 )
& ( in @ X916 @ X908 ) )
=> ( ex1 @ X908
@ ^ [V_18: $i] : ( X910 @ V_18 ) ) ) )
=> ( ! [X918: $i,X920: $i] :
( ( ( setadjoin @ X918 @ emptyset )
= ( setadjoin @ X920 @ emptyset ) )
=> ( X918 = X920 ) )
=> ( ! [X922: $i,X924: $i,X926: $i] :
( ( in @ ( setadjoin @ X926 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X922 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X922 @ ( setadjoin @ X924 @ emptyset ) ) @ emptyset ) ) )
=> ( X922 = X926 ) )
=> ( ! [X928: $i] :
( ( iskpair @ X928 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X928 )
@ ^ [V_19: $i] : ( in @ ( setadjoin @ V_19 @ emptyset ) @ X928 ) ) ) )
=> ( ! [X930: $i] :
( ( singleton @ X930 )
=> ( in @ ( setunion @ X930 ) @ X930 ) )
=> ( ! [X932: $i,X934: $i] :
( ( kfst @ ( kpair @ X932 @ X934 ) )
= X932 )
=> ( ! [X936: $i,X938: $i,X940: $i] :
( ( in @ X940 @ ( cartprod @ X936 @ X938 ) )
=> ( in @ ( kfst @ X940 ) @ X936 ) )
=> ( ! [X942: $i,X944: $i,X946: $i,X948: $i] :
( ( ( setadjoin @ ( setadjoin @ X942 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X942 @ ( setadjoin @ X944 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X946 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X946 @ ( setadjoin @ X948 @ emptyset ) ) @ emptyset ) ) )
=> ( X942 = X946 ) )
=> ( ! [X950: $i,X952: $i,X954: $i,X956: $i] :
( ( ( kpair @ X950 @ X952 )
= ( kpair @ X954 @ X956 ) )
=> ( X950 = X954 ) )
=> ( ! [X958: $i,X960: $i] :
( ( X958 = X960 )
=> ( ( setadjoin @ X958 @ ( setadjoin @ X960 @ emptyset ) )
= ( setadjoin @ X958 @ emptyset ) ) )
=> ( ! [X962: $i,X964: $i] :
( ( X962 = X964 )
=> ( ( setadjoin @ ( setadjoin @ X962 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X962 @ ( setadjoin @ X964 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X962 @ emptyset ) @ emptyset ) ) )
=> ( ! [X966: $i,X968: $i,X970: $i,X972: $i] :
( ( ( setadjoin @ ( setadjoin @ X966 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X966 @ ( setadjoin @ X968 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X970 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X970 @ ( setadjoin @ X972 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X970 = X972 )
=> ( X968 = X972 ) ) )
=> ( ! [X974: $i,X976: $i,X978: $i] :
( ( ( setadjoin @ X974 @ ( setadjoin @ X976 @ emptyset ) )
= ( setadjoin @ X978 @ emptyset ) )
=> ( X974 = X976 ) )
=> ( ! [X980: $i,X982: $i,X984: $i,X986: $i] :
( ( ( setadjoin @ ( setadjoin @ X980 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X980 @ ( setadjoin @ X982 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X984 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X984 @ ( setadjoin @ X986 @ emptyset ) ) @ emptyset ) ) )
=> ( X982 = X986 ) )
=> ( ! [X988: $i,X990: $i,X992: $i,X994: $i] :
( ( ( kpair @ X988 @ X990 )
= ( kpair @ X992 @ X994 ) )
=> ( X990 = X994 ) )
=> ( ! [X996: $i] :
( ( iskpair @ X996 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X996 )
@ ^ [V_20: $i] :
( X996
= ( kpair @ ( kfst @ X996 ) @ V_20 ) ) ) ) )
=> ( ! [X998: $i,X1000: $i] :
( ( ksnd @ ( kpair @ X998 @ X1000 ) )
= X1000 )
=> ( ! [X1002: $i] :
( ( iskpair @ X1002 )
=> ( ( kpair @ ( kfst @ X1002 ) @ ( ksnd @ X1002 ) )
= X1002 ) )
=> ( ! [X1004: $i,X1006: $i,X1008: $i] :
( ( in @ X1008 @ ( cartprod @ X1004 @ X1006 ) )
=> ( in @ ( ksnd @ X1008 ) @ X1006 ) )
=> ( ! [X1010: $i,X1012: $i,X1014: $i,X1016: $i] :
( ( in @ ( kpair @ X1014 @ X1016 ) @ ( cartprod @ X1010 @ X1012 ) )
=> ( in @ X1014 @ X1010 ) )
=> ( ! [X1018: $i,X1020: $i,X1022: $i,X1024: $i] :
( ( in @ ( kpair @ X1022 @ X1024 ) @ ( cartprod @ X1018 @ X1020 ) )
=> ( in @ X1024 @ X1020 ) )
=> ( ! [X1026: $i,X1028: $i,X1030: $i] :
( ( in @ X1030 @ X1026 )
=> ! [X1032: $i] :
( ( in @ X1032 @ X1028 )
=> ( ( kpair @ X1030 @ X1032 )
= ( kpair @ X1030 @ X1032 ) ) ) )
=> ( ! [X1034: $i,X1036: $i,X1038: $i] :
( ( in @ X1038 @ X1034 )
=> ! [X1040: $i] :
( ( in @ X1040 @ X1036 )
=> ( ( kfst @ ( kpair @ X1038 @ X1040 ) )
= X1038 ) ) )
=> ( ! [X1042: $i,X1044: $i,X1046: $i] :
( ( in @ X1046 @ X1042 )
=> ! [X1048: $i] :
( ( in @ X1048 @ X1044 )
=> ( ( ksnd @ ( kpair @ X1046 @ X1048 ) )
= X1048 ) ) )
=> ( ! [X1050: $i,X1052: $i,X1054: $i] :
( ( in @ X1054 @ ( cartprod @ X1050 @ X1052 ) )
=> ( ( kpair @ ( kfst @ X1054 ) @ ( ksnd @ X1054 ) )
= X1054 ) )
=> ( ! [X1056: $i,X1058: $i,X1060: $i > $i > $o,X1062: $i] :
( ( in @ X1062 @ X1056 )
=> ! [X1064: $i] :
( ( in @ X1064 @ X1058 )
=> ( ( X1060 @ X1062 @ X1064 )
=> ( in @ ( kpair @ X1062 @ X1064 )
@ ( dpsetconstr @ X1056 @ X1058
@ ^ [V_21: $i,V_22: $i] : ( X1060 @ V_21 @ V_22 ) ) ) ) ) )
=> ( ! [X1066: $i,X1068: $i,X1070: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1066 @ X1068
@ ^ [V_23: $i,V_24: $i] : ( X1070 @ V_23 @ V_24 ) )
@ ( cartprod @ X1066 @ X1068 ) )
=> ( ! [X1072: $i,X1074: $i,X1076: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1072 @ X1074
@ ^ [V_25: $i,V_26: $i] : ( X1076 @ V_25 @ V_26 ) )
@ ( cartprod @ X1072 @ X1074 ) )
=> ( ! [X1078: $i,X1080: $i,X1082: $i > $i > $o,X1084: $i] :
( ( in @ X1084 @ X1078 )
=> ! [X1086: $i] :
( ( in @ X1086 @ X1080 )
=> ( ( in @ ( kpair @ X1084 @ X1086 )
@ ( dpsetconstr @ X1078 @ X1080
@ ^ [V_27: $i,V_28: $i] : ( X1082 @ V_27 @ V_28 ) ) )
=> ( X1082 @ X1084 @ X1086 ) ) ) )
=> ( ! [X1088: $i,X1090: $i,X1092: $i > $i > $o,X1094: $i,X1096: $i] :
( ( in @ ( kpair @ X1094 @ X1096 )
@ ( dpsetconstr @ X1088 @ X1090
@ ^ [V_29: $i,V_30: $i] : ( X1092 @ V_29 @ V_30 ) ) )
=> ( in @ X1094 @ X1088 ) )
=> ( ! [X1098: $i,X1100: $i,X1102: $i > $i > $o,X1104: $i,X1106: $i] :
( ( in @ ( kpair @ X1104 @ X1106 )
@ ( dpsetconstr @ X1098 @ X1100
@ ^ [V_31: $i,V_32: $i] : ( X1102 @ V_31 @ V_32 ) ) )
=> ( in @ X1106 @ X1100 ) )
=> ( ! [X1108: $i,X1110: $i,X1112: $i > $i > $o,X1114: $i,X1116: $i] :
( ( in @ ( kpair @ X1114 @ X1116 )
@ ( dpsetconstr @ X1108 @ X1110
@ ^ [V_33: $i,V_34: $i] : ( X1112 @ V_33 @ V_34 ) ) )
=> ( X1112 @ X1114 @ X1116 ) )
=> ( ! [X1118: $i,X1120: $i,X1122: $i] :
( ( func @ X1118 @ X1120 @ X1122 )
=> ! [X1124: $i] :
( ( in @ X1124 @ X1118 )
=> ( singleton
@ ( dsetconstr @ X1120
@ ^ [V_35: $i] : ( in @ ( kpair @ X1124 @ V_35 ) @ X1122 ) ) ) ) )
=> ( ! [X1126: $i,X1128: $i,X1130: $i] :
( ( func @ X1126 @ X1128 @ X1130 )
=> ! [X1132: $i] :
( ( in @ X1132 @ X1126 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X1128
@ ^ [V_36: $i] : ( in @ ( kpair @ X1132 @ V_36 ) @ X1130 ) ) )
@ X1128 ) ) )
=> ( ! [X1134: $i,X1136: $i,X1138: $i] :
( ( func @ X1134 @ X1136 @ X1138 )
=> ! [X1140: $i] :
( ( in @ X1140 @ X1134 )
=> ( in @ ( ap @ X1134 @ X1136 @ X1138 @ X1140 ) @ X1136 ) ) )
=> ( ! [X1142: $i,X1144: $i,X1146: $i] :
( ( in @ X1146 @ ( funcSet @ X1142 @ X1144 ) )
=> ( func @ X1142 @ X1144 @ X1146 ) )
=> ( ! [X1148: $i,X1150: $i,X1152: $i] :
( ( in @ X1152 @ ( funcSet @ X1148 @ X1150 ) )
=> ! [X1154: $i] :
( ( in @ X1154 @ X1148 )
=> ( in @ ( ap @ X1148 @ X1150 @ X1152 @ X1154 ) @ X1150 ) ) )
=> ( ! [X1156: $i,X1158: $i,X1160: $i] :
( ( func @ X1156 @ X1158 @ X1160 )
=> ( in @ X1160 @ ( funcSet @ X1156 @ X1158 ) ) )
=> ( ! [X1162: $i,X1164: $i,X1166: $i > $i] :
( ! [X1168: $i] :
( ( in @ X1168 @ X1162 )
=> ( in @ ( X1166 @ X1168 ) @ X1164 ) )
=> ( func @ X1162 @ X1164
@ ( dpsetconstr @ X1162 @ X1164
@ ^ [V_37: $i,V_38: $i] :
( ( X1166 @ V_37 )
= V_38 ) ) ) )
=> ( ! [X1170: $i,X1172: $i,X1174: $i > $i] :
( ! [X1176: $i] :
( ( in @ X1176 @ X1170 )
=> ( in @ ( X1174 @ X1176 ) @ X1172 ) )
=> ( func @ X1170 @ X1172
@ ( lam @ X1170 @ X1172
@ ^ [V_39: $i] : ( X1174 @ V_39 ) ) ) )
=> ( ! [X1178: $i,X1180: $i,X1182: $i > $i] :
( ! [X1184: $i] :
( ( in @ X1184 @ X1178 )
=> ( in @ ( X1182 @ X1184 ) @ X1180 ) )
=> ( in
@ ( lam @ X1178 @ X1180
@ ^ [V_40: $i] : ( X1182 @ V_40 ) )
@ ( funcSet @ X1178 @ X1180 ) ) )
=> ( ! [X1186: $i,X1188: $i,X1190: $i] :
( ( subset @ X1190 @ ( cartprod @ X1186 @ X1188 ) )
=> ! [X1192: $i > $o] :
( ! [X1194: $i] :
( ( in @ X1194 @ X1186 )
=> ! [X1196: $i] :
( ( in @ X1196 @ X1188 )
=> ( ( in @ ( kpair @ X1194 @ X1196 ) @ X1190 )
=> ( X1192 @ ( kpair @ X1194 @ X1196 ) ) ) ) )
=> ! [X1198: $i] :
( ( in @ X1198 @ X1190 )
=> ( X1192 @ X1198 ) ) ) )
=> ( ! [X1200: $i,X1202: $i,X1204: $i] :
( ( subset @ X1204 @ ( cartprod @ X1200 @ X1202 ) )
=> ! [X1206: $i > $o] :
( ! [X1208: $i] :
( ( in @ X1208 @ X1200 )
=> ! [X1210: $i] :
( ( in @ X1210 @ X1202 )
=> ( ( in @ ( kpair @ X1208 @ X1210 ) @ X1204 )
=> ( X1206 @ ( kpair @ X1208 @ X1210 ) ) ) ) )
=> ! [X1212: $i] :
( ( in @ X1212 @ X1204 )
=> ( X1206 @ X1212 ) ) ) )
=> ( ! [X1214: $i,X1216: $i > $o] :
( ( ex1 @ X1214
@ ^ [V_41: $i] : ( X1216 @ V_41 ) )
=> ! [X1218: $i] :
( ( in @ X1218 @ X1214 )
=> ! [X1220: $i] :
( ( in @ X1220 @ X1214 )
=> ( ( X1216 @ X1218 )
=> ( ( X1216 @ X1220 )
=> ( X1218 = X1220 ) ) ) ) ) )
=> ( ! [X1222: $i,X1224: $i,X1226: $i] :
( ( func @ X1222 @ X1224 @ X1226 )
=> ! [X1228: $i] :
( ( in @ X1228 @ X1222 )
=> ( in @ ( kpair @ X1228 @ ( ap @ X1222 @ X1224 @ X1226 @ X1228 ) ) @ X1226 ) ) )
=> ( ! [X1230: $i,X1232: $i,X1234: $i] :
( ( in @ X1234 @ ( funcSet @ X1230 @ X1232 ) )
=> ! [X1236: $i] :
( ( in @ X1236 @ X1230 )
=> ( in @ ( kpair @ X1236 @ ( ap @ X1230 @ X1232 @ X1234 @ X1236 ) ) @ X1234 ) ) )
=> ( ! [X1238: $i,X1240: $i,X1242: $i] :
( ( func @ X1238 @ X1240 @ X1242 )
=> ! [X1244: $i] :
( ( in @ X1244 @ X1238 )
=> ! [X1246: $i] :
( ( in @ X1246 @ X1240 )
=> ( ( in @ ( kpair @ X1244 @ X1246 ) @ X1242 )
=> ( ( ap @ X1238 @ X1240 @ X1242 @ X1244 )
= X1246 ) ) ) ) )
=> ( ! [X1248: $i,X1250: $i,X1252: $i] :
( ( func @ X1248 @ X1250 @ X1252 )
=> ! [X1254: $i] :
( ( func @ X1248 @ X1250 @ X1254 )
=> ( ! [X1256: $i] :
( ( in @ X1256 @ X1248 )
=> ( ( ap @ X1248 @ X1250 @ X1252 @ X1256 )
= ( ap @ X1248 @ X1250 @ X1254 @ X1256 ) ) )
=> ! [X1258: $i] :
( ( in @ X1258 @ X1248 )
=> ! [X1260: $i] :
( ( in @ X1260 @ X1250 )
=> ( ( in @ ( kpair @ X1258 @ X1260 ) @ X1254 )
=> ( in @ ( kpair @ X1258 @ X1260 ) @ X1252 ) ) ) ) ) ) )
=> ( ! [X1262: $i,X1264: $i,X1266: $i] :
( ( in @ X1266 @ ( funcSet @ X1262 @ X1264 ) )
=> ! [X1268: $i] :
( ( in @ X1268 @ X1262 )
=> ! [X1270: $i] :
( ( in @ X1270 @ X1264 )
=> ( ( in @ ( kpair @ X1268 @ X1270 ) @ X1266 )
=> ( ( ap @ X1262 @ X1264 @ X1266 @ X1268 )
= X1270 ) ) ) ) )
=> ( ! [X1272: $i,X1274: $i,X1276: $i] :
( ( subset @ X1276 @ ( cartprod @ X1272 @ X1274 ) )
=> ! [X1278: $i] :
( ( subset @ X1278 @ ( cartprod @ X1272 @ X1274 ) )
=> ( ! [X1280: $i] :
( ( in @ X1280 @ X1272 )
=> ! [X1282: $i] :
( ( in @ X1282 @ X1274 )
=> ( ( in @ ( kpair @ X1280 @ X1282 ) @ X1276 )
=> ( in @ ( kpair @ X1280 @ X1282 ) @ X1278 ) ) ) )
=> ( subset @ X1276 @ X1278 ) ) ) )
=> ( ! [X1284: $i,X1286: $i,X1288: $i] :
( ( subset @ X1288 @ ( cartprod @ X1284 @ X1286 ) )
=> ! [X1290: $i] :
( ( subset @ X1290 @ ( cartprod @ X1284 @ X1286 ) )
=> ( ! [X1292: $i] :
( ( in @ X1292 @ X1284 )
=> ! [X1294: $i] :
( ( in @ X1294 @ X1286 )
=> ( ( in @ ( kpair @ X1292 @ X1294 ) @ X1288 )
=> ( in @ ( kpair @ X1292 @ X1294 ) @ X1290 ) ) ) )
=> ( ! [X1296: $i] :
( ( in @ X1296 @ X1284 )
=> ! [X1298: $i] :
( ( in @ X1298 @ X1286 )
=> ( ( in @ ( kpair @ X1296 @ X1298 ) @ X1290 )
=> ( in @ ( kpair @ X1296 @ X1298 ) @ X1288 ) ) ) )
=> ( X1288 = X1290 ) ) ) ) )
=> ( ! [X1300: $i,X1302: $i,X1304: $i] :
( ( func @ X1300 @ X1302 @ X1304 )
=> ! [X1306: $i] :
( ( func @ X1300 @ X1302 @ X1306 )
=> ( ! [X1308: $i] :
( ( in @ X1308 @ X1300 )
=> ( ( ap @ X1300 @ X1302 @ X1304 @ X1308 )
= ( ap @ X1300 @ X1302 @ X1306 @ X1308 ) ) )
=> ( X1304 = X1306 ) ) ) )
=> ( ! [X1310: $i,X1312: $i,X1314: $i] :
( ( in @ X1314 @ ( funcSet @ X1310 @ X1312 ) )
=> ! [X1316: $i] :
( ( in @ X1316 @ ( funcSet @ X1310 @ X1312 ) )
=> ( ! [X1318: $i] :
( ( in @ X1318 @ X1310 )
=> ( ( ap @ X1310 @ X1312 @ X1314 @ X1318 )
= ( ap @ X1310 @ X1312 @ X1316 @ X1318 ) ) )
=> ( X1314 = X1316 ) ) ) )
=> ( ! [X1320: $i,X1322: $i,X1324: $i] :
( ( in @ X1324 @ ( funcSet @ X1320 @ X1322 ) )
=> ! [X1326: $i] :
( ( in @ X1326 @ X1320 )
=> ( ( ap @ X1320 @ X1322 @ X1324 @ X1326 )
= ( ap @ X1320 @ X1322 @ X1324 @ X1326 ) ) ) )
=> ( ! [X1328: $i,X1330: $i,X1332: $i] :
( ( func @ X1328 @ X1330 @ X1332 )
=> ! [X1334: $i] :
( ( in @ X1334 @ X1328 )
=> ( ( ap @ X1328 @ X1330 @ X1332 @ X1334 )
= ( ap @ X1328 @ X1330 @ X1332 @ X1334 ) ) ) )
=> ( ! [X1336: $i,X1338: $i,X1340: $i > $i] :
( ! [X1342: $i] :
( ( in @ X1342 @ X1336 )
=> ( in @ ( X1340 @ X1342 ) @ X1338 ) )
=> ! [X1344: $i] :
( ( in @ X1344 @ X1336 )
=> ( ( ap @ X1336 @ X1338
@ ( lam @ X1336 @ X1338
@ ^ [V_42: $i] : ( X1340 @ V_42 ) )
@ X1344 )
= ( X1340 @ X1344 ) ) ) )
=> ( ! [X1346: $i,X1348: $i,X1350: $i] :
( ( func @ X1346 @ X1348 @ X1350 )
=> ( ( lam @ X1346 @ X1348
@ ^ [V_43: $i] : ( ap @ X1346 @ X1348 @ X1350 @ V_43 ) )
= X1350 ) )
=> ( ! [X1352: $i,X1354: $i,X1356: $i > $i] :
( ! [X1358: $i] :
( ( in @ X1358 @ X1352 )
=> ( in @ ( X1356 @ X1358 ) @ X1354 ) )
=> ( ( lam @ X1352 @ X1354
@ ^ [V_44: $i] : ( X1356 @ V_44 ) )
= ( lam @ X1352 @ X1354
@ ^ [V_45: $i] : ( X1356 @ V_45 ) ) ) )
=> ( ! [X1360: $i,X1362: $i,X1364: $i > $i] :
( ! [X1366: $i] :
( ( in @ X1366 @ X1360 )
=> ( in @ ( X1364 @ X1366 ) @ X1362 ) )
=> ! [X1368: $i] :
( ( in @ X1368 @ X1360 )
=> ( ( ap @ X1360 @ X1362
@ ( lam @ X1360 @ X1362
@ ^ [V_46: $i] : ( X1364 @ V_46 ) )
@ X1368 )
= ( X1364 @ X1368 ) ) ) )
=> ( ! [X1370: $i,X1372: $i,X1374: $i] :
( ( in @ X1374 @ ( funcSet @ X1370 @ X1372 ) )
=> ( ( lam @ X1370 @ X1372
@ ^ [V_47: $i] : ( ap @ X1370 @ X1372 @ X1374 @ V_47 ) )
= X1374 ) )
=> ( ! [X1376: $i,X1378: $o,X1380: $i] :
( ( in @ X1380 @ X1376 )
=> ! [X1382: $i] :
( ( in @ X1382 @ X1376 )
=> ( ~ X1378
=> ( in @ X1382
@ ( dsetconstr @ X1376
@ ^ [V_48: $i] :
( ( ( V_48 = X1382 )
& ~ X1378 )
| ( ( V_48 = X1380 )
& X1378 ) ) ) ) ) ) )
=> ( ! [X1384: $i,X1386: $o,X1388: $i] :
( ( in @ X1388 @ X1384 )
=> ! [X1390: $i] :
( ( in @ X1390 @ X1384 )
=> ( ~ X1386
=> ( ( dsetconstr @ X1384
@ ^ [V_49: $i] :
( ( ( V_49 = X1390 )
& ~ X1386 )
| ( ( V_49 = X1388 )
& X1386 ) ) )
= ( setadjoin @ X1390 @ emptyset ) ) ) ) )
=> ( ! [X1392: $i,X1394: $o,X1396: $i] :
( ( in @ X1396 @ X1392 )
=> ! [X1398: $i] :
( ( in @ X1398 @ X1392 )
=> ( X1394
=> ( in @ X1396
@ ( dsetconstr @ X1392
@ ^ [V_50: $i] :
( ( ( V_50 = X1398 )
& ~ X1394 )
| ( ( V_50 = X1396 )
& X1394 ) ) ) ) ) ) )
=> ( ! [X1400: $i,X1402: $o,X1404: $i] :
( ( in @ X1404 @ X1400 )
=> ! [X1406: $i] :
( ( in @ X1406 @ X1400 )
=> ( X1402
=> ( ( dsetconstr @ X1400
@ ^ [V_51: $i] :
( ( ( V_51 = X1406 )
& ~ X1402 )
| ( ( V_51 = X1404 )
& X1402 ) ) )
= ( setadjoin @ X1404 @ emptyset ) ) ) ) )
=> ( ! [X1408: $i,X1410: $o,X1412: $i] :
( ( in @ X1412 @ X1408 )
=> ! [X1414: $i] :
( ( in @ X1414 @ X1408 )
=> ( singleton
@ ( dsetconstr @ X1408
@ ^ [V_52: $i] :
( ( ( V_52 = X1414 )
& ~ X1410 )
| ( ( V_52 = X1412 )
& X1410 ) ) ) ) ) )
=> ( ! [X1416: $i,X1418: $o,X1420: $i] :
( ( in @ X1420 @ X1416 )
=> ! [X1422: $i] :
( ( in @ X1422 @ X1416 )
=> ( in @ ( if @ X1416 @ X1418 @ X1420 @ X1422 ) @ X1416 ) ) )
=> ( ! [X1424: $i] :
( ( singleton @ X1424 )
=> ! [X1426: $i] :
( ( in @ X1426 @ X1424 )
=> ( ( setunion @ X1424 )
= X1426 ) ) )
=> ( ! [X1428: $i,X1430: $o,X1432: $i] :
( ( in @ X1432 @ X1428 )
=> ! [X1434: $i] :
( ( in @ X1434 @ X1428 )
=> ( X1430
=> ( ( if @ X1428 @ X1430 @ X1432 @ X1434 )
= X1432 ) ) ) )
=> ( ! [X1436: $i,X1438: $o,X1440: $i] :
( ( in @ X1440 @ X1436 )
=> ! [X1442: $i] :
( ( in @ X1442 @ X1436 )
=> ( ~ X1438
=> ( ( if @ X1436 @ X1438 @ X1440 @ X1442 )
= X1442 ) ) ) )
=> ( ! [X1444: $i,X1446: $o,X1448: $i] :
( ( in @ X1448 @ X1444 )
=> ! [X1450: $i] :
( ( in @ X1450 @ X1444 )
=> ( in @ ( if @ X1444 @ X1446 @ X1448 @ X1450 ) @ ( setadjoin @ X1448 @ ( setadjoin @ X1450 @ emptyset ) ) ) ) )
=> ( ! [X1452: $i,X1454: $i] :
( ( in @ X1454 @ ( powerset @ X1452 ) )
=> ! [X1456: $i] :
( ( in @ X1456 @ ( powerset @ X1452 ) )
=> ( in @ ( binintersect @ X1454 @ X1456 ) @ ( powerset @ X1452 ) ) ) )
=> ( ! [X1458: $i,X1460: $i] :
( ( in @ X1460 @ ( powerset @ X1458 ) )
=> ! [X1462: $i] :
( ( in @ X1462 @ ( powerset @ X1458 ) )
=> ( in @ ( binunion @ X1460 @ X1462 ) @ ( powerset @ X1458 ) ) ) )
=> ( ! [X1464: $i,X1466: $i] :
( ( in @ X1466 @ ( powerset @ X1464 ) )
=> ( in @ ( powerset @ X1466 ) @ ( powerset @ ( powerset @ X1464 ) ) ) )
=> ( ! [X1468: $i,X1470: $i] :
( ( in @ X1470 @ ( powerset @ X1468 ) )
=> ! [X1472: $i] :
( ( in @ X1472 @ ( powerset @ X1468 ) )
=> ( in @ ( setminus @ X1470 @ X1472 ) @ ( powerset @ X1468 ) ) ) )
=> ( ! [X1474: $i,X1476: $i] :
( ( in @ X1476 @ ( powerset @ X1474 ) )
=> ( in @ ( setminus @ X1474 @ X1476 ) @ ( powerset @ X1474 ) ) )
=> ( ! [X1478: $i,X1480: $i] :
( ( in @ X1480 @ ( powerset @ X1478 ) )
=> ! [X1482: $i] :
( ( in @ X1482 @ ( powerset @ X1478 ) )
=> ( ! [X1484: $i] :
( ( in @ X1484 @ X1478 )
=> ( ( in @ X1484 @ X1480 )
=> ( in @ X1484 @ X1482 ) ) )
=> ( ! [X1486: $i] :
( ( in @ X1486 @ X1478 )
=> ( ( in @ X1486 @ X1482 )
=> ( in @ X1486 @ X1480 ) ) )
=> ( X1480 = X1482 ) ) ) ) )
=> ( ! [X1488: $i,X1490: $i] :
( ( in @ X1490 @ ( powerset @ X1488 ) )
=> ! [X1492: $i] :
( ( in @ X1492 @ ( powerset @ X1488 ) )
=> ( ! [X1494: $i] :
( ( in @ X1494 @ X1488 )
=> ( ( in @ X1494 @ X1490 )
=> ( in @ X1494 @ X1492 ) ) )
=> ( subset @ X1490 @ X1492 ) ) ) )
=> ( ! [X1496: $i,X1498: $i] :
( ( in @ X1498 @ ( powerset @ X1496 ) )
=> ! [X1500: $i] :
( ( in @ X1500 @ ( powerset @ X1496 ) )
=> ( ! [X1502: $i] :
( ( in @ X1502 @ X1496 )
=> ( ( in @ X1502 @ X1498 )
=> ( in @ X1502 @ X1500 ) ) )
=> ( in @ X1498 @ ( powerset @ X1500 ) ) ) ) )
=> ( ! [X1504: $i,X1506: $i] :
( ( in @ X1506 @ ( powerset @ X1504 ) )
=> ! [X1508: $i] :
( ( in @ X1508 @ ( powerset @ X1504 ) )
=> ! [X1510: $i] :
( ( in @ X1510 @ X1504 )
=> ( ( in @ X1506 @ ( powerset @ X1508 ) )
=> ( ( in @ X1510 @ X1506 )
=> ( in @ X1510 @ X1508 ) ) ) ) ) )
=> ( ! [X1512: $i,X1514: $i] :
( ( in @ X1514 @ ( powerset @ X1512 ) )
=> ! [X1516: $i] :
( ( in @ X1516 @ X1512 )
=> ( ( in @ X1516 @ X1514 )
=> ~ ( in @ X1516 @ ( setminus @ X1512 @ X1514 ) ) ) ) )
=> ( ! [X1518: $i,X1520: $i] :
( ( in @ X1520 @ ( powerset @ X1518 ) )
=> ! [X1522: $i] :
( ( in @ X1522 @ X1518 )
=> ( ~ ( in @ X1522 @ ( setminus @ X1518 @ X1520 ) )
=> ( in @ X1522 @ X1520 ) ) ) )
=> ( ! [X1524: $i,X1526: $i] :
( ( in @ X1526 @ ( powerset @ X1524 ) )
=> ! [X1528: $i] :
( ( in @ X1528 @ ( powerset @ X1524 ) )
=> ! [X1530: $i] :
( ( in @ X1530 @ X1524 )
=> ( ~ ( in @ X1530 @ X1526 )
=> ~ ( in @ X1530 @ ( binintersect @ X1526 @ X1528 ) ) ) ) ) )
=> ( ! [X1532: $i,X1534: $i] :
( ( in @ X1534 @ ( powerset @ X1532 ) )
=> ! [X1536: $i] :
( ( in @ X1536 @ ( powerset @ X1532 ) )
=> ! [X1538: $i] :
( ( in @ X1538 @ X1532 )
=> ( ~ ( in @ X1538 @ X1536 )
=> ~ ( in @ X1538 @ ( binintersect @ X1534 @ X1536 ) ) ) ) ) )
=> ( ! [X1540: $i,X1542: $i] :
( ( in @ X1542 @ ( powerset @ X1540 ) )
=> ! [X1544: $i] :
( ( in @ X1544 @ ( powerset @ X1540 ) )
=> ! [X1546: $i] :
( ( in @ X1546 @ X1540 )
=> ( ( subset @ X1542 @ ( setminus @ X1540 @ X1544 ) )
=> ( ( in @ X1546 @ X1544 )
=> ~ ( in @ X1546 @ X1542 ) ) ) ) ) )
=> ( ! [X1548: $i,X1550: $i] :
( ( in @ X1550 @ ( powerset @ X1548 ) )
=> ! [X1552: $i] :
( ( in @ X1552 @ ( powerset @ X1548 ) )
=> ! [X1554: $i] :
( ( in @ X1554 @ X1548 )
=> ( ( subset @ X1550 @ X1552 )
=> ( ~ ( in @ X1554 @ X1552 )
=> ~ ( in @ X1554 @ X1550 ) ) ) ) ) )
=> ( ! [X1556: $i,X1558: $i] :
( ( in @ X1558 @ ( powerset @ X1556 ) )
=> ! [X1560: $i] :
( ( in @ X1560 @ ( powerset @ X1556 ) )
=> ( ( subset @ X1558 @ X1560 )
=> ( subset @ ( setminus @ X1556 @ X1560 ) @ ( setminus @ X1556 @ X1558 ) ) ) ) )
=> ( ! [X1562: $i,X1564: $i] :
( ( in @ X1564 @ ( powerset @ X1562 ) )
=> ! [X1566: $i] :
( ( in @ X1566 @ ( powerset @ X1562 ) )
=> ( ( subset @ ( setminus @ X1562 @ X1566 ) @ ( setminus @ X1562 @ X1564 ) )
=> ( subset @ X1564 @ X1566 ) ) ) )
=> ( ! [X1568: $i,X1570: $i] :
( ( in @ X1570 @ ( powerset @ X1568 ) )
=> ! [X1572: $i] :
( ( in @ X1572 @ X1568 )
=> ( ( in @ X1572 @ X1570 )
=> ( in @ X1572 @ ( setminus @ X1568 @ ( setminus @ X1568 @ X1570 ) ) ) ) ) )
=> ( ! [X1574: $i,X1576: $i] :
( ( in @ X1576 @ ( powerset @ X1574 ) )
=> ! [X1578: $i] :
( ( in @ X1578 @ X1574 )
=> ( ( in @ X1578 @ ( setminus @ X1574 @ ( setminus @ X1574 @ X1576 ) ) )
=> ( in @ X1578 @ X1576 ) ) ) )
=> ( ! [X1580: $i,X1582: $i] :
( ( in @ X1582 @ ( powerset @ X1580 ) )
=> ( subset @ X1582 @ ( setminus @ X1580 @ ( setminus @ X1580 @ X1582 ) ) ) )
=> ( ! [X1584: $i,X1586: $i] :
( ( in @ X1586 @ ( powerset @ X1584 ) )
=> ( subset @ ( setminus @ X1584 @ ( setminus @ X1584 @ X1586 ) ) @ X1586 ) )
=> ( ! [X1588: $i,X1590: $i] :
( ( in @ X1590 @ ( powerset @ X1588 ) )
=> ( X1590
= ( setminus @ X1588 @ ( setminus @ X1588 @ X1590 ) ) ) )
=> ( ! [X1592: $i,X1594: $i] :
( ( in @ X1594 @ ( powerset @ X1592 ) )
=> ! [X1596: $i] :
( ( in @ X1596 @ ( powerset @ X1592 ) )
=> ! [X1598: $i] :
( ( in @ X1598 @ X1592 )
=> ( ( in @ X1598 @ ( setminus @ X1592 @ X1594 ) )
=> ~ ( in @ X1598 @ ( binintersect @ X1594 @ X1596 ) ) ) ) ) )
=> ( ! [X1600: $i,X1602: $i] :
( ( in @ X1602 @ ( powerset @ X1600 ) )
=> ! [X1604: $i] :
( ( in @ X1604 @ ( powerset @ X1600 ) )
=> ! [X1606: $i] :
( ( in @ X1606 @ X1600 )
=> ( ( in @ X1606 @ ( setminus @ X1600 @ X1602 ) )
=> ( in @ X1606 @ ( setminus @ X1600 @ ( binintersect @ X1602 @ X1604 ) ) ) ) ) ) )
=> ( ! [X1608: $i,X1610: $i] :
( ( in @ X1610 @ ( powerset @ X1608 ) )
=> ! [X1612: $i] :
( ( in @ X1612 @ ( powerset @ X1608 ) )
=> ( subset @ ( setminus @ X1608 @ X1610 ) @ ( setminus @ X1608 @ ( binintersect @ X1610 @ X1612 ) ) ) ) )
=> ( ! [X1614: $i,X1616: $i] :
( ( in @ X1616 @ ( powerset @ X1614 ) )
=> ! [X1618: $i] :
( ( in @ X1618 @ ( powerset @ X1614 ) )
=> ( in @ ( setminus @ X1614 @ X1616 ) @ ( powerset @ ( setminus @ X1614 @ ( binintersect @ X1616 @ X1618 ) ) ) ) ) )
=> ( ! [X1620: $i,X1622: $i] :
( ( in @ X1622 @ ( powerset @ X1620 ) )
=> ! [X1624: $i] :
( ( in @ X1624 @ ( powerset @ X1620 ) )
=> ( ( subset @ X1622 @ ( setminus @ X1620 @ X1624 ) )
=> ! [X1626: $i] :
( ( in @ X1626 @ X1620 )
=> ( ( in @ X1626 @ X1624 )
=> ( in @ X1626 @ ( setminus @ X1620 @ X1622 ) ) ) ) ) ) )
=> ( ! [X1628: $i,X1630: $i] :
( ( in @ X1630 @ ( powerset @ X1628 ) )
=> ! [X1632: $i] :
( ( in @ X1632 @ ( powerset @ X1628 ) )
=> ( ( subset @ X1630 @ ( setminus @ X1628 @ X1632 ) )
=> ( subset @ X1632 @ ( setminus @ X1628 @ X1630 ) ) ) ) )
=> ( ! [X1634: $i,X1636: $i] :
( ( in @ X1636 @ ( powerset @ X1634 ) )
=> ! [X1638: $i] :
( ( in @ X1638 @ ( powerset @ X1634 ) )
=> ! [X1640: $i] :
( ( in @ X1640 @ X1634 )
=> ( ~ ( in @ X1640 @ ( binunion @ X1636 @ X1638 ) )
=> ~ ( in @ X1640 @ X1636 ) ) ) ) )
=> ( ! [X1642: $i,X1644: $i] :
( ( in @ X1644 @ ( powerset @ X1642 ) )
=> ! [X1646: $i] :
( ( in @ X1646 @ ( powerset @ X1642 ) )
=> ! [X1648: $i] :
( ( in @ X1648 @ X1642 )
=> ( ~ ( in @ X1648 @ ( binunion @ X1644 @ X1646 ) )
=> ~ ( in @ X1648 @ X1646 ) ) ) ) )
=> ( ! [X1650: $i,X1652: $i] :
( ( in @ X1652 @ ( powerset @ X1650 ) )
=> ! [X1654: $i] :
( ( in @ X1654 @ ( powerset @ X1650 ) )
=> ! [X1656: $i] :
( ( in @ X1656 @ ( powerset @ X1650 ) )
=> ! [X1658: $i] :
( ( in @ X1658 @ X1650 )
=> ( ( in @ X1658 @ ( binintersect @ X1652 @ X1654 ) )
=> ( in @ X1658 @ ( binunion @ X1652 @ X1656 ) ) ) ) ) ) )
=> ( ! [X1660: $i,X1662: $i] :
( ( in @ X1662 @ ( powerset @ X1660 ) )
=> ! [X1664: $i] :
( ( in @ X1664 @ ( powerset @ X1660 ) )
=> ! [X1666: $i] :
( ( in @ X1666 @ ( powerset @ X1660 ) )
=> ! [X1668: $i] :
( ( in @ X1668 @ X1660 )
=> ( ( in @ X1668 @ ( binintersect @ X1662 @ X1664 ) )
=> ( in @ X1668 @ ( binunion @ X1664 @ X1666 ) ) ) ) ) ) )
=> ( ! [X1670: $i,X1672: $i] :
( ( in @ X1672 @ ( powerset @ X1670 ) )
=> ! [X1674: $i] :
( ( in @ X1674 @ ( powerset @ X1670 ) )
=> ! [X1676: $i] :
( ( in @ X1676 @ ( powerset @ X1670 ) )
=> ! [X1678: $i] :
( ( in @ X1678 @ X1670 )
=> ( ( in @ X1678 @ ( binintersect @ X1672 @ X1674 ) )
=> ( in @ X1678 @ ( binintersect @ ( binunion @ X1672 @ X1676 ) @ ( binunion @ X1674 @ X1676 ) ) ) ) ) ) ) )
=> ( ! [X1680: $i,X1682: $i] :
( ( in @ X1682 @ ( powerset @ X1680 ) )
=> ! [X1684: $i] :
( ( in @ X1684 @ ( powerset @ X1680 ) )
=> ! [X1686: $i] :
( ( in @ X1686 @ ( powerset @ X1680 ) )
=> ( in @ ( binintersect @ X1682 @ X1684 ) @ ( powerset @ ( binintersect @ ( binunion @ X1682 @ X1686 ) @ ( binunion @ X1684 @ X1686 ) ) ) ) ) ) )
=> ( ! [X1688: $i,X1690: $i] :
( ( in @ X1690 @ ( powerset @ X1688 ) )
=> ! [X1692: $i] :
( ( in @ X1692 @ ( powerset @ X1688 ) )
=> ! [X1694: $i] :
( ( in @ X1694 @ X1688 )
=> ( ( in @ X1694 @ ( setminus @ X1688 @ ( binunion @ X1690 @ X1692 ) ) )
=> ~ ( in @ X1694 @ X1690 ) ) ) ) )
=> ( ! [X1696: $i,X1698: $i] :
( ( in @ X1698 @ ( powerset @ X1696 ) )
=> ! [X1700: $i] :
( ( in @ X1700 @ ( powerset @ X1696 ) )
=> ! [X1702: $i] :
( ( in @ X1702 @ X1696 )
=> ( ( in @ X1702 @ ( setminus @ X1696 @ ( binunion @ X1698 @ X1700 ) ) )
=> ( in @ X1702 @ ( setminus @ X1696 @ X1698 ) ) ) ) ) )
=> ( ! [X1704: $i,X1706: $i] :
( ( in @ X1706 @ ( powerset @ X1704 ) )
=> ! [X1708: $i] :
( ( in @ X1708 @ ( powerset @ X1704 ) )
=> ! [X1710: $o,X1712: $i] :
( ( in @ X1712 @ X1704 )
=> ( ( in @ X1712 @ ( binunion @ X1706 @ X1708 ) )
=> ( ( ( in @ X1712 @ X1706 )
=> X1710 )
=> ( ( ( in @ X1712 @ X1708 )
=> X1710 )
=> X1710 ) ) ) ) ) )
=> ( ! [X1714: $i,X1716: $i] :
( ( in @ X1716 @ ( powerset @ X1714 ) )
=> ! [X1718: $i] :
( ( in @ X1718 @ ( powerset @ X1714 ) )
=> ! [X1720: $i] :
( ( in @ X1720 @ X1714 )
=> ( ~ ( in @ X1720 @ X1716 )
=> ( ~ ( in @ X1720 @ X1718 )
=> ~ ( in @ X1720 @ ( binunion @ X1716 @ X1718 ) ) ) ) ) ) )
=> ( ! [X1722: $i,X1724: $i] :
( ( in @ X1724 @ ( powerset @ X1722 ) )
=> ! [X1726: $i] :
( ( in @ X1726 @ ( powerset @ X1722 ) )
=> ! [X1728: $i] :
( ( in @ X1728 @ X1722 )
=> ( ( in @ X1728 @ ( setminus @ X1722 @ ( binunion @ X1724 @ X1726 ) ) )
=> ( in @ X1728 @ ( setminus @ X1722 @ X1724 ) ) ) ) ) )
=> ( ! [X1730: $i,X1732: $i] :
( ( in @ X1732 @ ( powerset @ X1730 ) )
=> ! [X1734: $i] :
( ( in @ X1734 @ ( powerset @ X1730 ) )
=> ( in @ ( setminus @ X1730 @ ( binunion @ X1732 @ X1734 ) ) @ ( powerset @ ( setminus @ X1730 @ X1732 ) ) ) ) )
=> ( ! [X1736: $i,X1738: $i] :
( ( in @ X1738 @ ( powerset @ X1736 ) )
=> ! [X1740: $i] :
( ( in @ X1740 @ ( powerset @ X1736 ) )
=> ! [X1742: $i] :
( ( in @ X1742 @ X1736 )
=> ( ( in @ X1742 @ ( setminus @ X1736 @ ( binunion @ X1738 @ X1740 ) ) )
=> ( in @ X1742 @ ( setminus @ X1736 @ X1740 ) ) ) ) ) )
=> ( ! [X1744: $i,X1746: $i] :
( ( in @ X1746 @ ( powerset @ X1744 ) )
=> ! [X1748: $i] :
( ( in @ X1748 @ ( powerset @ X1744 ) )
=> ! [X1750: $i] :
( ( in @ X1750 @ X1744 )
=> ( ( in @ X1750 @ ( setminus @ X1744 @ ( binintersect @ X1746 @ X1748 ) ) )
=> ( in @ X1750 @ ( binunion @ ( setminus @ X1744 @ X1746 ) @ ( setminus @ X1744 @ X1748 ) ) ) ) ) ) )
=> ( ! [X1752: $i,X1754: $i] :
( ( in @ X1754 @ ( powerset @ X1752 ) )
=> ! [X1756: $i] :
( ( in @ X1756 @ ( powerset @ X1752 ) )
=> ! [X1758: $i] :
( ( in @ X1758 @ X1752 )
=> ( ( in @ X1758 @ ( binunion @ ( setminus @ X1752 @ X1754 ) @ ( setminus @ X1752 @ X1756 ) ) )
=> ( in @ X1758 @ ( setminus @ X1752 @ ( binintersect @ X1754 @ X1756 ) ) ) ) ) ) )
=> ( ! [X1760: $i,X1762: $i] :
( ( in @ X1762 @ ( powerset @ X1760 ) )
=> ! [X1764: $i] :
( ( in @ X1764 @ ( powerset @ X1760 ) )
=> ( ( setminus @ X1760 @ ( binintersect @ X1762 @ X1764 ) )
= ( binunion @ ( setminus @ X1760 @ X1762 ) @ ( setminus @ X1760 @ X1764 ) ) ) ) )
=> ( ! [X1766: $i,X1768: $i] :
( ( in @ X1768 @ ( powerset @ X1766 ) )
=> ! [X1770: $i] :
( ( in @ X1770 @ ( powerset @ X1766 ) )
=> ! [X1772: $i] :
( ( in @ X1772 @ X1766 )
=> ( ( in @ X1772 @ ( setminus @ X1766 @ ( binunion @ X1768 @ X1770 ) ) )
=> ( in @ X1772 @ ( binintersect @ ( setminus @ X1766 @ X1768 ) @ ( setminus @ X1766 @ X1770 ) ) ) ) ) ) )
=> ( ! [X1774: $i,X1776: $i] :
( ( in @ X1776 @ ( powerset @ X1774 ) )
=> ! [X1778: $i] :
( ( in @ X1778 @ ( powerset @ X1774 ) )
=> ! [X1780: $i] :
( ( in @ X1780 @ X1774 )
=> ( ( in @ X1780 @ ( setminus @ X1774 @ X1776 ) )
=> ( ( in @ X1780 @ ( setminus @ X1774 @ X1778 ) )
=> ( in @ X1780 @ ( setminus @ X1774 @ ( binunion @ X1776 @ X1778 ) ) ) ) ) ) ) )
=> ( ! [X1782: $i,X1784: $i] :
( ( in @ X1784 @ ( powerset @ X1782 ) )
=> ! [X1786: $i] :
( ( in @ X1786 @ ( powerset @ X1782 ) )
=> ! [X1788: $i] :
( ( in @ X1788 @ X1782 )
=> ( ( in @ X1788 @ ( binintersect @ ( setminus @ X1782 @ X1784 ) @ ( setminus @ X1782 @ X1786 ) ) )
=> ( in @ X1788 @ ( setminus @ X1782 @ ( binunion @ X1784 @ X1786 ) ) ) ) ) ) )
=> ( ! [X1790: $i,X1792: $i] :
( ( in @ X1792 @ ( powerset @ X1790 ) )
=> ! [X1794: $i] :
( ( in @ X1794 @ ( powerset @ X1790 ) )
=> ( ( setminus @ X1790 @ ( binunion @ X1792 @ X1794 ) )
= ( binintersect @ ( setminus @ X1790 @ X1792 ) @ ( setminus @ X1790 @ X1794 ) ) ) ) )
=> ( ! [X1796: $i,X1798: $i] :
( ( in @ X1798 @ ( powerset @ X1796 ) )
=> ! [X1800: $i] :
( ( in @ X1800 @ ( powerset @ X1796 ) )
=> ! [X1802: $i] :
( ( in @ X1802 @ ( binintersect @ X1798 @ X1800 ) )
=> ( in @ X1802 @ X1796 ) ) ) )
=> ( ! [X1804: $i,X1806: $i] :
( ( in @ X1806 @ ( powerset @ X1804 ) )
=> ! [X1808: $i] :
( ( in @ X1808 @ ( powerset @ X1804 ) )
=> ! [X1810: $i] :
( ( in @ X1810 @ ( powerset @ X1804 ) )
=> ( ( subset @ X1806 @ X1810 )
=> ( subset @ ( binintersect @ X1806 @ X1808 ) @ X1810 ) ) ) ) )
=> ( ! [X1812: $i,X1814: $i] :
( ( in @ X1814 @ ( powerset @ X1812 ) )
=> ! [X1816: $i] :
( ( in @ X1816 @ ( powerset @ X1812 ) )
=> ! [X1818: $i] :
( ( in @ X1818 @ ( powerset @ X1812 ) )
=> ( ( subset @ X1816 @ X1818 )
=> ( subset @ ( binintersect @ X1814 @ X1816 ) @ X1818 ) ) ) ) )
=> ( ! [X1820: $i,X1822: $i] :
( ( in @ X1822 @ ( powerset @ X1820 ) )
=> ! [X1824: $i] :
( ( in @ X1824 @ ( powerset @ X1820 ) )
=> ! [X1826: $i] :
( ( in @ X1826 @ ( powerset @ X1820 ) )
=> ( ( subset @ X1822 @ X1824 )
=> ( ( subset @ X1822 @ X1826 )
=> ( subset @ X1822 @ ( binintersect @ X1824 @ X1826 ) ) ) ) ) ) )
=> ( ! [X1828: $i,X1830: $i] :
( ( in @ X1830 @ ( powerset @ X1828 ) )
=> ! [X1832: $i] :
( ( in @ X1832 @ ( powerset @ X1828 ) )
=> ! [X1834: $i] :
( ( in @ X1834 @ ( powerset @ X1828 ) )
=> ! [X1836: $i] :
( ( in @ X1836 @ ( powerset @ X1828 ) )
=> ( ( subset @ X1830 @ X1834 )
=> ( ( subset @ X1832 @ X1836 )
=> ( subset @ ( binintersect @ X1830 @ X1832 ) @ ( binintersect @ X1834 @ X1836 ) ) ) ) ) ) ) )
=> ( ! [X1838: $i,X1840: $i] :
( ( in @ X1840 @ ( powerset @ X1838 ) )
=> ! [X1842: $i] :
( ( in @ X1842 @ ( powerset @ X1838 ) )
=> ( in @ ( setminus @ X1838 @ X1840 ) @ ( powerset @ ( setminus @ X1838 @ ( binintersect @ X1840 @ X1842 ) ) ) ) ) )
=> ( ! [X1844: $i,X1846: $i] :
( ( in @ X1846 @ ( powerset @ X1844 ) )
=> ! [X1848: $i] :
( ( in @ X1848 @ ( powerset @ X1844 ) )
=> ! [X1850: $i] :
( ( in @ X1850 @ ( powerset @ X1844 ) )
=> ! [X1852: $i] :
( ( in @ X1852 @ ( powerset @ X1844 ) )
=> ( ( setminus @ X1844 @ ( binintersect @ ( binunion @ X1846 @ X1848 ) @ ( binunion @ X1850 @ X1852 ) ) )
= ( binunion @ ( binintersect @ ( setminus @ X1844 @ X1846 ) @ ( setminus @ X1844 @ X1848 ) ) @ ( binintersect @ ( setminus @ X1844 @ X1850 ) @ ( setminus @ X1844 @ X1852 ) ) ) ) ) ) ) )
=> ( ! [X1854: $i,X1856: $i] :
( ( in @ X1856 @ ( powerset @ X1854 ) )
=> ! [X1858: $i] :
( ( in @ X1858 @ ( powerset @ X1854 ) )
=> ! [X1860: $i] :
( ( in @ X1860 @ ( powerset @ X1854 ) )
=> ( in @ ( binintersect @ X1856 @ X1858 ) @ ( powerset @ ( binintersect @ ( binunion @ X1856 @ X1860 ) @ ( binunion @ X1858 @ X1860 ) ) ) ) ) ) )
=> ( ! [X1862: $i,X1864: $i] :
( ( in @ X1864 @ ( powerset @ X1862 ) )
=> ! [X1866: $i] :
( ( in @ X1866 @ ( powerset @ X1862 ) )
=> ( ( subset @ X1864 @ ( setminus @ X1862 @ X1866 ) )
=> ( subset @ X1866 @ ( setminus @ X1862 @ X1864 ) ) ) ) )
=> ( ! [X1868: $i,X1870: $i] :
( ( in @ X1870 @ ( powerset @ X1868 ) )
=> ! [X1872: $i] :
( ( in @ X1872 @ ( powerset @ X1868 ) )
=> ( in @ ( setminus @ X1868 @ ( binunion @ X1870 @ X1872 ) ) @ ( powerset @ ( setminus @ X1868 @ X1870 ) ) ) ) )
=> ( ! [X1874: $i,X1876: $i] :
( ( subset @ X1876 @ ( cartprod @ X1874 @ X1874 ) )
=> ! [X1878: $i > $o] :
( ! [X1880: $i] :
( ( in @ X1880 @ X1874 )
=> ! [X1882: $i] :
( ( in @ X1882 @ X1874 )
=> ( ( in @ ( kpair @ X1880 @ X1882 ) @ X1876 )
=> ( X1878 @ ( kpair @ X1880 @ X1882 ) ) ) ) )
=> ! [X1884: $i] :
( ( in @ X1884 @ X1876 )
=> ( X1878 @ X1884 ) ) ) )
=> ( ! [X1886: $i,X1888: $i] :
( ( in @ X1888 @ ( breln1Set @ X1886 ) )
=> ( subset @ X1888 @ ( cartprod @ X1886 @ X1886 ) ) )
=> ( ! [X1890: $i,X1892: $i,X1894: $i > $i > $o] :
( ! [X1896: $i] :
( ( in @ X1896 @ X1890 )
=> ? [X1898: $i] :
( ( X1894 @ X1896 @ X1898 )
& ( in @ X1898 @ X1892 ) ) )
=> ! [X1900: $i] :
( ( in @ X1900 @ ( breln1Set @ X1892 ) )
=> ( ( reflwellordering @ X1892 @ X1900 )
=> ! [X1902: $i] :
( ( in @ X1902 @ X1890 )
=> ( singleton
@ ( dsetconstr @ X1892
@ ^ [V_53: $i] :
( ! [X1904: $i] :
( ( in @ X1904 @ X1892 )
=> ( ( X1894 @ X1902 @ X1904 )
=> ( in @ ( kpair @ V_53 @ X1904 ) @ X1900 ) ) )
& ( X1894 @ X1902 @ V_53 ) ) ) ) ) ) ) )
=> ( ! [X1906: $i,X1908: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1906 @ X1906
@ ^ [V_54: $i,V_55: $i] : ( X1908 @ V_54 @ V_55 ) )
@ ( cartprod @ X1906 @ X1906 ) )
=> ( ! [X1910: $i,X1912: $i] :
( ( subset @ X1912 @ ( cartprod @ X1910 @ X1910 ) )
=> ! [X1914: $i > $o] :
( ! [X1916: $i] :
( ( in @ X1916 @ X1910 )
=> ! [X1918: $i] :
( ( in @ X1918 @ X1910 )
=> ( ( in @ ( kpair @ X1916 @ X1918 ) @ X1912 )
=> ( X1914 @ ( kpair @ X1916 @ X1918 ) ) ) ) )
=> ! [X1920: $i] :
( ( in @ X1920 @ X1912 )
=> ( X1914 @ X1920 ) ) ) )
=> ( ! [X1922: $i,X1924: $i] :
( ( subset @ X1924 @ ( cartprod @ X1922 @ X1922 ) )
=> ! [X1926: $i] :
( ( subset @ X1926 @ ( cartprod @ X1922 @ X1922 ) )
=> ( ! [X1928: $i] :
( ( in @ X1928 @ X1922 )
=> ! [X1930: $i] :
( ( in @ X1930 @ X1922 )
=> ( ( in @ ( kpair @ X1928 @ X1930 ) @ X1924 )
=> ( in @ ( kpair @ X1928 @ X1930 ) @ X1926 ) ) ) )
=> ( subset @ X1924 @ X1926 ) ) ) )
=> ( ! [X1932: $i,X1934: $i] :
( ( subset @ X1934 @ ( cartprod @ X1932 @ X1932 ) )
=> ! [X1936: $i] :
( ( subset @ X1936 @ ( cartprod @ X1932 @ X1932 ) )
=> ( ! [X1938: $i] :
( ( in @ X1938 @ X1932 )
=> ! [X1940: $i] :
( ( in @ X1940 @ X1932 )
=> ( ( in @ ( kpair @ X1938 @ X1940 ) @ X1934 )
=> ( in @ ( kpair @ X1938 @ X1940 ) @ X1936 ) ) ) )
=> ( ! [X1942: $i] :
( ( in @ X1942 @ X1932 )
=> ! [X1944: $i] :
( ( in @ X1944 @ X1932 )
=> ( ( in @ ( kpair @ X1942 @ X1944 ) @ X1936 )
=> ( in @ ( kpair @ X1942 @ X1944 ) @ X1934 ) ) ) )
=> ( X1934 = X1936 ) ) ) ) )
=> ( ! [X1946: $i,X1948: $i] :
( ( subset @ X1948 @ ( cartprod @ X1946 @ X1946 ) )
=> ( subset
@ ( dpsetconstr @ X1946 @ X1946
@ ^ [V_56: $i,V_57: $i] : ( in @ ( kpair @ V_57 @ V_56 ) @ X1948 ) )
@ ( cartprod @ X1946 @ X1946 ) ) )
=> ( ! [X1950: $i,X1952: $i] :
( ( subset @ X1952 @ ( cartprod @ X1950 @ X1950 ) )
=> ! [X1954: $i] :
( ( in @ X1954 @ X1950 )
=> ! [X1956: $i] :
( ( in @ X1956 @ X1950 )
=> ( ( in @ ( kpair @ X1954 @ X1956 ) @ X1952 )
=> ( in @ ( kpair @ X1956 @ X1954 )
@ ( dpsetconstr @ X1950 @ X1950
@ ^ [V_58: $i,V_59: $i] : ( in @ ( kpair @ V_59 @ V_58 ) @ X1952 ) ) ) ) ) ) )
=> ! [X1958: $i,X1960: $i] :
( ( subset @ X1960 @ ( cartprod @ X1958 @ X1958 ) )
=> ! [X1962: $i] :
( ( in @ X1962 @ X1958 )
=> ! [X1964: $i] :
( ( in @ X1964 @ X1958 )
=> ( ( in @ ( kpair @ X1964 @ X1962 )
@ ( dpsetconstr @ X1958 @ X1958
@ ^ [V_60: $i,V_61: $i] : ( in @ ( kpair @ V_61 @ V_60 ) @ X1960 ) ) )
=> ( in @ ( kpair @ X1962 @ X1964 ) @ X1960 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference('cnf.neg',[status(esa)],[zf_stmt_13]) ).
thf(zip_derived_cl266,plain,
~ ( in @ ( kpair @ sk__1001 @ sk__1002 ) @ sk__1000 ),
inference(cnf,[status(esa)],[zf_stmt_14]) ).
thf(zip_derived_cl265,plain,
( in @ ( kpair @ sk__1002 @ sk__1001 )
@ ( dpsetconstr @ sk__999 @ sk__999
@ ^ [Y0: $i,Y1: $i] : ( in @ ( kpair @ Y1 @ Y0 ) @ sk__1000 ) ) ),
inference(cnf,[status(esa)],[zf_stmt_14]) ).
thf(zip_derived_cl263,plain,
! [X464: $i,X465: $i,X466: $i,X467: $i] :
( ~ ( in @ X464 @ X465 )
| ~ ( in @ ( kpair @ X464 @ X466 ) @ X467 )
| ( in @ ( kpair @ X466 @ X464 )
@ ( dpsetconstr @ X465 @ X465
@ ^ [Y0: $i,Y1: $i] : ( in @ ( kpair @ Y1 @ Y0 ) @ X467 ) ) )
| ~ ( in @ X466 @ X465 )
| ~ ( subset @ X467 @ ( cartprod @ X465 @ X465 ) ) ),
inference(cnf,[status(esa)],[zf_stmt_14]) ).
thf(zip_derived_cl11348,plain,
! [X467: $i,X468: $i,X469: $i] :
( ( '#l_lift13430' @ X467 @ X468 @ X469 )
= ( in @ ( kpair @ X469 @ X468 ) @ X467 ) ),
define([status(thm)]) ).
thf(zip_derived_cl11350,plain,
in @ ( kpair @ sk__1002 @ sk__1001 ) @ ( dpsetconstr @ sk__999 @ sk__999 @ ( '#l_lift13430' @ sk__1000 ) ),
inference(lambda_lifting,[status(thm)],[zip_derived_cl265,zip_derived_cl11348]) ).
thf(zip_derived_cl11348_001,plain,
! [X467: $i,X468: $i,X469: $i] :
( ( '#l_lift13430' @ X467 @ X468 @ X469 )
= ( in @ ( kpair @ X469 @ X468 ) @ X467 ) ),
define([status(thm)]) ).
thf(zip_derived_cl11364,plain,
! [X650: $i > $i > $o,X651: $i,X652: $i,X653: $i,X654: $i] :
( ( X650 @ X651 @ X652 )
| ~ ( in @ ( kpair @ X651 @ X652 ) @ ( dpsetconstr @ X653 @ X654 @ X650 ) ) ),
inference(cnf,[status(esa)],[zf_stmt_14]) ).
thf(zip_derived_cl11400,plain,
$false,
inference(eprover,[status(thm)],[zip_derived_cl266,zip_derived_cl11350,zip_derived_cl11348,zip_derived_cl11364]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU777^1 : TPTP v8.1.2. Released v3.7.0.
% 0.00/0.13 % Command : python3 /export/starexec/sandbox/solver/bin/portfolio.lams.parallel.py %s %d /export/starexec/sandbox/tmp/tmp.Z6OcAbSkpJ true
% 0.13/0.34 % Computer : n026.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Wed Aug 23 16:45:31 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.13/0.34 % Running portfolio for 300 s
% 0.13/0.34 % File : /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.13/0.35 % Number of cores: 8
% 0.13/0.35 % Python version: Python 3.6.8
% 0.13/0.35 % Running in HO mode
% 0.21/0.67 % Total configuration time : 828
% 0.21/0.67 % Estimated wc time : 1656
% 0.21/0.67 % Estimated cpu time (8 cpus) : 207.0
% 0.21/0.73 % /export/starexec/sandbox/solver/bin/lams/40_c.s.sh running for 80s
% 0.21/0.75 % /export/starexec/sandbox/solver/bin/lams/35_full_unif4.sh running for 80s
% 0.21/0.77 % /export/starexec/sandbox/solver/bin/lams/40_c_ic.sh running for 80s
% 0.21/0.77 % /export/starexec/sandbox/solver/bin/lams/40_noforms.sh running for 90s
% 0.21/0.77 % /export/starexec/sandbox/solver/bin/lams/40_b.comb.sh running for 70s
% 0.21/0.77 % /export/starexec/sandbox/solver/bin/lams/15_e_short1.sh running for 30s
% 1.27/0.79 % /export/starexec/sandbox/solver/bin/lams/20_acsne_simpl.sh running for 40s
% 1.27/0.79 % /export/starexec/sandbox/solver/bin/lams/30_sp5.sh running for 60s
% 132.10/17.64 % /export/starexec/sandbox/solver/bin/lams/30_b.l.sh running for 90s
% 132.10/17.65 % Solved by lams/40_c.s.sh.
% 132.10/17.65 % done 736 iterations in 16.586s
% 132.10/17.65 % SZS status Theorem for '/export/starexec/sandbox/benchmark/theBenchmark.p'
% 132.10/17.65 % SZS output start Refutation
% See solution above
% 132.57/17.72
% 132.57/17.72
% 132.57/17.72 % Terminating...
% 133.16/17.94 % Runner terminated.
% 133.16/17.95 % Zipperpin 1.5 exiting
%------------------------------------------------------------------------------