%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : SEU788^1 : TPTP v9.2.0. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Oct 9 02:49:59 PM UTC 2025
% Result : Theorem 211.26s 210.58s
% Output : Proof 211.26s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 1017
% Syntax : Number of formulae : 1025 ( 319 unt; 0 typ; 308 def)
% Number of atoms : 5388 ( 480 equ; 1 cnn)
% Maximal formula atoms : 174 ( 5 avg)
% Number of connectives : 6832 ( 541 ~; 368 |; 0 &;3213 @)
% ( 354 <=>;2355 =>; 0 <=; 0 <~>)
% Maximal formula depth : 174 ( 4 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 66 ( 66 >; 0 *; 0 +; 0 <<)
% Number of symbols : 703 ( 699 usr; 699 con; 0-2 aty)
% ( 1 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 1074 ( 16 ^;1058 !; 0 ?;1074 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_nonemptyImpWitness,type,
nonemptyImpWitness: $o ).
thf(ty_inIntersectImpInIntersectUnions,type,
inIntersectImpInIntersectUnions: $o ).
thf(ty_demorgan1a,type,
demorgan1a: $o ).
thf(ty_binunionIL,type,
binunionIL: $o ).
thf(ty_quantDeMorgan4,type,
quantDeMorgan4: $o ).
thf(ty_subsetTI,type,
subsetTI: $o ).
thf(ty_complementUnionInPowersetComplement,type,
complementUnionInPowersetComplement: $o ).
thf(ty_upairequniteq,type,
upairequniteq: $o ).
thf(ty_iftrueProp2,type,
iftrueProp2: $o ).
thf(ty_emptysetsubset,type,
emptysetsubset: $o ).
thf(ty_subsetI1,type,
subsetI1: $o ).
thf(ty_kpairp,type,
kpairp: $o ).
thf(ty_wellorderingAx,type,
wellorderingAx: $o ).
thf(ty_setunionE,type,
setunionE: $o ).
thf(ty_singletonsubset,type,
singletonsubset: $o ).
thf(ty_setukpairIL,type,
setukpairIL: $o ).
thf(ty_breln1compI,type,
breln1compI: $o ).
thf(ty_binunionEcases,type,
binunionEcases: $o ).
thf(ty_complementTE1,type,
complementTE1: $o ).
thf(ty_breln1SetBreln1,type,
breln1SetBreln1: $o ).
thf(ty_setminusT_lem,type,
setminusT_lem: $o ).
thf(ty_setminusSubset2,type,
setminusSubset2: $o ).
thf(ty_breln1compprop,type,
breln1compprop: $o ).
thf(ty_exu__Cong,type,
exu__Cong: $o ).
thf(ty_setminusEL,type,
setminusEL: $o ).
thf(ty_emptysetE,type,
emptysetE: $o ).
thf(ty_upairsetIL,type,
upairsetIL: $o ).
thf(ty_subsetE,type,
subsetE: $o ).
thf(ty_funcGraphProp4,type,
funcGraphProp4: $o ).
thf(ty_exuE3e,type,
exuE3e: $o ).
thf(ty_eta2,type,
eta2: $o ).
thf(ty_inComplementUnionImpInComplement1,type,
inComplementUnionImpInComplement1: $o ).
thf(ty_dpsetconstrI,type,
dpsetconstrI: $o ).
thf(ty_quantDeMorgan2,type,
quantDeMorgan2: $o ).
thf(ty_symdiffIneg1,type,
symdiffIneg1: $o ).
thf(ty_dpsetconstrSub,type,
dpsetconstrSub: $o ).
thf(ty_powersetTE1,type,
powersetTE1: $o ).
thf(ty_doubleComplementSub2,type,
doubleComplementSub2: $o ).
thf(ty_apProp,type,
apProp: $o ).
thf(ty_emptyInPowerset,type,
emptyInPowerset: $o ).
thf(ty_funcGraphProp1,type,
funcGraphProp1: $o ).
thf(ty_binunionLsub,type,
binunionLsub: $o ).
thf(ty_doubleComplementI1,type,
doubleComplementI1: $o ).
thf(ty_breln1unionprop,type,
breln1unionprop: $o ).
thf(ty_binunionTE,type,
binunionTE: $o ).
thf(ty_singletonprop,type,
singletonprop: $o ).
thf(ty_demorgan2b2,type,
demorgan2b2: $o ).
thf(ty_doubleComplementE1,type,
doubleComplementE1: $o ).
thf(ty_emptysetimpfalse,type,
emptysetimpfalse: $o ).
thf(ty_exuE1,type,
exuE1: $o ).
thf(ty_prop2set2propI,type,
prop2set2propI: $o ).
thf(ty_binintersectSubset5,type,
binintersectSubset5: $o ).
thf(ty_sepSubset,type,
sepSubset: $o ).
thf(ty_ex1I2,type,
ex1I2: $o ).
thf(ty_woz1_3,type,
woz1_3: $o ).
thf(ty_iftrue,type,
iftrue: $o ).
thf(ty_woz1_5,type,
woz1_5: $o ).
thf(ty_ap2p,type,
ap2p: $o ).
thf(ty_funcext,type,
funcext: $o ).
thf(ty_setbeta,type,
setbeta: $o ).
thf(ty_setukpairinjR11,type,
setukpairinjR11: $o ).
thf(ty_kpairsurjEq,type,
kpairsurjEq: $o ).
thf(ty_breln1,type,
breln1: $i > $i > $o ).
thf(ty_binunionRsub,type,
binunionRsub: $o ).
thf(ty_binintersectEL,type,
binintersectEL: $o ).
thf(ty_demorgan1,type,
demorgan1: $o ).
thf(ty_upairsetE,type,
upairsetE: $o ).
thf(ty_singletonsswitch,type,
singletonsswitch: $o ).
thf(ty_emptyE1,type,
emptyE1: $o ).
thf(ty_prop2setE,type,
prop2setE: $o ).
thf(ty_woz1_1,type,
woz1_1: $o ).
thf(ty_setukpairinjL,type,
setukpairinjL: $o ).
thf(ty_lamp,type,
lamp: $o ).
thf(ty_powerset__Cong,type,
powerset__Cong: $o ).
thf(ty_symdiffI1,type,
symdiffI1: $o ).
thf(ty_cartprodsndpairEq,type,
cartprodsndpairEq: $o ).
thf(ty_eqbreln,type,
eqbreln: $o ).
thf(ty_setadjoinAx,type,
setadjoinAx: $o ).
thf(ty_setadjoinIL,type,
setadjoinIL: $o ).
thf(ty_complementInPowersetComplementIntersect,type,
complementInPowersetComplementIntersect: $o ).
thf(ty_subbreln1,type,
subbreln1: $o ).
thf(ty_setoftrueEq,type,
setoftrueEq: $o ).
thf(ty_notinsingleton,type,
notinsingleton: $o ).
thf(ty_setadjoinSub,type,
setadjoinSub: $o ).
thf(ty_setminusIRneg,type,
setminusIRneg: $o ).
thf(ty_contrasubsetT3,type,
contrasubsetT3: $o ).
thf(ty_setminusERneg,type,
setminusERneg: $o ).
thf(ty_setunionE2,type,
setunionE2: $o ).
thf(ty_binintersectT_lem,type,
binintersectT_lem: $o ).
thf(ty_binintersectTELcontra,type,
binintersectTELcontra: $o ).
thf(ty_secondinupair,type,
secondinupair: $o ).
thf(ty_cartprodsndin,type,
cartprodsndin: $o ).
thf(ty_binintersectER,type,
binintersectER: $o ).
thf(ty_setukpairIR,type,
setukpairIR: $o ).
thf(ty_exuE2,type,
exuE2: $o ).
thf(ty_cartprodpairin,type,
cartprodpairin: $o ).
thf(ty_inIntersectImpInUnion,type,
inIntersectImpInUnion: $o ).
thf(ty_setOfPairsIsBReln1,type,
setOfPairsIsBReln1: $o ).
thf(ty_funcGraphProp3,type,
funcGraphProp3: $o ).
thf(ty_funcextLem,type,
funcextLem: $o ).
thf(ty_notdallE,type,
notdallE: $o ).
thf(ty_ex1E2,type,
ex1E2: $o ).
thf(ty_complementTI1,type,
complementTI1: $o ).
thf(ty_noeltsimpempty,type,
noeltsimpempty: $o ).
thf(ty_beta2,type,
beta2: $o ).
thf(ty_setukpairinjR,type,
setukpairinjR: $o ).
thf(ty_binintersectRsub,type,
binintersectRsub: $o ).
thf(ty_setminusLsub,type,
setminusLsub: $o ).
thf(ty_powersetE,type,
powersetE: $o ).
thf(ty_subsetE2,type,
subsetE2: $o ).
thf(ty_exuEu,type,
exuEu: $o ).
thf(ty_theprop,type,
theprop: $o ).
thf(ty_binunionTILcontra,type,
binunionTILcontra: $o ).
thf(ty_breln1all2,type,
breln1all2: $o ).
thf(ty_binunionE,type,
binunionE: $o ).
thf(ty_binunionTEcontra,type,
binunionTEcontra: $o ).
thf(ty_dsetconstr__Cong,type,
dsetconstr__Cong: $o ).
thf(ty_inIntersectImpInUnion2,type,
inIntersectImpInUnion2: $o ).
thf(ty_woz13rule2,type,
woz13rule2: $o ).
thf(ty_nonemptyE1,type,
nonemptyE1: $o ).
thf(ty_vacuousDall,type,
vacuousDall: $o ).
thf(ty_notsubsetI,type,
notsubsetI: $o ).
thf(ty_binintersectSubset1,type,
binintersectSubset1: $o ).
thf(ty_setOfPairsIsBReln,type,
setOfPairsIsBReln: $o ).
thf(ty_emptyinunitempty,type,
emptyinunitempty: $o ).
thf(ty_descr__Cong,type,
descr__Cong: $o ).
thf(ty_kpairiskpair,type,
kpairiskpair: $o ).
thf(ty_emptyinPowerset,type,
emptyinPowerset: $o ).
thf(ty_setukpairinjL2,type,
setukpairinjL2: $o ).
thf(ty_cartprodpairmemEL,type,
cartprodpairmemEL: $o ).
thf(ty_emptysetAx,type,
emptysetAx: $o ).
thf(ty_singletoninpowerset,type,
singletoninpowerset: $o ).
thf(ty_dpsetconstrER,type,
dpsetconstrER: $o ).
thf(ty_setminusER,type,
setminusER: $o ).
thf(ty_eigen__22,type,
eigen__22: $i ).
thf(ty_bs114d,type,
bs114d: $o ).
thf(ty_cartprodmempair,type,
cartprodmempair: $o ).
thf(ty_contrasubsetT,type,
contrasubsetT: $o ).
thf(ty_dpsetconstrEL1,type,
dpsetconstrEL1: $o ).
thf(ty_quantDeMorgan1,type,
quantDeMorgan1: $o ).
thf(ty_eqbreln1,type,
eqbreln1: $o ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_ubforcartprodlem3,type,
ubforcartprodlem3: $o ).
thf(ty_subsetTrans,type,
subsetTrans: $o ).
thf(ty_inCongP,type,
inCongP: $o ).
thf(ty_disjointsetsI1,type,
disjointsetsI1: $o ).
thf(ty_setunionsingleton1,type,
setunionsingleton1: $o ).
thf(ty_woz13rule4,type,
woz13rule4: $o ).
thf(ty_uniqinunit,type,
uniqinunit: $o ).
thf(ty_iffalseProp2,type,
iffalseProp2: $o ).
thf(ty_demorgan2b,type,
demorgan2b: $o ).
thf(ty_dsetconstrEL,type,
dsetconstrEL: $o ).
thf(ty_exuI2,type,
exuI2: $o ).
thf(ty_omega0Ax,type,
omega0Ax: $o ).
thf(ty_binunionTIRcontra,type,
binunionTIRcontra: $o ).
thf(ty_woz1_2,type,
woz1_2: $o ).
thf(ty_binunion,type,
binunion: $i > $i > $i ).
thf(ty_complementSubsetComplementIntersect,type,
complementSubsetComplementIntersect: $o ).
thf(ty_eqinunit,type,
eqinunit: $o ).
thf(ty_powersetE1,type,
powersetE1: $o ).
thf(ty_setunionsingleton,type,
setunionsingleton: $o ).
thf(ty_breln1invprop,type,
breln1invprop: $o ).
thf(ty_exuE3u,type,
exuE3u: $o ).
thf(ty_setextsub,type,
setextsub: $o ).
thf(ty_setminusSubset1,type,
setminusSubset1: $o ).
thf(ty_setukpairinjR2,type,
setukpairinjR2: $o ).
thf(ty_setadjoinE,type,
setadjoinE: $o ).
thf(ty_prop2setI,type,
prop2setI: $o ).
thf(ty_woz13rule1,type,
woz13rule1: $o ).
thf(ty_breln1invE,type,
breln1invE: $o ).
thf(ty_breln1unionIR,type,
breln1unionIR: $o ).
thf(ty_exuI1,type,
exuI1: $o ).
thf(ty_inPowerset,type,
inPowerset: $o ).
thf(ty_ifp,type,
ifp: $o ).
thf(ty_powersetT_lem,type,
powersetT_lem: $o ).
thf(ty_demorgan2a1,type,
demorgan2a1: $o ).
thf(ty_foundationAx,type,
foundationAx: $o ).
thf(ty_emptyset__Cong,type,
emptyset__Cong: $o ).
thf(ty_lam2p,type,
lam2p: $o ).
thf(ty_binintersectTERcontra,type,
binintersectTERcontra: $o ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_demorgan2a2,type,
demorgan2a2: $o ).
thf(ty_ex1I,type,
ex1I: $o ).
thf(ty_exuI3,type,
exuI3: $o ).
thf(ty_binintersectI,type,
binintersectI: $o ).
thf(ty_setunionsingleton2,type,
setunionsingleton2: $o ).
thf(ty_symdiffIneg2,type,
symdiffIneg2: $o ).
thf(ty_cartprodfstin,type,
cartprodfstin: $o ).
thf(ty_ap2apEq2,type,
ap2apEq2: $o ).
thf(ty_beta1,type,
beta1: $o ).
thf(ty_setextAx,type,
setextAx: $o ).
thf(ty_setext,type,
setext: $o ).
thf(ty_contraSubsetComplement,type,
contraSubsetComplement: $o ).
thf(ty_demorgan1b,type,
demorgan1b: $o ).
thf(ty_setunionAx,type,
setunionAx: $o ).
thf(ty_quantDeMorgan3,type,
quantDeMorgan3: $o ).
thf(ty_binintersectSubset4,type,
binintersectSubset4: $o ).
thf(ty_powersetI,type,
powersetI: $o ).
thf(ty_notequalI2,type,
notequalI2: $o ).
thf(ty_upairset2IR,type,
upairset2IR: $o ).
thf(ty_cartprodfstpairEq,type,
cartprodfstpairEq: $o ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_binintersectSubset3,type,
binintersectSubset3: $o ).
thf(ty_powersetAx,type,
powersetAx: $o ).
thf(ty_setextT,type,
setextT: $o ).
thf(ty_setukpairinjR12,type,
setukpairinjR12: $o ).
thf(ty_dsetconstrI,type,
dsetconstrI: $o ).
thf(ty_doubleComplementSub1,type,
doubleComplementSub1: $o ).
thf(ty_contrasubsetT2,type,
contrasubsetT2: $o ).
thf(ty_upairsetIR,type,
upairsetIR: $o ).
thf(ty_replAx,type,
replAx: $o ).
thf(ty_ubforcartprodlem1,type,
ubforcartprodlem1: $o ).
thf(ty_binunionT_lem,type,
binunionT_lem: $o ).
thf(ty_setminusELneg,type,
setminusELneg: $o ).
thf(ty_complementT_lem,type,
complementT_lem: $o ).
thf(ty_powersetTI1,type,
powersetTI1: $o ).
thf(ty_omega__Cong,type,
omega__Cong: $o ).
thf(ty_intersectInPowersetIntersectUnions,type,
intersectInPowersetIntersectUnions: $o ).
thf(ty_setadjoin__Cong,type,
setadjoin__Cong: $o ).
thf(ty_powersetsubset,type,
powersetsubset: $o ).
thf(ty_brelnall1,type,
brelnall1: $o ).
thf(ty_ifSingleton,type,
ifSingleton: $o ).
thf(ty_notequalI1,type,
notequalI1: $o ).
thf(ty_singletonsuniq,type,
singletonsuniq: $o ).
thf(ty_iftrueProp1,type,
iftrueProp1: $o ).
thf(ty_subsetemptysetimpeq,type,
subsetemptysetimpeq: $o ).
thf(ty_setunion__Cong,type,
setunion__Cong: $o ).
thf(ty_demorgan2a,type,
demorgan2a: $o ).
thf(ty_singletoninpowunion,type,
singletoninpowunion: $o ).
thf(ty_funcinfuncset,type,
funcinfuncset: $o ).
thf(ty_demorgan2,type,
demorgan2: $o ).
thf(ty_breln1unionEcases,type,
breln1unionEcases: $o ).
thf(ty_woz13rule0,type,
woz13rule0: $o ).
thf(ty_powersetI1,type,
powersetI1: $o ).
thf(ty_cartprodmempair1,type,
cartprodmempair1: $o ).
thf(ty_ksndpairEq,type,
ksndpairEq: $o ).
thf(ty_cartprodpairsurjEq,type,
cartprodpairsurjEq: $o ).
thf(ty_breln1compE,type,
breln1compE: $o ).
thf(ty_kfstpairEq,type,
kfstpairEq: $o ).
thf(ty_omegaSAx,type,
omegaSAx: $o ).
thf(ty_omegaIndAx,type,
omegaIndAx: $o ).
thf(ty_woz13rule3,type,
woz13rule3: $o ).
thf(ty_infuncsetfunc,type,
infuncsetfunc: $o ).
thf(ty_iffalseProp1,type,
iffalseProp1: $o ).
thf(ty_subsetRefl,type,
subsetRefl: $o ).
thf(ty_symdiffI2,type,
symdiffI2: $o ).
thf(ty_lam2lamEq,type,
lam2lamEq: $o ).
thf(ty_upairset2E,type,
upairset2E: $o ).
thf(ty_dpsetconstrEL2,type,
dpsetconstrEL2: $o ).
thf(ty_emptyI,type,
emptyI: $o ).
thf(ty_lamProp,type,
lamProp: $o ).
thf(ty_funcext2,type,
funcext2: $o ).
thf(ty_setminusILneg,type,
setminusILneg: $o ).
thf(ty_upairinpowunion,type,
upairinpowunion: $o ).
thf(ty_breln1all1,type,
breln1all1: $o ).
thf(ty_funcImageSingleton,type,
funcImageSingleton: $o ).
thf(ty_subPowSU,type,
subPowSU: $o ).
thf(ty_brelnall2,type,
brelnall2: $o ).
thf(ty_setukpairinjR1,type,
setukpairinjR1: $o ).
thf(ty_eta1,type,
eta1: $o ).
thf(ty_subsetI2,type,
subsetI2: $o ).
thf(ty_ex1E1,type,
ex1E1: $o ).
thf(ty_breln1unionE,type,
breln1unionE: $o ).
thf(ty_funcGraphProp2,type,
funcGraphProp2: $o ).
thf(ty_setunionI,type,
setunionI: $o ).
thf(ty_sepInPowerset,type,
sepInPowerset: $o ).
thf(ty_dsetconstrER,type,
dsetconstrER: $o ).
thf(ty_nonemptyI1,type,
nonemptyI1: $o ).
thf(ty_breln1unionIL,type,
breln1unionIL: $o ).
thf(ty_kfstsingleton,type,
kfstsingleton: $o ).
thf(ty_subset2powerset,type,
subset2powerset: $o ).
thf(ty_iffalse,type,
iffalse: $o ).
thf(ty_ubforcartprodlem2,type,
ubforcartprodlem2: $o ).
thf(ty_complementTnotintersectT,type,
complementTnotintersectT: $o ).
thf(ty_descrp,type,
descrp: $o ).
thf(ty_setadjoinSub2,type,
setadjoinSub2: $o ).
thf(ty_eqimpsubset2,type,
eqimpsubset2: $o ).
thf(ty_cartprodmempaircEq,type,
cartprodmempaircEq: $o ).
thf(ty_choice2fnsingleton,type,
choice2fnsingleton: $o ).
thf(ty_in__Cong,type,
in__Cong: $o ).
thf(ty_setadjoinIR,type,
setadjoinIR: $o ).
thf(ty_binintersectSubset2,type,
binintersectSubset2: $o ).
thf(ty_upairsubunion,type,
upairsubunion: $o ).
thf(ty_doubleComplementEq,type,
doubleComplementEq: $o ).
thf(ty_subbreln,type,
subbreln: $o ).
thf(ty_breln1invI,type,
breln1invI: $o ).
thf(ty_complementTcontraSubset,type,
complementTcontraSubset: $o ).
thf(ty_inComplementUnionImpNotIn1,type,
inComplementUnionImpNotIn1: $o ).
thf(ty_binunionIR,type,
binunionIR: $o ).
thf(ty_setminusI,type,
setminusI: $o ).
thf(ty_ap2apEq1,type,
ap2apEq1: $o ).
thf(ty_eqimpsubset1,type,
eqimpsubset1: $o ).
thf(ty_cartprodpairmemER,type,
cartprodpairmemER: $o ).
thf(ty_nonemptyI,type,
nonemptyI: $o ).
thf(ty_symdiffE,type,
symdiffE: $o ).
thf(ty_binintersectLsub,type,
binintersectLsub: $o ).
thf(ty_contrasubsetT1,type,
contrasubsetT1: $o ).
thf(ty_woz1_4,type,
woz1_4: $o ).
thf(ty_breln1unionI,type,
breln1unionI: $o ).
thf(ty_setadjoinOr,type,
setadjoinOr: $o ).
thf(ty_iftrueorfalse,type,
iftrueorfalse: $o ).
thf(ty_notdexE,type,
notdexE: $o ).
thf(ty_setukpairinjL1,type,
setukpairinjL1: $o ).
thf(ty_ksndsingleton,type,
ksndsingleton: $o ).
thf(ty_breln1compEex,type,
breln1compEex: $o ).
thf(ty_theeq,type,
theeq: $o ).
thf(ty_notinemptyset,type,
notinemptyset: $o ).
thf(ty_dpsetconstrERa,type,
dpsetconstrERa: $o ).
thf(ty_app,type,
app: $o ).
thf(ty_complementImpComplementIntersect,type,
complementImpComplementIntersect: $o ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__1,definition,
( eigen__1
= ( eps__0
@ ^ [X1: $i] :
~ ( ( breln1 @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( breln1 @ eigen__0 @ X2 )
=> ( ( binunion @ X1 @ X2 )
= ( binunion @ X2 @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(eigendef_eigen__0,definition,
( eigen__0
= ( eps__0
@ ^ [X1: $i] :
~ ! [X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( ( binunion @ X2 @ X3 )
= ( binunion @ X3 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__0])]) ).
thf(eigendef_eigen__22,definition,
( eigen__22
= ( eps__0
@ ^ [X1: $i] :
( ( in @ X1 @ ( binunion @ eigen__2 @ eigen__1 ) )
!= ( in @ X1 @ ( binunion @ eigen__1 @ eigen__2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__22])]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: $i] :
~ ( ( breln1 @ eigen__0 @ X1 )
=> ( ( binunion @ eigen__1 @ X1 )
= ( binunion @ X1 @ eigen__1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(sP1,plain,
( sP1
<=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( ( binunion @ X2 @ X3 )
= ( binunion @ X3 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( woz1_2
=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( ( binunion @ X2 @ X3 )
= ( binunion @ X3 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( setextAx
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( in @ eigen__22 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( 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
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( 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
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> sP6 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( 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
=> sP2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ! [X1: $i] :
( ( in @ X1 @ ( binunion @ eigen__1 @ eigen__2 ) )
=> ( ~ ( in @ X1 @ eigen__1 )
=> ( in @ X1 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( 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
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( 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
=> sP5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( 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
=> sP10 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( omega__Cong
=> ( exuEu
=> sP5 ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( 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
=> sP7 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( breln1invE
=> ( breln1compprop
=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> ( breln1unionE
=> ( breln1unionEcases
=> ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( ( binunion @ X2 @ X3 )
= ( binunion @ X3 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( breln1unionE
=> ( breln1unionEcases
=> ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( ( binunion @ X2 @ X3 )
= ( binunion @ X3 @ X2 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( 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
=> sP7 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> sP7 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> sP12 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> sP1 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> sP12 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> sP12 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( breln1compI
=> ( breln1compE
=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionI
=> sP16 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> sP12 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> sP11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> sP14 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( 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
=> sP18 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ( 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
=> sP10 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> sP18 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ( ( in @ eigen__22 @ ( binunion @ eigen__2 @ eigen__1 ) )
= ( in @ eigen__22 @ ( binunion @ eigen__1 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(sP31,plain,
( sP31
<=> ( breln1all2
=> ( breln1SetBreln1
=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> ( subbreln1
=> ( eqbreln1
=> ( breln1invprop
=> ( breln1invI
=> sP15 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP31])]) ).
thf(sP32,plain,
( sP32
<=> binunionE ),
introduced(definition,[new_symbols(definition,[sP32])]) ).
thf(sP33,plain,
( sP33
<=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> sP10 ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP33])]) ).
thf(sP34,plain,
( sP34
<=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> sP10 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP34])]) ).
thf(sP35,plain,
( sP35
<=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> sP20 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP35])]) ).
thf(sP36,plain,
( sP36
<=> ( 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
=> sP13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP36])]) ).
thf(sP37,plain,
( sP37
<=> ( demorgan2
=> ( woz13rule0
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> sP2 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP37])]) ).
thf(sP38,plain,
( sP38
<=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> sP25 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP38])]) ).
thf(sP39,plain,
( sP39
<=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> sP1 ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP39])]) ).
thf(sP40,plain,
( sP40
<=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> sP2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP40])]) ).
thf(sP41,plain,
( sP41
<=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> sP26 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP41])]) ).
thf(sP42,plain,
( sP42
<=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> sP12 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP42])]) ).
thf(sP43,plain,
( sP43
<=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> sP33 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP43])]) ).
thf(sP44,plain,
( sP44
<=> ( woz1_3
=> ( woz1_4
=> ( woz1_5
=> sP31 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP44])]) ).
thf(sP45,plain,
( sP45
<=> ( ( in @ eigen__22 @ ( binunion @ eigen__2 @ eigen__1 ) )
=> ( ~ sP4
=> ( in @ eigen__22 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP45])]) ).
thf(sP46,plain,
( sP46
<=> ( ( binunion @ eigen__2 @ eigen__1 )
= ( binunion @ eigen__1 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP46])]) ).
thf(sP47,plain,
( sP47
<=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> sP21 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP47])]) ).
thf(sP48,plain,
( sP48
<=> ( binunionIR
=> ( binunionEcases
=> ( sP32
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> sP17 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP48])]) ).
thf(sP49,plain,
( sP49
<=> ( funcGraphProp2
=> ( funcextLem
=> sP43 ) ) ),
introduced(definition,[new_symbols(definition,[sP49])]) ).
thf(sP50,plain,
( sP50
<=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> sP8 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP50])]) ).
thf(sP51,plain,
( sP51
<=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> sP48 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP51])]) ).
thf(sP52,plain,
( sP52
<=> ( binintersectRsub
=> sP17 ) ),
introduced(definition,[new_symbols(definition,[sP52])]) ).
thf(sP53,plain,
( sP53
<=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> sP51 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP53])]) ).
thf(sP54,plain,
( sP54
<=> ( breln1unionI
=> sP16 ) ),
introduced(definition,[new_symbols(definition,[sP54])]) ).
thf(sP55,plain,
( sP55
<=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> sP18 ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP55])]) ).
thf(sP56,plain,
( sP56
<=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> sP49 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP56])]) ).
thf(sP57,plain,
( sP57
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__1 )
=> ( in @ X1 @ ( binunion @ eigen__2 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP57])]) ).
thf(sP58,plain,
( sP58
<=> ( ubforcartprodlem3
=> sP18 ) ),
introduced(definition,[new_symbols(definition,[sP58])]) ).
thf(sP59,plain,
( sP59
<=> ( dpsetconstrEL1
=> sP12 ) ),
introduced(definition,[new_symbols(definition,[sP59])]) ).
thf(sP60,plain,
( sP60
<=> ( ( in @ eigen__22 @ eigen__1 )
=> ( in @ eigen__22 @ ( binunion @ eigen__1 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP60])]) ).
thf(sP61,plain,
( sP61
<=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> sP29 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP61])]) ).
thf(sP62,plain,
( sP62
<=> ( subsetE2
=> sP14 ) ),
introduced(definition,[new_symbols(definition,[sP62])]) ).
thf(sP63,plain,
( sP63
<=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> sP51 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP63])]) ).
thf(sP64,plain,
( sP64
<=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> sP28 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP64])]) ).
thf(sP65,plain,
( sP65
<=> ( sP4
=> ( in @ eigen__22 @ ( binunion @ eigen__1 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP65])]) ).
thf(sP66,plain,
( sP66
<=> ( 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
=> sP13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP66])]) ).
thf(sP67,plain,
( sP67
<=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> sP6 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP67])]) ).
thf(sP68,plain,
( sP68
<=> ( 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
=> sP13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP68])]) ).
thf(sP69,plain,
( sP69
<=> ( funcImageSingleton
=> ( apProp
=> ( app
=> sP28 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP69])]) ).
thf(sP70,plain,
( sP70
<=> ( ( breln1 @ eigen__0 @ eigen__1 )
=> ! [X1: $i] :
( ( breln1 @ eigen__0 @ X1 )
=> ( ( binunion @ eigen__1 @ X1 )
= ( binunion @ X1 @ eigen__1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP70])]) ).
thf(sP71,plain,
( sP71
<=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> sP25 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP71])]) ).
thf(sP72,plain,
( sP72
<=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> sP59 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP72])]) ).
thf(sP73,plain,
( sP73
<=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> sP1 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP73])]) ).
thf(sP74,plain,
( sP74
<=> ( contraSubsetComplement
=> sP8 ) ),
introduced(definition,[new_symbols(definition,[sP74])]) ).
thf(sP75,plain,
( sP75
<=> ( setunion__Cong
=> sP13 ) ),
introduced(definition,[new_symbols(definition,[sP75])]) ).
thf(sP76,plain,
( sP76
<=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> sP72 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP76])]) ).
thf(sP77,plain,
( sP77
<=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> sP11 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP77])]) ).
thf(sP78,plain,
( sP78
<=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> sP11 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP78])]) ).
thf(sP79,plain,
( sP79
<=> ( binintersectER
=> ( disjointsetsI1
=> sP52 ) ) ),
introduced(definition,[new_symbols(definition,[sP79])]) ).
thf(sP80,plain,
( sP80
<=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> sP72 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP80])]) ).
thf(sP81,plain,
( sP81
<=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> sP37 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP81])]) ).
thf(sP82,plain,
( sP82
<=> ( app
=> sP28 ) ),
introduced(definition,[new_symbols(definition,[sP82])]) ).
thf(sP83,plain,
( sP83
<=> ( cartprodmempair1
=> ( cartprodmempair
=> sP7 ) ) ),
introduced(definition,[new_symbols(definition,[sP83])]) ).
thf(sP84,plain,
( sP84
<=> ( powersetTE1
=> sP39 ) ),
introduced(definition,[new_symbols(definition,[sP84])]) ).
thf(sP85,plain,
( sP85
<=> ( breln1invprop
=> ( breln1invI
=> sP15 ) ) ),
introduced(definition,[new_symbols(definition,[sP85])]) ).
thf(sP86,plain,
( sP86
<=> ( lamProp
=> ( lamp
=> sP56 ) ) ),
introduced(definition,[new_symbols(definition,[sP86])]) ).
thf(sP87,plain,
( sP87
<=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> sP77 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP87])]) ).
thf(sP88,plain,
( sP88
<=> ( cartprodmempair
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP88])]) ).
thf(sP89,plain,
( sP89
<=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> sP6 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP89])]) ).
thf(sP90,plain,
( sP90
<=> ( in @ eigen__22 @ ( binunion @ eigen__2 @ eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP90])]) ).
thf(sP91,plain,
( sP91
<=> ( eqimpsubset2
=> ( eqimpsubset1
=> sP26 ) ) ),
introduced(definition,[new_symbols(definition,[sP91])]) ).
thf(sP92,plain,
( sP92
<=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> sP34 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP92])]) ).
thf(sP93,plain,
( sP93
<=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> sP81 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP93])]) ).
thf(sP94,plain,
( sP94
<=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> sP63 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP94])]) ).
thf(sP95,plain,
( sP95
<=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> sP11 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP95])]) ).
thf(sP96,plain,
( sP96
<=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> sP66 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP96])]) ).
thf(sP97,plain,
( sP97
<=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> sP79 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP97])]) ).
thf(sP98,plain,
( sP98
<=> ( eta2
=> ( iffalseProp1
=> sP34 ) ) ),
introduced(definition,[new_symbols(definition,[sP98])]) ).
thf(sP99,plain,
( sP99
<=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ eigen__2 )
=> ( in @ X2 @ ( binunion @ eigen__2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP99])]) ).
thf(sP100,plain,
( sP100
<=> ( funcext
=> ( funcext2
=> sP33 ) ) ),
introduced(definition,[new_symbols(definition,[sP100])]) ).
thf(sP101,plain,
( sP101
<=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> sP51 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP101])]) ).
thf(sP102,plain,
( sP102
<=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> sP27 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP102])]) ).
thf(sP103,plain,
( sP103
<=> ( funcextLem
=> sP43 ) ),
introduced(definition,[new_symbols(definition,[sP103])]) ).
thf(sP104,plain,
( sP104
<=> ( subsetTI
=> sP20 ) ),
introduced(definition,[new_symbols(definition,[sP104])]) ).
thf(sP105,plain,
( sP105
<=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> sP35 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP105])]) ).
thf(sP106,plain,
( sP106
<=> ( disjointsetsI1
=> sP52 ) ),
introduced(definition,[new_symbols(definition,[sP106])]) ).
thf(sP107,plain,
( sP107
<=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> sP59 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP107])]) ).
thf(sP108,plain,
( sP108
<=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> sP37 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP108])]) ).
thf(sP109,plain,
( sP109
<=> ( breln1compEex
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> sP54 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP109])]) ).
thf(sP110,plain,
( sP110
<=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> sP75 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP110])]) ).
thf(sP111,plain,
( sP111
<=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> sP72 ) ) ),
introduced(definition,[new_symbols(definition,[sP111])]) ).
thf(sP112,plain,
( sP112
<=> ( demorgan2b2
=> ( demorgan2b
=> sP37 ) ) ),
introduced(definition,[new_symbols(definition,[sP112])]) ).
thf(sP113,plain,
( sP113
<=> ! [X1: $i] :
( ( breln1 @ eigen__0 @ X1 )
=> ( ( binunion @ eigen__1 @ X1 )
= ( binunion @ X1 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP113])]) ).
thf(sP114,plain,
( sP114
<=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> sP110 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP114])]) ).
thf(sP115,plain,
( sP115
<=> ( binunionEcases
=> ( sP32
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> sP97 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP115])]) ).
thf(sP116,plain,
( sP116
<=> ( setminusER
=> ( setminusSubset2
=> sP27 ) ) ),
introduced(definition,[new_symbols(definition,[sP116])]) ).
thf(sP117,plain,
( sP117
<=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( X2 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP117])]) ).
thf(sP118,plain,
( sP118
<=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> sP68 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP118])]) ).
thf(sP119,plain,
( sP119
<=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> sP114 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP119])]) ).
thf(sP120,plain,
( sP120
<=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> sP11 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP120])]) ).
thf(sP121,plain,
( sP121
<=> ( prop2setE
=> sP77 ) ),
introduced(definition,[new_symbols(definition,[sP121])]) ).
thf(sP122,plain,
( sP122
<=> ( upairsetE
=> sP68 ) ),
introduced(definition,[new_symbols(definition,[sP122])]) ).
thf(sP123,plain,
( sP123
<=> ( sP32
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> sP97 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP123])]) ).
thf(sP124,plain,
( sP124
<=> ( ! [X1: $i] :
( ( in @ X1 @ ( binunion @ eigen__2 @ eigen__1 ) )
= ( in @ X1 @ ( binunion @ eigen__1 @ eigen__2 ) ) )
=> sP46 ) ),
introduced(definition,[new_symbols(definition,[sP124])]) ).
thf(sP125,plain,
( sP125
<=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( binunion @ eigen__1 @ X1 ) )
=> ( ~ ( in @ X2 @ eigen__1 )
=> ( in @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP125])]) ).
thf(sP126,plain,
( sP126
<=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> sP114 ) ) ),
introduced(definition,[new_symbols(definition,[sP126])]) ).
thf(sP127,plain,
( sP127
<=> ( powersetT_lem
=> ( setminusT_lem
=> sP35 ) ) ),
introduced(definition,[new_symbols(definition,[sP127])]) ).
thf(sP128,plain,
( sP128
<=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> sP108 ) ) ),
introduced(definition,[new_symbols(definition,[sP128])]) ).
thf(sP129,plain,
( sP129
<=> ( cartprodpairsurjEq
=> sP72 ) ),
introduced(definition,[new_symbols(definition,[sP129])]) ).
thf(sP130,plain,
( sP130
<=> ( demorgan2b
=> sP37 ) ),
introduced(definition,[new_symbols(definition,[sP130])]) ).
thf(sP131,plain,
( sP131
<=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> sP49 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP131])]) ).
thf(sP132,plain,
( sP132
<=> ( ~ sP4
=> ( in @ eigen__22 @ eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP132])]) ).
thf(sP133,plain,
( sP133
<=> ( subbreln
=> ( eqbreln
=> sP100 ) ) ),
introduced(definition,[new_symbols(definition,[sP133])]) ).
thf(sP134,plain,
( sP134
<=> ( sP4
=> sP90 ) ),
introduced(definition,[new_symbols(definition,[sP134])]) ).
thf(sP135,plain,
( sP135
<=> ( breln1compprop
=> sP23 ) ),
introduced(definition,[new_symbols(definition,[sP135])]) ).
thf(sP136,plain,
( sP136
<=> ( emptysetimpfalse
=> ( notinemptyset
=> sP78 ) ) ),
introduced(definition,[new_symbols(definition,[sP136])]) ).
thf(sP137,plain,
( sP137
<=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( binunion @ eigen__1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP137])]) ).
thf(sP138,plain,
( sP138
<=> ( powerset__Cong
=> sP75 ) ),
introduced(definition,[new_symbols(definition,[sP138])]) ).
thf(sP139,plain,
( sP139
<=> ( setukpairIR
=> sP55 ) ),
introduced(definition,[new_symbols(definition,[sP139])]) ).
thf(sP140,plain,
( sP140
<=> ( woz13rule2
=> sP40 ) ),
introduced(definition,[new_symbols(definition,[sP140])]) ).
thf(sP141,plain,
( sP141
<=> ( demorgan2a1
=> sP128 ) ),
introduced(definition,[new_symbols(definition,[sP141])]) ).
thf(sP142,plain,
( sP142
<=> ! [X1: $i] :
( ( breln1 @ eigen__0 @ X1 )
=> ! [X2: $i] :
( ( breln1 @ eigen__0 @ X2 )
=> ( ( binunion @ X1 @ X2 )
= ( binunion @ X2 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP142])]) ).
thf(sP143,plain,
( sP143
<=> ( inPowerset
=> sP51 ) ),
introduced(definition,[new_symbols(definition,[sP143])]) ).
thf(sP144,plain,
( sP144
<=> ( lamp
=> sP56 ) ),
introduced(definition,[new_symbols(definition,[sP144])]) ).
thf(sP145,plain,
( sP145
<=> ( inComplementUnionImpNotIn1
=> sP81 ) ),
introduced(definition,[new_symbols(definition,[sP145])]) ).
thf(sP146,plain,
( sP146
<=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> sP143 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP146])]) ).
thf(sP147,plain,
( sP147
<=> ( ( binunion @ eigen__1 @ eigen__2 )
= ( binunion @ eigen__2 @ eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP147])]) ).
thf(sP148,plain,
( sP148
<=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> sP98 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP148])]) ).
thf(sP149,plain,
( sP149
<=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP149])]) ).
thf(sP150,plain,
( sP150
<=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> sP29 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP150])]) ).
thf(sP151,plain,
( sP151
<=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> sP1 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP151])]) ).
thf(sP152,plain,
( sP152
<=> ( woz1_4
=> ( woz1_5
=> sP31 ) ) ),
introduced(definition,[new_symbols(definition,[sP152])]) ).
thf(sP153,plain,
( sP153
<=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ eigen__1 )
=> ( in @ X2 @ ( binunion @ eigen__1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP153])]) ).
thf(sP154,plain,
( sP154
<=> ( funcGraphProp3
=> sP49 ) ),
introduced(definition,[new_symbols(definition,[sP154])]) ).
thf(sP155,plain,
( sP155
<=> ( subbreln1
=> ( eqbreln1
=> sP85 ) ) ),
introduced(definition,[new_symbols(definition,[sP155])]) ).
thf(sP156,plain,
( sP156
<=> ( symdiffIneg1
=> sP29 ) ),
introduced(definition,[new_symbols(definition,[sP156])]) ).
thf(sP157,plain,
( sP157
<=> ( dsetconstrER
=> ( exuE1
=> sP121 ) ) ),
introduced(definition,[new_symbols(definition,[sP157])]) ).
thf(sP158,plain,
( sP158
<=> ( ex1E2
=> ( funcGraphProp1
=> sP154 ) ) ),
introduced(definition,[new_symbols(definition,[sP158])]) ).
thf(sP159,plain,
( sP159
<=> ! [X1: $i] :
( ( ( binunion @ eigen__2 @ eigen__1 )
= X1 )
=> ( X1
= ( binunion @ eigen__2 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP159])]) ).
thf(sP160,plain,
( sP160
<=> ( setadjoinOr
=> ( setoftrueEq
=> sP36 ) ) ),
introduced(definition,[new_symbols(definition,[sP160])]) ).
thf(sP161,plain,
( sP161
<=> ( powersetE1
=> sP143 ) ),
introduced(definition,[new_symbols(definition,[sP161])]) ).
thf(sP162,plain,
( sP162
<=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> sP1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP162])]) ).
thf(sP163,plain,
( sP163
<=> ( setadjoinSub
=> ( setadjoinSub2
=> sP63 ) ) ),
introduced(definition,[new_symbols(definition,[sP163])]) ).
thf(sP164,plain,
( sP164
<=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> sP19 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP164])]) ).
thf(sP165,plain,
( sP165
<=> ( cartprodsndin
=> ( cartprodpairmemEL
=> sP76 ) ) ),
introduced(definition,[new_symbols(definition,[sP165])]) ).
thf(sP166,plain,
( sP166
<=> ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) ) ),
introduced(definition,[new_symbols(definition,[sP166])]) ).
thf(sP167,plain,
( sP167
<=> ( setoftrueEq
=> sP36 ) ),
introduced(definition,[new_symbols(definition,[sP167])]) ).
thf(sP168,plain,
( sP168
<=> ( singletonsswitch
=> sP122 ) ),
introduced(definition,[new_symbols(definition,[sP168])]) ).
thf(sP169,plain,
( sP169
<=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> sP58 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP169])]) ).
thf(sP170,plain,
( sP170
<=> ( choice2fnsingleton
=> ( setOfPairsIsBReln1
=> ( breln1all1
=> sP155 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP170])]) ).
thf(sP171,plain,
( sP171
<=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP171])]) ).
thf(sP172,plain,
( sP172
<=> ( in @ eigen__22 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP172])]) ).
thf(sP173,plain,
( sP173
<=> ( quantDeMorgan1
=> sP126 ) ),
introduced(definition,[new_symbols(definition,[sP173])]) ).
thf(sP174,plain,
( sP174
<=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> sP19 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP174])]) ).
thf(sP175,plain,
( sP175
<=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> sP71 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP175])]) ).
thf(sP176,plain,
( sP176
<=> ( setukpairinjR12
=> ( setukpairinjR1
=> sP21 ) ) ),
introduced(definition,[new_symbols(definition,[sP176])]) ).
thf(sP177,plain,
( sP177
<=> ( complementInPowersetComplementIntersect
=> sP74 ) ),
introduced(definition,[new_symbols(definition,[sP177])]) ).
thf(sP178,plain,
( sP178
<=> ( dsetconstrI
=> sP87 ) ),
introduced(definition,[new_symbols(definition,[sP178])]) ).
thf(sP179,plain,
( sP179
<=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> sP58 ) ) ),
introduced(definition,[new_symbols(definition,[sP179])]) ).
thf(sP180,plain,
( sP180
<=> ( cartprodpairmemEL
=> sP76 ) ),
introduced(definition,[new_symbols(definition,[sP180])]) ).
thf(sP181,plain,
( sP181
<=> ( ap2p
=> ( funcinfuncset
=> sP86 ) ) ),
introduced(definition,[new_symbols(definition,[sP181])]) ).
thf(sP182,plain,
( sP182
<=> ( setadjoinE
=> sP160 ) ),
introduced(definition,[new_symbols(definition,[sP182])]) ).
thf(sP183,plain,
( sP183
<=> ( funcext2
=> sP33 ) ),
introduced(definition,[new_symbols(definition,[sP183])]) ).
thf(sP184,plain,
( sP184
<=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> sP179 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP184])]) ).
thf(sP185,plain,
( sP185
<=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ ( binunion @ X1 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP185])]) ).
thf(sP186,plain,
( sP186
<=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( binunion @ eigen__2 @ X1 ) )
=> ( ~ ( in @ X2 @ eigen__2 )
=> ( in @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP186])]) ).
thf(sP187,plain,
( sP187
<=> ( binunionIL
=> ( upairset2IR
=> sP48 ) ) ),
introduced(definition,[new_symbols(definition,[sP187])]) ).
thf(sP188,plain,
( sP188
<=> ( setunionsingleton1
=> ( setunionsingleton2
=> sP67 ) ) ),
introduced(definition,[new_symbols(definition,[sP188])]) ).
thf(sP189,plain,
( sP189
<=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> sP59 ) ) ),
introduced(definition,[new_symbols(definition,[sP189])]) ).
thf(sP190,plain,
( sP190
<=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> sP118 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP190])]) ).
thf(sP191,plain,
( sP191
<=> ( beta1
=> sP148 ) ),
introduced(definition,[new_symbols(definition,[sP191])]) ).
thf(sP192,plain,
( sP192
<=> ( inIntersectImpInUnion2
=> sP93 ) ),
introduced(definition,[new_symbols(definition,[sP192])]) ).
thf(sP193,plain,
( sP193
<=> ( prop2setI
=> ( prop2set2propI
=> sP110 ) ) ),
introduced(definition,[new_symbols(definition,[sP193])]) ).
thf(sP194,plain,
( sP194
<=> ( inIntersectImpInUnion
=> sP192 ) ),
introduced(definition,[new_symbols(definition,[sP194])]) ).
thf(sP195,plain,
( sP195
<=> ( setadjoinIL
=> sP11 ) ),
introduced(definition,[new_symbols(definition,[sP195])]) ).
thf(sP196,plain,
( sP196
<=> ( setukpairinjL
=> sP47 ) ),
introduced(definition,[new_symbols(definition,[sP196])]) ).
thf(sP197,plain,
( sP197
<=> ( cartprodfstpairEq
=> sP111 ) ),
introduced(definition,[new_symbols(definition,[sP197])]) ).
thf(sP198,plain,
( sP198
<=> ( setukpairIL
=> sP139 ) ),
introduced(definition,[new_symbols(definition,[sP198])]) ).
thf(sP199,plain,
( sP199
<=> ( setminusT_lem
=> sP35 ) ),
introduced(definition,[new_symbols(definition,[sP199])]) ).
thf(sP200,plain,
( sP200
<=> ( woz13rule0
=> ( woz13rule1
=> sP140 ) ) ),
introduced(definition,[new_symbols(definition,[sP200])]) ).
thf(sP201,plain,
( sP201
<=> ( setadjoinAx
=> sP38 ) ),
introduced(definition,[new_symbols(definition,[sP201])]) ).
thf(sP202,plain,
( sP202
<=> ( eqbreln1
=> sP85 ) ),
introduced(definition,[new_symbols(definition,[sP202])]) ).
thf(sP203,plain,
( sP203
<=> ( kpairsurjEq
=> sP165 ) ),
introduced(definition,[new_symbols(definition,[sP203])]) ).
thf(sP204,plain,
( sP204
<=> ( brelnall2
=> sP158 ) ),
introduced(definition,[new_symbols(definition,[sP204])]) ).
thf(sP205,plain,
( sP205
<=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> sP112 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP205])]) ).
thf(sP206,plain,
( sP206
<=> ( in @ eigen__22 @ ( binunion @ eigen__1 @ eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP206])]) ).
thf(sP207,plain,
( sP207
<=> ( ubforcartprodlem2
=> sP58 ) ),
introduced(definition,[new_symbols(definition,[sP207])]) ).
thf(sP208,plain,
( sP208
<=> ( dpsetconstrERa
=> sP59 ) ),
introduced(definition,[new_symbols(definition,[sP208])]) ).
thf(sP209,plain,
( sP209
<=> ( setminusI
=> ( setminusEL
=> sP116 ) ) ),
introduced(definition,[new_symbols(definition,[sP209])]) ).
thf(sP210,plain,
( sP210
<=> ( demorgan2a
=> sP112 ) ),
introduced(definition,[new_symbols(definition,[sP210])]) ).
thf(sP211,plain,
( sP211
<=> ( ~ sP172
=> sP4 ) ),
introduced(definition,[new_symbols(definition,[sP211])]) ).
thf(sP212,plain,
( sP212
<=> ( setadjoinIR
=> sP182 ) ),
introduced(definition,[new_symbols(definition,[sP212])]) ).
thf(sP213,plain,
( sP213
<=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> sP138 ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP213])]) ).
thf(sP214,plain,
( sP214
<=> ( demorgan2a2
=> sP108 ) ),
introduced(definition,[new_symbols(definition,[sP214])]) ).
thf(sP215,plain,
( sP215
<=> ( wellorderingAx
=> sP25 ) ),
introduced(definition,[new_symbols(definition,[sP215])]) ).
thf(sP216,plain,
( sP216
<=> ( powersetE
=> ( setunionI
=> sP66 ) ) ),
introduced(definition,[new_symbols(definition,[sP216])]) ).
thf(sP217,plain,
( sP217
<=> ( setadjoin__Cong
=> sP138 ) ),
introduced(definition,[new_symbols(definition,[sP217])]) ).
thf(sP218,plain,
( sP218
<=> ( binunionTE
=> ( binunionTEcontra
=> sP141 ) ) ),
introduced(definition,[new_symbols(definition,[sP218])]) ).
thf(sP219,plain,
( sP219
<=> ( breln1unionIL
=> ( breln1unionIR
=> sP54 ) ) ),
introduced(definition,[new_symbols(definition,[sP219])]) ).
thf(sP220,plain,
( sP220
<=> ( ex1I2
=> ( singletonsuniq
=> sP6 ) ) ),
introduced(definition,[new_symbols(definition,[sP220])]) ).
thf(sP221,plain,
( sP221
<=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> sP195 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP221])]) ).
thf(sP222,plain,
( sP222
<=> ( binunionTILcontra
=> ( binunionTIRcontra
=> sP194 ) ) ),
introduced(definition,[new_symbols(definition,[sP222])]) ).
thf(sP223,plain,
( sP223
<=> ( omegaSAx
=> ( omegaIndAx
=> sP71 ) ) ),
introduced(definition,[new_symbols(definition,[sP223])]) ).
thf(sP224,plain,
( sP224
<=> ( quantDeMorgan3
=> sP114 ) ),
introduced(definition,[new_symbols(definition,[sP224])]) ).
thf(sP225,plain,
( sP225
<=> ( complementSubsetComplementIntersect
=> sP177 ) ),
introduced(definition,[new_symbols(definition,[sP225])]) ).
thf(sP226,plain,
( sP226
<=> ( nonemptyI1
=> sP195 ) ),
introduced(definition,[new_symbols(definition,[sP226])]) ).
thf(sP227,plain,
( sP227
<=> ( funcGraphProp1
=> sP154 ) ),
introduced(definition,[new_symbols(definition,[sP227])]) ).
thf(sP228,plain,
( sP228
<=> ( woz13rule1
=> sP140 ) ),
introduced(definition,[new_symbols(definition,[sP228])]) ).
thf(sP229,plain,
( sP229
<=> ( binunionTIRcontra
=> sP194 ) ),
introduced(definition,[new_symbols(definition,[sP229])]) ).
thf(sP230,plain,
( sP230
<=> ( apProp
=> sP82 ) ),
introduced(definition,[new_symbols(definition,[sP230])]) ).
thf(sP231,plain,
( sP231
<=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> sP127 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP231])]) ).
thf(sP232,plain,
( sP232
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__1 )
=> ( in @ X1 @ ( binunion @ eigen__1 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP232])]) ).
thf(sP233,plain,
( sP233
<=> ( omegaIndAx
=> sP71 ) ),
introduced(definition,[new_symbols(definition,[sP233])]) ).
thf(sP234,plain,
( sP234
<=> ( eqinunit
=> sP168 ) ),
introduced(definition,[new_symbols(definition,[sP234])]) ).
thf(sP235,plain,
( sP235
<=> ! [X1: $i] :
( ( in @ X1 @ ( binunion @ eigen__2 @ eigen__1 ) )
=> ( ~ ( in @ X1 @ eigen__2 )
=> ( in @ X1 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP235])]) ).
thf(sP236,plain,
( sP236
<=> ( cartprodfstin
=> sP19 ) ),
introduced(definition,[new_symbols(definition,[sP236])]) ).
thf(sP237,plain,
( sP237
<=> ( iftrueProp1
=> ( iftrueProp2
=> sP10 ) ) ),
introduced(definition,[new_symbols(definition,[sP237])]) ).
thf(sP238,plain,
( sP238
<=> ( setukpairinjR
=> sP24 ) ),
introduced(definition,[new_symbols(definition,[sP238])]) ).
thf(sP239,plain,
( sP239
<=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> sP127 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP239])]) ).
thf(sP240,plain,
( sP240
<=> ( nonemptyImpWitness
=> ( uniqinunit
=> sP118 ) ) ),
introduced(definition,[new_symbols(definition,[sP240])]) ).
thf(sP241,plain,
( sP241
<=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> sP97 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP241])]) ).
thf(sP242,plain,
( sP242
<=> ( sP46
=> sP147 ) ),
introduced(definition,[new_symbols(definition,[sP242])]) ).
thf(sP243,plain,
( sP243
<=> ( binunionT_lem
=> sP127 ) ),
introduced(definition,[new_symbols(definition,[sP243])]) ).
thf(sP244,plain,
( sP244
<=> ( breln1unionEcases
=> ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( ( binunion @ X2 @ X3 )
= ( binunion @ X3 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP244])]) ).
thf(sP245,plain,
( sP245
<=> ( setunionI
=> sP66 ) ),
introduced(definition,[new_symbols(definition,[sP245])]) ).
thf(sP246,plain,
( sP246
<=> ( emptyE1
=> sP119 ) ),
introduced(definition,[new_symbols(definition,[sP246])]) ).
thf(sP247,plain,
( sP247
<=> ( upairsubunion
=> ( upairinpowunion
=> sP179 ) ) ),
introduced(definition,[new_symbols(definition,[sP247])]) ).
thf(sP248,plain,
( sP248
<=> ( setunionsingleton2
=> sP67 ) ),
introduced(definition,[new_symbols(definition,[sP248])]) ).
thf(sP249,plain,
( sP249
<=> ( sepSubset
=> sP187 ) ),
introduced(definition,[new_symbols(definition,[sP249])]) ).
thf(sP250,plain,
( sP250
<=> ( setbeta
=> sP221 ) ),
introduced(definition,[new_symbols(definition,[sP250])]) ).
thf(sP251,plain,
( sP251
<=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> sP1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP251])]) ).
thf(sP252,plain,
( sP252
<=> ( binintersectSubset3
=> sP79 ) ),
introduced(definition,[new_symbols(definition,[sP252])]) ).
thf(sP253,plain,
( sP253
<=> ( upairset2IR
=> sP48 ) ),
introduced(definition,[new_symbols(definition,[sP253])]) ).
thf(sP254,plain,
( sP254
<=> ( setunionAx
=> sP175 ) ),
introduced(definition,[new_symbols(definition,[sP254])]) ).
thf(sP255,plain,
( sP255
<=> ( setminusSubset2
=> sP27 ) ),
introduced(definition,[new_symbols(definition,[sP255])]) ).
thf(sP256,plain,
( sP256
<=> ( breln1invI
=> sP15 ) ),
introduced(definition,[new_symbols(definition,[sP256])]) ).
thf(sP257,plain,
( sP257
<=> ( sP206
=> sP211 ) ),
introduced(definition,[new_symbols(definition,[sP257])]) ).
thf(sP258,plain,
( sP258
<=> ( emptyI
=> sP120 ) ),
introduced(definition,[new_symbols(definition,[sP258])]) ).
thf(sP259,plain,
( sP259
<=> ( sepInPowerset
=> sP249 ) ),
introduced(definition,[new_symbols(definition,[sP259])]) ).
thf(sP260,plain,
( sP260
<=> ( uniqinunit
=> sP118 ) ),
introduced(definition,[new_symbols(definition,[sP260])]) ).
thf(sP261,plain,
( sP261
<=> ( kfstpairEq
=> sP236 ) ),
introduced(definition,[new_symbols(definition,[sP261])]) ).
thf(sP262,plain,
( sP262
<=> ( binintersectI
=> sP97 ) ),
introduced(definition,[new_symbols(definition,[sP262])]) ).
thf(sP263,plain,
( sP263
<=> ( secondinupair
=> sP198 ) ),
introduced(definition,[new_symbols(definition,[sP263])]) ).
thf(sP264,plain,
( sP264
<=> ( symdiffI1
=> ( symdiffI2
=> sP156 ) ) ),
introduced(definition,[new_symbols(definition,[sP264])]) ).
thf(sP265,plain,
( sP265
<=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> sP217 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP265])]) ).
thf(sP266,plain,
( sP266
<=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> sP264 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP266])]) ).
thf(sP267,plain,
( sP267
<=> ( ex1I
=> sP220 ) ),
introduced(definition,[new_symbols(definition,[sP267])]) ).
thf(sP268,plain,
( sP268
<=> ( breln1all1
=> sP155 ) ),
introduced(definition,[new_symbols(definition,[sP268])]) ).
thf(sP269,plain,
( sP269
<=> ( doubleComplementE1
=> sP1 ) ),
introduced(definition,[new_symbols(definition,[sP269])]) ).
thf(sP270,plain,
( sP270
<=> ( nonemptyI
=> sP226 ) ),
introduced(definition,[new_symbols(definition,[sP270])]) ).
thf(sP271,plain,
( sP271
<=> ( emptysetsubset
=> ( subsetE
=> sP62 ) ) ),
introduced(definition,[new_symbols(definition,[sP271])]) ).
thf(sP272,plain,
( sP272
<=> ( binunionRsub
=> sP262 ) ),
introduced(definition,[new_symbols(definition,[sP272])]) ).
thf(sP273,plain,
( sP273
<=> ( emptysetAx
=> sP201 ) ),
introduced(definition,[new_symbols(definition,[sP273])]) ).
thf(sP274,plain,
( sP274
<=> ( singletonsuniq
=> sP6 ) ),
introduced(definition,[new_symbols(definition,[sP274])]) ).
thf(sP275,plain,
( sP275
<=> ( emptyInPowerset
=> sP216 ) ),
introduced(definition,[new_symbols(definition,[sP275])]) ).
thf(sP276,plain,
( sP276
<=> ( breln1SetBreln1
=> sP170 ) ),
introduced(definition,[new_symbols(definition,[sP276])]) ).
thf(sP277,plain,
( sP277
<=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> sP217 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP277])]) ).
thf(sP278,plain,
( sP278
<=> ( woz13rule4
=> ( woz1_1
=> sP2 ) ) ),
introduced(definition,[new_symbols(definition,[sP278])]) ).
thf(sP279,plain,
( sP279
<=> ( woz1_5
=> sP31 ) ),
introduced(definition,[new_symbols(definition,[sP279])]) ).
thf(sP280,plain,
( sP280
<=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> sP252 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP280])]) ).
thf(sP281,plain,
( sP281
<=> ( symdiffI2
=> sP156 ) ),
introduced(definition,[new_symbols(definition,[sP281])]) ).
thf(sP282,plain,
( sP282
<=> ( binunionIL = sP185 ) ),
introduced(definition,[new_symbols(definition,[sP282])]) ).
thf(sP283,plain,
( sP283
<=> ( setextT
=> sP104 ) ),
introduced(definition,[new_symbols(definition,[sP283])]) ).
thf(sP284,plain,
( sP284
<=> setextAx ),
introduced(definition,[new_symbols(definition,[sP284])]) ).
thf(sP285,plain,
( sP285
<=> ( foundationAx
=> sP215 ) ),
introduced(definition,[new_symbols(definition,[sP285])]) ).
thf(sP286,plain,
( sP286
<=> ( eqbreln
=> sP100 ) ),
introduced(definition,[new_symbols(definition,[sP286])]) ).
thf(sP287,plain,
( sP287
<=> ( binintersectLsub
=> ( binintersectSubset2
=> sP252 ) ) ),
introduced(definition,[new_symbols(definition,[sP287])]) ).
thf(sP288,plain,
( sP288
<=> binunionIR ),
introduced(definition,[new_symbols(definition,[sP288])]) ).
thf(sP289,plain,
( sP289
<=> ( setminusEL
=> sP116 ) ),
introduced(definition,[new_symbols(definition,[sP289])]) ).
thf(sP290,plain,
( sP290
<=> ( prop2set2propI
=> sP110 ) ),
introduced(definition,[new_symbols(definition,[sP290])]) ).
thf(sP291,plain,
( sP291
<=> ( iffalse
=> sP239 ) ),
introduced(definition,[new_symbols(definition,[sP291])]) ).
thf(sP292,plain,
( sP292
<=> ( binintersectTERcontra
=> sP151 ) ),
introduced(definition,[new_symbols(definition,[sP292])]) ).
thf(sP293,plain,
( sP293
<=> ( demorgan1
=> sP210 ) ),
introduced(definition,[new_symbols(definition,[sP293])]) ).
thf(sP294,plain,
( sP294
<=> ( notinemptyset
=> sP78 ) ),
introduced(definition,[new_symbols(definition,[sP294])]) ).
thf(sP295,plain,
( sP295
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__2 )
=> ( in @ X1 @ ( binunion @ eigen__2 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP295])]) ).
thf(sP296,plain,
( sP296
<=> ( ex1E1
=> sP267 ) ),
introduced(definition,[new_symbols(definition,[sP296])]) ).
thf(sP297,plain,
( sP297
<=> ( exuEu
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP297])]) ).
thf(sP298,plain,
( sP298
<=> ( sP284
=> sP273 ) ),
introduced(definition,[new_symbols(definition,[sP298])]) ).
thf(sP299,plain,
( sP299
<=> ( funcinfuncset
=> sP86 ) ),
introduced(definition,[new_symbols(definition,[sP299])]) ).
thf(sP300,plain,
( sP300
<=> ( binintersectT_lem
=> sP243 ) ),
introduced(definition,[new_symbols(definition,[sP300])]) ).
thf(sP301,plain,
( sP301
<=> ( upairsetIR
=> sP246 ) ),
introduced(definition,[new_symbols(definition,[sP301])]) ).
thf(sP302,plain,
( sP302
<=> ( singletoninpowerset
=> sP184 ) ),
introduced(definition,[new_symbols(definition,[sP302])]) ).
thf(sP303,plain,
( sP303
<=> ( complementImpComplementIntersect
=> sP225 ) ),
introduced(definition,[new_symbols(definition,[sP303])]) ).
thf(sP304,plain,
( sP304
<=> ( doubleComplementI1
=> sP269 ) ),
introduced(definition,[new_symbols(definition,[sP304])]) ).
thf(sP305,plain,
( sP305
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__2 )
=> ( in @ X1 @ ( binunion @ eigen__1 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP305])]) ).
thf(sP306,plain,
( sP306
<=> ( setminusIRneg
=> sP266 ) ),
introduced(definition,[new_symbols(definition,[sP306])]) ).
thf(sP307,plain,
( sP307
<=> ( lam2lamEq
=> ( beta2
=> sP98 ) ) ),
introduced(definition,[new_symbols(definition,[sP307])]) ).
thf(sP308,plain,
( sP308
<=> ( sP172
=> sP90 ) ),
introduced(definition,[new_symbols(definition,[sP308])]) ).
thf(sP309,plain,
( sP309
<=> ( inCongP
=> sP265 ) ),
introduced(definition,[new_symbols(definition,[sP309])]) ).
thf(sP310,plain,
( sP310
<=> ( subsetRefl
=> ( subsetTrans
=> sP163 ) ) ),
introduced(definition,[new_symbols(definition,[sP310])]) ).
thf(sP311,plain,
( sP311
<=> ( singletonsubset
=> sP302 ) ),
introduced(definition,[new_symbols(definition,[sP311])]) ).
thf(sP312,plain,
( sP312
<=> ( binunionTEcontra
=> sP141 ) ),
introduced(definition,[new_symbols(definition,[sP312])]) ).
thf(sP313,plain,
( sP313
<=> ( exuE1
=> sP121 ) ),
introduced(definition,[new_symbols(definition,[sP313])]) ).
thf(sP314,plain,
( sP314
<=> ( exu__Cong
=> ( emptyset__Cong
=> sP217 ) ) ),
introduced(definition,[new_symbols(definition,[sP314])]) ).
thf(sP315,plain,
( sP315
<=> ( exuI2
=> sP309 ) ),
introduced(definition,[new_symbols(definition,[sP315])]) ).
thf(sP316,plain,
( sP316
<=> binunionIL ),
introduced(definition,[new_symbols(definition,[sP316])]) ).
thf(sP317,plain,
( sP317
<=> ( subPowSU
=> sP190 ) ),
introduced(definition,[new_symbols(definition,[sP317])]) ).
thf(sP318,plain,
( sP318
<=> ( powersetI1
=> sP161 ) ),
introduced(definition,[new_symbols(definition,[sP318])]) ).
thf(sP319,plain,
( sP319
<=> ( ( breln1 @ eigen__0 @ eigen__2 )
=> sP147 ) ),
introduced(definition,[new_symbols(definition,[sP319])]) ).
thf(sP320,plain,
( sP320
<=> ( exuI1
=> ( exuI3
=> sP315 ) ) ),
introduced(definition,[new_symbols(definition,[sP320])]) ).
thf(sP321,plain,
( sP321
<=> ( setadjoinSub2
=> sP63 ) ),
introduced(definition,[new_symbols(definition,[sP321])]) ).
thf(sP322,plain,
( sP322
<=> ( setukpairinjR1
=> sP21 ) ),
introduced(definition,[new_symbols(definition,[sP322])]) ).
thf(sP323,plain,
( sP323
<=> ( binintersectSubset2
=> sP252 ) ),
introduced(definition,[new_symbols(definition,[sP323])]) ).
thf(sP324,plain,
( sP324
<=> ( breln1unionprop
=> sP219 ) ),
introduced(definition,[new_symbols(definition,[sP324])]) ).
thf(sP325,plain,
( sP325
<=> ! [X1: $i] :
( ( in @ X1 @ ( binunion @ eigen__2 @ eigen__1 ) )
= ( in @ X1 @ ( binunion @ eigen__1 @ eigen__2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP325])]) ).
thf(sP326,plain,
( sP326
<=> ( subsetE
=> sP62 ) ),
introduced(definition,[new_symbols(definition,[sP326])]) ).
thf(sP327,plain,
( sP327
<=> ( iftrueProp2
=> sP10 ) ),
introduced(definition,[new_symbols(definition,[sP327])]) ).
thf(sP328,plain,
( sP328
<=> ( setOfPairsIsBReln1
=> sP268 ) ),
introduced(definition,[new_symbols(definition,[sP328])]) ).
thf(sP329,plain,
( sP329
<=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( binunion @ eigen__2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP329])]) ).
thf(sP330,plain,
( sP330
<=> ( upairset2E
=> sP247 ) ),
introduced(definition,[new_symbols(definition,[sP330])]) ).
thf(sP331,plain,
( sP331
<=> ( subsetTrans
=> sP163 ) ),
introduced(definition,[new_symbols(definition,[sP331])]) ).
thf(sP332,plain,
( sP332
<=> ( emptyset__Cong
=> sP217 ) ),
introduced(definition,[new_symbols(definition,[sP332])]) ).
thf(sP333,plain,
( sP333
<=> ( symdiffE
=> sP264 ) ),
introduced(definition,[new_symbols(definition,[sP333])]) ).
thf(sP334,plain,
( sP334
<=> ( beta2
=> sP98 ) ),
introduced(definition,[new_symbols(definition,[sP334])]) ).
thf(sP335,plain,
( sP335
<=> ( contrasubsetT3
=> sP304 ) ),
introduced(definition,[new_symbols(definition,[sP335])]) ).
thf(sP336,plain,
( sP336
<=> ( doubleComplementEq
=> sP50 ) ),
introduced(definition,[new_symbols(definition,[sP336])]) ).
thf(sP337,plain,
( sP337
<=> ( sP288 = sP149 ) ),
introduced(definition,[new_symbols(definition,[sP337])]) ).
thf(sP338,plain,
( sP338
<=> ( dsetconstr__Cong
=> sP41 ) ),
introduced(definition,[new_symbols(definition,[sP338])]) ).
thf(sP339,plain,
( sP339
<=> ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( ( binunion @ X2 @ X3 )
= ( binunion @ X3 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP339])]) ).
thf(sP340,plain,
( sP340
<=> ( eqimpsubset1
=> sP26 ) ),
introduced(definition,[new_symbols(definition,[sP340])]) ).
thf(sP341,plain,
( sP341
<=> ( theeq
=> sP231 ) ),
introduced(definition,[new_symbols(definition,[sP341])]) ).
thf(sP342,plain,
( sP342
<=> ( complementTE1
=> sP73 ) ),
introduced(definition,[new_symbols(definition,[sP342])]) ).
thf(sP343,plain,
( sP343
<=> ( woz1_1
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP343])]) ).
thf(sP344,plain,
( sP344
<=> ( breln1compE
=> sP109 ) ),
introduced(definition,[new_symbols(definition,[sP344])]) ).
thf(sP345,plain,
( sP345
<=> ( doubleComplementSub2
=> sP336 ) ),
introduced(definition,[new_symbols(definition,[sP345])]) ).
thf(sP346,plain,
( sP346
<=> ( bs114d
=> sP209 ) ),
introduced(definition,[new_symbols(definition,[sP346])]) ).
thf(sP347,plain,
( sP347
<=> ( setminusSubset1
=> sP333 ) ),
introduced(definition,[new_symbols(definition,[sP347])]) ).
thf(sP348,plain,
( sP348
<=> ( sP32 = sP171 ) ),
introduced(definition,[new_symbols(definition,[sP348])]) ).
thf(sP349,plain,
( sP349
<=> ( breln1unionIR
=> sP54 ) ),
introduced(definition,[new_symbols(definition,[sP349])]) ).
thf(sP350,plain,
( sP350
<=> ( intersectInPowersetIntersectUnions
=> sP145 ) ),
introduced(definition,[new_symbols(definition,[sP350])]) ).
thf(sP351,plain,
( sP351
<=> ( exuI3
=> sP315 ) ),
introduced(definition,[new_symbols(definition,[sP351])]) ).
thf(sP352,plain,
( sP352
<=> ( upairinpowunion
=> sP179 ) ),
introduced(definition,[new_symbols(definition,[sP352])]) ).
thf(sP353,plain,
( sP353
<=> ( iffalseProp1
=> sP34 ) ),
introduced(definition,[new_symbols(definition,[sP353])]) ).
thf(sP354,plain,
( sP354
<=> ! [X1: $i] :
( ! [X2: $i] :
( ( in @ X2 @ ( binunion @ eigen__2 @ eigen__1 ) )
= ( in @ X2 @ X1 ) )
=> ( ( binunion @ eigen__2 @ eigen__1 )
= X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP354])]) ).
thf(def_setextAx,definition,
setextAx = sP166 ).
thf(def_emptysetAx,definition,
( emptysetAx
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_setadjoinAx,definition,
( setadjoinAx
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_powersetAx,definition,
( powersetAx
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_setunionAx,definition,
( setunionAx
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_omega0Ax,definition,
( omega0Ax
= ( in @ emptyset @ omega ) ) ).
thf(def_omegaSAx,definition,
( omegaSAx
= ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ) ) ) ).
thf(def_omegaIndAx,definition,
( omegaIndAx
= ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) ) ) ) ).
thf(def_replAx,definition,
( replAx
= ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( exu @ ( X1 @ X3 ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) ) ) ) ).
thf(def_foundationAx,definition,
( foundationAx
= ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_wellorderingAx,definition,
( wellorderingAx
= ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) ) ) ) ).
thf(def_descrp,definition,
( descrp
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ( X1 @ ( descr @ X1 ) ) ) ) ) ).
thf(def_dsetconstrI,definition,
( dsetconstrI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_dsetconstrEL,definition,
( dsetconstrEL
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ) ).
thf(def_dsetconstrER,definition,
( dsetconstrER
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) ) ) ) ).
thf(def_exuE1,definition,
( exuE1
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ).
thf(def_prop2setE,definition,
( prop2setE
= ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 ) ) ) ).
thf(def_emptysetE,definition,
( emptysetE
= ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 ) ) ) ).
thf(def_emptysetimpfalse,definition,
( emptysetimpfalse
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_notinemptyset,definition,
( notinemptyset
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_exuE3e,definition,
( exuE3e
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) ) ).
thf(def_setext,definition,
( setext
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) ) ) ) ).
thf(def_emptyI,definition,
( emptyI
= ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_noeltsimpempty,definition,
( noeltsimpempty
= ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_setbeta,definition,
( setbeta
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) ) ) ) ).
thf(def_nonemptyE1,definition,
( nonemptyE1
= ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ) ) ) ).
thf(def_nonemptyI,definition,
( nonemptyI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( nonempty @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_nonemptyI1,definition,
( nonemptyI1
= ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( nonempty @ X1 ) ) ) ) ).
thf(def_setadjoinIL,definition,
( setadjoinIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ) ).
thf(def_emptyinunitempty,definition,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
thf(def_setadjoinIR,definition,
( setadjoinIR
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) ).
thf(def_setadjoinE,definition,
( setadjoinE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ) ) ) ).
thf(def_setadjoinOr,definition,
( setadjoinOr
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_setoftrueEq,definition,
( setoftrueEq
= ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 ) ) ) ).
thf(def_powersetI,definition,
( powersetI
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_emptyinPowerset,definition,
( emptyinPowerset
= ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ).
thf(def_emptyInPowerset,definition,
( emptyInPowerset
= ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ).
thf(def_powersetE,definition,
( powersetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_setunionI,definition,
( setunionI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) ) ) ) ).
thf(def_setunionE,definition,
( setunionE
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) ) ) ) ).
thf(def_subPowSU,definition,
( subPowSU
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ) ).
thf(def_exuE2,definition,
( exuE2
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) ) ) ) ).
thf(def_nonemptyImpWitness,definition,
( nonemptyImpWitness
= ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ) ) ) ).
thf(def_uniqinunit,definition,
( uniqinunit
= ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_notinsingleton,definition,
( notinsingleton
= ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_eqinunit,definition,
( eqinunit
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_singletonsswitch,definition,
( singletonsswitch
= ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_upairsetE,definition,
( upairsetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) ) ) ) ).
thf(def_upairsetIL,definition,
( upairsetIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_upairsetIR,definition,
( upairsetIR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_emptyE1,definition,
( emptyE1
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) ) ) ).
thf(def_vacuousDall,definition,
( vacuousDall
= ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) ) ) ) ).
thf(def_quantDeMorgan1,definition,
( quantDeMorgan1
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_quantDeMorgan2,definition,
( quantDeMorgan2
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_quantDeMorgan3,definition,
( quantDeMorgan3
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_quantDeMorgan4,definition,
( quantDeMorgan4
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_prop2setI,definition,
( prop2setI
= ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) ) ) ) ).
thf(def_prop2set2propI,definition,
( prop2set2propI
= ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) ) ) ) ).
thf(def_notdexE,definition,
( notdexE
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_notdallE,definition,
( notdallE
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_exuI1,definition,
( exuI1
= ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) ) ) ).
thf(def_exuI3,definition,
( exuI3
= ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) ) ) ) ).
thf(def_exuI2,definition,
( exuI2
= ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ( exu @ X1 ) ) ) ) ).
thf(def_inCongP,definition,
( inCongP
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X4 @ X2 ) ) ) ) ) ) ).
thf(def_in__Cong,definition,
( in__Cong
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( in @ X3 @ X1 )
= ( in @ X4 @ X2 ) ) ) ) ) ) ).
thf(def_exuE3u,definition,
( exuE3u
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ).
thf(def_exu__Cong,definition,
( exu__Cong
= ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( exu @ X1 )
= ( exu @ X2 ) ) ) ) ) ).
thf(def_emptyset__Cong,definition,
( emptyset__Cong
= ( emptyset = emptyset ) ) ).
thf(def_setadjoin__Cong,definition,
( setadjoin__Cong
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( setadjoin @ X1 @ X3 )
= ( setadjoin @ X2 @ X4 ) ) ) ) ) ) ).
thf(def_powerset__Cong,definition,
( powerset__Cong
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( powerset @ X1 )
= ( powerset @ X2 ) ) ) ) ) ).
thf(def_setunion__Cong,definition,
( setunion__Cong
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setunion @ X1 )
= ( setunion @ X2 ) ) ) ) ) ).
thf(def_omega__Cong,definition,
( omega__Cong
= ( omega = omega ) ) ).
thf(def_exuEu,definition,
( exuEu
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ).
thf(def_descr__Cong,definition,
( descr__Cong
= ( ! [X1: $i > $o,X2: $i > $o] :
( ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( X1 @ X3 )
= ( X2 @ X4 ) ) )
=> ( ( exu @ X1 )
=> ( ( exu @ X2 )
=> ( ( descr @ X1 )
= ( descr @ X2 ) ) ) ) ) ) ) ).
thf(def_dsetconstr__Cong,definition,
( dsetconstr__Cong
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ! [X3: $i > $o,X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( X5 = X6 )
=> ( ( X3 @ X5 )
= ( X4 @ X6 ) ) ) ) )
=> ( ( dsetconstr @ X1 @ X3 )
= ( dsetconstr @ X2 @ X4 ) ) ) ) ) ) ).
thf(def_subsetI1,definition,
( subsetI1
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_eqimpsubset2,definition,
( eqimpsubset2
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X2 @ X1 ) ) ) ) ).
thf(def_eqimpsubset1,definition,
( eqimpsubset1
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_subsetI2,definition,
( subsetI2
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_emptysetsubset,definition,
( emptysetsubset
= ( !! @ ( subset @ emptyset ) ) ) ).
thf(def_subsetE,definition,
( subsetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_subsetE2,definition,
( subsetE2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_notsubsetI,definition,
( notsubsetI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( subset @ X1 @ X2 ) ) ) ) ) ).
thf(def_notequalI1,definition,
( notequalI1
= ( ! [X1: $i,X2: $i] :
( ~ ( subset @ X1 @ X2 )
=> ( X1 != X2 ) ) ) ) ).
thf(def_notequalI2,definition,
( notequalI2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( X1 != X2 ) ) ) ) ) ).
thf(def_subsetRefl,definition,
( subsetRefl
= ( ! [X1: $i] : ( subset @ X1 @ X1 ) ) ) ).
thf(def_subsetTrans,definition,
( subsetTrans
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ X1 @ X3 ) ) ) ) ) ).
thf(def_setadjoinSub,definition,
( setadjoinSub
= ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( setadjoin @ X1 @ X2 ) ) ) ) ).
thf(def_setadjoinSub2,definition,
( setadjoinSub2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X1 @ X3 )
=> ( subset @ X1 @ ( setadjoin @ X2 @ X3 ) ) ) ) ) ).
thf(def_subset2powerset,definition,
( subset2powerset
= ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( in @ X1 @ ( powerset @ X2 ) ) ) ) ) ).
thf(def_setextsub,definition,
( setextsub
= ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( X1 = X2 ) ) ) ) ) ).
thf(def_subsetemptysetimpeq,definition,
( subsetemptysetimpeq
= ( ! [X1: $i] :
( ( subset @ X1 @ emptyset )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_powersetI1,definition,
( powersetI1
= ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_powersetE1,definition,
( powersetE1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ X1 ) ) ) ) ).
thf(def_inPowerset,definition,
( inPowerset
= ( ! [X1: $i] : ( in @ X1 @ ( powerset @ X1 ) ) ) ) ).
thf(def_powersetsubset,definition,
( powersetsubset
= ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( subset @ ( powerset @ X1 ) @ ( powerset @ X2 ) ) ) ) ) ).
thf(def_sepInPowerset,definition,
( sepInPowerset
= ( ! [X1: $i,X2: $i > $o] : ( in @ ( dsetconstr @ X1 @ X2 ) @ ( powerset @ X1 ) ) ) ) ).
thf(def_sepSubset,definition,
( sepSubset
= ( ! [X1: $i,X2: $i > $o] : ( subset @ ( dsetconstr @ X1 @ X2 ) @ X1 ) ) ) ).
thf(def_binunionIL,definition,
binunionIL = sP185 ).
thf(def_upairset2IR,definition,
( upairset2IR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_binunionIR,definition,
binunionIR = sP149 ).
thf(def_binunionEcases,definition,
( binunionEcases
= ( ! [X1: $i,X2: $i,X3: $i,X4: $o] :
( ( in @ X3 @ ( binunion @ X1 @ X2 ) )
=> ( ( ( in @ X3 @ X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ) ) ) ).
thf(def_binunionE,definition,
binunionE = sP171 ).
thf(def_binunionLsub,definition,
( binunionLsub
= ( ! [X1: $i,X2: $i] : ( subset @ X1 @ ( binunion @ X1 @ X2 ) ) ) ) ).
thf(def_binunionRsub,definition,
( binunionRsub
= ( ! [X1: $i,X2: $i] : ( subset @ X2 @ ( binunion @ X1 @ X2 ) ) ) ) ).
thf(def_binintersectI,definition,
( binintersectI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_binintersectSubset5,definition,
( binintersectSubset5
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( subset @ X3 @ X1 )
=> ( ( subset @ X3 @ X2 )
=> ( subset @ X3 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_binintersectEL,definition,
( binintersectEL
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ) ).
thf(def_binintersectLsub,definition,
( binintersectLsub
= ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X1 ) ) ) ).
thf(def_binintersectSubset2,definition,
( binintersectSubset2
= ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( binintersect @ X1 @ X2 )
= X1 ) ) ) ) ).
thf(def_binintersectSubset3,definition,
( binintersectSubset3
= ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X2 )
=> ( subset @ X2 @ X1 ) ) ) ) ).
thf(def_binintersectER,definition,
( binintersectER
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) ) ) ) ).
thf(def_disjointsetsI1,definition,
( disjointsetsI1
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ X2 ) )
=> ( ( binintersect @ X1 @ X2 )
= emptyset ) ) ) ) ).
thf(def_binintersectRsub,definition,
( binintersectRsub
= ( ! [X1: $i,X2: $i] : ( subset @ ( binintersect @ X1 @ X2 ) @ X2 ) ) ) ).
thf(def_binintersectSubset4,definition,
( binintersectSubset4
= ( ! [X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( binintersect @ X1 @ X2 )
= X2 ) ) ) ) ).
thf(def_binintersectSubset1,definition,
( binintersectSubset1
= ( ! [X1: $i,X2: $i] :
( ( ( binintersect @ X1 @ X2 )
= X1 )
=> ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_bs114d,definition,
( bs114d
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( binintersect @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binunion @ ( binintersect @ X1 @ X2 ) @ ( binintersect @ X1 @ X3 ) ) ) ) ) ).
thf(def_setminusI,definition,
( setminusI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_setminusEL,definition,
( setminusEL
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ) ).
thf(def_setminusER,definition,
( setminusER
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X3 @ X2 ) ) ) ) ).
thf(def_setminusSubset2,definition,
( setminusSubset2
= ( ! [X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ( setminus @ X1 @ X2 )
= emptyset ) ) ) ) ).
thf(def_setminusERneg,definition,
( setminusERneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_setminusELneg,definition,
( setminusELneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_setminusILneg,definition,
( setminusILneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ).
thf(def_setminusIRneg,definition,
( setminusIRneg
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ).
thf(def_setminusLsub,definition,
( setminusLsub
= ( ! [X1: $i,X2: $i] : ( subset @ ( setminus @ X1 @ X2 ) @ X1 ) ) ) ).
thf(def_setminusSubset1,definition,
( setminusSubset1
= ( ! [X1: $i,X2: $i] :
( ( ( setminus @ X1 @ X2 )
= emptyset )
=> ( subset @ X1 @ X2 ) ) ) ) ).
thf(def_symdiffE,definition,
( symdiffE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( symdiff @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> X4 ) )
=> ( ( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> X4 ) )
=> X4 ) ) ) ) ) ).
thf(def_symdiffI1,definition,
( symdiffI1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_symdiffI2,definition,
( symdiffI2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_symdiffIneg1,definition,
( symdiffIneg1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_symdiffIneg2,definition,
( symdiffIneg2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ~ ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( symdiff @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_secondinupair,definition,
( secondinupair
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_setukpairIL,definition,
( setukpairIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf(def_setukpairIR,definition,
( setukpairIR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setunion @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf(def_kpairiskpair,definition,
( kpairiskpair
= ( ! [X1: $i,X2: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) ) ) ) ).
thf(def_kpairp,definition,
( kpairp
= ( ! [X1: $i,X2: $i] : ( iskpair @ ( kpair @ X1 @ X2 ) ) ) ) ).
thf(def_singletonsubset,definition,
( singletonsubset
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( subset @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) ) ).
thf(def_singletoninpowerset,definition,
( singletoninpowerset
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ ( setadjoin @ X2 @ emptyset ) @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_singletoninpowunion,definition,
( singletoninpowunion
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ ( setadjoin @ X3 @ emptyset ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_upairset2E,definition,
( upairset2E
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) ) ) ) ).
thf(def_upairsubunion,definition,
( upairsubunion
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_upairinpowunion,definition,
( upairinpowunion
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_ubforcartprodlem1,definition,
( ubforcartprodlem1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_ubforcartprodlem2,definition,
( ubforcartprodlem2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_ubforcartprodlem3,definition,
( ubforcartprodlem3
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( powerset @ ( powerset @ ( binunion @ X1 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_cartprodpairin,definition,
( cartprodpairin
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_cartprodmempair1,definition,
( cartprodmempair1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ~ ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( X3
!= ( kpair @ X4 @ X5 ) ) ) ) ) ) ) ).
thf(def_cartprodmempair,definition,
( cartprodmempair
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( iskpair @ X3 ) ) ) ) ).
thf(def_setunionE2,definition,
( setunionE2
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( in @ X2 @ X3 ) ) ) ) ) ).
thf(def_setunionsingleton1,definition,
( setunionsingleton1
= ( ! [X1: $i] : ( subset @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) @ X1 ) ) ) ).
thf(def_setunionsingleton2,definition,
( setunionsingleton2
= ( ! [X1: $i] : ( subset @ X1 @ ( setunion @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_setunionsingleton,definition,
( setunionsingleton
= ( ! [X1: $i] :
( ( setunion @ ( setadjoin @ X1 @ emptyset ) )
= X1 ) ) ) ).
thf(def_singletonprop,definition,
( singletonprop
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( singleton @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_ex1E1,definition,
( ex1E1
= ( ! [X1: $i,X2: $i > $o] :
( ( ex1 @ X1 @ X2 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_ex1I,definition,
( ex1I
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X4 )
=> ( X4 = X3 ) ) )
=> ( ex1 @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_ex1I2,definition,
( ex1I2
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) )
=> ( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ex1 @ X1 @ X2 ) ) ) ) ) ).
thf(def_singletonsuniq,definition,
( singletonsuniq
= ( ! [X1: $i,X2: $i] :
( ( ( setadjoin @ X1 @ emptyset )
= ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_setukpairinjL1,definition,
( setukpairinjL1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) ) ) ) ).
thf(def_kfstsingleton,definition,
( kfstsingleton
= ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X1 ) ) ) ) ) ) ).
thf(def_theprop,definition,
( theprop
= ( ! [X1: $i] :
( ( singleton @ X1 )
=> ( in @ ( setunion @ X1 ) @ X1 ) ) ) ) ).
thf(def_kfstpairEq,definition,
( kfstpairEq
= ( ! [X1: $i,X2: $i] :
( ( kfst @ ( kpair @ X1 @ X2 ) )
= X1 ) ) ) ).
thf(def_cartprodfstin,definition,
( cartprodfstin
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( kfst @ X3 ) @ X1 ) ) ) ) ).
thf(def_setukpairinjL2,definition,
( setukpairinjL2
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X1 = X3 ) ) ) ) ).
thf(def_setukpairinjL,definition,
( setukpairinjL
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X1 = X3 ) ) ) ) ).
thf(def_setukpairinjR11,definition,
( setukpairinjR11
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_setukpairinjR12,definition,
( setukpairinjR12
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ emptyset ) ) ) ) ) ).
thf(def_setukpairinjR1,definition,
( setukpairinjR1
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X3 = X4 )
=> ( X2 = X4 ) ) ) ) ) ).
thf(def_upairequniteq,definition,
( upairequniteq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) )
= ( setadjoin @ X3 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_setukpairinjR2,definition,
( setukpairinjR2
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( setadjoin @ ( setadjoin @ X1 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X3 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) @ emptyset ) ) )
=> ( X2 = X4 ) ) ) ) ).
thf(def_setukpairinjR,definition,
( setukpairinjR
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( ( kpair @ X1 @ X2 )
= ( kpair @ X3 @ X4 ) )
=> ( X2 = X4 ) ) ) ) ).
thf(def_ksndsingleton,definition,
( ksndsingleton
= ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X1 )
@ ^ [X2: $i] :
( X1
= ( kpair @ ( kfst @ X1 ) @ X2 ) ) ) ) ) ) ) ).
thf(def_ksndpairEq,definition,
( ksndpairEq
= ( ! [X1: $i,X2: $i] :
( ( ksnd @ ( kpair @ X1 @ X2 ) )
= X2 ) ) ) ).
thf(def_kpairsurjEq,definition,
( kpairsurjEq
= ( ! [X1: $i] :
( ( iskpair @ X1 )
=> ( ( kpair @ ( kfst @ X1 ) @ ( ksnd @ X1 ) )
= X1 ) ) ) ) ).
thf(def_cartprodsndin,definition,
( cartprodsndin
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( in @ ( ksnd @ X3 ) @ X2 ) ) ) ) ).
thf(def_cartprodpairmemEL,definition,
( cartprodpairmemEL
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ) ).
thf(def_cartprodpairmemER,definition,
( cartprodpairmemER
= ( ! [X1: $i,X2: $i,X3: $i,X4: $i] :
( ( in @ ( kpair @ X3 @ X4 ) @ ( cartprod @ X1 @ X2 ) )
=> ( in @ X4 @ X2 ) ) ) ) ).
thf(def_cartprodmempaircEq,definition,
( cartprodmempaircEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kpair @ X3 @ X4 )
= ( kpair @ X3 @ X4 ) ) ) ) ) ) ).
thf(def_cartprodfstpairEq,definition,
( cartprodfstpairEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( kfst @ ( kpair @ X3 @ X4 ) )
= X3 ) ) ) ) ) ).
thf(def_cartprodsndpairEq,definition,
( cartprodsndpairEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( ( ksnd @ ( kpair @ X3 @ X4 ) )
= X4 ) ) ) ) ) ).
thf(def_cartprodpairsurjEq,definition,
( cartprodpairsurjEq
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( cartprod @ X1 @ X2 ) )
=> ( ( kpair @ ( kfst @ X3 ) @ ( ksnd @ X3 ) )
= X3 ) ) ) ) ).
thf(def_dpsetconstrI,definition,
( dpsetconstrI
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( X3 @ X4 @ X5 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) ) ) ) ).
thf(def_dpsetconstrSub,definition,
( dpsetconstrSub
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( subset @ ( dpsetconstr @ X1 @ X2 @ X3 ) @ ( cartprod @ X1 @ X2 ) ) ) ) ).
thf(def_setOfPairsIsBReln,definition,
( setOfPairsIsBReln
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o] : ( breln @ X1 @ X2 @ ( dpsetconstr @ X1 @ X2 @ X3 ) ) ) ) ).
thf(def_dpsetconstrERa,definition,
( dpsetconstrERa
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) ) ) ) ).
thf(def_dpsetconstrEL1,definition,
( dpsetconstrEL1
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) ) ) ) ).
thf(def_dpsetconstrEL2,definition,
( dpsetconstrEL2
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( in @ X5 @ X2 ) ) ) ) ).
thf(def_dpsetconstrER,definition,
( dpsetconstrER
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o,X4: $i,X5: $i] :
( ( in @ ( kpair @ X4 @ X5 ) @ ( dpsetconstr @ X1 @ X2 @ X3 ) )
=> ( X3 @ X4 @ X5 ) ) ) ) ).
thf(def_funcImageSingleton,definition,
( funcImageSingleton
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) ) ) ).
thf(def_apProp,definition,
( apProp
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X2
@ ^ [X5: $i] : ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) )
@ X2 ) ) ) ) ) ).
thf(def_app,definition,
( app
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) ) ) ) ).
thf(def_infuncsetfunc,definition,
( infuncsetfunc
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( func @ X1 @ X2 @ X3 ) ) ) ) ).
thf(def_ap2p,definition,
( ap2p
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X1 @ X2 @ X3 @ X4 ) @ X2 ) ) ) ) ) ).
thf(def_funcinfuncset,definition,
( funcinfuncset
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( in @ X3 @ ( funcSet @ X1 @ X2 ) ) ) ) ) ).
thf(def_lamProp,definition,
( lamProp
= ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2
@ ( dpsetconstr @ X1 @ X2
@ ^ [X4: $i] : ( (=) @ ( X3 @ X4 ) ) ) ) ) ) ) ).
thf(def_lamp,definition,
( lamp
= ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( func @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) ) ) ) ) ).
thf(def_lam2p,definition,
( lam2p
= ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( in @ ( lam @ X1 @ X2 @ X3 ) @ ( funcSet @ X1 @ X2 ) ) ) ) ) ).
thf(def_brelnall1,definition,
( brelnall1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( breln @ X1 @ X2 @ X3 )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) ) ) ) ).
thf(def_brelnall2,definition,
( brelnall2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( breln @ X1 @ X2 @ X3 )
=> ! [X4: $i > $o] :
( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( X4 @ ( kpair @ X5 @ X6 ) ) ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( X4 @ X5 ) ) ) ) ) ) ).
thf(def_ex1E2,definition,
( ex1E2
= ( ! [X1: $i,X2: $i > $o] :
( ( ex1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( X2 @ X4 )
=> ( X3 = X4 ) ) ) ) ) ) ) ) ).
thf(def_funcGraphProp1,definition,
( funcGraphProp1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) ) ) ) ).
thf(def_funcGraphProp3,definition,
( funcGraphProp3
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( kpair @ X4 @ ( ap @ X1 @ X2 @ X3 @ X4 ) ) @ X3 ) ) ) ) ) ).
thf(def_funcGraphProp2,definition,
( funcGraphProp2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) ) ) ) ).
thf(def_funcextLem,definition,
( funcextLem
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_funcGraphProp4,definition,
( funcGraphProp4
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= X5 ) ) ) ) ) ) ) ).
thf(def_subbreln,definition,
( subbreln
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( breln @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( breln @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( subset @ X3 @ X4 ) ) ) ) ) ) ).
thf(def_eqbreln,definition,
( eqbreln
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( breln @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( breln @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X3 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X4 ) ) ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ( in @ ( kpair @ X5 @ X6 ) @ X4 )
=> ( in @ ( kpair @ X5 @ X6 ) @ X3 ) ) ) )
=> ( X3 = X4 ) ) ) ) ) ) ) ).
thf(def_funcext,definition,
( funcext
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( func @ X1 @ X2 @ X4 )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) ) ) ) ).
thf(def_funcext2,definition,
( funcext2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( funcSet @ X1 @ X2 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X5 )
= ( ap @ X1 @ X2 @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) ) ) ) ).
thf(def_ap2apEq1,definition,
( ap2apEq1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= ( ap @ X1 @ X2 @ X3 @ X4 ) ) ) ) ) ) ).
thf(def_ap2apEq2,definition,
( ap2apEq2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ X3 @ X4 )
= ( ap @ X1 @ X2 @ X3 @ X4 ) ) ) ) ) ) ).
thf(def_beta1,definition,
( beta1
= ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) ) ) ) ).
thf(def_eta1,definition,
( eta1
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( func @ X1 @ X2 @ X3 )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) ) ) ) ).
thf(def_lam2lamEq,definition,
( lam2lamEq
= ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ( ( lam @ X1 @ X2 @ X3 )
= ( lam @ X1 @ X2 @ X3 ) ) ) ) ) ).
thf(def_beta2,definition,
( beta2
= ( ! [X1: $i,X2: $i,X3: $i > $i] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( X3 @ X4 ) @ X2 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( ap @ X1 @ X2 @ ( lam @ X1 @ X2 @ X3 ) @ X4 )
= ( X3 @ X4 ) ) ) ) ) ) ).
thf(def_eta2,definition,
( eta2
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( funcSet @ X1 @ X2 ) )
=> ( ( lam @ X1 @ X2 @ ( ap @ X1 @ X2 @ X3 ) )
= X3 ) ) ) ) ).
thf(def_iffalseProp1,definition,
( iffalseProp1
= ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( in @ X4
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) ) ) ) ).
thf(def_iffalseProp2,definition,
( iffalseProp2
= ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ).
thf(def_iftrueProp1,definition,
( iftrueProp1
= ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) ) ) ) ).
thf(def_iftrueProp2,definition,
( iftrueProp2
= ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) )
= ( setadjoin @ X3 @ emptyset ) ) ) ) ) ) ) ).
thf(def_ifSingleton,definition,
( ifSingleton
= ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( singleton
@ ( dsetconstr @ X1
@ ^ [X5: $i] :
( ( X2
=> ( X5 != X3 ) )
=> ~ ( ~ X2
=> ( X5 != X4 ) ) ) ) ) ) ) ) ) ).
thf(def_ifp,definition,
( ifp
= ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ X1 ) ) ) ) ) ).
thf(def_theeq,definition,
( theeq
= ( ! [X1: $i] :
( ( singleton @ X1 )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ( setunion @ X1 )
= X2 ) ) ) ) ) ).
thf(def_iftrue,definition,
( iftrue
= ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( X2
=> ( ( if @ X1 @ X2 @ X3 @ X4 )
= X3 ) ) ) ) ) ) ).
thf(def_iffalse,definition,
( iffalse
= ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ X2
=> ( ( if @ X1 @ X2 @ X3 @ X4 )
= X4 ) ) ) ) ) ) ).
thf(def_iftrueorfalse,definition,
( iftrueorfalse
= ( ! [X1: $i,X2: $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( if @ X1 @ X2 @ X3 @ X4 ) @ ( setadjoin @ X3 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ) ) ) ).
thf(def_binintersectT_lem,definition,
( binintersectT_lem
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binintersect @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ).
thf(def_binunionT_lem,definition,
( binunionT_lem
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( binunion @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ).
thf(def_powersetT_lem,definition,
( powersetT_lem
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( in @ ( powerset @ X2 ) @ ( powerset @ ( powerset @ X1 ) ) ) ) ) ) ).
thf(def_setminusT_lem,definition,
( setminusT_lem
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( setminus @ X2 @ X3 ) @ ( powerset @ X1 ) ) ) ) ) ) ).
thf(def_complementT_lem,definition,
( complementT_lem
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( in @ ( setminus @ X1 @ X2 ) @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_setextT,definition,
( setextT
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ X2 )
=> ( in @ X4 @ X3 ) ) )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ X3 )
=> ( in @ X4 @ X2 ) ) )
=> ( X2 = X3 ) ) ) ) ) ) ) ).
thf(def_subsetTI,definition,
( subsetTI
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ X2 )
=> ( in @ X4 @ X3 ) ) )
=> ( subset @ X2 @ X3 ) ) ) ) ) ) ).
thf(def_powersetTI1,definition,
( powersetTI1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ X2 )
=> ( in @ X4 @ X3 ) ) )
=> ( in @ X2 @ ( powerset @ X3 ) ) ) ) ) ) ) ).
thf(def_powersetTE1,definition,
( powersetTE1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X2 @ ( powerset @ X3 ) )
=> ( ( in @ X4 @ X2 )
=> ( in @ X4 @ X3 ) ) ) ) ) ) ) ) ).
thf(def_complementTI1,definition,
( complementTI1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_complementTE1,definition,
( complementTE1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ~ ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ X2 ) ) ) ) ) ) ).
thf(def_binintersectTELcontra,definition,
( binintersectTELcontra
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ ( in @ X4 @ X2 )
=> ~ ( in @ X4 @ ( binintersect @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(def_binintersectTERcontra,definition,
( binintersectTERcontra
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ ( in @ X4 @ X3 )
=> ~ ( in @ X4 @ ( binintersect @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(def_contrasubsetT,definition,
( contrasubsetT
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( subset @ X2 @ ( setminus @ X1 @ X3 ) )
=> ( ( in @ X4 @ X3 )
=> ~ ( in @ X4 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_contrasubsetT1,definition,
( contrasubsetT1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( subset @ X2 @ X3 )
=> ( ~ ( in @ X4 @ X3 )
=> ~ ( in @ X4 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_contrasubsetT2,definition,
( contrasubsetT2
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ( subset @ X2 @ X3 )
=> ( subset @ ( setminus @ X1 @ X3 ) @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_contrasubsetT3,definition,
( contrasubsetT3
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ( subset @ ( setminus @ X1 @ X3 ) @ ( setminus @ X1 @ X2 ) )
=> ( subset @ X2 @ X3 ) ) ) ) ) ) ).
thf(def_doubleComplementI1,definition,
( doubleComplementI1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setminus @ X1 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_doubleComplementE1,definition,
( doubleComplementE1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( setminus @ X1 @ ( setminus @ X1 @ X2 ) ) )
=> ( in @ X3 @ X2 ) ) ) ) ) ) ).
thf(def_doubleComplementSub1,definition,
( doubleComplementSub1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ X2 @ ( setminus @ X1 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_doubleComplementSub2,definition,
( doubleComplementSub2
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( subset @ ( setminus @ X1 @ ( setminus @ X1 @ X2 ) ) @ X2 ) ) ) ) ).
thf(def_doubleComplementEq,definition,
( doubleComplementEq
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( X2
= ( setminus @ X1 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_complementTnotintersectT,definition,
( complementTnotintersectT
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( setminus @ X1 @ X2 ) )
=> ~ ( in @ X4 @ ( binintersect @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(def_complementImpComplementIntersect,definition,
( complementImpComplementIntersect
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X4 @ ( setminus @ X1 @ ( binintersect @ X2 @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_complementSubsetComplementIntersect,definition,
( complementSubsetComplementIntersect
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( subset @ ( setminus @ X1 @ X2 ) @ ( setminus @ X1 @ ( binintersect @ X2 @ X3 ) ) ) ) ) ) ) ).
thf(def_complementInPowersetComplementIntersect,definition,
( complementInPowersetComplementIntersect
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( setminus @ X1 @ X2 ) @ ( powerset @ ( setminus @ X1 @ ( binintersect @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(def_contraSubsetComplement,definition,
( contraSubsetComplement
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ( subset @ X2 @ ( setminus @ X1 @ X3 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ X3 )
=> ( in @ X4 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ) ) ).
thf(def_complementTcontraSubset,definition,
( complementTcontraSubset
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ( subset @ X2 @ ( setminus @ X1 @ X3 ) )
=> ( subset @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_binunionTILcontra,definition,
( binunionTILcontra
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ ( in @ X4 @ ( binunion @ X2 @ X3 ) )
=> ~ ( in @ X4 @ X2 ) ) ) ) ) ) ) ).
thf(def_binunionTIRcontra,definition,
( binunionTIRcontra
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ ( in @ X4 @ ( binunion @ X2 @ X3 ) )
=> ~ ( in @ X4 @ X3 ) ) ) ) ) ) ) ).
thf(def_inIntersectImpInUnion,definition,
( inIntersectImpInUnion
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ X5 @ ( binintersect @ X2 @ X3 ) )
=> ( in @ X5 @ ( binunion @ X2 @ X4 ) ) ) ) ) ) ) ) ) ).
thf(def_inIntersectImpInUnion2,definition,
( inIntersectImpInUnion2
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ X5 @ ( binintersect @ X2 @ X3 ) )
=> ( in @ X5 @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ) ) ) ).
thf(def_inIntersectImpInIntersectUnions,definition,
( inIntersectImpInIntersectUnions
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ X5 @ ( binintersect @ X2 @ X3 ) )
=> ( in @ X5 @ ( binintersect @ ( binunion @ X2 @ X4 ) @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ) ) ) ) ).
thf(def_intersectInPowersetIntersectUnions,definition,
( intersectInPowersetIntersectUnions
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ( in @ ( binintersect @ X2 @ X3 ) @ ( powerset @ ( binintersect @ ( binunion @ X2 @ X4 ) @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ) ) ) ).
thf(def_inComplementUnionImpNotIn1,definition,
( inComplementUnionImpNotIn1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( setminus @ X1 @ ( binunion @ X2 @ X3 ) ) )
=> ~ ( in @ X4 @ X2 ) ) ) ) ) ) ) ).
thf(def_inComplementUnionImpInComplement1,definition,
( inComplementUnionImpInComplement1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( setminus @ X1 @ ( binunion @ X2 @ X3 ) ) )
=> ( in @ X4 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_binunionTE,definition,
( binunionTE
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $o,X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ X5 @ ( binunion @ X2 @ X3 ) )
=> ( ( ( in @ X5 @ X2 )
=> X4 )
=> ( ( ( in @ X5 @ X3 )
=> X4 )
=> X4 ) ) ) ) ) ) ) ) ).
thf(def_binunionTEcontra,definition,
( binunionTEcontra
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ~ ( in @ X4 @ X2 )
=> ( ~ ( in @ X4 @ X3 )
=> ~ ( in @ X4 @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_demorgan2a1,definition,
( demorgan2a1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( setminus @ X1 @ ( binunion @ X2 @ X3 ) ) )
=> ( in @ X4 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_complementUnionInPowersetComplement,definition,
( complementUnionInPowersetComplement
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( setminus @ X1 @ ( binunion @ X2 @ X3 ) ) @ ( powerset @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_demorgan2a2,definition,
( demorgan2a2
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( setminus @ X1 @ ( binunion @ X2 @ X3 ) ) )
=> ( in @ X4 @ ( setminus @ X1 @ X3 ) ) ) ) ) ) ) ) ).
thf(def_demorgan1a,definition,
( demorgan1a
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( setminus @ X1 @ ( binintersect @ X2 @ X3 ) ) )
=> ( in @ X4 @ ( binunion @ ( setminus @ X1 @ X2 ) @ ( setminus @ X1 @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_demorgan1b,definition,
( demorgan1b
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( binunion @ ( setminus @ X1 @ X2 ) @ ( setminus @ X1 @ X3 ) ) )
=> ( in @ X4 @ ( setminus @ X1 @ ( binintersect @ X2 @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_demorgan1,definition,
( demorgan1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ( setminus @ X1 @ ( binintersect @ X2 @ X3 ) )
= ( binunion @ ( setminus @ X1 @ X2 ) @ ( setminus @ X1 @ X3 ) ) ) ) ) ) ) ).
thf(def_demorgan2a,definition,
( demorgan2a
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( setminus @ X1 @ ( binunion @ X2 @ X3 ) ) )
=> ( in @ X4 @ ( binintersect @ ( setminus @ X1 @ X2 ) @ ( setminus @ X1 @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_demorgan2b2,definition,
( demorgan2b2
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X4 @ ( setminus @ X1 @ X3 ) )
=> ( in @ X4 @ ( setminus @ X1 @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ).
thf(def_demorgan2b,definition,
( demorgan2b
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ X4 @ ( binintersect @ ( setminus @ X1 @ X2 ) @ ( setminus @ X1 @ X3 ) ) )
=> ( in @ X4 @ ( setminus @ X1 @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_demorgan2,definition,
( demorgan2
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ( setminus @ X1 @ ( binunion @ X2 @ X3 ) )
= ( binintersect @ ( setminus @ X1 @ X2 ) @ ( setminus @ X1 @ X3 ) ) ) ) ) ) ) ).
thf(def_woz13rule0,definition,
( woz13rule0
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( binintersect @ X2 @ X3 ) )
=> ( in @ X4 @ X1 ) ) ) ) ) ) ).
thf(def_woz13rule1,definition,
( woz13rule1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ( ( subset @ X2 @ X4 )
=> ( subset @ ( binintersect @ X2 @ X3 ) @ X4 ) ) ) ) ) ) ) ).
thf(def_woz13rule2,definition,
( woz13rule2
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ( ( subset @ X3 @ X4 )
=> ( subset @ ( binintersect @ X2 @ X3 ) @ X4 ) ) ) ) ) ) ) ).
thf(def_woz13rule3,definition,
( woz13rule3
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ( ( subset @ X2 @ X3 )
=> ( ( subset @ X2 @ X4 )
=> ( subset @ X2 @ ( binintersect @ X3 @ X4 ) ) ) ) ) ) ) ) ) ).
thf(def_woz13rule4,definition,
( woz13rule4
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X1 ) )
=> ( ( subset @ X2 @ X4 )
=> ( ( subset @ X3 @ X5 )
=> ( subset @ ( binintersect @ X2 @ X3 ) @ ( binintersect @ X4 @ X5 ) ) ) ) ) ) ) ) ) ) ).
thf(def_woz1_1,definition,
( woz1_1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( setminus @ X1 @ X2 ) @ ( powerset @ ( setminus @ X1 @ ( binintersect @ X2 @ X3 ) ) ) ) ) ) ) ) ).
thf(def_woz1_2,definition,
( woz1_2
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ! [X5: $i] :
( ( in @ X5 @ ( powerset @ X1 ) )
=> ( ( setminus @ X1 @ ( binintersect @ ( binunion @ X2 @ X3 ) @ ( binunion @ X4 @ X5 ) ) )
= ( binunion @ ( binintersect @ ( setminus @ X1 @ X2 ) @ ( setminus @ X1 @ X3 ) ) @ ( binintersect @ ( setminus @ X1 @ X4 ) @ ( setminus @ X1 @ X5 ) ) ) ) ) ) ) ) ) ) ).
thf(def_woz1_3,definition,
( woz1_3
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( powerset @ X1 ) )
=> ( in @ ( binintersect @ X2 @ X3 ) @ ( powerset @ ( binintersect @ ( binunion @ X2 @ X4 ) @ ( binunion @ X3 @ X4 ) ) ) ) ) ) ) ) ) ).
thf(def_woz1_4,definition,
( woz1_4
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ( subset @ X2 @ ( setminus @ X1 @ X3 ) )
=> ( subset @ X3 @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_woz1_5,definition,
( woz1_5
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( in @ ( setminus @ X1 @ ( binunion @ X2 @ X3 ) ) @ ( powerset @ ( setminus @ X1 @ X2 ) ) ) ) ) ) ) ).
thf(def_breln1all2,definition,
( breln1all2
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i > $o] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X2 )
=> ( X3 @ ( kpair @ X4 @ X5 ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( X3 @ X4 ) ) ) ) ) ) ).
thf(def_breln1SetBreln1,definition,
( breln1SetBreln1
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( breln1Set @ X1 ) )
=> ( breln1 @ X1 @ X2 ) ) ) ) ).
thf(def_choice2fnsingleton,definition,
( choice2fnsingleton
= ( ! [X1: $i,X2: $i,X3: $i > $i > $o] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X3 @ X4 @ X5 ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ ( breln1Set @ X2 ) )
=> ( ( reflwellordering @ X2 @ X4 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( singleton
@ ( dsetconstr @ X2
@ ^ [X6: $i] :
~ ( ( X3 @ X5 @ X6 )
=> ~ ! [X7: $i] :
( ( in @ X7 @ X2 )
=> ( ( X3 @ X5 @ X7 )
=> ( in @ ( kpair @ X6 @ X7 ) @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(def_setOfPairsIsBReln1,definition,
( setOfPairsIsBReln1
= ( ! [X1: $i,X2: $i > $i > $o] : ( breln1 @ X1 @ ( dpsetconstr @ X1 @ X1 @ X2 ) ) ) ) ).
thf(def_breln1all1,definition,
( breln1all1
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i > $o] :
( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X2 )
=> ( X3 @ ( kpair @ X4 @ X5 ) ) ) ) )
=> ! [X4: $i] :
( ( in @ X4 @ X2 )
=> ( X3 @ X4 ) ) ) ) ) ) ).
thf(def_subbreln1,definition,
( subbreln1
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X2 )
=> ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ( subset @ X2 @ X3 ) ) ) ) ) ) ).
thf(def_eqbreln1,definition,
( eqbreln1
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X2 )
=> ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) )
=> ( ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( in @ ( kpair @ X4 @ X5 ) @ X2 ) ) ) )
=> ( X2 = X3 ) ) ) ) ) ) ) ).
thf(def_breln1invprop,definition,
( breln1invprop
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ( breln1 @ X1 @ ( breln1invset @ X1 @ X2 ) ) ) ) ) ).
thf(def_breln1invI,definition,
( breln1invI
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ ( kpair @ X3 @ X4 ) @ X2 )
=> ( in @ ( kpair @ X4 @ X3 ) @ ( breln1invset @ X1 @ X2 ) ) ) ) ) ) ) ) ).
thf(def_breln1invE,definition,
( breln1invE
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X3 ) @ ( breln1invset @ X1 @ X2 ) )
=> ( in @ ( kpair @ X3 @ X4 ) @ X2 ) ) ) ) ) ) ) ).
thf(def_breln1compprop,definition,
( breln1compprop
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( breln1 @ X1 @ ( breln1compset @ X1 @ X2 @ X3 ) ) ) ) ) ) ).
thf(def_breln1compI,definition,
( breln1compI
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ! [X6: $i] :
( ( in @ X6 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X6 ) @ X2 )
=> ( ( in @ ( kpair @ X6 @ X5 ) @ X3 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( breln1compset @ X1 @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ) ).
thf(def_breln1compE,definition,
( breln1compE
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( breln1compset @ X1 @ X2 @ X3 ) )
=> ~ ! [X6: $i] :
( ~ ( ( in @ X6 @ X1 )
=> ~ ( in @ ( kpair @ X4 @ X6 ) @ X2 ) )
=> ~ ( in @ ( kpair @ X6 @ X5 ) @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_breln1compEex,definition,
( breln1compEex
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( breln1compset @ X1 @ X2 @ X3 ) )
=> ! [X6: $o] :
( ! [X7: $i] :
( ( in @ X7 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X7 ) @ X2 )
=> ( ( in @ ( kpair @ X7 @ X5 ) @ X3 )
=> X6 ) ) )
=> X6 ) ) ) ) ) ) ) ) ).
thf(def_breln1unionprop,definition,
( breln1unionprop
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ( breln1 @ X1 @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ).
thf(def_breln1unionIL,definition,
( breln1unionIL
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X2 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_breln1unionIR,definition,
( breln1unionIR
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_breln1unionI,definition,
( breln1unionI
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( ~ ( in @ ( kpair @ X4 @ X5 ) @ X2 )
=> ( in @ ( kpair @ X4 @ X5 ) @ X3 ) )
=> ( in @ ( kpair @ X4 @ X5 ) @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_breln1unionE,definition,
( breln1unionE
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( binunion @ X2 @ X3 ) )
=> ( ~ ( in @ ( kpair @ X4 @ X5 ) @ X2 )
=> ( in @ ( kpair @ X4 @ X5 ) @ X3 ) ) ) ) ) ) ) ) ) ).
thf(def_breln1unionEcases,definition,
( breln1unionEcases
= ( ! [X1: $i,X2: $i] :
( ( breln1 @ X1 @ X2 )
=> ! [X3: $i] :
( ( breln1 @ X1 @ X3 )
=> ! [X4: $i] :
( ( in @ X4 @ X1 )
=> ! [X5: $i] :
( ( in @ X5 @ X1 )
=> ( ( in @ ( kpair @ X4 @ X5 ) @ ( binunion @ X2 @ X3 ) )
=> ! [X6: $o] :
( ( ( in @ ( kpair @ X4 @ X5 ) @ X2 )
=> X6 )
=> ( ( ( in @ ( kpair @ X4 @ X5 ) @ X3 )
=> X6 )
=> X6 ) ) ) ) ) ) ) ) ) ).
thf(breln1unionCommutes,conjecture,
sP298 ).
thf(h1,negated_conjecture,
~ sP298,
inference(assume_negation,[status(cth)],[breln1unionCommutes]) ).
thf(1,plain,
( ~ sP235
| sP45 ),
inference(all_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP45
| ~ sP90
| sP132 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP132
| sP4
| sP172 ),
inference(prop_rule,[status(thm)],[]) ).
thf(4,plain,
( ~ sP9
| sP257 ),
inference(all_rule,[status(thm)],[]) ).
thf(5,plain,
( ~ sP257
| ~ sP206
| sP211 ),
inference(prop_rule,[status(thm)],[]) ).
thf(6,plain,
( ~ sP211
| sP172
| sP4 ),
inference(prop_rule,[status(thm)],[]) ).
thf(7,plain,
( ~ sP171
| sP125 ),
inference(all_rule,[status(thm)],[]) ).
thf(8,plain,
( ~ sP125
| sP9 ),
inference(all_rule,[status(thm)],[]) ).
thf(9,plain,
( ~ sP171
| sP186 ),
inference(all_rule,[status(thm)],[]) ).
thf(10,plain,
( ~ sP186
| sP235 ),
inference(all_rule,[status(thm)],[]) ).
thf(11,plain,
( ~ sP57
| sP308 ),
inference(all_rule,[status(thm)],[]) ).
thf(12,plain,
( ~ sP308
| ~ sP172
| sP90 ),
inference(prop_rule,[status(thm)],[]) ).
thf(13,plain,
( ~ sP305
| sP65 ),
inference(all_rule,[status(thm)],[]) ).
thf(14,plain,
( ~ sP65
| ~ sP4
| sP206 ),
inference(prop_rule,[status(thm)],[]) ).
thf(15,plain,
( ~ sP149
| sP137 ),
inference(all_rule,[status(thm)],[]) ).
thf(16,plain,
( ~ sP137
| sP305 ),
inference(all_rule,[status(thm)],[]) ).
thf(17,plain,
( ~ sP149
| sP329 ),
inference(all_rule,[status(thm)],[]) ).
thf(18,plain,
( ~ sP329
| sP57 ),
inference(all_rule,[status(thm)],[]) ).
thf(19,plain,
( ~ sP295
| sP134 ),
inference(all_rule,[status(thm)],[]) ).
thf(20,plain,
( ~ sP134
| ~ sP4
| sP90 ),
inference(prop_rule,[status(thm)],[]) ).
thf(21,plain,
( ~ sP232
| sP60 ),
inference(all_rule,[status(thm)],[]) ).
thf(22,plain,
( ~ sP60
| ~ sP172
| sP206 ),
inference(prop_rule,[status(thm)],[]) ).
thf(23,plain,
( ~ sP185
| sP153 ),
inference(all_rule,[status(thm)],[]) ).
thf(24,plain,
( ~ sP153
| sP232 ),
inference(all_rule,[status(thm)],[]) ).
thf(25,plain,
( ~ sP185
| sP99 ),
inference(all_rule,[status(thm)],[]) ).
thf(26,plain,
( ~ sP99
| sP295 ),
inference(all_rule,[status(thm)],[]) ).
thf(27,plain,
( ~ sP348
| ~ sP32
| sP171 ),
inference(prop_rule,[status(thm)],[]) ).
thf(28,plain,
( ~ sP337
| ~ sP288
| sP149 ),
inference(prop_rule,[status(thm)],[]) ).
thf(29,plain,
( ~ sP282
| ~ sP316
| sP185 ),
inference(prop_rule,[status(thm)],[]) ).
thf(30,plain,
( sP30
| ~ sP90
| ~ sP206 ),
inference(prop_rule,[status(thm)],[]) ).
thf(31,plain,
( sP30
| sP90
| sP206 ),
inference(prop_rule,[status(thm)],[]) ).
thf(32,plain,
( sP325
| ~ sP30 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__22]) ).
thf(33,plain,
( ~ sP124
| ~ sP325
| sP46 ),
inference(prop_rule,[status(thm)],[]) ).
thf(34,plain,
( ~ sP354
| sP124 ),
inference(all_rule,[status(thm)],[]) ).
thf(35,plain,
( ~ sP166
| sP354 ),
inference(all_rule,[status(thm)],[]) ).
thf(36,plain,
( ~ sP3
| ~ sP284
| sP166 ),
inference(prop_rule,[status(thm)],[]) ).
thf(37,plain,
( ~ sP242
| ~ sP46
| sP147 ),
inference(prop_rule,[status(thm)],[]) ).
thf(38,plain,
( ~ sP159
| sP242 ),
inference(all_rule,[status(thm)],[]) ).
thf(39,plain,
( ~ sP117
| sP159 ),
inference(all_rule,[status(thm)],[]) ).
thf(40,plain,
sP117,
inference(eq_sym,[status(thm)],[]) ).
thf(41,plain,
( sP319
| ~ sP147 ),
inference(prop_rule,[status(thm)],[]) ).
thf(42,plain,
( sP113
| ~ sP319 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(43,plain,
( sP70
| ~ sP113 ),
inference(prop_rule,[status(thm)],[]) ).
thf(44,plain,
( sP142
| ~ sP70 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(45,plain,
( sP339
| ~ sP142 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0]) ).
thf(46,plain,
( sP244
| ~ sP339 ),
inference(prop_rule,[status(thm)],[]) ).
thf(47,plain,
( sP16
| ~ sP244 ),
inference(prop_rule,[status(thm)],[]) ).
thf(48,plain,
( sP54
| ~ sP16 ),
inference(prop_rule,[status(thm)],[]) ).
thf(49,plain,
( sP349
| ~ sP54 ),
inference(prop_rule,[status(thm)],[]) ).
thf(50,plain,
( sP219
| ~ sP349 ),
inference(prop_rule,[status(thm)],[]) ).
thf(51,plain,
( sP324
| ~ sP219 ),
inference(prop_rule,[status(thm)],[]) ).
thf(52,plain,
( sP109
| ~ sP324 ),
inference(prop_rule,[status(thm)],[]) ).
thf(53,plain,
( sP344
| ~ sP109 ),
inference(prop_rule,[status(thm)],[]) ).
thf(54,plain,
( sP23
| ~ sP344 ),
inference(prop_rule,[status(thm)],[]) ).
thf(55,plain,
( sP135
| ~ sP23 ),
inference(prop_rule,[status(thm)],[]) ).
thf(56,plain,
( sP15
| ~ sP135 ),
inference(prop_rule,[status(thm)],[]) ).
thf(57,plain,
( sP256
| ~ sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(58,plain,
( sP85
| ~ sP256 ),
inference(prop_rule,[status(thm)],[]) ).
thf(59,plain,
( sP202
| ~ sP85 ),
inference(prop_rule,[status(thm)],[]) ).
thf(60,plain,
( sP155
| ~ sP202 ),
inference(prop_rule,[status(thm)],[]) ).
thf(61,plain,
( sP268
| ~ sP155 ),
inference(prop_rule,[status(thm)],[]) ).
thf(62,plain,
( sP328
| ~ sP268 ),
inference(prop_rule,[status(thm)],[]) ).
thf(63,plain,
( sP170
| ~ sP328 ),
inference(prop_rule,[status(thm)],[]) ).
thf(64,plain,
( sP276
| ~ sP170 ),
inference(prop_rule,[status(thm)],[]) ).
thf(65,plain,
( sP31
| ~ sP276 ),
inference(prop_rule,[status(thm)],[]) ).
thf(66,plain,
( sP279
| ~ sP31 ),
inference(prop_rule,[status(thm)],[]) ).
thf(67,plain,
( sP152
| ~ sP279 ),
inference(prop_rule,[status(thm)],[]) ).
thf(68,plain,
( sP44
| ~ sP152 ),
inference(prop_rule,[status(thm)],[]) ).
thf(69,plain,
( sP2
| ~ sP44 ),
inference(prop_rule,[status(thm)],[]) ).
thf(70,plain,
( sP343
| ~ sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(71,plain,
( sP278
| ~ sP343 ),
inference(prop_rule,[status(thm)],[]) ).
thf(72,plain,
( sP40
| ~ sP278 ),
inference(prop_rule,[status(thm)],[]) ).
thf(73,plain,
( sP140
| ~ sP40 ),
inference(prop_rule,[status(thm)],[]) ).
thf(74,plain,
( sP228
| ~ sP140 ),
inference(prop_rule,[status(thm)],[]) ).
thf(75,plain,
( sP200
| ~ sP228 ),
inference(prop_rule,[status(thm)],[]) ).
thf(76,plain,
( sP37
| ~ sP200 ),
inference(prop_rule,[status(thm)],[]) ).
thf(77,plain,
( sP130
| ~ sP37 ),
inference(prop_rule,[status(thm)],[]) ).
thf(78,plain,
( sP112
| ~ sP130 ),
inference(prop_rule,[status(thm)],[]) ).
thf(79,plain,
( sP210
| ~ sP112 ),
inference(prop_rule,[status(thm)],[]) ).
thf(80,plain,
( sP293
| ~ sP210 ),
inference(prop_rule,[status(thm)],[]) ).
thf(81,plain,
( sP205
| ~ sP293 ),
inference(prop_rule,[status(thm)],[]) ).
thf(82,plain,
( sP108
| ~ sP205 ),
inference(prop_rule,[status(thm)],[]) ).
thf(83,plain,
( sP214
| ~ sP108 ),
inference(prop_rule,[status(thm)],[]) ).
thf(84,plain,
( sP128
| ~ sP214 ),
inference(prop_rule,[status(thm)],[]) ).
thf(85,plain,
( sP141
| ~ sP128 ),
inference(prop_rule,[status(thm)],[]) ).
thf(86,plain,
( sP312
| ~ sP141 ),
inference(prop_rule,[status(thm)],[]) ).
thf(87,plain,
( sP218
| ~ sP312 ),
inference(prop_rule,[status(thm)],[]) ).
thf(88,plain,
( sP81
| ~ sP218 ),
inference(prop_rule,[status(thm)],[]) ).
thf(89,plain,
( sP145
| ~ sP81 ),
inference(prop_rule,[status(thm)],[]) ).
thf(90,plain,
( sP350
| ~ sP145 ),
inference(prop_rule,[status(thm)],[]) ).
thf(91,plain,
( sP93
| ~ sP350 ),
inference(prop_rule,[status(thm)],[]) ).
thf(92,plain,
( sP192
| ~ sP93 ),
inference(prop_rule,[status(thm)],[]) ).
thf(93,plain,
( sP194
| ~ sP192 ),
inference(prop_rule,[status(thm)],[]) ).
thf(94,plain,
( sP229
| ~ sP194 ),
inference(prop_rule,[status(thm)],[]) ).
thf(95,plain,
( sP222
| ~ sP229 ),
inference(prop_rule,[status(thm)],[]) ).
thf(96,plain,
( sP8
| ~ sP222 ),
inference(prop_rule,[status(thm)],[]) ).
thf(97,plain,
( sP74
| ~ sP8 ),
inference(prop_rule,[status(thm)],[]) ).
thf(98,plain,
( sP177
| ~ sP74 ),
inference(prop_rule,[status(thm)],[]) ).
thf(99,plain,
( sP225
| ~ sP177 ),
inference(prop_rule,[status(thm)],[]) ).
thf(100,plain,
( sP303
| ~ sP225 ),
inference(prop_rule,[status(thm)],[]) ).
thf(101,plain,
( sP50
| ~ sP303 ),
inference(prop_rule,[status(thm)],[]) ).
thf(102,plain,
( sP336
| ~ sP50 ),
inference(prop_rule,[status(thm)],[]) ).
thf(103,plain,
( sP345
| ~ sP336 ),
inference(prop_rule,[status(thm)],[]) ).
thf(104,plain,
( sP1
| ~ sP345 ),
inference(prop_rule,[status(thm)],[]) ).
thf(105,plain,
( sP269
| ~ sP1 ),
inference(prop_rule,[status(thm)],[]) ).
thf(106,plain,
( sP304
| ~ sP269 ),
inference(prop_rule,[status(thm)],[]) ).
thf(107,plain,
( sP335
| ~ sP304 ),
inference(prop_rule,[status(thm)],[]) ).
thf(108,plain,
( sP251
| ~ sP335 ),
inference(prop_rule,[status(thm)],[]) ).
thf(109,plain,
( sP162
| ~ sP251 ),
inference(prop_rule,[status(thm)],[]) ).
thf(110,plain,
( sP151
| ~ sP162 ),
inference(prop_rule,[status(thm)],[]) ).
thf(111,plain,
( sP292
| ~ sP151 ),
inference(prop_rule,[status(thm)],[]) ).
thf(112,plain,
( sP73
| ~ sP292 ),
inference(prop_rule,[status(thm)],[]) ).
thf(113,plain,
( sP342
| ~ sP73 ),
inference(prop_rule,[status(thm)],[]) ).
thf(114,plain,
( sP39
| ~ sP342 ),
inference(prop_rule,[status(thm)],[]) ).
thf(115,plain,
( sP84
| ~ sP39 ),
inference(prop_rule,[status(thm)],[]) ).
thf(116,plain,
( sP20
| ~ sP84 ),
inference(prop_rule,[status(thm)],[]) ).
thf(117,plain,
( sP104
| ~ sP20 ),
inference(prop_rule,[status(thm)],[]) ).
thf(118,plain,
( sP283
| ~ sP104 ),
inference(prop_rule,[status(thm)],[]) ).
thf(119,plain,
( sP35
| ~ sP283 ),
inference(prop_rule,[status(thm)],[]) ).
thf(120,plain,
( sP199
| ~ sP35 ),
inference(prop_rule,[status(thm)],[]) ).
thf(121,plain,
( sP127
| ~ sP199 ),
inference(prop_rule,[status(thm)],[]) ).
thf(122,plain,
( sP243
| ~ sP127 ),
inference(prop_rule,[status(thm)],[]) ).
thf(123,plain,
( sP300
| ~ sP243 ),
inference(prop_rule,[status(thm)],[]) ).
thf(124,plain,
( sP239
| ~ sP300 ),
inference(prop_rule,[status(thm)],[]) ).
thf(125,plain,
( sP291
| ~ sP239 ),
inference(prop_rule,[status(thm)],[]) ).
thf(126,plain,
( sP231
| ~ sP291 ),
inference(prop_rule,[status(thm)],[]) ).
thf(127,plain,
( sP341
| ~ sP231 ),
inference(prop_rule,[status(thm)],[]) ).
thf(128,plain,
( sP105
| ~ sP341 ),
inference(prop_rule,[status(thm)],[]) ).
thf(129,plain,
( sP10
| ~ sP105 ),
inference(prop_rule,[status(thm)],[]) ).
thf(130,plain,
( sP327
| ~ sP10 ),
inference(prop_rule,[status(thm)],[]) ).
thf(131,plain,
( sP237
| ~ sP327 ),
inference(prop_rule,[status(thm)],[]) ).
thf(132,plain,
( sP34
| ~ sP237 ),
inference(prop_rule,[status(thm)],[]) ).
thf(133,plain,
( sP353
| ~ sP34 ),
inference(prop_rule,[status(thm)],[]) ).
thf(134,plain,
( sP98
| ~ sP353 ),
inference(prop_rule,[status(thm)],[]) ).
thf(135,plain,
( sP334
| ~ sP98 ),
inference(prop_rule,[status(thm)],[]) ).
thf(136,plain,
( sP307
| ~ sP334 ),
inference(prop_rule,[status(thm)],[]) ).
thf(137,plain,
( sP148
| ~ sP307 ),
inference(prop_rule,[status(thm)],[]) ).
thf(138,plain,
( sP191
| ~ sP148 ),
inference(prop_rule,[status(thm)],[]) ).
thf(139,plain,
( sP92
| ~ sP191 ),
inference(prop_rule,[status(thm)],[]) ).
thf(140,plain,
( sP33
| ~ sP92 ),
inference(prop_rule,[status(thm)],[]) ).
thf(141,plain,
( sP183
| ~ sP33 ),
inference(prop_rule,[status(thm)],[]) ).
thf(142,plain,
( sP100
| ~ sP183 ),
inference(prop_rule,[status(thm)],[]) ).
thf(143,plain,
( sP286
| ~ sP100 ),
inference(prop_rule,[status(thm)],[]) ).
thf(144,plain,
( sP133
| ~ sP286 ),
inference(prop_rule,[status(thm)],[]) ).
thf(145,plain,
( sP43
| ~ sP133 ),
inference(prop_rule,[status(thm)],[]) ).
thf(146,plain,
( sP103
| ~ sP43 ),
inference(prop_rule,[status(thm)],[]) ).
thf(147,plain,
( sP49
| ~ sP103 ),
inference(prop_rule,[status(thm)],[]) ).
thf(148,plain,
( sP154
| ~ sP49 ),
inference(prop_rule,[status(thm)],[]) ).
thf(149,plain,
( sP227
| ~ sP154 ),
inference(prop_rule,[status(thm)],[]) ).
thf(150,plain,
( sP158
| ~ sP227 ),
inference(prop_rule,[status(thm)],[]) ).
thf(151,plain,
( sP204
| ~ sP158 ),
inference(prop_rule,[status(thm)],[]) ).
thf(152,plain,
( sP131
| ~ sP204 ),
inference(prop_rule,[status(thm)],[]) ).
thf(153,plain,
( sP56
| ~ sP131 ),
inference(prop_rule,[status(thm)],[]) ).
thf(154,plain,
( sP144
| ~ sP56 ),
inference(prop_rule,[status(thm)],[]) ).
thf(155,plain,
( sP86
| ~ sP144 ),
inference(prop_rule,[status(thm)],[]) ).
thf(156,plain,
( sP299
| ~ sP86 ),
inference(prop_rule,[status(thm)],[]) ).
thf(157,plain,
( sP181
| ~ sP299 ),
inference(prop_rule,[status(thm)],[]) ).
thf(158,plain,
( sP28
| ~ sP181 ),
inference(prop_rule,[status(thm)],[]) ).
thf(159,plain,
( sP82
| ~ sP28 ),
inference(prop_rule,[status(thm)],[]) ).
thf(160,plain,
( sP230
| ~ sP82 ),
inference(prop_rule,[status(thm)],[]) ).
thf(161,plain,
( sP69
| ~ sP230 ),
inference(prop_rule,[status(thm)],[]) ).
thf(162,plain,
( sP64
| ~ sP69 ),
inference(prop_rule,[status(thm)],[]) ).
thf(163,plain,
( sP12
| ~ sP64 ),
inference(prop_rule,[status(thm)],[]) ).
thf(164,plain,
( sP59
| ~ sP12 ),
inference(prop_rule,[status(thm)],[]) ).
thf(165,plain,
( sP208
| ~ sP59 ),
inference(prop_rule,[status(thm)],[]) ).
thf(166,plain,
( sP189
| ~ sP208 ),
inference(prop_rule,[status(thm)],[]) ).
thf(167,plain,
( sP107
| ~ sP189 ),
inference(prop_rule,[status(thm)],[]) ).
thf(168,plain,
( sP72
| ~ sP107 ),
inference(prop_rule,[status(thm)],[]) ).
thf(169,plain,
( sP129
| ~ sP72 ),
inference(prop_rule,[status(thm)],[]) ).
thf(170,plain,
( sP111
| ~ sP129 ),
inference(prop_rule,[status(thm)],[]) ).
thf(171,plain,
( sP197
| ~ sP111 ),
inference(prop_rule,[status(thm)],[]) ).
thf(172,plain,
( sP80
| ~ sP197 ),
inference(prop_rule,[status(thm)],[]) ).
thf(173,plain,
( sP76
| ~ sP80 ),
inference(prop_rule,[status(thm)],[]) ).
thf(174,plain,
( sP180
| ~ sP76 ),
inference(prop_rule,[status(thm)],[]) ).
thf(175,plain,
( sP165
| ~ sP180 ),
inference(prop_rule,[status(thm)],[]) ).
thf(176,plain,
( sP203
| ~ sP165 ),
inference(prop_rule,[status(thm)],[]) ).
thf(177,plain,
( sP42
| ~ sP203 ),
inference(prop_rule,[status(thm)],[]) ).
thf(178,plain,
( sP24
| ~ sP42 ),
inference(prop_rule,[status(thm)],[]) ).
thf(179,plain,
( sP238
| ~ sP24 ),
inference(prop_rule,[status(thm)],[]) ).
thf(180,plain,
( sP22
| ~ sP238 ),
inference(prop_rule,[status(thm)],[]) ).
thf(181,plain,
( sP21
| ~ sP22 ),
inference(prop_rule,[status(thm)],[]) ).
thf(182,plain,
( sP322
| ~ sP21 ),
inference(prop_rule,[status(thm)],[]) ).
thf(183,plain,
( sP176
| ~ sP322 ),
inference(prop_rule,[status(thm)],[]) ).
thf(184,plain,
( sP47
| ~ sP176 ),
inference(prop_rule,[status(thm)],[]) ).
thf(185,plain,
( sP196
| ~ sP47 ),
inference(prop_rule,[status(thm)],[]) ).
thf(186,plain,
( sP19
| ~ sP196 ),
inference(prop_rule,[status(thm)],[]) ).
thf(187,plain,
( sP236
| ~ sP19 ),
inference(prop_rule,[status(thm)],[]) ).
thf(188,plain,
( sP261
| ~ sP236 ),
inference(prop_rule,[status(thm)],[]) ).
thf(189,plain,
( sP174
| ~ sP261 ),
inference(prop_rule,[status(thm)],[]) ).
thf(190,plain,
( sP164
| ~ sP174 ),
inference(prop_rule,[status(thm)],[]) ).
thf(191,plain,
( sP6
| ~ sP164 ),
inference(prop_rule,[status(thm)],[]) ).
thf(192,plain,
( sP274
| ~ sP6 ),
inference(prop_rule,[status(thm)],[]) ).
thf(193,plain,
( sP220
| ~ sP274 ),
inference(prop_rule,[status(thm)],[]) ).
thf(194,plain,
( sP267
| ~ sP220 ),
inference(prop_rule,[status(thm)],[]) ).
thf(195,plain,
( sP296
| ~ sP267 ),
inference(prop_rule,[status(thm)],[]) ).
thf(196,plain,
( sP89
| ~ sP296 ),
inference(prop_rule,[status(thm)],[]) ).
thf(197,plain,
( sP67
| ~ sP89 ),
inference(prop_rule,[status(thm)],[]) ).
thf(198,plain,
( sP248
| ~ sP67 ),
inference(prop_rule,[status(thm)],[]) ).
thf(199,plain,
( sP188
| ~ sP248 ),
inference(prop_rule,[status(thm)],[]) ).
thf(200,plain,
( sP7
| ~ sP188 ),
inference(prop_rule,[status(thm)],[]) ).
thf(201,plain,
( sP88
| ~ sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(202,plain,
( sP83
| ~ sP88 ),
inference(prop_rule,[status(thm)],[]) ).
thf(203,plain,
( sP18
| ~ sP83 ),
inference(prop_rule,[status(thm)],[]) ).
thf(204,plain,
( sP58
| ~ sP18 ),
inference(prop_rule,[status(thm)],[]) ).
thf(205,plain,
( sP207
| ~ sP58 ),
inference(prop_rule,[status(thm)],[]) ).
thf(206,plain,
( sP179
| ~ sP207 ),
inference(prop_rule,[status(thm)],[]) ).
thf(207,plain,
( sP352
| ~ sP179 ),
inference(prop_rule,[status(thm)],[]) ).
thf(208,plain,
( sP247
| ~ sP352 ),
inference(prop_rule,[status(thm)],[]) ).
thf(209,plain,
( sP330
| ~ sP247 ),
inference(prop_rule,[status(thm)],[]) ).
thf(210,plain,
( sP184
| ~ sP330 ),
inference(prop_rule,[status(thm)],[]) ).
thf(211,plain,
( sP302
| ~ sP184 ),
inference(prop_rule,[status(thm)],[]) ).
thf(212,plain,
( sP311
| ~ sP302 ),
inference(prop_rule,[status(thm)],[]) ).
thf(213,plain,
( sP169
| ~ sP311 ),
inference(prop_rule,[status(thm)],[]) ).
thf(214,plain,
( sP55
| ~ sP169 ),
inference(prop_rule,[status(thm)],[]) ).
thf(215,plain,
( sP139
| ~ sP55 ),
inference(prop_rule,[status(thm)],[]) ).
thf(216,plain,
( sP198
| ~ sP139 ),
inference(prop_rule,[status(thm)],[]) ).
thf(217,plain,
( sP263
| ~ sP198 ),
inference(prop_rule,[status(thm)],[]) ).
thf(218,plain,
( sP29
| ~ sP263 ),
inference(prop_rule,[status(thm)],[]) ).
thf(219,plain,
( sP156
| ~ sP29 ),
inference(prop_rule,[status(thm)],[]) ).
thf(220,plain,
( sP281
| ~ sP156 ),
inference(prop_rule,[status(thm)],[]) ).
thf(221,plain,
( sP264
| ~ sP281 ),
inference(prop_rule,[status(thm)],[]) ).
thf(222,plain,
( sP333
| ~ sP264 ),
inference(prop_rule,[status(thm)],[]) ).
thf(223,plain,
( sP347
| ~ sP333 ),
inference(prop_rule,[status(thm)],[]) ).
thf(224,plain,
( sP266
| ~ sP347 ),
inference(prop_rule,[status(thm)],[]) ).
thf(225,plain,
( sP306
| ~ sP266 ),
inference(prop_rule,[status(thm)],[]) ).
thf(226,plain,
( sP150
| ~ sP306 ),
inference(prop_rule,[status(thm)],[]) ).
thf(227,plain,
( sP61
| ~ sP150 ),
inference(prop_rule,[status(thm)],[]) ).
thf(228,plain,
( sP27
| ~ sP61 ),
inference(prop_rule,[status(thm)],[]) ).
thf(229,plain,
( sP255
| ~ sP27 ),
inference(prop_rule,[status(thm)],[]) ).
thf(230,plain,
( sP116
| ~ sP255 ),
inference(prop_rule,[status(thm)],[]) ).
thf(231,plain,
( sP289
| ~ sP116 ),
inference(prop_rule,[status(thm)],[]) ).
thf(232,plain,
( sP209
| ~ sP289 ),
inference(prop_rule,[status(thm)],[]) ).
thf(233,plain,
( sP346
| ~ sP209 ),
inference(prop_rule,[status(thm)],[]) ).
thf(234,plain,
( sP102
| ~ sP346 ),
inference(prop_rule,[status(thm)],[]) ).
thf(235,plain,
( sP17
| ~ sP102 ),
inference(prop_rule,[status(thm)],[]) ).
thf(236,plain,
( sP52
| ~ sP17 ),
inference(prop_rule,[status(thm)],[]) ).
thf(237,plain,
( sP106
| ~ sP52 ),
inference(prop_rule,[status(thm)],[]) ).
thf(238,plain,
( sP79
| ~ sP106 ),
inference(prop_rule,[status(thm)],[]) ).
thf(239,plain,
( sP252
| ~ sP79 ),
inference(prop_rule,[status(thm)],[]) ).
thf(240,plain,
( sP323
| ~ sP252 ),
inference(prop_rule,[status(thm)],[]) ).
thf(241,plain,
( sP287
| ~ sP323 ),
inference(prop_rule,[status(thm)],[]) ).
thf(242,plain,
( sP280
| ~ sP287 ),
inference(prop_rule,[status(thm)],[]) ).
thf(243,plain,
( sP97
| ~ sP280 ),
inference(prop_rule,[status(thm)],[]) ).
thf(244,plain,
( sP262
| ~ sP97 ),
inference(prop_rule,[status(thm)],[]) ).
thf(245,plain,
( sP272
| ~ sP262 ),
inference(prop_rule,[status(thm)],[]) ).
thf(246,plain,
( sP241
| ~ sP272 ),
inference(prop_rule,[status(thm)],[]) ).
thf(247,plain,
( sP123
| ~ sP241 ),
inference(prop_rule,[status(thm)],[]) ).
thf(248,plain,
( sP123
| sP32 ),
inference(prop_rule,[status(thm)],[]) ).
thf(249,plain,
( sP115
| ~ sP123 ),
inference(prop_rule,[status(thm)],[]) ).
thf(250,plain,
( sP48
| ~ sP115 ),
inference(prop_rule,[status(thm)],[]) ).
thf(251,plain,
( sP48
| sP288 ),
inference(prop_rule,[status(thm)],[]) ).
thf(252,plain,
( sP253
| ~ sP48 ),
inference(prop_rule,[status(thm)],[]) ).
thf(253,plain,
( sP187
| ~ sP253 ),
inference(prop_rule,[status(thm)],[]) ).
thf(254,plain,
( sP187
| sP316 ),
inference(prop_rule,[status(thm)],[]) ).
thf(255,plain,
( sP249
| ~ sP187 ),
inference(prop_rule,[status(thm)],[]) ).
thf(256,plain,
( sP259
| ~ sP249 ),
inference(prop_rule,[status(thm)],[]) ).
thf(257,plain,
( sP51
| ~ sP259 ),
inference(prop_rule,[status(thm)],[]) ).
thf(258,plain,
( sP143
| ~ sP51 ),
inference(prop_rule,[status(thm)],[]) ).
thf(259,plain,
( sP161
| ~ sP143 ),
inference(prop_rule,[status(thm)],[]) ).
thf(260,plain,
( sP318
| ~ sP161 ),
inference(prop_rule,[status(thm)],[]) ).
thf(261,plain,
( sP146
| ~ sP318 ),
inference(prop_rule,[status(thm)],[]) ).
thf(262,plain,
( sP101
| ~ sP146 ),
inference(prop_rule,[status(thm)],[]) ).
thf(263,plain,
( sP63
| ~ sP101 ),
inference(prop_rule,[status(thm)],[]) ).
thf(264,plain,
( sP321
| ~ sP63 ),
inference(prop_rule,[status(thm)],[]) ).
thf(265,plain,
( sP163
| ~ sP321 ),
inference(prop_rule,[status(thm)],[]) ).
thf(266,plain,
( sP331
| ~ sP163 ),
inference(prop_rule,[status(thm)],[]) ).
thf(267,plain,
( sP310
| ~ sP331 ),
inference(prop_rule,[status(thm)],[]) ).
thf(268,plain,
( sP94
| ~ sP310 ),
inference(prop_rule,[status(thm)],[]) ).
thf(269,plain,
( sP53
| ~ sP94 ),
inference(prop_rule,[status(thm)],[]) ).
thf(270,plain,
( sP14
| ~ sP53 ),
inference(prop_rule,[status(thm)],[]) ).
thf(271,plain,
( sP62
| ~ sP14 ),
inference(prop_rule,[status(thm)],[]) ).
thf(272,plain,
( sP326
| ~ sP62 ),
inference(prop_rule,[status(thm)],[]) ).
thf(273,plain,
( sP271
| ~ sP326 ),
inference(prop_rule,[status(thm)],[]) ).
thf(274,plain,
( sP26
| ~ sP271 ),
inference(prop_rule,[status(thm)],[]) ).
thf(275,plain,
( sP340
| ~ sP26 ),
inference(prop_rule,[status(thm)],[]) ).
thf(276,plain,
( sP91
| ~ sP340 ),
inference(prop_rule,[status(thm)],[]) ).
thf(277,plain,
( sP41
| ~ sP91 ),
inference(prop_rule,[status(thm)],[]) ).
thf(278,plain,
( sP338
| ~ sP41 ),
inference(prop_rule,[status(thm)],[]) ).
thf(279,plain,
( sP5
| ~ sP338 ),
inference(prop_rule,[status(thm)],[]) ).
thf(280,plain,
( sP297
| ~ sP5 ),
inference(prop_rule,[status(thm)],[]) ).
thf(281,plain,
( sP13
| ~ sP297 ),
inference(prop_rule,[status(thm)],[]) ).
thf(282,plain,
( sP75
| ~ sP13 ),
inference(prop_rule,[status(thm)],[]) ).
thf(283,plain,
( sP138
| ~ sP75 ),
inference(prop_rule,[status(thm)],[]) ).
thf(284,plain,
( sP217
| ~ sP138 ),
inference(prop_rule,[status(thm)],[]) ).
thf(285,plain,
( sP332
| ~ sP217 ),
inference(prop_rule,[status(thm)],[]) ).
thf(286,plain,
( sP314
| ~ sP332 ),
inference(prop_rule,[status(thm)],[]) ).
thf(287,plain,
( sP277
| ~ sP314 ),
inference(prop_rule,[status(thm)],[]) ).
thf(288,plain,
( sP265
| ~ sP277 ),
inference(prop_rule,[status(thm)],[]) ).
thf(289,plain,
( sP309
| ~ sP265 ),
inference(prop_rule,[status(thm)],[]) ).
thf(290,plain,
( sP315
| ~ sP309 ),
inference(prop_rule,[status(thm)],[]) ).
thf(291,plain,
( sP351
| ~ sP315 ),
inference(prop_rule,[status(thm)],[]) ).
thf(292,plain,
( sP320
| ~ sP351 ),
inference(prop_rule,[status(thm)],[]) ).
thf(293,plain,
( sP213
| ~ sP320 ),
inference(prop_rule,[status(thm)],[]) ).
thf(294,plain,
( sP110
| ~ sP213 ),
inference(prop_rule,[status(thm)],[]) ).
thf(295,plain,
( sP290
| ~ sP110 ),
inference(prop_rule,[status(thm)],[]) ).
thf(296,plain,
( sP193
| ~ sP290 ),
inference(prop_rule,[status(thm)],[]) ).
thf(297,plain,
( sP114
| ~ sP193 ),
inference(prop_rule,[status(thm)],[]) ).
thf(298,plain,
( sP224
| ~ sP114 ),
inference(prop_rule,[status(thm)],[]) ).
thf(299,plain,
( sP126
| ~ sP224 ),
inference(prop_rule,[status(thm)],[]) ).
thf(300,plain,
( sP173
| ~ sP126 ),
inference(prop_rule,[status(thm)],[]) ).
thf(301,plain,
( sP119
| ~ sP173 ),
inference(prop_rule,[status(thm)],[]) ).
thf(302,plain,
( sP246
| ~ sP119 ),
inference(prop_rule,[status(thm)],[]) ).
thf(303,plain,
( sP301
| ~ sP246 ),
inference(prop_rule,[status(thm)],[]) ).
thf(304,plain,
( sP68
| ~ sP301 ),
inference(prop_rule,[status(thm)],[]) ).
thf(305,plain,
( sP122
| ~ sP68 ),
inference(prop_rule,[status(thm)],[]) ).
thf(306,plain,
( sP168
| ~ sP122 ),
inference(prop_rule,[status(thm)],[]) ).
thf(307,plain,
( sP234
| ~ sP168 ),
inference(prop_rule,[status(thm)],[]) ).
thf(308,plain,
( sP118
| ~ sP234 ),
inference(prop_rule,[status(thm)],[]) ).
thf(309,plain,
( sP260
| ~ sP118 ),
inference(prop_rule,[status(thm)],[]) ).
thf(310,plain,
( sP240
| ~ sP260 ),
inference(prop_rule,[status(thm)],[]) ).
thf(311,plain,
( sP190
| ~ sP240 ),
inference(prop_rule,[status(thm)],[]) ).
thf(312,plain,
( sP317
| ~ sP190 ),
inference(prop_rule,[status(thm)],[]) ).
thf(313,plain,
( sP66
| ~ sP317 ),
inference(prop_rule,[status(thm)],[]) ).
thf(314,plain,
( sP245
| ~ sP66 ),
inference(prop_rule,[status(thm)],[]) ).
thf(315,plain,
( sP216
| ~ sP245 ),
inference(prop_rule,[status(thm)],[]) ).
thf(316,plain,
( sP275
| ~ sP216 ),
inference(prop_rule,[status(thm)],[]) ).
thf(317,plain,
( sP96
| ~ sP275 ),
inference(prop_rule,[status(thm)],[]) ).
thf(318,plain,
( sP36
| ~ sP96 ),
inference(prop_rule,[status(thm)],[]) ).
thf(319,plain,
( sP167
| ~ sP36 ),
inference(prop_rule,[status(thm)],[]) ).
thf(320,plain,
( sP160
| ~ sP167 ),
inference(prop_rule,[status(thm)],[]) ).
thf(321,plain,
( sP182
| ~ sP160 ),
inference(prop_rule,[status(thm)],[]) ).
thf(322,plain,
( sP212
| ~ sP182 ),
inference(prop_rule,[status(thm)],[]) ).
thf(323,plain,
( sP11
| ~ sP212 ),
inference(prop_rule,[status(thm)],[]) ).
thf(324,plain,
( sP195
| ~ sP11 ),
inference(prop_rule,[status(thm)],[]) ).
thf(325,plain,
( sP226
| ~ sP195 ),
inference(prop_rule,[status(thm)],[]) ).
thf(326,plain,
( sP270
| ~ sP226 ),
inference(prop_rule,[status(thm)],[]) ).
thf(327,plain,
( sP221
| ~ sP270 ),
inference(prop_rule,[status(thm)],[]) ).
thf(328,plain,
( sP250
| ~ sP221 ),
inference(prop_rule,[status(thm)],[]) ).
thf(329,plain,
( sP120
| ~ sP250 ),
inference(prop_rule,[status(thm)],[]) ).
thf(330,plain,
( sP258
| ~ sP120 ),
inference(prop_rule,[status(thm)],[]) ).
thf(331,plain,
( sP95
| ~ sP258 ),
inference(prop_rule,[status(thm)],[]) ).
thf(332,plain,
( sP78
| ~ sP95 ),
inference(prop_rule,[status(thm)],[]) ).
thf(333,plain,
( sP294
| ~ sP78 ),
inference(prop_rule,[status(thm)],[]) ).
thf(334,plain,
( sP136
| ~ sP294 ),
inference(prop_rule,[status(thm)],[]) ).
thf(335,plain,
( sP77
| ~ sP136 ),
inference(prop_rule,[status(thm)],[]) ).
thf(336,plain,
( sP121
| ~ sP77 ),
inference(prop_rule,[status(thm)],[]) ).
thf(337,plain,
( sP313
| ~ sP121 ),
inference(prop_rule,[status(thm)],[]) ).
thf(338,plain,
( sP157
| ~ sP313 ),
inference(prop_rule,[status(thm)],[]) ).
thf(339,plain,
( sP87
| ~ sP157 ),
inference(prop_rule,[status(thm)],[]) ).
thf(340,plain,
( sP178
| ~ sP87 ),
inference(prop_rule,[status(thm)],[]) ).
thf(341,plain,
( sP25
| ~ sP178 ),
inference(prop_rule,[status(thm)],[]) ).
thf(342,plain,
( sP215
| ~ sP25 ),
inference(prop_rule,[status(thm)],[]) ).
thf(343,plain,
( sP285
| ~ sP215 ),
inference(prop_rule,[status(thm)],[]) ).
thf(344,plain,
( sP71
| ~ sP285 ),
inference(prop_rule,[status(thm)],[]) ).
thf(345,plain,
( sP233
| ~ sP71 ),
inference(prop_rule,[status(thm)],[]) ).
thf(346,plain,
( sP223
| ~ sP233 ),
inference(prop_rule,[status(thm)],[]) ).
thf(347,plain,
( sP175
| ~ sP223 ),
inference(prop_rule,[status(thm)],[]) ).
thf(348,plain,
( sP254
| ~ sP175 ),
inference(prop_rule,[status(thm)],[]) ).
thf(349,plain,
( sP38
| ~ sP254 ),
inference(prop_rule,[status(thm)],[]) ).
thf(350,plain,
( sP201
| ~ sP38 ),
inference(prop_rule,[status(thm)],[]) ).
thf(351,plain,
( sP273
| ~ sP201 ),
inference(prop_rule,[status(thm)],[]) ).
thf(352,plain,
( sP298
| ~ sP273 ),
inference(prop_rule,[status(thm)],[]) ).
thf(353,plain,
( sP298
| sP284 ),
inference(prop_rule,[status(thm)],[]) ).
thf(binunionE,axiom,
sP348 ).
thf(binunionIR,axiom,
sP337 ).
thf(binunionIL,axiom,
sP282 ).
thf(setextAx,axiom,
sP3 ).
thf(354,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,binunionE,binunionIR,binunionIL,setextAx,h1]) ).
thf(355,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[354,h0]) ).
thf(0,theorem,
sP298,
inference(contra,[status(thm),contra(discharge,[h1])],[354,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : SEU788^1 : TPTP v9.2.0. Released v3.7.0.
% 0.06/0.12 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.12/0.33 % Computer : n029.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Wed Oct 8 21:51:53 EDT 2025
% 0.12/0.33 % CPUTime :
% 184.95/184.19 slave returned with unknown status
% 211.26/210.58 % SZS status Theorem
% 211.26/210.58 % Mode: mode494:USE_SINE=true:SINE_TOLERANCE=5.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=1.:SINE_DEPTH=0
% 211.26/210.58 % Inferences: 6010
% 211.26/210.58 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------