%------------------------------------------------------------------------------
% File : Zipperpin---2.1.9999
% Problem : SEU675^1 : TPTP v8.1.2. Released v3.7.0.
% Transfm : NO INFORMATION
% Format : NO INFORMATION
% Command : python3 /export/starexec/sandbox2/solver/bin/portfolio.lams.parallel.py %s %d /export/starexec/sandbox2/tmp/tmp.0eGY29e6zI true
% Computer : n031.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 19:16:01 EDT 2023
% Result : Theorem 59.31s 8.29s
% Output : Refutation 59.31s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 620
% Syntax : Number of formulae : 633 ( 409 unt; 220 typ; 0 def)
% Number of atoms : 4035 ( 963 equ; 44 cnn)
% Maximal formula atoms : 1112 ( 9 avg)
% Number of connectives : 11051 ( 253 ~; 35 |; 265 &;7678 @)
% ( 70 <=>;2160 =>; 0 <=; 0 <~>)
% Maximal formula depth : 217 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 310 ( 310 >; 0 *; 0 +; 0 <<)
% Number of symbols : 226 ( 220 usr; 198 con; 0-4 aty)
% ( 529 !!; 61 ??; 0 @@+; 0 @@-)
% Number of variables : 3160 ( 883 ^;2081 !; 196 ?;3160 :)
% Comments :
%------------------------------------------------------------------------------
thf(setadjoinAx_type,type,
setadjoinAx: $o ).
thf(cartprodmempaircEq_type,type,
cartprodmempaircEq: $o ).
thf(notdexE_type,type,
notdexE: $o ).
thf(setunionAx_type,type,
setunionAx: $o ).
thf(setminusELneg_type,type,
setminusELneg: $o ).
thf(kpairsurjEq_type,type,
kpairsurjEq: $o ).
thf(powersetE_type,type,
powersetE: $o ).
thf(omegaSAx_type,type,
omegaSAx: $o ).
thf(exuI1_type,type,
exuI1: $o ).
thf(exuI2_type,type,
exuI2: $o ).
thf(notequalI2_type,type,
notequalI2: $o ).
thf(kpairiskpair_type,type,
kpairiskpair: $o ).
thf(upairsubunion_type,type,
upairsubunion: $o ).
thf(powerset__Cong_type,type,
powerset__Cong: $o ).
thf(setminusILneg_type,type,
setminusILneg: $o ).
thf(wellorderingAx_type,type,
wellorderingAx: $o ).
thf(setadjoinSub2_type,type,
setadjoinSub2: $o ).
thf(binintersectSubset5_type,type,
binintersectSubset5: $o ).
thf(upairinpowunion_type,type,
upairinpowunion: $o ).
thf(setoftrueEq_type,type,
setoftrueEq: $o ).
thf(func_type,type,
func: $i > $i > $i > $o ).
thf(emptyset__Cong_type,type,
emptyset__Cong: $o ).
thf(cartprodpairmemER_type,type,
cartprodpairmemER: $o ).
thf(dpsetconstrI_type,type,
dpsetconstrI: $o ).
thf(breln_type,type,
breln: $i > $i > $i > $o ).
thf(dpsetconstrER_type,type,
dpsetconstrER: $o ).
thf(sepInPowerset_type,type,
sepInPowerset: $o ).
thf(setukpairIR_type,type,
setukpairIR: $o ).
thf(kpair_type,type,
kpair: $i > $i > $i ).
thf(binunionE_type,type,
binunionE: $o ).
thf(setext_type,type,
setext: $o ).
thf(emptyinPowerset_type,type,
emptyinPowerset: $o ).
thf(ksnd_type,type,
ksnd: $i > $i ).
thf(ubforcartprodlem1_type,type,
ubforcartprodlem1: $o ).
thf(setukpairinjR12_type,type,
setukpairinjR12: $o ).
thf(descr__Cong_type,type,
descr__Cong: $o ).
thf(binintersectSubset1_type,type,
binintersectSubset1: $o ).
thf(in_type,type,
in: $i > $i > $o ).
thf(app_type,type,
app: $o ).
thf(dpsetconstrSub_type,type,
dpsetconstrSub: $o ).
thf(cartprodfstpairEq_type,type,
cartprodfstpairEq: $o ).
thf(quantDeMorgan1_type,type,
quantDeMorgan1: $o ).
thf(binunion_type,type,
binunion: $i > $i > $i ).
thf(setadjoinIR_type,type,
setadjoinIR: $o ).
thf(setadjoin__Cong_type,type,
setadjoin__Cong: $o ).
thf(symdiffI2_type,type,
symdiffI2: $o ).
thf(upairset2IR_type,type,
upairset2IR: $o ).
thf(subsetE_type,type,
subsetE: $o ).
thf(prop2set2propI_type,type,
prop2set2propI: $o ).
thf(dsetconstrEL_type,type,
dsetconstrEL: $o ).
thf(setbeta_type,type,
setbeta: $o ).
thf(eqinunit_type,type,
eqinunit: $o ).
thf(upairsetIL_type,type,
upairsetIL: $o ).
thf(powersetI1_type,type,
powersetI1: $o ).
thf(binintersectSubset3_type,type,
binintersectSubset3: $o ).
thf(funcImageSingleton_type,type,
funcImageSingleton: $o ).
thf(setukpairinjR_type,type,
setukpairinjR: $o ).
thf(cartprodfstin_type,type,
cartprodfstin: $o ).
thf(setminusIRneg_type,type,
setminusIRneg: $o ).
thf(kfst_type,type,
kfst: $i > $i ).
thf(powersetAx_type,type,
powersetAx: $o ).
thf(singletonprop_type,type,
singletonprop: $o ).
thf(binintersectSubset4_type,type,
binintersectSubset4: $o ).
thf(singletoninpowunion_type,type,
singletoninpowunion: $o ).
thf(powersetI_type,type,
powersetI: $o ).
thf(cartprodmempair_type,type,
cartprodmempair: $o ).
thf(upairsetIR_type,type,
upairsetIR: $o ).
thf(cartprodsndpairEq_type,type,
cartprodsndpairEq: $o ).
thf(inCongP_type,type,
inCongP: $o ).
thf(setukpairinjL_type,type,
setukpairinjL: $o ).
thf(setadjoinSub_type,type,
setadjoinSub: $o ).
thf(exuE3e_type,type,
exuE3e: $o ).
thf(emptysetE_type,type,
emptysetE: $o ).
thf(upairequniteq_type,type,
upairequniteq: $o ).
thf(binintersectSubset2_type,type,
binintersectSubset2: $o ).
thf(setunionI_type,type,
setunionI: $o ).
thf(symdiff_type,type,
symdiff: $i > $i > $i ).
thf(setunion__Cong_type,type,
setunion__Cong: $o ).
thf(prop2setI_type,type,
prop2setI: $o ).
thf(nonemptyImpWitness_type,type,
nonemptyImpWitness: $o ).
thf(setminusER_type,type,
setminusER: $o ).
thf(exuE2_type,type,
exuE2: $o ).
thf(emptyset_type,type,
emptyset: $i ).
thf(setminus_type,type,
setminus: $i > $i > $i ).
thf(powersetsubset_type,type,
powersetsubset: $o ).
thf(setukpairinjR1_type,type,
setukpairinjR1: $o ).
thf(dpsetconstrERa_type,type,
dpsetconstrERa: $o ).
thf(subPowSU_type,type,
subPowSU: $o ).
thf(ex1_type,type,
ex1: $i > ( $i > $o ) > $o ).
thf(notequalI1_type,type,
notequalI1: $o ).
thf(notinemptyset_type,type,
notinemptyset: $o ).
thf(nonemptyI1_type,type,
nonemptyI1: $o ).
thf(subsetTrans_type,type,
subsetTrans: $o ).
thf(setukpairinjR2_type,type,
setukpairinjR2: $o ).
thf(kpairp_type,type,
kpairp: $o ).
thf(setminusI_type,type,
setminusI: $o ).
thf(binunionIL_type,type,
binunionIL: $o ).
thf(setunionsingleton_type,type,
setunionsingleton: $o ).
thf(subsetE2_type,type,
subsetE2: $o ).
thf(singletonsswitch_type,type,
singletonsswitch: $o ).
thf(kfstpairEq_type,type,
kfstpairEq: $o ).
thf(ubforcartprodlem3_type,type,
ubforcartprodlem3: $o ).
thf(setextsub_type,type,
setextsub: $o ).
thf(binunionLsub_type,type,
binunionLsub: $o ).
thf(setunionsingleton1_type,type,
setunionsingleton1: $o ).
thf(binintersect_type,type,
binintersect: $i > $i > $i ).
thf(emptysetsubset_type,type,
emptysetsubset: $o ).
thf(quantDeMorgan3_type,type,
quantDeMorgan3: $o ).
thf(setunionE_type,type,
setunionE: $o ).
thf(setminusSubset1_type,type,
setminusSubset1: $o ).
thf(emptyinunitempty_type,type,
emptyinunitempty: $o ).
thf(binunionIR_type,type,
binunionIR: $o ).
thf(setadjoinE_type,type,
setadjoinE: $o ).
thf(descr_type,type,
descr: ( $i > $o ) > $i ).
thf(cartprodpairmemEL_type,type,
cartprodpairmemEL: $o ).
thf(singletonsuniq_type,type,
singletonsuniq: $o ).
thf(dpsetconstrEL2_type,type,
dpsetconstrEL2: $o ).
thf(ap_type,type,
ap: $i > $i > $i > $i > $i ).
thf(setadjoinIL_type,type,
setadjoinIL: $o ).
thf(exuE1_type,type,
exuE1: $o ).
thf(ubforcartprodlem2_type,type,
ubforcartprodlem2: $o ).
thf(binintersectRsub_type,type,
binintersectRsub: $o ).
thf(setminusLsub_type,type,
setminusLsub: $o ).
thf(prop2set_type,type,
prop2set: $o > $i ).
thf(notinsingleton_type,type,
notinsingleton: $o ).
thf(dpsetconstrEL1_type,type,
dpsetconstrEL1: $o ).
thf(eqimpsubset2_type,type,
eqimpsubset2: $o ).
thf(exuEu_type,type,
exuEu: $o ).
thf(emptysetimpfalse_type,type,
emptysetimpfalse: $o ).
thf(ex1I2_type,type,
ex1I2: $o ).
thf(upairsetE_type,type,
upairsetE: $o ).
thf(omega0Ax_type,type,
omega0Ax: $o ).
thf(notsubsetI_type,type,
notsubsetI: $o ).
thf(quantDeMorgan4_type,type,
quantDeMorgan4: $o ).
thf(setextAx_type,type,
setextAx: $o ).
thf(powersetE1_type,type,
powersetE1: $o ).
thf(bs114d_type,type,
bs114d: $o ).
thf(noeltsimpempty_type,type,
noeltsimpempty: $o ).
thf(setunion_type,type,
setunion: $i > $i ).
thf(iskpair_type,type,
iskpair: $i > $o ).
thf(binintersectLsub_type,type,
binintersectLsub: $o ).
thf(subsetI1_type,type,
subsetI1: $o ).
thf(symdiffE_type,type,
symdiffE: $o ).
thf(descrp_type,type,
descrp: $o ).
thf(dsetconstr__Cong_type,type,
dsetconstr__Cong: $o ).
thf(foundationAx_type,type,
foundationAx: $o ).
thf(emptysetAx_type,type,
emptysetAx: $o ).
thf(emptyI_type,type,
emptyI: $o ).
thf(setadjoinOr_type,type,
setadjoinOr: $o ).
thf(binintersectEL_type,type,
binintersectEL: $o ).
thf(emptyE1_type,type,
emptyE1: $o ).
thf(emptyInPowerset_type,type,
emptyInPowerset: $o ).
thf(vacuousDall_type,type,
vacuousDall: $o ).
thf(dsetconstr_type,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(dsetconstrER_type,type,
dsetconstrER: $o ).
thf(exu_type,type,
exu: ( $i > $o ) > $o ).
thf(binintersectI_type,type,
binintersectI: $o ).
thf(cartprodpairin_type,type,
cartprodpairin: $o ).
thf(ex1I_type,type,
ex1I: $o ).
thf(cartprodmempair1_type,type,
cartprodmempair1: $o ).
thf(cartprodsndin_type,type,
cartprodsndin: $o ).
thf(apProp_type,type,
apProp: $o ).
thf(setukpairinjR11_type,type,
setukpairinjR11: $o ).
thf(dpsetconstr_type,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(singleton_type,type,
singleton: $i > $o ).
thf(symdiffI1_type,type,
symdiffI1: $o ).
thf(subsetI2_type,type,
subsetI2: $o ).
thf(setOfPairsIsBReln_type,type,
setOfPairsIsBReln: $o ).
thf(symdiffIneg1_type,type,
symdiffIneg1: $o ).
thf(theprop_type,type,
theprop: $o ).
thf(setminusERneg_type,type,
setminusERneg: $o ).
thf(symdiffIneg2_type,type,
symdiffIneg2: $o ).
thf(omega__Cong_type,type,
omega__Cong: $o ).
thf(subsetRefl_type,type,
subsetRefl: $o ).
thf(uniqinunit_type,type,
uniqinunit: $o ).
thf(setminusSubset2_type,type,
setminusSubset2: $o ).
thf(prop2setE_type,type,
prop2setE: $o ).
thf(binunionEcases_type,type,
binunionEcases: $o ).
thf(omega_type,type,
omega: $i ).
thf(nonemptyI_type,type,
nonemptyI: $o ).
thf(disjointsetsI1_type,type,
disjointsetsI1: $o ).
thf(exuE3u_type,type,
exuE3u: $o ).
thf(setminusEL_type,type,
setminusEL: $o ).
thf(setunionsingleton2_type,type,
setunionsingleton2: $o ).
thf(sepSubset_type,type,
sepSubset: $o ).
thf(upairset2E_type,type,
upairset2E: $o ).
thf(eqimpsubset1_type,type,
eqimpsubset1: $o ).
thf(exuI3_type,type,
exuI3: $o ).
thf(funcSet_type,type,
funcSet: $i > $i > $i ).
thf(notdallE_type,type,
notdallE: $o ).
thf(binintersectER_type,type,
binintersectER: $o ).
thf(ksndsingleton_type,type,
ksndsingleton: $o ).
thf(powerset_type,type,
powerset: $i > $i ).
thf(kfstsingleton_type,type,
kfstsingleton: $o ).
thf(dsetconstrI_type,type,
dsetconstrI: $o ).
thf(cartprodpairsurjEq_type,type,
cartprodpairsurjEq: $o ).
thf(nonempty_type,type,
nonempty: $i > $o ).
thf(set2prop_type,type,
set2prop: $i > $o ).
thf(setadjoin_type,type,
setadjoin: $i > $i > $i ).
thf(exu__Cong_type,type,
exu__Cong: $o ).
thf(ksndpairEq_type,type,
ksndpairEq: $o ).
thf(quantDeMorgan2_type,type,
quantDeMorgan2: $o ).
thf(ex1E1_type,type,
ex1E1: $o ).
thf(secondinupair_type,type,
secondinupair: $o ).
thf(singletonsubset_type,type,
singletonsubset: $o ).
thf(singletoninpowerset_type,type,
singletoninpowerset: $o ).
thf(subsetemptysetimpeq_type,type,
subsetemptysetimpeq: $o ).
thf(replAx_type,type,
replAx: $o ).
thf(nonemptyE1_type,type,
nonemptyE1: $o ).
thf(omegaIndAx_type,type,
omegaIndAx: $o ).
thf(setukpairIL_type,type,
setukpairIL: $o ).
thf(setunionE2_type,type,
setunionE2: $o ).
thf(inPowerset_type,type,
inPowerset: $o ).
thf(cartprod_type,type,
cartprod: $i > $i > $i ).
thf(subset_type,type,
subset: $i > $i > $o ).
thf(in__Cong_type,type,
in__Cong: $o ).
thf(setukpairinjL2_type,type,
setukpairinjL2: $o ).
thf(binunionRsub_type,type,
binunionRsub: $o ).
thf(subset2powerset_type,type,
subset2powerset: $o ).
thf(setukpairinjL1_type,type,
setukpairinjL1: $o ).
thf(app,axiom,
( app
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
thf('0',plain,
( app
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in @ ( ap @ X4 @ X6 @ X8 @ X10 ) @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(ap,axiom,
( ap
= ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
( setunion
@ ( dsetconstr @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
thf('1',plain,
( ap
= ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
( setunion
@ ( dsetconstr @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[ap]) ).
thf('2',plain,
( ap
= ( ^ [V_1: $i,V_2: $i,V_3: $i,V_4: $i] :
( setunion
@ ( dsetconstr @ V_2
@ ^ [V_5: $i] : ( in @ ( kpair @ V_4 @ V_5 ) @ V_3 ) ) ) ) ),
define([status(thm)]) ).
thf(apProp,axiom,
( apProp
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in
@ ( setunion
@ ( dsetconstr @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
@ B ) ) ) ) ) ).
thf('3',plain,
( apProp
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X6
@ ^ [V_1: $i] : ( in @ ( kpair @ X10 @ V_1 ) @ X8 ) ) )
@ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(funcImageSingleton,axiom,
( funcImageSingleton
= ( ! [A: $i,B: $i,Xf: $i] :
( ( func @ A @ B @ Xf )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( singleton
@ ( dsetconstr @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
thf('4',plain,
( funcImageSingleton
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( func @ X4 @ X6 @ X8 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( singleton
@ ( dsetconstr @ X6
@ ^ [V_1: $i] : ( in @ ( kpair @ X10 @ V_1 ) @ X8 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(funcSet,axiom,
( funcSet
= ( ^ [A: $i,B: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
@ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ) ).
thf(func,axiom,
( func
= ( ^ [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
& ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ex1 @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
thf(breln,axiom,
( breln
= ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
thf('5',plain,
( breln
= ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[breln]) ).
thf('6',plain,
( breln
= ( ^ [V_1: $i,V_2: $i,V_3: $i] : ( subset @ V_3 @ ( cartprod @ V_1 @ V_2 ) ) ) ),
define([status(thm)]) ).
thf(ex1,axiom,
( ex1
= ( ^ [A: $i,Xphi: $i > $o] :
( singleton
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
thf(singleton,axiom,
( singleton
= ( ^ [A: $i] :
? [Xx: $i] :
( ( A
= ( setadjoin @ Xx @ emptyset ) )
& ( in @ Xx @ A ) ) ) ) ).
thf('7',plain,
( singleton
= ( ^ [A: $i] :
? [Xx: $i] :
( ( A
= ( setadjoin @ Xx @ emptyset ) )
& ( in @ Xx @ A ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[singleton]) ).
thf('8',plain,
( singleton
= ( ^ [V_1: $i] :
? [X4: $i] :
( ( V_1
= ( setadjoin @ X4 @ emptyset ) )
& ( in @ X4 @ V_1 ) ) ) ),
define([status(thm)]) ).
thf('9',plain,
( ex1
= ( ^ [A: $i,Xphi: $i > $o] :
( singleton
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[ex1,'8']) ).
thf('10',plain,
( ex1
= ( ^ [V_1: $i,V_2: $i > $o] :
( singleton
@ ( dsetconstr @ V_1
@ ^ [V_3: $i] : ( V_2 @ V_3 ) ) ) ) ),
define([status(thm)]) ).
thf('11',plain,
( func
= ( ^ [A: $i,B: $i,R: $i] :
( ( breln @ A @ B @ R )
& ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( ex1 @ B
@ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[func,'6','10','8']) ).
thf('12',plain,
( func
= ( ^ [V_1: $i,V_2: $i,V_3: $i] :
( ( breln @ V_1 @ V_2 @ V_3 )
& ! [X4: $i] :
( ( in @ X4 @ V_1 )
=> ( ex1 @ V_2
@ ^ [V_4: $i] : ( in @ ( kpair @ X4 @ V_4 ) @ V_3 ) ) ) ) ) ),
define([status(thm)]) ).
thf('13',plain,
( funcSet
= ( ^ [A: $i,B: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
@ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[funcSet,'12','10','8']) ).
thf('14',plain,
( funcSet
= ( ^ [V_1: $i,V_2: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ V_1 @ V_2 ) )
@ ^ [V_3: $i] : ( func @ V_1 @ V_2 @ V_3 ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrER,axiom,
( dpsetconstrER
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( Xphi @ Xx @ Xy ) ) ) ) ).
thf('15',plain,
( dpsetconstrER
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i,X12: $i] :
( ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) )
=> ( X8 @ X10 @ X12 ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrEL2,axiom,
( dpsetconstrEL2
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( in @ Xy @ B ) ) ) ) ).
thf('16',plain,
( dpsetconstrEL2
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i,X12: $i] :
( ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) )
=> ( in @ X12 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrEL1,axiom,
( dpsetconstrEL1
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('17',plain,
( dpsetconstrEL1
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i,X12: $i] :
( ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) )
=> ( in @ X10 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrERa,axiom,
( dpsetconstrERa
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
=> ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
thf('18',plain,
( dpsetconstrERa
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X6 )
=> ( ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) )
=> ( X8 @ X10 @ X12 ) ) ) ) ) ),
define([status(thm)]) ).
thf(setOfPairsIsBReln,axiom,
( setOfPairsIsBReln
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
( breln @ A @ B
@ ( dpsetconstr @ A @ B
@ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
thf('19',plain,
( setOfPairsIsBReln
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o] :
( breln @ X4 @ X6
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrSub,axiom,
( dpsetconstrSub
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
( subset
@ ( dpsetconstr @ A @ B
@ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
@ ( cartprod @ A @ B ) ) ) ) ).
thf('20',plain,
( dpsetconstrSub
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) )
@ ( cartprod @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(dpsetconstrI,axiom,
( dpsetconstrI
= ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( Xphi @ Xx @ Xy )
=> ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
thf('21',plain,
( dpsetconstrI
= ( ! [X4: $i,X6: $i,X8: $i > $i > $o,X10: $i] :
( ( in @ X10 @ X4 )
=> ! [X12: $i] :
( ( in @ X12 @ X6 )
=> ( ( X8 @ X10 @ X12 )
=> ( in @ ( kpair @ X10 @ X12 )
@ ( dpsetconstr @ X4 @ X6
@ ^ [V_1: $i,V_2: $i] : ( X8 @ V_1 @ V_2 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodpairsurjEq,axiom,
( cartprodpairsurjEq
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
= Xu ) ) ) ) ).
thf('22',plain,
( cartprodpairsurjEq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ( ( kpair @ ( kfst @ X8 ) @ ( ksnd @ X8 ) )
= X8 ) ) ) ),
define([status(thm)]) ).
thf(cartprodsndpairEq,axiom,
( cartprodsndpairEq
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( ksnd @ ( kpair @ Xx @ Xy ) )
= Xy ) ) ) ) ) ).
thf('23',plain,
( cartprodsndpairEq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( ( ksnd @ ( kpair @ X8 @ X10 ) )
= X10 ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodfstpairEq,axiom,
( cartprodfstpairEq
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( kfst @ ( kpair @ Xx @ Xy ) )
= Xx ) ) ) ) ) ).
thf('24',plain,
( cartprodfstpairEq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( ( kfst @ ( kpair @ X8 @ X10 ) )
= X8 ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodmempaircEq,axiom,
( cartprodmempaircEq
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( kpair @ Xx @ Xy )
= ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
thf('25',plain,
( cartprodmempaircEq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( ( kpair @ X8 @ X10 )
= ( kpair @ X8 @ X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodpairmemER,axiom,
( cartprodpairmemER
= ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
=> ( in @ Xy @ B ) ) ) ) ).
thf('26',plain,
( cartprodpairmemER
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( in @ ( kpair @ X8 @ X10 ) @ ( cartprod @ X4 @ X6 ) )
=> ( in @ X10 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(cartprodpairmemEL,axiom,
( cartprodpairmemEL
= ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('27',plain,
( cartprodpairmemEL
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( in @ ( kpair @ X8 @ X10 ) @ ( cartprod @ X4 @ X6 ) )
=> ( in @ X8 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(cartprodsndin,axiom,
( cartprodsndin
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
thf('28',plain,
( cartprodsndin
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ( in @ ( ksnd @ X8 ) @ X6 ) ) ) ),
define([status(thm)]) ).
thf(kpairsurjEq,axiom,
( kpairsurjEq
= ( ! [Xu: $i] :
( ( iskpair @ Xu )
=> ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
= Xu ) ) ) ) ).
thf('29',plain,
( kpairsurjEq
= ( ! [X4: $i] :
( ( iskpair @ X4 )
=> ( ( kpair @ ( kfst @ X4 ) @ ( ksnd @ X4 ) )
= X4 ) ) ) ),
define([status(thm)]) ).
thf(ksndpairEq,axiom,
( ksndpairEq
= ( ! [Xx: $i,Xy: $i] :
( ( ksnd @ ( kpair @ Xx @ Xy ) )
= Xy ) ) ) ).
thf('30',plain,
( ksndpairEq
= ( ! [X4: $i,X6: $i] :
( ( ksnd @ ( kpair @ X4 @ X6 ) )
= X6 ) ) ),
define([status(thm)]) ).
thf(ksndsingleton,axiom,
( ksndsingleton
= ( ! [Xu: $i] :
( ( iskpair @ Xu )
=> ( singleton
@ ( dsetconstr @ ( setunion @ Xu )
@ ^ [Xx: $i] :
( Xu
= ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
thf('31',plain,
( ksndsingleton
= ( ! [X4: $i] :
( ( iskpair @ X4 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X4 )
@ ^ [V_1: $i] :
( X4
= ( kpair @ ( kfst @ X4 ) @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR,axiom,
( setukpairinjR
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( kpair @ Xx @ Xy )
= ( kpair @ Xz @ Xu ) )
=> ( Xy = Xu ) ) ) ) ).
thf('32',plain,
( setukpairinjR
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( kpair @ X4 @ X6 )
= ( kpair @ X8 @ X10 ) )
=> ( X6 = X10 ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR2,axiom,
( setukpairinjR2
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
=> ( Xy = Xu ) ) ) ) ).
thf('33',plain,
( setukpairinjR2
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) )
=> ( X6 = X10 ) ) ) ),
define([status(thm)]) ).
thf(upairequniteq,axiom,
( upairequniteq
= ( ! [Xx: $i,Xy: $i,Xz: $i] :
( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
= ( setadjoin @ Xz @ emptyset ) )
=> ( Xx = Xy ) ) ) ) ).
thf('34',plain,
( upairequniteq
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) )
= ( setadjoin @ X8 @ emptyset ) )
=> ( X4 = X6 ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR1,axiom,
( setukpairinjR1
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
=> ( ( Xz = Xu )
=> ( Xy = Xu ) ) ) ) ) ).
thf('35',plain,
( setukpairinjR1
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X8 = X10 )
=> ( X6 = X10 ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR12,axiom,
( setukpairinjR12
= ( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
thf('36',plain,
( setukpairinjR12
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjR11,axiom,
( setukpairinjR11
= ( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
= ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
thf('37',plain,
( setukpairinjR11
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) )
= ( setadjoin @ X4 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjL,axiom,
( setukpairinjL
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( kpair @ Xx @ Xy )
= ( kpair @ Xz @ Xu ) )
=> ( Xx = Xz ) ) ) ) ).
thf('38',plain,
( setukpairinjL
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( kpair @ X4 @ X6 )
= ( kpair @ X8 @ X10 ) )
=> ( X4 = X8 ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjL2,axiom,
( setukpairinjL2
= ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
=> ( Xx = Xz ) ) ) ) ).
thf('39',plain,
( setukpairinjL2
= ( ! [X4: $i,X6: $i,X8: $i,X10: $i] :
( ( ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) )
=> ( X4 = X8 ) ) ) ),
define([status(thm)]) ).
thf(cartprodfstin,axiom,
( cartprodfstin
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
thf('40',plain,
( cartprodfstin
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ( in @ ( kfst @ X8 ) @ X4 ) ) ) ),
define([status(thm)]) ).
thf(kfstpairEq,axiom,
( kfstpairEq
= ( ! [Xx: $i,Xy: $i] :
( ( kfst @ ( kpair @ Xx @ Xy ) )
= Xx ) ) ) ).
thf('41',plain,
( kfstpairEq
= ( ! [X4: $i,X6: $i] :
( ( kfst @ ( kpair @ X4 @ X6 ) )
= X4 ) ) ),
define([status(thm)]) ).
thf(theprop,axiom,
( theprop
= ( ! [X: $i] :
( ( singleton @ X )
=> ( in @ ( setunion @ X ) @ X ) ) ) ) ).
thf('42',plain,
( theprop
= ( ! [X4: $i] :
( ( singleton @ X4 )
=> ( in @ ( setunion @ X4 ) @ X4 ) ) ) ),
define([status(thm)]) ).
thf(kfstsingleton,axiom,
( kfstsingleton
= ( ! [Xu: $i] :
( ( iskpair @ Xu )
=> ( singleton
@ ( dsetconstr @ ( setunion @ Xu )
@ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
thf('43',plain,
( kfstsingleton
= ( ! [X4: $i] :
( ( iskpair @ X4 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X4 )
@ ^ [V_1: $i] : ( in @ ( setadjoin @ V_1 @ emptyset ) @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairinjL1,axiom,
( setukpairinjL1
= ( ! [Xx: $i,Xy: $i,Xz: $i] :
( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
=> ( Xx = Xz ) ) ) ) ).
thf('44',plain,
( setukpairinjL1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) )
=> ( X4 = X8 ) ) ) ),
define([status(thm)]) ).
thf(singletonsuniq,axiom,
( singletonsuniq
= ( ! [Xx: $i,Xy: $i] :
( ( ( setadjoin @ Xx @ emptyset )
= ( setadjoin @ Xy @ emptyset ) )
=> ( Xx = Xy ) ) ) ) ).
thf('45',plain,
( singletonsuniq
= ( ! [X4: $i,X6: $i] :
( ( ( setadjoin @ X4 @ emptyset )
= ( setadjoin @ X6 @ emptyset ) )
=> ( X4 = X6 ) ) ) ),
define([status(thm)]) ).
thf(ex1I2,axiom,
( ex1I2
= ( ! [A: $i,Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) )
=> ( ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ( ex1 @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
thf('46',plain,
( ex1I2
= ( ! [X4: $i,X6: $i > $o] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( X6 @ X8 )
=> ( ( X6 @ X10 )
=> ( X8 = X10 ) ) ) ) )
=> ( ? [X12: $i] :
( ( X6 @ X12 )
& ( in @ X12 @ X4 ) )
=> ( ex1 @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ),
define([status(thm)]) ).
thf(ex1I,axiom,
( ex1I
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xy )
=> ( Xy = Xx ) ) )
=> ( ex1 @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
thf('47',plain,
( ex1I
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( X6 @ X8 )
=> ( ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( X6 @ X10 )
=> ( X10 = X8 ) ) )
=> ( ex1 @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(ex1E1,axiom,
( ex1E1
= ( ! [A: $i,Xphi: $i > $o] :
( ( ex1 @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('48',plain,
( ex1E1
= ( ! [X4: $i,X6: $i > $o] :
( ( ex1 @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) )
=> ? [X8: $i] :
( ( X6 @ X8 )
& ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(singletonprop,axiom,
( singletonprop
= ( ! [A: $i,Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) )
=> ( ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ( singleton
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
thf('49',plain,
( singletonprop
= ( ! [X4: $i,X6: $i > $o] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( ( X6 @ X8 )
=> ( ( X6 @ X10 )
=> ( X8 = X10 ) ) ) ) )
=> ( ? [X12: $i] :
( ( X6 @ X12 )
& ( in @ X12 @ X4 ) )
=> ( singleton
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(setunionsingleton,axiom,
( setunionsingleton
= ( ! [Xx: $i] :
( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
= Xx ) ) ) ).
thf('50',plain,
( setunionsingleton
= ( ! [X4: $i] :
( ( setunion @ ( setadjoin @ X4 @ emptyset ) )
= X4 ) ) ),
define([status(thm)]) ).
thf(setunionsingleton2,axiom,
( setunionsingleton2
= ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
thf('51',plain,
( setunionsingleton2
= ( ! [X4: $i] : ( subset @ X4 @ ( setunion @ ( setadjoin @ X4 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(setunionsingleton1,axiom,
( setunionsingleton1
= ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
thf('52',plain,
( setunionsingleton1
= ( ! [X4: $i] : ( subset @ ( setunion @ ( setadjoin @ X4 @ emptyset ) ) @ X4 ) ) ),
define([status(thm)]) ).
thf(setunionE2,axiom,
( setunionE2
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ ( setunion @ A ) )
=> ? [X: $i] :
( ( in @ Xx @ X )
& ( in @ X @ A ) ) ) ) ) ).
thf('53',plain,
( setunionE2
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( setunion @ X4 ) )
=> ? [X8: $i] :
( ( in @ X6 @ X8 )
& ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodmempair,axiom,
( cartprodmempair
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ( iskpair @ Xu ) ) ) ) ).
thf('54',plain,
( cartprodmempair
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ( iskpair @ X8 ) ) ) ),
define([status(thm)]) ).
thf(cartprodmempair1,axiom,
( cartprodmempair1
= ( ! [A: $i,B: $i,Xu: $i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ? [Xx: $i] :
( ? [Xy: $i] :
( ( Xu
= ( kpair @ Xx @ Xy ) )
& ( in @ Xy @ B ) )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('55',plain,
( cartprodmempair1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( cartprod @ X4 @ X6 ) )
=> ? [X10: $i] :
( ? [X12: $i] :
( ( X8
= ( kpair @ X10 @ X12 ) )
& ( in @ X12 @ X6 ) )
& ( in @ X10 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(cartprodpairin,axiom,
( cartprodpairin
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
thf('56',plain,
( cartprodpairin
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ ( kpair @ X8 @ X10 ) @ ( cartprod @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(ubforcartprodlem3,axiom,
( ubforcartprodlem3
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
thf('57',plain,
( ubforcartprodlem3
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ ( kpair @ X8 @ X10 ) @ ( powerset @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(ubforcartprodlem2,axiom,
( ubforcartprodlem2
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
thf('58',plain,
( ubforcartprodlem2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(ubforcartprodlem1,axiom,
( ubforcartprodlem1
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
thf('59',plain,
( ubforcartprodlem1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X8 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(upairinpowunion,axiom,
( upairinpowunion
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
thf('60',plain,
( upairinpowunion
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(upairsubunion,axiom,
( upairsubunion
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
thf('61',plain,
( upairsubunion
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( subset @ ( setadjoin @ X8 @ ( setadjoin @ X10 @ emptyset ) ) @ ( binunion @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(upairset2E,axiom,
( upairset2E
= ( ! [Xx: $i,Xy: $i,Xz: $i] :
( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
=> ( ( Xz = Xx )
| ( Xz = Xy ) ) ) ) ) ).
thf('62',plain,
( upairset2E
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) )
=> ( ( X8 = X4 )
| ( X8 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(singletoninpowunion,axiom,
( singletoninpowunion
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
thf('63',plain,
( singletoninpowunion
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ ( setadjoin @ X8 @ emptyset ) @ ( powerset @ ( binunion @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(singletoninpowerset,axiom,
( singletoninpowerset
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
thf('64',plain,
( singletoninpowerset
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ X4 )
=> ( in @ ( setadjoin @ X6 @ emptyset ) @ ( powerset @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(singletonsubset,axiom,
( singletonsubset
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
thf('65',plain,
( singletonsubset
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ X4 )
=> ( subset @ ( setadjoin @ X6 @ emptyset ) @ X4 ) ) ) ),
define([status(thm)]) ).
thf(kpairp,axiom,
( kpairp
= ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
thf('66',plain,
( kpairp
= ( ! [X4: $i,X6: $i] : ( iskpair @ ( kpair @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(kpairiskpair,axiom,
( kpairiskpair
= ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
thf('67',plain,
( kpairiskpair
= ( ! [X4: $i,X6: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairIR,axiom,
( setukpairIR
= ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf('68',plain,
( setukpairIR
= ( ! [X4: $i,X6: $i] : ( in @ X6 @ ( setunion @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) ) ) ),
define([status(thm)]) ).
thf(setukpairIL,axiom,
( setukpairIL
= ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf('69',plain,
( setukpairIL
= ( ! [X4: $i,X6: $i] : ( in @ X4 @ ( setunion @ ( setadjoin @ ( setadjoin @ X4 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) @ emptyset ) ) ) ) ) ),
define([status(thm)]) ).
thf(secondinupair,axiom,
( secondinupair
= ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('70',plain,
( secondinupair
= ( ! [X4: $i,X6: $i] : ( in @ X6 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffIneg2,axiom,
( symdiffIneg2
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf('71',plain,
( symdiffIneg2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffIneg1,axiom,
( symdiffIneg1
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf('72',plain,
( symdiffIneg1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffI2,axiom,
( symdiffI2
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf('73',plain,
( symdiffI2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffI1,axiom,
( symdiffI1
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf('74',plain,
( symdiffI1
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ( in @ X8 @ ( symdiff @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(symdiffE,axiom,
( symdiffE
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( symdiff @ A @ B ) )
=> ! [Xphi: $o] :
( ( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> Xphi ) )
=> ( ( ~ ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> Xphi ) )
=> Xphi ) ) ) ) ) ).
thf('75',plain,
( symdiffE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( symdiff @ X4 @ X6 ) )
=> ! [X10: $o] :
( ( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> X10 ) )
=> ( ( ~ ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> X10 ) )
=> X10 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusSubset1,axiom,
( setminusSubset1
= ( ! [A: $i,B: $i] :
( ( ( setminus @ A @ B )
= emptyset )
=> ( subset @ A @ B ) ) ) ) ).
thf('76',plain,
( setminusSubset1
= ( ! [X4: $i,X6: $i] :
( ( ( setminus @ X4 @ X6 )
= emptyset )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(setminusLsub,axiom,
( setminusLsub
= ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
thf('77',plain,
( setminusLsub
= ( ! [X4: $i,X6: $i] : ( subset @ ( setminus @ X4 @ X6 ) @ X4 ) ) ),
define([status(thm)]) ).
thf(setminusIRneg,axiom,
( setminusIRneg
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ B )
=> ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
thf('78',plain,
( setminusIRneg
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusILneg,axiom,
( setminusILneg
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ A )
=> ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
thf('79',plain,
( setminusILneg
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ X4 )
=> ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusELneg,axiom,
( setminusELneg
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ ( setminus @ A @ B ) )
=> ( ~ ( in @ Xx @ B )
=> ~ ( in @ Xx @ A ) ) ) ) ) ).
thf('80',plain,
( setminusELneg
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) )
=> ( ~ ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusERneg,axiom,
( setminusERneg
= ( ! [A: $i,B: $i,Xx: $i] :
( ~ ( in @ Xx @ ( setminus @ A @ B ) )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ B ) ) ) ) ) ).
thf('81',plain,
( setminusERneg
= ( ! [X4: $i,X6: $i,X8: $i] :
( ~ ( in @ X8 @ ( setminus @ X4 @ X6 ) )
=> ( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setminusSubset2,axiom,
( setminusSubset2
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( setminus @ A @ B )
= emptyset ) ) ) ) ).
thf('82',plain,
( setminusSubset2
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( ( setminus @ X4 @ X6 )
= emptyset ) ) ) ),
define([status(thm)]) ).
thf(setminusER,axiom,
( setminusER
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( setminus @ A @ B ) )
=> ~ ( in @ Xx @ B ) ) ) ) ).
thf('83',plain,
( setminusER
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setminus @ X4 @ X6 ) )
=> ~ ( in @ X8 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(setminusEL,axiom,
( setminusEL
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( setminus @ A @ B ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('84',plain,
( setminusEL
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setminus @ X4 @ X6 ) )
=> ( in @ X8 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(setminusI,axiom,
( setminusI
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
thf('85',plain,
( setminusI
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ( in @ X8 @ ( setminus @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(bs114d,axiom,
( bs114d
= ( ! [A: $i,B: $i,C: $i] :
( ( binintersect @ A @ ( binunion @ B @ C ) )
= ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
thf('86',plain,
( bs114d
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( binintersect @ X4 @ ( binunion @ X6 @ X8 ) )
= ( binunion @ ( binintersect @ X4 @ X6 ) @ ( binintersect @ X4 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset1,axiom,
( binintersectSubset1
= ( ! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= A )
=> ( subset @ A @ B ) ) ) ) ).
thf('87',plain,
( binintersectSubset1
= ( ! [X4: $i,X6: $i] :
( ( ( binintersect @ X4 @ X6 )
= X4 )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset4,axiom,
( binintersectSubset4
= ( ! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( ( binintersect @ A @ B )
= B ) ) ) ) ).
thf('88',plain,
( binintersectSubset4
= ( ! [X4: $i,X6: $i] :
( ( subset @ X6 @ X4 )
=> ( ( binintersect @ X4 @ X6 )
= X6 ) ) ) ),
define([status(thm)]) ).
thf(binintersectRsub,axiom,
( binintersectRsub
= ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
thf('89',plain,
( binintersectRsub
= ( ! [X4: $i,X6: $i] : ( subset @ ( binintersect @ X4 @ X6 ) @ X6 ) ) ),
define([status(thm)]) ).
thf(disjointsetsI1,axiom,
( disjointsetsI1
= ( ! [A: $i,B: $i] :
( ~ ? [Xx: $i] :
( ( in @ Xx @ B )
& ( in @ Xx @ A ) )
=> ( ( binintersect @ A @ B )
= emptyset ) ) ) ) ).
thf('90',plain,
( disjointsetsI1
= ( ! [X4: $i,X6: $i] :
( ~ ? [X8: $i] :
( ( in @ X8 @ X6 )
& ( in @ X8 @ X4 ) )
=> ( ( binintersect @ X4 @ X6 )
= emptyset ) ) ) ),
define([status(thm)]) ).
thf(binintersectER,axiom,
( binintersectER
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( binintersect @ A @ B ) )
=> ( in @ Xx @ B ) ) ) ) ).
thf('91',plain,
( binintersectER
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( binintersect @ X4 @ X6 ) )
=> ( in @ X8 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset3,axiom,
( binintersectSubset3
= ( ! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= B )
=> ( subset @ B @ A ) ) ) ) ).
thf('92',plain,
( binintersectSubset3
= ( ! [X4: $i,X6: $i] :
( ( ( binintersect @ X4 @ X6 )
= X6 )
=> ( subset @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset2,axiom,
( binintersectSubset2
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( binintersect @ A @ B )
= A ) ) ) ) ).
thf('93',plain,
( binintersectSubset2
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( ( binintersect @ X4 @ X6 )
= X4 ) ) ) ),
define([status(thm)]) ).
thf(binintersectLsub,axiom,
( binintersectLsub
= ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
thf('94',plain,
( binintersectLsub
= ( ! [X4: $i,X6: $i] : ( subset @ ( binintersect @ X4 @ X6 ) @ X4 ) ) ),
define([status(thm)]) ).
thf(binintersectEL,axiom,
( binintersectEL
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( binintersect @ A @ B ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('95',plain,
( binintersectEL
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( binintersect @ X4 @ X6 ) )
=> ( in @ X8 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(binintersectSubset5,axiom,
( binintersectSubset5
= ( ! [A: $i,B: $i,C: $i] :
( ( subset @ C @ A )
=> ( ( subset @ C @ B )
=> ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
thf('96',plain,
( binintersectSubset5
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X8 @ X4 )
=> ( ( subset @ X8 @ X6 )
=> ( subset @ X8 @ ( binintersect @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(binintersectI,axiom,
( binintersectI
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx @ B )
=> ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
thf('97',plain,
( binintersectI
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( in @ X8 @ X6 )
=> ( in @ X8 @ ( binintersect @ X4 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(binunionRsub,axiom,
( binunionRsub
= ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
thf('98',plain,
( binunionRsub
= ( ! [X4: $i,X6: $i] : ( subset @ X6 @ ( binunion @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(binunionLsub,axiom,
( binunionLsub
= ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
thf('99',plain,
( binunionLsub
= ( ! [X4: $i,X6: $i] : ( subset @ X4 @ ( binunion @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(binunionE,axiom,
( binunionE
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ ( binunion @ A @ B ) )
=> ( ( in @ Xx @ A )
| ( in @ Xx @ B ) ) ) ) ) ).
thf('100',plain,
( binunionE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( binunion @ X4 @ X6 ) )
=> ( ( in @ X8 @ X4 )
| ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(binunionEcases,axiom,
( binunionEcases
= ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
( ( in @ Xx @ ( binunion @ A @ B ) )
=> ( ( ( in @ Xx @ A )
=> Xphi )
=> ( ( ( in @ Xx @ B )
=> Xphi )
=> Xphi ) ) ) ) ) ).
thf('101',plain,
( binunionEcases
= ( ! [X4: $i,X6: $i,X8: $i,X10: $o] :
( ( in @ X8 @ ( binunion @ X4 @ X6 ) )
=> ( ( ( in @ X8 @ X4 )
=> X10 )
=> ( ( ( in @ X8 @ X6 )
=> X10 )
=> X10 ) ) ) ) ),
define([status(thm)]) ).
thf(binunionIR,axiom,
( binunionIR
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
thf('102',plain,
( binunionIR
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X6 )
=> ( in @ X8 @ ( binunion @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(upairset2IR,axiom,
( upairset2IR
= ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('103',plain,
( upairset2IR
= ( ! [X4: $i,X6: $i] : ( in @ X6 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(binunionIL,axiom,
( binunionIL
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
thf('104',plain,
( binunionIL
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ X8 @ ( binunion @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(sepSubset,axiom,
( sepSubset
= ( ! [A: $i,Xphi: $i > $o] :
( subset
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
@ A ) ) ) ).
thf('105',plain,
( sepSubset
= ( ! [X4: $i,X6: $i > $o] :
( subset
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) )
@ X4 ) ) ),
define([status(thm)]) ).
thf(sepInPowerset,axiom,
( sepInPowerset
= ( ! [A: $i,Xphi: $i > $o] :
( in
@ ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
@ ( powerset @ A ) ) ) ) ).
thf('106',plain,
( sepInPowerset
= ( ! [X4: $i,X6: $i > $o] :
( in
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) )
@ ( powerset @ X4 ) ) ) ),
define([status(thm)]) ).
thf(powersetsubset,axiom,
( powersetsubset
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
thf('107',plain,
( powersetsubset
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( subset @ ( powerset @ X4 ) @ ( powerset @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(inPowerset,axiom,
( inPowerset
= ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
thf('108',plain,
( inPowerset
= ( ! [X4: $i] : ( in @ X4 @ ( powerset @ X4 ) ) ) ),
define([status(thm)]) ).
thf(powersetE1,axiom,
( powersetE1
= ( ! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( subset @ B @ A ) ) ) ) ).
thf('109',plain,
( powersetE1
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( subset @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(powersetI1,axiom,
( powersetI1
= ( ! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( in @ B @ ( powerset @ A ) ) ) ) ) ).
thf('110',plain,
( powersetI1
= ( ! [X4: $i,X6: $i] :
( ( subset @ X6 @ X4 )
=> ( in @ X6 @ ( powerset @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(subsetemptysetimpeq,axiom,
( subsetemptysetimpeq
= ( ! [A: $i] :
( ( subset @ A @ emptyset )
=> ( A = emptyset ) ) ) ) ).
thf('111',plain,
( subsetemptysetimpeq
= ( ! [X4: $i] :
( ( subset @ X4 @ emptyset )
=> ( X4 = emptyset ) ) ) ),
define([status(thm)]) ).
thf(setextsub,axiom,
( setextsub
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ A )
=> ( A = B ) ) ) ) ) ).
thf('112',plain,
( setextsub
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( ( subset @ X6 @ X4 )
=> ( X4 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(subset2powerset,axiom,
( subset2powerset
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( in @ A @ ( powerset @ B ) ) ) ) ) ).
thf('113',plain,
( subset2powerset
= ( ! [X4: $i,X6: $i] :
( ( subset @ X4 @ X6 )
=> ( in @ X4 @ ( powerset @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinSub2,axiom,
( setadjoinSub2
= ( ! [A: $i,Xx: $i,B: $i] :
( ( subset @ A @ B )
=> ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
thf('114',plain,
( setadjoinSub2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X4 @ X8 )
=> ( subset @ X4 @ ( setadjoin @ X6 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinSub,axiom,
( setadjoinSub
= ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
thf('115',plain,
( setadjoinSub
= ( ! [X4: $i,X6: $i] : ( subset @ X6 @ ( setadjoin @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(subsetTrans,axiom,
( subsetTrans
= ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ C )
=> ( subset @ A @ C ) ) ) ) ) ).
thf('116',plain,
( subsetTrans
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X4 @ X6 )
=> ( ( subset @ X6 @ X8 )
=> ( subset @ X4 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(subsetRefl,axiom,
( subsetRefl
= ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
thf('117',plain,
( subsetRefl
= ( ! [X4: $i] : ( subset @ X4 @ X4 ) ) ),
define([status(thm)]) ).
thf(notequalI2,axiom,
( notequalI2
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ( A != B ) ) ) ) ) ).
thf('118',plain,
( notequalI2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ( X4 != X6 ) ) ) ) ),
define([status(thm)]) ).
thf(notequalI1,axiom,
( notequalI1
= ( ! [A: $i,B: $i] :
( ~ ( subset @ A @ B )
=> ( A != B ) ) ) ) ).
thf('119',plain,
( notequalI1
= ( ! [X4: $i,X6: $i] :
( ~ ( subset @ X4 @ X6 )
=> ( X4 != X6 ) ) ) ),
define([status(thm)]) ).
thf(notsubsetI,axiom,
( notsubsetI
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( ~ ( in @ Xx @ B )
=> ~ ( subset @ A @ B ) ) ) ) ) ).
thf('120',plain,
( notsubsetI
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ~ ( in @ X8 @ X6 )
=> ~ ( subset @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(subsetE2,axiom,
( subsetE2
= ( ! [A: $i,B: $i,Xx: $i] :
( ( subset @ A @ B )
=> ( ~ ( in @ Xx @ B )
=> ~ ( in @ Xx @ A ) ) ) ) ) ).
thf('121',plain,
( subsetE2
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X4 @ X6 )
=> ( ~ ( in @ X8 @ X6 )
=> ~ ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(subsetE,axiom,
( subsetE
= ( ! [A: $i,B: $i,Xx: $i] :
( ( subset @ A @ B )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ B ) ) ) ) ) ).
thf('122',plain,
( subsetE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( subset @ X4 @ X6 )
=> ( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(emptysetsubset,axiom,
( emptysetsubset
= ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
thf('123',plain,
( emptysetsubset
= ( ! [X4: $i] : ( subset @ emptyset @ X4 ) ) ),
define([status(thm)]) ).
thf(subsetI2,axiom,
( subsetI2
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( subset @ A @ B ) ) ) ) ).
thf('124',plain,
( subsetI2
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(eqimpsubset1,axiom,
( eqimpsubset1
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ A @ B ) ) ) ) ).
thf('125',plain,
( eqimpsubset1
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(eqimpsubset2,axiom,
( eqimpsubset2
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ B @ A ) ) ) ) ).
thf('126',plain,
( eqimpsubset2
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( subset @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(subsetI1,axiom,
( subsetI1
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( subset @ A @ B ) ) ) ) ).
thf('127',plain,
( subsetI1
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) )
=> ( subset @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(dsetconstr__Cong,axiom,
( dsetconstr__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [Xphi: $i > $o,Xpsi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ! [Xy: $i] :
( ( in @ Xy @ B )
=> ( ( Xx = Xy )
=> ( ( Xphi @ Xx )
<=> ( Xpsi @ Xy ) ) ) ) )
=> ( ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
= ( dsetconstr @ B
@ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
thf('128',plain,
( dsetconstr__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ! [X8: $i > $o,X10: $i > $o] :
( ! [X12: $i] :
( ( in @ X12 @ X4 )
=> ! [X14: $i] :
( ( in @ X14 @ X6 )
=> ( ( X12 = X14 )
=> ( ( X8 @ X12 )
<=> ( X10 @ X14 ) ) ) ) )
=> ( ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X8 @ V_1 ) )
= ( dsetconstr @ X6
@ ^ [V_2: $i] : ( X10 @ V_2 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(descr__Cong,axiom,
( descr__Cong
= ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( Xphi @ Xx )
<=> ( Xpsi @ Xy ) ) )
=> ( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ( ( exu
@ ^ [Xx: $i] : ( Xpsi @ Xx ) )
=> ( ( descr
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
= ( descr
@ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
thf('129',plain,
( descr__Cong
= ( ! [X4: $i > $o,X6: $i > $o] :
( ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( X4 @ X8 )
<=> ( X6 @ X10 ) ) )
=> ( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ( ( exu
@ ^ [V_2: $i] : ( X6 @ V_2 ) )
=> ( ( descr
@ ^ [V_3: $i] : ( X4 @ V_3 ) )
= ( descr
@ ^ [V_4: $i] : ( X6 @ V_4 ) ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(exuEu,axiom,
( exuEu
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ! [Xx: $i,Xy: $i] :
( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ) ) ).
thf('130',plain,
( exuEu
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ! [X6: $i,X8: $i] :
( ( X4 @ X6 )
=> ( ( X4 @ X8 )
=> ( X6 = X8 ) ) ) ) ) ),
define([status(thm)]) ).
thf(omega__Cong,axiom,
( omega__Cong
= ( omega = omega ) ) ).
thf('131',plain,
( omega__Cong
= ( omega = omega ) ),
define([status(thm)]) ).
thf(setunion__Cong,axiom,
( setunion__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( setunion @ A )
= ( setunion @ B ) ) ) ) ) ).
thf('132',plain,
( setunion__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( ( setunion @ X4 )
= ( setunion @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(powerset__Cong,axiom,
( powerset__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( powerset @ A )
= ( powerset @ B ) ) ) ) ) ).
thf('133',plain,
( powerset__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( ( powerset @ X4 )
= ( powerset @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoin__Cong,axiom,
( setadjoin__Cong
= ( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ! [Xz: $i,Xu: $i] :
( ( Xz = Xu )
=> ( ( setadjoin @ Xx @ Xz )
= ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
thf('134',plain,
( setadjoin__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( setadjoin @ X4 @ X8 )
= ( setadjoin @ X6 @ X10 ) ) ) ) ) ),
define([status(thm)]) ).
thf(emptyset__Cong,axiom,
( emptyset__Cong
= ( emptyset = emptyset ) ) ).
thf('135',plain,
( emptyset__Cong
= ( emptyset = emptyset ) ),
define([status(thm)]) ).
thf(exu__Cong,axiom,
( exu__Cong
= ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( Xphi @ Xx )
<=> ( Xpsi @ Xy ) ) )
=> ( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
<=> ( exu
@ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
thf('136',plain,
( exu__Cong
= ( ! [X4: $i > $o,X6: $i > $o] :
( ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( X4 @ X8 )
<=> ( X6 @ X10 ) ) )
=> ( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
<=> ( exu
@ ^ [V_2: $i] : ( X6 @ V_2 ) ) ) ) ) ),
define([status(thm)]) ).
thf(exuE3u,axiom,
( exuE3u
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ! [Xx: $i,Xy: $i] :
( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) ) ) ) ) ).
thf('137',plain,
( exuE3u
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ! [X6: $i,X8: $i] :
( ( X4 @ X6 )
=> ( ( X4 @ X8 )
=> ( X6 = X8 ) ) ) ) ) ),
define([status(thm)]) ).
thf(in__Cong,axiom,
( in__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( in @ Xx @ A )
<=> ( in @ Xy @ B ) ) ) ) ) ) ).
thf('138',plain,
( in__Cong
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( in @ X8 @ X4 )
<=> ( in @ X10 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(inCongP,axiom,
( inCongP
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( ( in @ Xx @ A )
=> ( in @ Xy @ B ) ) ) ) ) ) ).
thf('139',plain,
( inCongP
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ! [X8: $i,X10: $i] :
( ( X8 = X10 )
=> ( ( in @ X8 @ X4 )
=> ( in @ X10 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(exuI2,axiom,
( exuI2
= ( ! [Xphi: $i > $o] :
( ? [Xx: $i] :
! [Xy: $i] :
( ( Xphi @ Xy )
<=> ( Xy = Xx ) )
=> ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
thf('140',plain,
( exuI2
= ( ! [X4: $i > $o] :
( ? [X6: $i] :
! [X8: $i] :
( ( X4 @ X8 )
<=> ( X8 = X6 ) )
=> ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) ) ) ) ),
define([status(thm)]) ).
thf(exuI3,axiom,
( exuI3
= ( ! [Xphi: $i > $o] :
( ? [Xx: $i] : ( Xphi @ Xx )
=> ( ! [Xx: $i,Xy: $i] :
( ( Xphi @ Xx )
=> ( ( Xphi @ Xy )
=> ( Xx = Xy ) ) )
=> ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
thf('141',plain,
( exuI3
= ( ! [X4: $i > $o] :
( ? [X6: $i] : ( X4 @ X6 )
=> ( ! [X8: $i,X10: $i] :
( ( X4 @ X8 )
=> ( ( X4 @ X10 )
=> ( X8 = X10 ) ) )
=> ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) ) ) ) ) ),
define([status(thm)]) ).
thf(exuI1,axiom,
( exuI1
= ( ! [Xphi: $i > $o] :
( ? [Xx: $i] :
( ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) )
& ( Xphi @ Xx ) )
=> ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
thf('142',plain,
( exuI1
= ( ! [X4: $i > $o] :
( ? [X6: $i] :
( ! [X8: $i] :
( ( X4 @ X8 )
=> ( X6 = X8 ) )
& ( X4 @ X6 ) )
=> ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) ) ) ) ),
define([status(thm)]) ).
thf(notdallE,axiom,
( notdallE
= ( ! [A: $i,Xphi: $i > $o] :
( ~ ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ~ ( Xphi @ Xx )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('143',plain,
( notdallE
= ( ! [X4: $i,X6: $i > $o] :
( ~ ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( X6 @ X8 ) )
=> ? [X10: $i] :
( ~ ( X6 @ X10 )
& ( in @ X10 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(notdexE,axiom,
( notdexE
= ( ! [A: $i,Xphi: $i > $o] :
( ~ ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ~ ( Xphi @ Xx ) ) ) ) ) ).
thf('144',plain,
( notdexE
= ( ! [X4: $i,X6: $i > $o] :
( ~ ? [X8: $i] :
( ( X6 @ X8 )
& ( in @ X8 @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ~ ( X6 @ X10 ) ) ) ) ),
define([status(thm)]) ).
thf(prop2set2propI,axiom,
( prop2set2propI
= ( ! [Xphi: $o] :
( Xphi
=> ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
thf('145',plain,
( prop2set2propI
= ( ! [X4: $o] :
( X4
=> ( set2prop @ ( prop2set @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(prop2setI,axiom,
( prop2setI
= ( ! [Xphi: $o] :
( Xphi
=> ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
thf('146',plain,
( prop2setI
= ( ! [X4: $o] :
( X4
=> ( in @ emptyset @ ( prop2set @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(quantDeMorgan4,axiom,
( quantDeMorgan4
= ( ! [A: $i,Xphi: $i > $o] :
( ? [Xx: $i] :
( ~ ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ~ ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( Xphi @ Xx ) ) ) ) ) ).
thf('147',plain,
( quantDeMorgan4
= ( ! [X4: $i,X6: $i > $o] :
( ? [X8: $i] :
( ~ ( X6 @ X8 )
& ( in @ X8 @ X4 ) )
=> ~ ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ( X6 @ X10 ) ) ) ) ),
define([status(thm)]) ).
thf(quantDeMorgan3,axiom,
( quantDeMorgan3
= ( ! [A: $i,Xphi: $i > $o] :
( ~ ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ! [Xx: $i] :
( ( in @ Xx @ A )
=> ~ ( Xphi @ Xx ) ) ) ) ) ).
thf('148',plain,
( quantDeMorgan3
= ( ! [X4: $i,X6: $i > $o] :
( ~ ? [X8: $i] :
( ( X6 @ X8 )
& ( in @ X8 @ X4 ) )
=> ! [X10: $i] :
( ( in @ X10 @ X4 )
=> ~ ( X6 @ X10 ) ) ) ) ),
define([status(thm)]) ).
thf(quantDeMorgan2,axiom,
( quantDeMorgan2
= ( ! [A: $i,Xphi: $i > $o] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ~ ( Xphi @ Xx ) )
=> ~ ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('149',plain,
( quantDeMorgan2
= ( ! [X4: $i,X6: $i > $o] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ~ ( X6 @ X8 ) )
=> ~ ? [X10: $i] :
( ( X6 @ X10 )
& ( in @ X10 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(quantDeMorgan1,axiom,
( quantDeMorgan1
= ( ! [A: $i,Xphi: $i > $o] :
( ~ ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ~ ( Xphi @ Xx )
& ( in @ Xx @ A ) ) ) ) ) ).
thf('150',plain,
( quantDeMorgan1
= ( ! [X4: $i,X6: $i > $o] :
( ~ ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( X6 @ X8 ) )
=> ? [X10: $i] :
( ~ ( X6 @ X10 )
& ( in @ X10 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(vacuousDall,axiom,
( vacuousDall
= ( ! [Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ emptyset )
=> ( Xphi @ Xx ) ) ) ) ).
thf('151',plain,
( vacuousDall
= ( ! [X4: $i > $o,X6: $i] :
( ( in @ X6 @ emptyset )
=> ( X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(emptyE1,axiom,
( emptyE1
= ( ! [A: $i,Xphi: $i > $o] :
( ? [Xx: $i] :
( ( Xphi @ Xx )
& ( in @ Xx @ A ) )
=> ( ( ( dsetconstr @ A
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
= emptyset )
=> $false ) ) ) ) ).
thf('152',plain,
( emptyE1
= ( ! [X4: $i,X6: $i > $o] :
( ? [X8: $i] :
( ( X6 @ X8 )
& ( in @ X8 @ X4 ) )
=> ( ( ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) )
= emptyset )
=> $false ) ) ) ),
define([status(thm)]) ).
thf(upairsetIR,axiom,
( upairsetIR
= ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('153',plain,
( upairsetIR
= ( ! [X4: $i,X6: $i] : ( in @ X6 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(upairsetIL,axiom,
( upairsetIL
= ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('154',plain,
( upairsetIL
= ( ! [X4: $i,X6: $i] : ( in @ X4 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(upairsetE,axiom,
( upairsetE
= ( ! [Xx: $i,Xy: $i,Xz: $i] :
( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
=> ( ( Xz = Xx )
| ( Xz = Xy ) ) ) ) ) ).
thf('155',plain,
( upairsetE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ ( setadjoin @ X6 @ emptyset ) ) )
=> ( ( X8 = X4 )
| ( X8 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(singletonsswitch,axiom,
( singletonsswitch
= ( ! [Xx: $i,Xy: $i] :
( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
=> ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
thf('156',plain,
( singletonsswitch
= ( ! [X4: $i,X6: $i] :
( ( in @ X4 @ ( setadjoin @ X6 @ emptyset ) )
=> ( in @ X6 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(eqinunit,axiom,
( eqinunit
= ( ! [Xx: $i,Xy: $i] :
( ( Xx = Xy )
=> ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
thf('157',plain,
( eqinunit
= ( ! [X4: $i,X6: $i] :
( ( X4 = X6 )
=> ( in @ X4 @ ( setadjoin @ X6 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(notinsingleton,axiom,
( notinsingleton
= ( ! [Xx: $i,Xy: $i] :
( ( Xx != Xy )
=> ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
thf('158',plain,
( notinsingleton
= ( ! [X4: $i,X6: $i] :
( ( X4 != X6 )
=> ~ ( in @ X6 @ ( setadjoin @ X4 @ emptyset ) ) ) ) ),
define([status(thm)]) ).
thf(uniqinunit,axiom,
( uniqinunit
= ( ! [Xx: $i,Xy: $i] :
( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
=> ( Xx = Xy ) ) ) ) ).
thf('159',plain,
( uniqinunit
= ( ! [X4: $i,X6: $i] :
( ( in @ X4 @ ( setadjoin @ X6 @ emptyset ) )
=> ( X4 = X6 ) ) ) ),
define([status(thm)]) ).
thf(nonemptyImpWitness,axiom,
( nonemptyImpWitness
= ( ! [A: $i] :
( ( nonempty @ A )
=> ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
thf('160',plain,
( nonemptyImpWitness
= ( ! [X4: $i] :
( ( nonempty @ X4 )
=> ? [X6: $i] : ( in @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(exuE2,axiom,
( exuE2
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] :
! [Xy: $i] :
( ( Xphi @ Xy )
<=> ( Xy = Xx ) ) ) ) ) ).
thf('161',plain,
( exuE2
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ? [X6: $i] :
! [X8: $i] :
( ( X4 @ X8 )
<=> ( X8 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(subPowSU,axiom,
( subPowSU
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
thf('162',plain,
( subPowSU
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ X4 )
=> ( in @ X6 @ ( powerset @ ( setunion @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(setunionE,axiom,
( setunionE
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ ( setunion @ A ) )
=> ! [Xphi: $o] :
( ! [B: $i] :
( ( in @ Xx @ B )
=> ( ( in @ B @ A )
=> Xphi ) )
=> Xphi ) ) ) ) ).
thf('163',plain,
( setunionE
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( setunion @ X4 ) )
=> ! [X8: $o] :
( ! [X10: $i] :
( ( in @ X6 @ X10 )
=> ( ( in @ X10 @ X4 )
=> X8 ) )
=> X8 ) ) ) ),
define([status(thm)]) ).
thf(setunionI,axiom,
( setunionI
= ( ! [A: $i,Xx: $i,B: $i] :
( ( in @ Xx @ B )
=> ( ( in @ B @ A )
=> ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
thf('164',plain,
( setunionI
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X6 @ X8 )
=> ( ( in @ X8 @ X4 )
=> ( in @ X6 @ ( setunion @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(powersetE,axiom,
( powersetE
= ( ! [A: $i,B: $i,Xx: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( ( in @ Xx @ B )
=> ( in @ Xx @ A ) ) ) ) ) ).
thf('165',plain,
( powersetE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
=> ( ( in @ X8 @ X6 )
=> ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(emptyInPowerset,axiom,
( emptyInPowerset
= ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
thf('166',plain,
( emptyInPowerset
= ( ! [X4: $i] : ( in @ emptyset @ ( powerset @ X4 ) ) ) ),
define([status(thm)]) ).
thf(emptyinPowerset,axiom,
( emptyinPowerset
= ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
thf('167',plain,
( emptyinPowerset
= ( ! [X4: $i] : ( in @ emptyset @ ( powerset @ X4 ) ) ) ),
define([status(thm)]) ).
thf(powersetI,axiom,
( powersetI
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) )
=> ( in @ B @ ( powerset @ A ) ) ) ) ) ).
thf('168',plain,
( powersetI
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X6 )
=> ( in @ X8 @ X4 ) )
=> ( in @ X6 @ ( powerset @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(setoftrueEq,axiom,
( setoftrueEq
= ( ! [A: $i] :
( ( dsetconstr @ A
@ ^ [Xx: $i] : $true )
= A ) ) ) ).
thf('169',plain,
( setoftrueEq
= ( ! [X4: $i] :
( ( dsetconstr @ X4
@ ^ [V_1: $i] : $true )
= X4 ) ) ),
define([status(thm)]) ).
thf(setadjoinOr,axiom,
( setadjoinOr
= ( ! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
=> ( ( Xy = Xx )
| ( in @ Xy @ A ) ) ) ) ) ).
thf('170',plain,
( setadjoinOr
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ X6 ) )
=> ( ( X8 = X4 )
| ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinE,axiom,
( setadjoinE
= ( ! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
=> ! [Xphi: $o] :
( ( ( Xy = Xx )
=> Xphi )
=> ( ( ( in @ Xy @ A )
=> Xphi )
=> Xphi ) ) ) ) ) ).
thf('171',plain,
( setadjoinE
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ X6 ) )
=> ! [X10: $o] :
( ( ( X8 = X4 )
=> X10 )
=> ( ( ( in @ X8 @ X6 )
=> X10 )
=> X10 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinIR,axiom,
( setadjoinIR
= ( ! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ A )
=> ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
thf('172',plain,
( setadjoinIR
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ X6 )
=> ( in @ X8 @ ( setadjoin @ X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(emptyinunitempty,axiom,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
thf('173',plain,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ),
inference(simplify_rw_rule,[status(thm)],[emptyinunitempty]) ).
thf('174',plain,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ),
define([status(thm)]) ).
thf(setadjoinIL,axiom,
( setadjoinIL
= ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
thf('175',plain,
( setadjoinIL
= ( ! [X4: $i,X6: $i] : ( in @ X4 @ ( setadjoin @ X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(nonemptyI1,axiom,
( nonemptyI1
= ( ! [A: $i] :
( ? [Xx: $i] : ( in @ Xx @ A )
=> ( nonempty @ A ) ) ) ) ).
thf('176',plain,
( nonemptyI1
= ( ! [X4: $i] :
( ? [X6: $i] : ( in @ X6 @ X4 )
=> ( nonempty @ X4 ) ) ) ),
define([status(thm)]) ).
thf(nonemptyI,axiom,
( nonemptyI
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( nonempty
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
thf('177',plain,
( nonemptyI
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( X6 @ X8 )
=> ( nonempty
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(nonemptyE1,axiom,
( nonemptyE1
= ( ! [A: $i] :
( ( nonempty @ A )
=> ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
thf('178',plain,
( nonemptyE1
= ( ! [X4: $i] :
( ( nonempty @ X4 )
=> ? [X6: $i] : ( in @ X6 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(nonempty,axiom,
( nonempty
= ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
thf('179',plain,
( nonempty
= ( ^ [Xx: $i] : ( Xx != emptyset ) ) ),
inference(simplify_rw_rule,[status(thm)],[nonempty]) ).
thf('180',plain,
( nonempty
= ( ^ [V_1: $i] : ( V_1 != emptyset ) ) ),
define([status(thm)]) ).
thf(setbeta,axiom,
( setbeta
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
<=> ( Xphi @ Xx ) ) ) ) ) ).
thf('181',plain,
( setbeta
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( in @ X8
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) )
<=> ( X6 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(noeltsimpempty,axiom,
( noeltsimpempty
= ( ! [A: $i] :
( ! [Xx: $i] :
~ ( in @ Xx @ A )
=> ( A = emptyset ) ) ) ) ).
thf('182',plain,
( noeltsimpempty
= ( ! [X4: $i] :
( ! [X6: $i] :
~ ( in @ X6 @ X4 )
=> ( X4 = emptyset ) ) ) ),
define([status(thm)]) ).
thf(emptyI,axiom,
( emptyI
= ( ! [A: $i] :
( ! [Xx: $i] :
~ ( in @ Xx @ A )
=> ( A = emptyset ) ) ) ) ).
thf('183',plain,
( emptyI
= ( ! [X4: $i] :
( ! [X6: $i] :
~ ( in @ X6 @ X4 )
=> ( X4 = emptyset ) ) ) ),
define([status(thm)]) ).
thf(setext,axiom,
( setext
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) )
=> ( A = B ) ) ) ) ) ).
thf('184',plain,
( setext
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
=> ( in @ X8 @ X6 ) )
=> ( ! [X10: $i] :
( ( in @ X10 @ X6 )
=> ( in @ X10 @ X4 ) )
=> ( X4 = X6 ) ) ) ) ),
define([status(thm)]) ).
thf(exuE3e,axiom,
( exuE3e
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
thf('185',plain,
( exuE3e
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ? [X6: $i] : ( X4 @ X6 ) ) ) ),
define([status(thm)]) ).
thf(notinemptyset,axiom,
( notinemptyset
= ( ! [Xx: $i] :
~ ( in @ Xx @ emptyset ) ) ) ).
thf('186',plain,
( notinemptyset
= ( ! [X4: $i] :
~ ( in @ X4 @ emptyset ) ) ),
define([status(thm)]) ).
thf(emptysetimpfalse,axiom,
( emptysetimpfalse
= ( ! [Xx: $i] :
( ( in @ Xx @ emptyset )
=> $false ) ) ) ).
thf('187',plain,
( emptysetimpfalse
= ( ! [X4: $i] :
( ( in @ X4 @ emptyset )
=> $false ) ) ),
define([status(thm)]) ).
thf(emptysetE,axiom,
( emptysetE
= ( ! [Xx: $i] :
( ( in @ Xx @ emptyset )
=> ! [Xphi: $o] : Xphi ) ) ) ).
thf('188',plain,
( emptysetE
= ( ! [X4: $i] :
( ( in @ X4 @ emptyset )
=> ! [X6: $o] : X6 ) ) ),
define([status(thm)]) ).
thf(prop2setE,axiom,
( prop2setE
= ( ! [Xphi: $o,Xx: $i] :
( ( in @ Xx @ ( prop2set @ Xphi ) )
=> Xphi ) ) ) ).
thf('189',plain,
( prop2setE
= ( ! [X4: $o,X6: $i] :
( ( in @ X6 @ ( prop2set @ X4 ) )
=> X4 ) ) ),
define([status(thm)]) ).
thf(exuE1,axiom,
( exuE1
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ? [Xx: $i] :
( ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) )
& ( Xphi @ Xx ) ) ) ) ) ).
thf('190',plain,
( exuE1
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ? [X6: $i] :
( ! [X8: $i] :
( ( X4 @ X8 )
=> ( X6 = X8 ) )
& ( X4 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(dsetconstrER,axiom,
( dsetconstrER
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
=> ( Xphi @ Xx ) ) ) ) ).
thf('191',plain,
( dsetconstrER
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) )
=> ( X6 @ X8 ) ) ) ),
define([status(thm)]) ).
thf(dsetconstrEL,axiom,
( dsetconstrEL
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
=> ( in @ Xx @ A ) ) ) ) ).
thf('192',plain,
( dsetconstrEL
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) )
=> ( in @ X8 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(dsetconstrI,axiom,
( dsetconstrI
= ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
thf('193',plain,
( dsetconstrI
= ( ! [X4: $i,X6: $i > $o,X8: $i] :
( ( in @ X8 @ X4 )
=> ( ( X6 @ X8 )
=> ( in @ X8
@ ( dsetconstr @ X4
@ ^ [V_1: $i] : ( X6 @ V_1 ) ) ) ) ) ) ),
define([status(thm)]) ).
thf(descrp,axiom,
( descrp
= ( ! [Xphi: $i > $o] :
( ( exu
@ ^ [Xx: $i] : ( Xphi @ Xx ) )
=> ( Xphi
@ ( descr
@ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
thf('194',plain,
( descrp
= ( ! [X4: $i > $o] :
( ( exu
@ ^ [V_1: $i] : ( X4 @ V_1 ) )
=> ( X4
@ ( descr
@ ^ [V_2: $i] : ( X4 @ V_2 ) ) ) ) ) ),
define([status(thm)]) ).
thf(wellorderingAx,axiom,
( wellorderingAx
= ( ! [A: $i] :
? [B: $i] :
( ! [C: $i] :
( ( ! [Xx: $i] :
( ( in @ Xx @ C )
=> ( in @ Xx @ A ) )
& ? [Xx: $i] : ( in @ Xx @ C ) )
=> ? [D: $i,Xx: $i] :
( ! [E: $i] :
( ( in @ E @ B )
=> ( ! [Xy: $i] :
( ( in @ Xy @ E )
=> ( in @ Xy @ D ) )
| ( in @ Xx @ E ) ) )
& ~ ? [Xy: $i] :
( ( in @ Xy @ C )
& ( in @ Xy @ D ) )
& ( in @ Xx @ C )
& ( in @ D @ B ) ) )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B ) )
=> ( ! [Xx: $i] :
( ( in @ Xx @ C )
=> ( in @ Xx @ D ) )
| ! [Xx: $i] :
( ( in @ Xx @ D )
=> ( in @ Xx @ C ) ) ) )
& ! [Xx: $i,Xy: $i] :
( ( ( in @ Xx @ A )
& ( in @ Xy @ A ) )
=> ( ! [C: $i] :
( ( in @ C @ B )
=> ( ( in @ Xx @ C )
<=> ( in @ Xy @ C ) ) )
=> ( Xx = Xy ) ) )
& ! [C: $i] :
( ( in @ C @ B )
=> ! [Xx: $i] :
( ( in @ Xx @ C )
=> ( in @ Xx @ A ) ) ) ) ) ) ).
thf('195',plain,
( wellorderingAx
= ( ! [X4: $i] :
? [X6: $i] :
( ! [X8: $i] :
( ( ! [X10: $i] :
( ( in @ X10 @ X8 )
=> ( in @ X10 @ X4 ) )
& ? [X12: $i] : ( in @ X12 @ X8 ) )
=> ? [X14: $i,X16: $i] :
( ! [X18: $i] :
( ( in @ X18 @ X6 )
=> ( ! [X20: $i] :
( ( in @ X20 @ X18 )
=> ( in @ X20 @ X14 ) )
| ( in @ X16 @ X18 ) ) )
& ~ ? [X22: $i] :
( ( in @ X22 @ X8 )
& ( in @ X22 @ X14 ) )
& ( in @ X16 @ X8 )
& ( in @ X14 @ X6 ) ) )
& ! [X24: $i,X26: $i] :
( ( ( in @ X24 @ X6 )
& ( in @ X26 @ X6 ) )
=> ( ! [X28: $i] :
( ( in @ X28 @ X24 )
=> ( in @ X28 @ X26 ) )
| ! [X30: $i] :
( ( in @ X30 @ X26 )
=> ( in @ X30 @ X24 ) ) ) )
& ! [X32: $i,X34: $i] :
( ( ( in @ X32 @ X4 )
& ( in @ X34 @ X4 ) )
=> ( ! [X36: $i] :
( ( in @ X36 @ X6 )
=> ( ( in @ X32 @ X36 )
<=> ( in @ X34 @ X36 ) ) )
=> ( X32 = X34 ) ) )
& ! [X38: $i] :
( ( in @ X38 @ X6 )
=> ! [X40: $i] :
( ( in @ X40 @ X38 )
=> ( in @ X40 @ X4 ) ) ) ) ) ),
define([status(thm)]) ).
thf(foundationAx,axiom,
( foundationAx
= ( ! [A: $i] :
( ? [Xx: $i] : ( in @ Xx @ A )
=> ? [B: $i] :
( ~ ? [Xx: $i] :
( ( in @ Xx @ A )
& ( in @ Xx @ B ) )
& ( in @ B @ A ) ) ) ) ) ).
thf('196',plain,
( foundationAx
= ( ! [X4: $i] :
( ? [X6: $i] : ( in @ X6 @ X4 )
=> ? [X8: $i] :
( ~ ? [X10: $i] :
( ( in @ X10 @ X4 )
& ( in @ X10 @ X8 ) )
& ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(replAx,axiom,
( replAx
= ( ! [Xphi: $i > $i > $o,A: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
=> ( exu
@ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
=> ? [B: $i] :
! [Xx: $i] :
( ( in @ Xx @ B )
<=> ? [Xy: $i] :
( ( Xphi @ Xy @ Xx )
& ( in @ Xy @ A ) ) ) ) ) ) ).
thf('197',plain,
( replAx
= ( ! [X4: $i > $i > $o,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X6 )
=> ( exu
@ ^ [V_1: $i] : ( X4 @ X8 @ V_1 ) ) )
=> ? [X10: $i] :
! [X12: $i] :
( ( in @ X12 @ X10 )
<=> ? [X14: $i] :
( ( X4 @ X14 @ X12 )
& ( in @ X14 @ X6 ) ) ) ) ) ),
define([status(thm)]) ).
thf(omegaIndAx,axiom,
( omegaIndAx
= ( ! [A: $i] :
( ( ( in @ emptyset @ A )
& ! [Xx: $i] :
( ( ( in @ Xx @ omega )
& ( in @ Xx @ A ) )
=> ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
=> ! [Xx: $i] :
( ( in @ Xx @ omega )
=> ( in @ Xx @ A ) ) ) ) ) ).
thf('198',plain,
( omegaIndAx
= ( ! [X4: $i] :
( ( ( in @ emptyset @ X4 )
& ! [X6: $i] :
( ( ( in @ X6 @ omega )
& ( in @ X6 @ X4 ) )
=> ( in @ ( setadjoin @ X6 @ X6 ) @ X4 ) ) )
=> ! [X8: $i] :
( ( in @ X8 @ omega )
=> ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(omegaSAx,axiom,
( omegaSAx
= ( ! [Xx: $i] :
( ( in @ Xx @ omega )
=> ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
thf('199',plain,
( omegaSAx
= ( ! [X4: $i] :
( ( in @ X4 @ omega )
=> ( in @ ( setadjoin @ X4 @ X4 ) @ omega ) ) ) ),
define([status(thm)]) ).
thf(omega0Ax,axiom,
( omega0Ax
= ( in @ emptyset @ omega ) ) ).
thf('200',plain,
( omega0Ax
= ( in @ emptyset @ omega ) ),
inference(simplify_rw_rule,[status(thm)],[omega0Ax]) ).
thf('201',plain,
( omega0Ax
= ( in @ emptyset @ omega ) ),
define([status(thm)]) ).
thf(setunionAx,axiom,
( setunionAx
= ( ! [A: $i,Xx: $i] :
( ( in @ Xx @ ( setunion @ A ) )
<=> ? [B: $i] :
( ( in @ B @ A )
& ( in @ Xx @ B ) ) ) ) ) ).
thf('202',plain,
( setunionAx
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( setunion @ X4 ) )
<=> ? [X8: $i] :
( ( in @ X8 @ X4 )
& ( in @ X6 @ X8 ) ) ) ) ),
define([status(thm)]) ).
thf(powersetAx,axiom,
( powersetAx
= ( ! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
<=> ! [Xx: $i] :
( ( in @ Xx @ B )
=> ( in @ Xx @ A ) ) ) ) ) ).
thf('203',plain,
( powersetAx
= ( ! [X4: $i,X6: $i] :
( ( in @ X6 @ ( powerset @ X4 ) )
<=> ! [X8: $i] :
( ( in @ X8 @ X6 )
=> ( in @ X8 @ X4 ) ) ) ) ),
define([status(thm)]) ).
thf(setadjoinAx,axiom,
( setadjoinAx
= ( ! [Xx: $i,A: $i,Xy: $i] :
( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
<=> ( ( Xy = Xx )
| ( in @ Xy @ A ) ) ) ) ) ).
thf('204',plain,
( setadjoinAx
= ( ! [X4: $i,X6: $i,X8: $i] :
( ( in @ X8 @ ( setadjoin @ X4 @ X6 ) )
<=> ( ( X8 = X4 )
| ( in @ X8 @ X6 ) ) ) ) ),
define([status(thm)]) ).
thf(emptysetAx,axiom,
( emptysetAx
= ( ! [Xx: $i] :
~ ( in @ Xx @ emptyset ) ) ) ).
thf('205',plain,
( emptysetAx
= ( ! [X4: $i] :
~ ( in @ X4 @ emptyset ) ) ),
define([status(thm)]) ).
thf(setextAx,axiom,
( setextAx
= ( ! [A: $i,B: $i] :
( ! [Xx: $i] :
( ( in @ Xx @ A )
<=> ( in @ Xx @ B ) )
=> ( A = B ) ) ) ) ).
thf('206',plain,
( setextAx
= ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
<=> ( in @ X8 @ X6 ) )
=> ( X4 = X6 ) ) ) ),
define([status(thm)]) ).
thf(exu,axiom,
( exu
= ( ^ [Xphi: $i > $o] :
? [Xx: $i] :
( ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) )
& ( Xphi @ Xx ) ) ) ) ).
thf('207',plain,
( exu
= ( ^ [Xphi: $i > $o] :
? [Xx: $i] :
( ! [Xy: $i] :
( ( Xphi @ Xy )
=> ( Xx = Xy ) )
& ( Xphi @ Xx ) ) ) ),
inference(simplify_rw_rule,[status(thm)],[exu]) ).
thf('208',plain,
( exu
= ( ^ [V_1: $i > $o] :
? [X4: $i] :
( ! [X6: $i] :
( ( V_1 @ X6 )
=> ( X4 = X6 ) )
& ( V_1 @ X4 ) ) ) ),
define([status(thm)]) ).
thf(infuncsetfunc,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ! [A: $i,B: $i,Xf: $i] :
( ( in @ Xf @ ( funcSet @ A @ B ) )
=> ( func @ A @ B @ Xf ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(zf_stmt_0,conjecture,
( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
<=> ( in @ X8 @ X6 ) )
=> ( X4 = X6 ) )
=> ( ! [X10: $i] :
~ ( in @ X10 @ emptyset )
=> ( ! [X12: $i,X14: $i,X16: $i] :
( ( in @ X16 @ ( setadjoin @ X12 @ X14 ) )
<=> ( ( in @ X16 @ X14 )
| ( X16 = X12 ) ) )
=> ( ! [X18: $i,X20: $i] :
( ( in @ X20 @ ( powerset @ X18 ) )
<=> ! [X22: $i] :
( ( in @ X22 @ X20 )
=> ( in @ X22 @ X18 ) ) )
=> ( ! [X24: $i,X26: $i] :
( ( in @ X26 @ ( setunion @ X24 ) )
<=> ? [X28: $i] :
( ( in @ X26 @ X28 )
& ( in @ X28 @ X24 ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X30: $i] :
( ( in @ X30 @ omega )
=> ( in @ ( setadjoin @ X30 @ X30 ) @ omega ) )
=> ( ! [X32: $i] :
( ( ! [X34: $i] :
( ( ( in @ X34 @ X32 )
& ( in @ X34 @ omega ) )
=> ( in @ ( setadjoin @ X34 @ X34 ) @ X32 ) )
& ( in @ emptyset @ X32 ) )
=> ! [X36: $i] :
( ( in @ X36 @ omega )
=> ( in @ X36 @ X32 ) ) )
=> ( ! [X38: $i > $i > $o,X40: $i] :
( ! [X42: $i] :
( ( in @ X42 @ X40 )
=> ? [X44: $i] :
( ( X38 @ X42 @ X44 )
& ! [X46: $i] :
( ( X38 @ X42 @ X46 )
=> ( X44 = X46 ) ) ) )
=> ? [X48: $i] :
! [X50: $i] :
( ( in @ X50 @ X48 )
<=> ? [X52: $i] :
( ( in @ X52 @ X40 )
& ( X38 @ X52 @ X50 ) ) ) )
=> ( ! [X54: $i] :
( ? [X56: $i] : ( in @ X56 @ X54 )
=> ? [X58: $i] :
( ( in @ X58 @ X54 )
& ~ ? [X60: $i] :
( ( in @ X60 @ X58 )
& ( in @ X60 @ X54 ) ) ) )
=> ( ! [X62: $i] :
? [X64: $i] :
( ! [X96: $i] :
( ( in @ X96 @ X64 )
=> ! [X98: $i] :
( ( in @ X98 @ X96 )
=> ( in @ X98 @ X62 ) ) )
& ! [X90: $i,X92: $i] :
( ( ( in @ X92 @ X62 )
& ( in @ X90 @ X62 ) )
=> ( ! [X94: $i] :
( ( in @ X94 @ X64 )
=> ( ( in @ X90 @ X94 )
<=> ( in @ X92 @ X94 ) ) )
=> ( X90 = X92 ) ) )
& ! [X82: $i,X84: $i] :
( ( ( in @ X84 @ X64 )
& ( in @ X82 @ X64 ) )
=> ( ! [X88: $i] :
( ( in @ X88 @ X84 )
=> ( in @ X88 @ X82 ) )
| ! [X86: $i] :
( ( in @ X86 @ X82 )
=> ( in @ X86 @ X84 ) ) ) )
& ! [X66: $i] :
( ( ? [X70: $i] : ( in @ X70 @ X66 )
& ! [X68: $i] :
( ( in @ X68 @ X66 )
=> ( in @ X68 @ X62 ) ) )
=> ? [X72: $i,X74: $i] :
( ( in @ X72 @ X64 )
& ( in @ X74 @ X66 )
& ~ ? [X80: $i] :
( ( in @ X80 @ X72 )
& ( in @ X80 @ X66 ) )
& ! [X76: $i] :
( ( in @ X76 @ X64 )
=> ( ( in @ X74 @ X76 )
| ! [X78: $i] :
( ( in @ X78 @ X76 )
=> ( in @ X78 @ X72 ) ) ) ) ) ) )
=> ( ! [X100: $i > $o] :
( ? [X102: $i] :
( ( X100 @ X102 )
& ! [X104: $i] :
( ( X100 @ X104 )
=> ( X102 = X104 ) ) )
=> ( X100
@ ( descr
@ ^ [V_1: $i] : ( X100 @ V_1 ) ) ) )
=> ( ! [X106: $i,X108: $i > $o,X110: $i] :
( ( in @ X110 @ X106 )
=> ( ( X108 @ X110 )
=> ( in @ X110
@ ( dsetconstr @ X106
@ ^ [V_2: $i] : ( X108 @ V_2 ) ) ) ) )
=> ( ! [X112: $i,X114: $i > $o,X116: $i] :
( ( in @ X116
@ ( dsetconstr @ X112
@ ^ [V_3: $i] : ( X114 @ V_3 ) ) )
=> ( in @ X116 @ X112 ) )
=> ( ! [X118: $i,X120: $i > $o,X122: $i] :
( ( in @ X122
@ ( dsetconstr @ X118
@ ^ [V_4: $i] : ( X120 @ V_4 ) ) )
=> ( X120 @ X122 ) )
=> ( ! [X124: $i > $o] :
( ? [X126: $i] :
( ( X124 @ X126 )
& ! [X128: $i] :
( ( X124 @ X128 )
=> ( X126 = X128 ) ) )
=> ? [X130: $i] :
( ( X124 @ X130 )
& ! [X132: $i] :
( ( X124 @ X132 )
=> ( X130 = X132 ) ) ) )
=> ( ! [X134: $o,X136: $i] :
( ( in @ X136 @ ( prop2set @ X134 ) )
=> X134 )
=> ( ! [X138: $i] :
( ( in @ X138 @ emptyset )
=> ! [X140: $o] : X140 )
=> ( ! [X142: $i] :
~ ( in @ X142 @ emptyset )
=> ( ! [X144: $i] :
~ ( in @ X144 @ emptyset )
=> ( ! [X146: $i > $o] :
( ? [X148: $i] :
( ( X146 @ X148 )
& ! [X150: $i] :
( ( X146 @ X150 )
=> ( X148 = X150 ) ) )
=> ? [X152: $i] : ( X146 @ X152 ) )
=> ( ! [X154: $i,X156: $i] :
( ! [X158: $i] :
( ( in @ X158 @ X154 )
=> ( in @ X158 @ X156 ) )
=> ( ! [X160: $i] :
( ( in @ X160 @ X156 )
=> ( in @ X160 @ X154 ) )
=> ( X154 = X156 ) ) )
=> ( ! [X162: $i] :
( ! [X164: $i] :
~ ( in @ X164 @ X162 )
=> ( X162 = emptyset ) )
=> ( ! [X166: $i] :
( ! [X168: $i] :
~ ( in @ X168 @ X166 )
=> ( X166 = emptyset ) )
=> ( ! [X170: $i,X172: $i > $o,X174: $i] :
( ( in @ X174 @ X170 )
=> ( ( in @ X174
@ ( dsetconstr @ X170
@ ^ [V_5: $i] : ( X172 @ V_5 ) ) )
<=> ( X172 @ X174 ) ) )
=> ( ! [X176: $i] :
( ( X176 != emptyset )
=> ? [X178: $i] : ( in @ X178 @ X176 ) )
=> ( ! [X180: $i,X182: $i > $o,X184: $i] :
( ( in @ X184 @ X180 )
=> ( ( X182 @ X184 )
=> ( ( dsetconstr @ X180
@ ^ [V_6: $i] : ( X182 @ V_6 ) )
!= emptyset ) ) )
=> ( ! [X186: $i] :
( ? [X188: $i] : ( in @ X188 @ X186 )
=> ( X186 != emptyset ) )
=> ( ! [X190: $i,X192: $i] : ( in @ X190 @ ( setadjoin @ X190 @ X192 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X194: $i,X196: $i,X198: $i] :
( ( in @ X198 @ X196 )
=> ( in @ X198 @ ( setadjoin @ X194 @ X196 ) ) )
=> ( ! [X200: $i,X202: $i,X204: $i] :
( ( in @ X204 @ ( setadjoin @ X200 @ X202 ) )
=> ! [X206: $o] :
( ( ( X204 = X200 )
=> X206 )
=> ( ( ( in @ X204 @ X202 )
=> X206 )
=> X206 ) ) )
=> ( ! [X208: $i,X210: $i,X212: $i] :
( ( in @ X212 @ ( setadjoin @ X208 @ X210 ) )
=> ( ( in @ X212 @ X210 )
| ( X212 = X208 ) ) )
=> ( ! [X214: $i] :
( ( dsetconstr @ X214
@ ^ [V_7: $i] : $true )
= X214 )
=> ( ! [X216: $i,X218: $i] :
( ! [X220: $i] :
( ( in @ X220 @ X218 )
=> ( in @ X220 @ X216 ) )
=> ( in @ X218 @ ( powerset @ X216 ) ) )
=> ( ! [X222: $i] : ( in @ emptyset @ ( powerset @ X222 ) )
=> ( ! [X224: $i] : ( in @ emptyset @ ( powerset @ X224 ) )
=> ( ! [X226: $i,X228: $i,X230: $i] :
( ( in @ X228 @ ( powerset @ X226 ) )
=> ( ( in @ X230 @ X228 )
=> ( in @ X230 @ X226 ) ) )
=> ( ! [X232: $i,X234: $i,X236: $i] :
( ( in @ X234 @ X236 )
=> ( ( in @ X236 @ X232 )
=> ( in @ X234 @ ( setunion @ X232 ) ) ) )
=> ( ! [X238: $i,X240: $i] :
( ( in @ X240 @ ( setunion @ X238 ) )
=> ! [X242: $o] :
( ! [X244: $i] :
( ( in @ X240 @ X244 )
=> ( ( in @ X244 @ X238 )
=> X242 ) )
=> X242 ) )
=> ( ! [X246: $i,X248: $i] :
( ( in @ X248 @ X246 )
=> ( in @ X248 @ ( powerset @ ( setunion @ X246 ) ) ) )
=> ( ! [X250: $i > $o] :
( ? [X252: $i] :
( ( X250 @ X252 )
& ! [X254: $i] :
( ( X250 @ X254 )
=> ( X252 = X254 ) ) )
=> ? [X256: $i] :
! [X258: $i] :
( ( X250 @ X258 )
<=> ( X258 = X256 ) ) )
=> ( ! [X260: $i] :
( ( X260 != emptyset )
=> ? [X262: $i] : ( in @ X262 @ X260 ) )
=> ( ! [X264: $i,X266: $i] :
( ( in @ X264 @ ( setadjoin @ X266 @ emptyset ) )
=> ( X264 = X266 ) )
=> ( ! [X268: $i,X270: $i] :
( ( X268 != X270 )
=> ~ ( in @ X270 @ ( setadjoin @ X268 @ emptyset ) ) )
=> ( ! [X272: $i,X274: $i] :
( ( X272 = X274 )
=> ( in @ X272 @ ( setadjoin @ X274 @ emptyset ) ) )
=> ( ! [X276: $i,X278: $i] :
( ( in @ X276 @ ( setadjoin @ X278 @ emptyset ) )
=> ( in @ X278 @ ( setadjoin @ X276 @ emptyset ) ) )
=> ( ! [X280: $i,X282: $i,X284: $i] :
( ( in @ X284 @ ( setadjoin @ X280 @ ( setadjoin @ X282 @ emptyset ) ) )
=> ( ( X284 = X282 )
| ( X284 = X280 ) ) )
=> ( ! [X286: $i,X288: $i] : ( in @ X286 @ ( setadjoin @ X286 @ ( setadjoin @ X288 @ emptyset ) ) )
=> ( ! [X290: $i,X292: $i] : ( in @ X292 @ ( setadjoin @ X290 @ ( setadjoin @ X292 @ emptyset ) ) )
=> ( ! [X294: $i,X296: $i > $o] :
( ? [X298: $i] :
( ( in @ X298 @ X294 )
& ( X296 @ X298 ) )
=> ( ( dsetconstr @ X294
@ ^ [V_8: $i] : ( X296 @ V_8 ) )
!= emptyset ) )
=> ( ! [X300: $i > $o,X302: $i] :
( ( in @ X302 @ emptyset )
=> ( X300 @ X302 ) )
=> ( ! [X304: $i,X306: $i > $o] :
( ~ ! [X308: $i] :
( ( in @ X308 @ X304 )
=> ( X306 @ X308 ) )
=> ? [X310: $i] :
( ( in @ X310 @ X304 )
& ~ ( X306 @ X310 ) ) )
=> ( ! [X312: $i,X314: $i > $o] :
( ! [X316: $i] :
( ( in @ X316 @ X312 )
=> ~ ( X314 @ X316 ) )
=> ~ ? [X318: $i] :
( ( in @ X318 @ X312 )
& ( X314 @ X318 ) ) )
=> ( ! [X320: $i,X322: $i > $o] :
( ~ ? [X324: $i] :
( ( in @ X324 @ X320 )
& ( X322 @ X324 ) )
=> ! [X326: $i] :
( ( in @ X326 @ X320 )
=> ~ ( X322 @ X326 ) ) )
=> ( ! [X328: $i,X330: $i > $o] :
( ? [X332: $i] :
( ( in @ X332 @ X328 )
& ~ ( X330 @ X332 ) )
=> ~ ! [X334: $i] :
( ( in @ X334 @ X328 )
=> ( X330 @ X334 ) ) )
=> ( ! [X336: $o] :
( X336
=> ( in @ emptyset @ ( prop2set @ X336 ) ) )
=> ( ! [X338: $o] :
( X338
=> ( set2prop @ ( prop2set @ X338 ) ) )
=> ( ! [X340: $i,X342: $i > $o] :
( ~ ? [X344: $i] :
( ( in @ X344 @ X340 )
& ( X342 @ X344 ) )
=> ! [X346: $i] :
( ( in @ X346 @ X340 )
=> ~ ( X342 @ X346 ) ) )
=> ( ! [X348: $i,X350: $i > $o] :
( ~ ! [X352: $i] :
( ( in @ X352 @ X348 )
=> ( X350 @ X352 ) )
=> ? [X354: $i] :
( ( in @ X354 @ X348 )
& ~ ( X350 @ X354 ) ) )
=> ( ! [X356: $i > $o] :
( ? [X358: $i] :
( ( X356 @ X358 )
& ! [X360: $i] :
( ( X356 @ X360 )
=> ( X358 = X360 ) ) )
=> ? [X362: $i] :
( ( X356 @ X362 )
& ! [X364: $i] :
( ( X356 @ X364 )
=> ( X362 = X364 ) ) ) )
=> ( ! [X366: $i > $o] :
( ? [X368: $i] : ( X366 @ X368 )
=> ( ! [X370: $i,X372: $i] :
( ( X366 @ X370 )
=> ( ( X366 @ X372 )
=> ( X370 = X372 ) ) )
=> ? [X374: $i] :
( ( X366 @ X374 )
& ! [X376: $i] :
( ( X366 @ X376 )
=> ( X374 = X376 ) ) ) ) )
=> ( ! [X378: $i > $o] :
( ? [X380: $i] :
! [X382: $i] :
( ( X378 @ X382 )
<=> ( X382 = X380 ) )
=> ? [X384: $i] :
( ( X378 @ X384 )
& ! [X386: $i] :
( ( X378 @ X386 )
=> ( X384 = X386 ) ) ) )
=> ( ! [X388: $i,X390: $i] :
( ( X388 = X390 )
=> ! [X392: $i,X394: $i] :
( ( X392 = X394 )
=> ( ( in @ X392 @ X388 )
=> ( in @ X394 @ X390 ) ) ) )
=> ( ! [X396: $i,X398: $i] :
( ( X396 = X398 )
=> ! [X400: $i,X402: $i] :
( ( X400 = X402 )
=> ( ( in @ X400 @ X396 )
<=> ( in @ X402 @ X398 ) ) ) )
=> ( ! [X404: $i > $o] :
( ? [X406: $i] :
( ( X404 @ X406 )
& ! [X408: $i] :
( ( X404 @ X408 )
=> ( X406 = X408 ) ) )
=> ! [X410: $i,X412: $i] :
( ( X404 @ X410 )
=> ( ( X404 @ X412 )
=> ( X410 = X412 ) ) ) )
=> ( ! [X414: $i > $o,X416: $i > $o] :
( ! [X418: $i,X420: $i] :
( ( X418 = X420 )
=> ( ( X414 @ X418 )
<=> ( X416 @ X420 ) ) )
=> ( ? [X422: $i] :
( ( X414 @ X422 )
& ! [X424: $i] :
( ( X414 @ X424 )
=> ( X422 = X424 ) ) )
<=> ? [X426: $i] :
( ( X416 @ X426 )
& ! [X428: $i] :
( ( X416 @ X428 )
=> ( X426 = X428 ) ) ) ) )
=> ( ( emptyset = emptyset )
=> ( ! [X430: $i,X432: $i] :
( ( X430 = X432 )
=> ! [X434: $i,X436: $i] :
( ( X434 = X436 )
=> ( ( setadjoin @ X430 @ X434 )
= ( setadjoin @ X432 @ X436 ) ) ) )
=> ( ! [X438: $i,X440: $i] :
( ( X438 = X440 )
=> ( ( powerset @ X438 )
= ( powerset @ X440 ) ) )
=> ( ! [X442: $i,X444: $i] :
( ( X442 = X444 )
=> ( ( setunion @ X442 )
= ( setunion @ X444 ) ) )
=> ( ( omega = omega )
=> ( ! [X446: $i > $o] :
( ? [X448: $i] :
( ( X446 @ X448 )
& ! [X450: $i] :
( ( X446 @ X450 )
=> ( X448 = X450 ) ) )
=> ! [X452: $i,X454: $i] :
( ( X446 @ X452 )
=> ( ( X446 @ X454 )
=> ( X452 = X454 ) ) ) )
=> ( ! [X456: $i > $o,X458: $i > $o] :
( ! [X460: $i,X462: $i] :
( ( X460 = X462 )
=> ( ( X456 @ X460 )
<=> ( X458 @ X462 ) ) )
=> ( ? [X464: $i] :
( ( X456 @ X464 )
& ! [X466: $i] :
( ( X456 @ X466 )
=> ( X464 = X466 ) ) )
=> ( ? [X468: $i] :
( ( X458 @ X468 )
& ! [X470: $i] :
( ( X458 @ X470 )
=> ( X468 = X470 ) ) )
=> ( ( descr
@ ^ [V_9: $i] : ( X456 @ V_9 ) )
= ( descr
@ ^ [V_10: $i] : ( X458 @ V_10 ) ) ) ) ) )
=> ( ! [X472: $i,X474: $i] :
( ( X472 = X474 )
=> ! [X476: $i > $o,X478: $i > $o] :
( ! [X480: $i] :
( ( in @ X480 @ X472 )
=> ! [X482: $i] :
( ( in @ X482 @ X474 )
=> ( ( X480 = X482 )
=> ( ( X476 @ X480 )
<=> ( X478 @ X482 ) ) ) ) )
=> ( ( dsetconstr @ X472
@ ^ [V_11: $i] : ( X476 @ V_11 ) )
= ( dsetconstr @ X474
@ ^ [V_12: $i] : ( X478 @ V_12 ) ) ) ) )
=> ( ! [X484: $i,X486: $i] :
( ! [X488: $i] :
( ( in @ X488 @ X484 )
=> ( in @ X488 @ X486 ) )
=> ( subset @ X484 @ X486 ) )
=> ( ! [X490: $i,X492: $i] :
( ( X490 = X492 )
=> ( subset @ X492 @ X490 ) )
=> ( ! [X494: $i,X496: $i] :
( ( X494 = X496 )
=> ( subset @ X494 @ X496 ) )
=> ( ! [X498: $i,X500: $i] :
( ! [X502: $i] :
( ( in @ X502 @ X498 )
=> ( in @ X502 @ X500 ) )
=> ( subset @ X498 @ X500 ) )
=> ( ! [X504: $i] : ( subset @ emptyset @ X504 )
=> ( ! [X506: $i,X508: $i,X510: $i] :
( ( subset @ X506 @ X508 )
=> ( ( in @ X510 @ X506 )
=> ( in @ X510 @ X508 ) ) )
=> ( ! [X512: $i,X514: $i,X516: $i] :
( ( subset @ X512 @ X514 )
=> ( ~ ( in @ X516 @ X514 )
=> ~ ( in @ X516 @ X512 ) ) )
=> ( ! [X518: $i,X520: $i,X522: $i] :
( ( in @ X522 @ X518 )
=> ( ~ ( in @ X522 @ X520 )
=> ~ ( subset @ X518 @ X520 ) ) )
=> ( ! [X524: $i,X526: $i] :
( ~ ( subset @ X524 @ X526 )
=> ( X524 != X526 ) )
=> ( ! [X528: $i,X530: $i,X532: $i] :
( ( in @ X532 @ X528 )
=> ( ~ ( in @ X532 @ X530 )
=> ( X528 != X530 ) ) )
=> ( ! [X534: $i] : ( subset @ X534 @ X534 )
=> ( ! [X536: $i,X538: $i,X540: $i] :
( ( subset @ X536 @ X538 )
=> ( ( subset @ X538 @ X540 )
=> ( subset @ X536 @ X540 ) ) )
=> ( ! [X542: $i,X544: $i] : ( subset @ X544 @ ( setadjoin @ X542 @ X544 ) )
=> ( ! [X546: $i,X548: $i,X550: $i] :
( ( subset @ X546 @ X550 )
=> ( subset @ X546 @ ( setadjoin @ X548 @ X550 ) ) )
=> ( ! [X552: $i,X554: $i] :
( ( subset @ X552 @ X554 )
=> ( in @ X552 @ ( powerset @ X554 ) ) )
=> ( ! [X556: $i,X558: $i] :
( ( subset @ X556 @ X558 )
=> ( ( subset @ X558 @ X556 )
=> ( X556 = X558 ) ) )
=> ( ! [X560: $i] :
( ( subset @ X560 @ emptyset )
=> ( X560 = emptyset ) )
=> ( ! [X562: $i,X564: $i] :
( ( subset @ X564 @ X562 )
=> ( in @ X564 @ ( powerset @ X562 ) ) )
=> ( ! [X566: $i,X568: $i] :
( ( in @ X568 @ ( powerset @ X566 ) )
=> ( subset @ X568 @ X566 ) )
=> ( ! [X570: $i] : ( in @ X570 @ ( powerset @ X570 ) )
=> ( ! [X572: $i,X574: $i] :
( ( subset @ X572 @ X574 )
=> ( subset @ ( powerset @ X572 ) @ ( powerset @ X574 ) ) )
=> ( ! [X576: $i,X578: $i > $o] :
( in
@ ( dsetconstr @ X576
@ ^ [V_13: $i] : ( X578 @ V_13 ) )
@ ( powerset @ X576 ) )
=> ( ! [X580: $i,X582: $i > $o] :
( subset
@ ( dsetconstr @ X580
@ ^ [V_14: $i] : ( X582 @ V_14 ) )
@ X580 )
=> ( ! [X584: $i,X586: $i,X588: $i] :
( ( in @ X588 @ X584 )
=> ( in @ X588 @ ( binunion @ X584 @ X586 ) ) )
=> ( ! [X590: $i,X592: $i] : ( in @ X592 @ ( setadjoin @ X590 @ ( setadjoin @ X592 @ emptyset ) ) )
=> ( ! [X594: $i,X596: $i,X598: $i] :
( ( in @ X598 @ X596 )
=> ( in @ X598 @ ( binunion @ X594 @ X596 ) ) )
=> ( ! [X600: $i,X602: $i,X604: $i,X606: $o] :
( ( in @ X604 @ ( binunion @ X600 @ X602 ) )
=> ( ( ( in @ X604 @ X600 )
=> X606 )
=> ( ( ( in @ X604 @ X602 )
=> X606 )
=> X606 ) ) )
=> ( ! [X608: $i,X610: $i,X612: $i] :
( ( in @ X612 @ ( binunion @ X608 @ X610 ) )
=> ( ( in @ X612 @ X610 )
| ( in @ X612 @ X608 ) ) )
=> ( ! [X614: $i,X616: $i] : ( subset @ X614 @ ( binunion @ X614 @ X616 ) )
=> ( ! [X618: $i,X620: $i] : ( subset @ X620 @ ( binunion @ X618 @ X620 ) )
=> ( ! [X622: $i,X624: $i,X626: $i] :
( ( in @ X626 @ X622 )
=> ( ( in @ X626 @ X624 )
=> ( in @ X626 @ ( binintersect @ X622 @ X624 ) ) ) )
=> ( ! [X628: $i,X630: $i,X632: $i] :
( ( subset @ X632 @ X628 )
=> ( ( subset @ X632 @ X630 )
=> ( subset @ X632 @ ( binintersect @ X628 @ X630 ) ) ) )
=> ( ! [X634: $i,X636: $i,X638: $i] :
( ( in @ X638 @ ( binintersect @ X634 @ X636 ) )
=> ( in @ X638 @ X634 ) )
=> ( ! [X640: $i,X642: $i] : ( subset @ ( binintersect @ X640 @ X642 ) @ X640 )
=> ( ! [X644: $i,X646: $i] :
( ( subset @ X644 @ X646 )
=> ( ( binintersect @ X644 @ X646 )
= X644 ) )
=> ( ! [X648: $i,X650: $i] :
( ( ( binintersect @ X648 @ X650 )
= X650 )
=> ( subset @ X650 @ X648 ) )
=> ( ! [X652: $i,X654: $i,X656: $i] :
( ( in @ X656 @ ( binintersect @ X652 @ X654 ) )
=> ( in @ X656 @ X654 ) )
=> ( ! [X658: $i,X660: $i] :
( ~ ? [X662: $i] :
( ( in @ X662 @ X658 )
& ( in @ X662 @ X660 ) )
=> ( ( binintersect @ X658 @ X660 )
= emptyset ) )
=> ( ! [X664: $i,X666: $i] : ( subset @ ( binintersect @ X664 @ X666 ) @ X666 )
=> ( ! [X668: $i,X670: $i] :
( ( subset @ X670 @ X668 )
=> ( ( binintersect @ X668 @ X670 )
= X670 ) )
=> ( ! [X672: $i,X674: $i] :
( ( ( binintersect @ X672 @ X674 )
= X672 )
=> ( subset @ X672 @ X674 ) )
=> ( ! [X676: $i,X678: $i,X680: $i] :
( ( binintersect @ X676 @ ( binunion @ X678 @ X680 ) )
= ( binunion @ ( binintersect @ X676 @ X678 ) @ ( binintersect @ X676 @ X680 ) ) )
=> ( ! [X682: $i,X684: $i,X686: $i] :
( ( in @ X686 @ X682 )
=> ( ~ ( in @ X686 @ X684 )
=> ( in @ X686 @ ( setminus @ X682 @ X684 ) ) ) )
=> ( ! [X688: $i,X690: $i,X692: $i] :
( ( in @ X692 @ ( setminus @ X688 @ X690 ) )
=> ( in @ X692 @ X688 ) )
=> ( ! [X694: $i,X696: $i,X698: $i] :
( ( in @ X698 @ ( setminus @ X694 @ X696 ) )
=> ~ ( in @ X698 @ X696 ) )
=> ( ! [X700: $i,X702: $i] :
( ( subset @ X700 @ X702 )
=> ( ( setminus @ X700 @ X702 )
= emptyset ) )
=> ( ! [X704: $i,X706: $i,X708: $i] :
( ~ ( in @ X708 @ ( setminus @ X704 @ X706 ) )
=> ( ( in @ X708 @ X704 )
=> ( in @ X708 @ X706 ) ) )
=> ( ! [X710: $i,X712: $i,X714: $i] :
( ~ ( in @ X714 @ ( setminus @ X710 @ X712 ) )
=> ( ~ ( in @ X714 @ X712 )
=> ~ ( in @ X714 @ X710 ) ) )
=> ( ! [X716: $i,X718: $i,X720: $i] :
( ~ ( in @ X720 @ X716 )
=> ~ ( in @ X720 @ ( setminus @ X716 @ X718 ) ) )
=> ( ! [X722: $i,X724: $i,X726: $i] :
( ( in @ X726 @ X724 )
=> ~ ( in @ X726 @ ( setminus @ X722 @ X724 ) ) )
=> ( ! [X728: $i,X730: $i] : ( subset @ ( setminus @ X728 @ X730 ) @ X728 )
=> ( ! [X732: $i,X734: $i] :
( ( ( setminus @ X732 @ X734 )
= emptyset )
=> ( subset @ X732 @ X734 ) )
=> ( ! [X736: $i,X738: $i,X740: $i] :
( ( in @ X740 @ ( symdiff @ X736 @ X738 ) )
=> ! [X742: $o] :
( ( ( in @ X740 @ X736 )
=> ( ~ ( in @ X740 @ X738 )
=> X742 ) )
=> ( ( ~ ( in @ X740 @ X736 )
=> ( ( in @ X740 @ X738 )
=> X742 ) )
=> X742 ) ) )
=> ( ! [X744: $i,X746: $i,X748: $i] :
( ( in @ X748 @ X744 )
=> ( ~ ( in @ X748 @ X746 )
=> ( in @ X748 @ ( symdiff @ X744 @ X746 ) ) ) )
=> ( ! [X750: $i,X752: $i,X754: $i] :
( ~ ( in @ X754 @ X750 )
=> ( ( in @ X754 @ X752 )
=> ( in @ X754 @ ( symdiff @ X750 @ X752 ) ) ) )
=> ( ! [X756: $i,X758: $i,X760: $i] :
( ( in @ X760 @ X756 )
=> ( ( in @ X760 @ X758 )
=> ~ ( in @ X760 @ ( symdiff @ X756 @ X758 ) ) ) )
=> ( ! [X762: $i,X764: $i,X766: $i] :
( ~ ( in @ X766 @ X762 )
=> ( ~ ( in @ X766 @ X764 )
=> ~ ( in @ X766 @ ( symdiff @ X762 @ X764 ) ) ) )
=> ( ! [X768: $i,X770: $i] : ( in @ X770 @ ( setadjoin @ X768 @ ( setadjoin @ X770 @ emptyset ) ) )
=> ( ! [X772: $i,X774: $i] : ( in @ X772 @ ( setunion @ ( setadjoin @ ( setadjoin @ X772 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X772 @ ( setadjoin @ X774 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X776: $i,X778: $i] : ( in @ X778 @ ( setunion @ ( setadjoin @ ( setadjoin @ X776 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X776 @ ( setadjoin @ X778 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X780: $i,X782: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X780 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X780 @ ( setadjoin @ X782 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X784: $i,X786: $i] : ( iskpair @ ( kpair @ X784 @ X786 ) )
=> ( ! [X788: $i,X790: $i] :
( ( in @ X790 @ X788 )
=> ( subset @ ( setadjoin @ X790 @ emptyset ) @ X788 ) )
=> ( ! [X792: $i,X794: $i] :
( ( in @ X794 @ X792 )
=> ( in @ ( setadjoin @ X794 @ emptyset ) @ ( powerset @ X792 ) ) )
=> ( ! [X796: $i,X798: $i,X800: $i] :
( ( in @ X800 @ X796 )
=> ( in @ ( setadjoin @ X800 @ emptyset ) @ ( powerset @ ( binunion @ X796 @ X798 ) ) ) )
=> ( ! [X802: $i,X804: $i,X806: $i] :
( ( in @ X806 @ ( setadjoin @ X802 @ ( setadjoin @ X804 @ emptyset ) ) )
=> ( ( X806 = X804 )
| ( X806 = X802 ) ) )
=> ( ! [X808: $i,X810: $i,X812: $i] :
( ( in @ X812 @ X808 )
=> ! [X814: $i] :
( ( in @ X814 @ X810 )
=> ( subset @ ( setadjoin @ X812 @ ( setadjoin @ X814 @ emptyset ) ) @ ( binunion @ X808 @ X810 ) ) ) )
=> ( ! [X816: $i,X818: $i,X820: $i] :
( ( in @ X820 @ X816 )
=> ! [X822: $i] :
( ( in @ X822 @ X818 )
=> ( in @ ( setadjoin @ X820 @ ( setadjoin @ X822 @ emptyset ) ) @ ( powerset @ ( binunion @ X816 @ X818 ) ) ) ) )
=> ( ! [X824: $i,X826: $i,X828: $i] :
( ( in @ X828 @ X824 )
=> ! [X830: $i] :
( ( in @ X830 @ X826 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X828 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X828 @ ( setadjoin @ X830 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X824 @ X826 ) ) ) ) )
=> ( ! [X832: $i,X834: $i,X836: $i] :
( ( in @ X836 @ X832 )
=> ! [X838: $i] :
( ( in @ X838 @ X834 )
=> ( in @ ( setadjoin @ ( setadjoin @ X836 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X836 @ ( setadjoin @ X838 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X832 @ X834 ) ) ) ) ) )
=> ( ! [X840: $i,X842: $i,X844: $i] :
( ( in @ X844 @ X840 )
=> ! [X846: $i] :
( ( in @ X846 @ X842 )
=> ( in @ ( kpair @ X844 @ X846 ) @ ( powerset @ ( powerset @ ( binunion @ X840 @ X842 ) ) ) ) ) )
=> ( ! [X848: $i,X850: $i,X852: $i] :
( ( in @ X852 @ X848 )
=> ! [X854: $i] :
( ( in @ X854 @ X850 )
=> ( in @ ( kpair @ X852 @ X854 ) @ ( cartprod @ X848 @ X850 ) ) ) )
=> ( ! [X856: $i,X858: $i,X860: $i] :
( ( in @ X860 @ ( cartprod @ X856 @ X858 ) )
=> ? [X862: $i] :
( ( in @ X862 @ X856 )
& ? [X864: $i] :
( ( in @ X864 @ X858 )
& ( X860
= ( kpair @ X862 @ X864 ) ) ) ) )
=> ( ! [X866: $i,X868: $i,X870: $i] :
( ( in @ X870 @ ( cartprod @ X866 @ X868 ) )
=> ( iskpair @ X870 ) )
=> ( ! [X872: $i,X874: $i] :
( ( in @ X874 @ ( setunion @ X872 ) )
=> ? [X876: $i] :
( ( in @ X876 @ X872 )
& ( in @ X874 @ X876 ) ) )
=> ( ! [X878: $i] : ( subset @ ( setunion @ ( setadjoin @ X878 @ emptyset ) ) @ X878 )
=> ( ! [X880: $i] : ( subset @ X880 @ ( setunion @ ( setadjoin @ X880 @ emptyset ) ) )
=> ( ! [X882: $i] :
( ( setunion @ ( setadjoin @ X882 @ emptyset ) )
= X882 )
=> ( ! [X884: $i,X886: $i > $o] :
( ! [X888: $i] :
( ( in @ X888 @ X884 )
=> ! [X890: $i] :
( ( in @ X890 @ X884 )
=> ( ( X886 @ X888 )
=> ( ( X886 @ X890 )
=> ( X888 = X890 ) ) ) ) )
=> ( ? [X892: $i] :
( ( in @ X892 @ X884 )
& ( X886 @ X892 ) )
=> ? [X894: $i] :
( ( in @ X894
@ ( dsetconstr @ X884
@ ^ [V_16: $i] : ( X886 @ V_16 ) ) )
& ( ( dsetconstr @ X884
@ ^ [V_15: $i] : ( X886 @ V_15 ) )
= ( setadjoin @ X894 @ emptyset ) ) ) ) )
=> ( ! [X896: $i,X898: $i > $o] :
( ? [X900: $i] :
( ( in @ X900
@ ( dsetconstr @ X896
@ ^ [V_18: $i] : ( X898 @ V_18 ) ) )
& ( ( dsetconstr @ X896
@ ^ [V_17: $i] : ( X898 @ V_17 ) )
= ( setadjoin @ X900 @ emptyset ) ) )
=> ? [X902: $i] :
( ( in @ X902 @ X896 )
& ( X898 @ X902 ) ) )
=> ( ! [X904: $i,X906: $i > $o,X908: $i] :
( ( in @ X908 @ X904 )
=> ( ( X906 @ X908 )
=> ( ! [X910: $i] :
( ( in @ X910 @ X904 )
=> ( ( X906 @ X910 )
=> ( X910 = X908 ) ) )
=> ? [X912: $i] :
( ( in @ X912
@ ( dsetconstr @ X904
@ ^ [V_20: $i] : ( X906 @ V_20 ) ) )
& ( ( dsetconstr @ X904
@ ^ [V_19: $i] : ( X906 @ V_19 ) )
= ( setadjoin @ X912 @ emptyset ) ) ) ) ) )
=> ( ! [X914: $i,X916: $i > $o] :
( ! [X918: $i] :
( ( in @ X918 @ X914 )
=> ! [X920: $i] :
( ( in @ X920 @ X914 )
=> ( ( X916 @ X918 )
=> ( ( X916 @ X920 )
=> ( X918 = X920 ) ) ) ) )
=> ( ? [X922: $i] :
( ( in @ X922 @ X914 )
& ( X916 @ X922 ) )
=> ? [X924: $i] :
( ( in @ X924
@ ( dsetconstr @ X914
@ ^ [V_22: $i] : ( X916 @ V_22 ) ) )
& ( ( dsetconstr @ X914
@ ^ [V_21: $i] : ( X916 @ V_21 ) )
= ( setadjoin @ X924 @ emptyset ) ) ) ) )
=> ( ! [X926: $i,X928: $i] :
( ( ( setadjoin @ X926 @ emptyset )
= ( setadjoin @ X928 @ emptyset ) )
=> ( X926 = X928 ) )
=> ( ! [X930: $i,X932: $i,X934: $i] :
( ( in @ ( setadjoin @ X934 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X930 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X930 @ ( setadjoin @ X932 @ emptyset ) ) @ emptyset ) ) )
=> ( X930 = X934 ) )
=> ( ! [X936: $i] :
( ( iskpair @ X936 )
=> ? [X938: $i] :
( ( in @ X938
@ ( dsetconstr @ ( setunion @ X936 )
@ ^ [V_24: $i] : ( in @ ( setadjoin @ V_24 @ emptyset ) @ X936 ) ) )
& ( ( dsetconstr @ ( setunion @ X936 )
@ ^ [V_23: $i] : ( in @ ( setadjoin @ V_23 @ emptyset ) @ X936 ) )
= ( setadjoin @ X938 @ emptyset ) ) ) )
=> ( ! [X940: $i] :
( ? [X942: $i] :
( ( in @ X942 @ X940 )
& ( X940
= ( setadjoin @ X942 @ emptyset ) ) )
=> ( in @ ( setunion @ X940 ) @ X940 ) )
=> ( ! [X944: $i,X946: $i] :
( ( kfst @ ( kpair @ X944 @ X946 ) )
= X944 )
=> ( ! [X948: $i,X950: $i,X952: $i] :
( ( in @ X952 @ ( cartprod @ X948 @ X950 ) )
=> ( in @ ( kfst @ X952 ) @ X948 ) )
=> ( ! [X954: $i,X956: $i,X958: $i,X960: $i] :
( ( ( setadjoin @ ( setadjoin @ X954 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X954 @ ( setadjoin @ X956 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X958 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X958 @ ( setadjoin @ X960 @ emptyset ) ) @ emptyset ) ) )
=> ( X954 = X958 ) )
=> ( ! [X962: $i,X964: $i,X966: $i,X968: $i] :
( ( ( kpair @ X962 @ X964 )
= ( kpair @ X966 @ X968 ) )
=> ( X962 = X966 ) )
=> ( ! [X970: $i,X972: $i] :
( ( X970 = X972 )
=> ( ( setadjoin @ X970 @ ( setadjoin @ X972 @ emptyset ) )
= ( setadjoin @ X970 @ emptyset ) ) )
=> ( ! [X974: $i,X976: $i] :
( ( X974 = X976 )
=> ( ( setadjoin @ ( setadjoin @ X974 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X974 @ ( setadjoin @ X976 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X974 @ emptyset ) @ emptyset ) ) )
=> ( ! [X978: $i,X980: $i,X982: $i,X984: $i] :
( ( ( setadjoin @ ( setadjoin @ X978 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X978 @ ( setadjoin @ X980 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X982 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X982 @ ( setadjoin @ X984 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X982 = X984 )
=> ( X980 = X984 ) ) )
=> ( ! [X986: $i,X988: $i,X990: $i] :
( ( ( setadjoin @ X986 @ ( setadjoin @ X988 @ emptyset ) )
= ( setadjoin @ X990 @ emptyset ) )
=> ( X986 = X988 ) )
=> ( ! [X992: $i,X994: $i,X996: $i,X998: $i] :
( ( ( setadjoin @ ( setadjoin @ X992 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X992 @ ( setadjoin @ X994 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X996 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X996 @ ( setadjoin @ X998 @ emptyset ) ) @ emptyset ) ) )
=> ( X994 = X998 ) )
=> ( ! [X1000: $i,X1002: $i,X1004: $i,X1006: $i] :
( ( ( kpair @ X1000 @ X1002 )
= ( kpair @ X1004 @ X1006 ) )
=> ( X1002 = X1006 ) )
=> ( ! [X1008: $i] :
( ( iskpair @ X1008 )
=> ? [X1010: $i] :
( ( in @ X1010
@ ( dsetconstr @ ( setunion @ X1008 )
@ ^ [V_26: $i] :
( X1008
= ( kpair @ ( kfst @ X1008 ) @ V_26 ) ) ) )
& ( ( dsetconstr @ ( setunion @ X1008 )
@ ^ [V_25: $i] :
( X1008
= ( kpair @ ( kfst @ X1008 ) @ V_25 ) ) )
= ( setadjoin @ X1010 @ emptyset ) ) ) )
=> ( ! [X1012: $i,X1014: $i] :
( ( ksnd @ ( kpair @ X1012 @ X1014 ) )
= X1014 )
=> ( ! [X1016: $i] :
( ( iskpair @ X1016 )
=> ( ( kpair @ ( kfst @ X1016 ) @ ( ksnd @ X1016 ) )
= X1016 ) )
=> ( ! [X1018: $i,X1020: $i,X1022: $i] :
( ( in @ X1022 @ ( cartprod @ X1018 @ X1020 ) )
=> ( in @ ( ksnd @ X1022 ) @ X1020 ) )
=> ( ! [X1024: $i,X1026: $i,X1028: $i,X1030: $i] :
( ( in @ ( kpair @ X1028 @ X1030 ) @ ( cartprod @ X1024 @ X1026 ) )
=> ( in @ X1028 @ X1024 ) )
=> ( ! [X1032: $i,X1034: $i,X1036: $i,X1038: $i] :
( ( in @ ( kpair @ X1036 @ X1038 ) @ ( cartprod @ X1032 @ X1034 ) )
=> ( in @ X1038 @ X1034 ) )
=> ( ! [X1040: $i,X1042: $i,X1044: $i] :
( ( in @ X1044 @ X1040 )
=> ! [X1046: $i] :
( ( in @ X1046 @ X1042 )
=> ( ( kpair @ X1044 @ X1046 )
= ( kpair @ X1044 @ X1046 ) ) ) )
=> ( ! [X1048: $i,X1050: $i,X1052: $i] :
( ( in @ X1052 @ X1048 )
=> ! [X1054: $i] :
( ( in @ X1054 @ X1050 )
=> ( ( kfst @ ( kpair @ X1052 @ X1054 ) )
= X1052 ) ) )
=> ( ! [X1056: $i,X1058: $i,X1060: $i] :
( ( in @ X1060 @ X1056 )
=> ! [X1062: $i] :
( ( in @ X1062 @ X1058 )
=> ( ( ksnd @ ( kpair @ X1060 @ X1062 ) )
= X1062 ) ) )
=> ( ! [X1064: $i,X1066: $i,X1068: $i] :
( ( in @ X1068 @ ( cartprod @ X1064 @ X1066 ) )
=> ( ( kpair @ ( kfst @ X1068 ) @ ( ksnd @ X1068 ) )
= X1068 ) )
=> ( ! [X1070: $i,X1072: $i,X1074: $i > $i > $o,X1076: $i] :
( ( in @ X1076 @ X1070 )
=> ! [X1078: $i] :
( ( in @ X1078 @ X1072 )
=> ( ( X1074 @ X1076 @ X1078 )
=> ( in @ ( kpair @ X1076 @ X1078 )
@ ( dpsetconstr @ X1070 @ X1072
@ ^ [V_27: $i,V_28: $i] : ( X1074 @ V_27 @ V_28 ) ) ) ) ) )
=> ( ! [X1080: $i,X1082: $i,X1084: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1080 @ X1082
@ ^ [V_29: $i,V_30: $i] : ( X1084 @ V_29 @ V_30 ) )
@ ( cartprod @ X1080 @ X1082 ) )
=> ( ! [X1086: $i,X1088: $i,X1090: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1086 @ X1088
@ ^ [V_31: $i,V_32: $i] : ( X1090 @ V_31 @ V_32 ) )
@ ( cartprod @ X1086 @ X1088 ) )
=> ( ! [X1092: $i,X1094: $i,X1096: $i > $i > $o,X1098: $i] :
( ( in @ X1098 @ X1092 )
=> ! [X1100: $i] :
( ( in @ X1100 @ X1094 )
=> ( ( in @ ( kpair @ X1098 @ X1100 )
@ ( dpsetconstr @ X1092 @ X1094
@ ^ [V_33: $i,V_34: $i] : ( X1096 @ V_33 @ V_34 ) ) )
=> ( X1096 @ X1098 @ X1100 ) ) ) )
=> ( ! [X1102: $i,X1104: $i,X1106: $i > $i > $o,X1108: $i,X1110: $i] :
( ( in @ ( kpair @ X1108 @ X1110 )
@ ( dpsetconstr @ X1102 @ X1104
@ ^ [V_35: $i,V_36: $i] : ( X1106 @ V_35 @ V_36 ) ) )
=> ( in @ X1108 @ X1102 ) )
=> ( ! [X1112: $i,X1114: $i,X1116: $i > $i > $o,X1118: $i,X1120: $i] :
( ( in @ ( kpair @ X1118 @ X1120 )
@ ( dpsetconstr @ X1112 @ X1114
@ ^ [V_37: $i,V_38: $i] : ( X1116 @ V_37 @ V_38 ) ) )
=> ( in @ X1120 @ X1114 ) )
=> ( ! [X1122: $i,X1124: $i,X1126: $i > $i > $o,X1128: $i,X1130: $i] :
( ( in @ ( kpair @ X1128 @ X1130 )
@ ( dpsetconstr @ X1122 @ X1124
@ ^ [V_39: $i,V_40: $i] : ( X1126 @ V_39 @ V_40 ) ) )
=> ( X1126 @ X1128 @ X1130 ) )
=> ( ! [X1132: $i,X1134: $i,X1136: $i] :
( ( ! [X1138: $i] :
( ( in @ X1138 @ X1132 )
=> ? [X1140: $i] :
( ( in @ X1140
@ ( dsetconstr @ X1134
@ ^ [V_42: $i] : ( in @ ( kpair @ X1138 @ V_42 ) @ X1136 ) ) )
& ( ( dsetconstr @ X1134
@ ^ [V_41: $i] : ( in @ ( kpair @ X1138 @ V_41 ) @ X1136 ) )
= ( setadjoin @ X1140 @ emptyset ) ) ) )
& ( subset @ X1136 @ ( cartprod @ X1132 @ X1134 ) ) )
=> ! [X1142: $i] :
( ( in @ X1142 @ X1132 )
=> ? [X1144: $i] :
( ( in @ X1144
@ ( dsetconstr @ X1134
@ ^ [V_44: $i] : ( in @ ( kpair @ X1142 @ V_44 ) @ X1136 ) ) )
& ( ( dsetconstr @ X1134
@ ^ [V_43: $i] : ( in @ ( kpair @ X1142 @ V_43 ) @ X1136 ) )
= ( setadjoin @ X1144 @ emptyset ) ) ) ) )
=> ( ! [X1146: $i,X1148: $i,X1150: $i] :
( ( ! [X1152: $i] :
( ( in @ X1152 @ X1146 )
=> ? [X1154: $i] :
( ( in @ X1154
@ ( dsetconstr @ X1148
@ ^ [V_46: $i] : ( in @ ( kpair @ X1152 @ V_46 ) @ X1150 ) ) )
& ( ( dsetconstr @ X1148
@ ^ [V_45: $i] : ( in @ ( kpair @ X1152 @ V_45 ) @ X1150 ) )
= ( setadjoin @ X1154 @ emptyset ) ) ) )
& ( subset @ X1150 @ ( cartprod @ X1146 @ X1148 ) ) )
=> ! [X1156: $i] :
( ( in @ X1156 @ X1146 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X1148
@ ^ [V_47: $i] : ( in @ ( kpair @ X1156 @ V_47 ) @ X1150 ) ) )
@ X1148 ) ) )
=> ( ! [X1158: $i,X1160: $i,X1162: $i] :
( ( ! [X1164: $i] :
( ( in @ X1164 @ X1158 )
=> ? [X1166: $i] :
( ( in @ X1166
@ ( dsetconstr @ X1160
@ ^ [V_49: $i] : ( in @ ( kpair @ X1164 @ V_49 ) @ X1162 ) ) )
& ( ( dsetconstr @ X1160
@ ^ [V_48: $i] : ( in @ ( kpair @ X1164 @ V_48 ) @ X1162 ) )
= ( setadjoin @ X1166 @ emptyset ) ) ) )
& ( subset @ X1162 @ ( cartprod @ X1158 @ X1160 ) ) )
=> ! [X1168: $i] :
( ( in @ X1168 @ X1158 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X1160
@ ^ [V_50: $i] : ( in @ ( kpair @ X1168 @ V_50 ) @ X1162 ) ) )
@ X1160 ) ) )
=> ! [X1170: $i,X1172: $i,X1174: $i] :
( ( in @ X1174
@ ( dsetconstr @ ( powerset @ ( cartprod @ X1170 @ X1172 ) )
@ ^ [V_51: $i] :
( ! [X1176: $i] :
( ( in @ X1176 @ X1170 )
=> ? [X1178: $i] :
( ( in @ X1178
@ ( dsetconstr @ X1172
@ ^ [V_53: $i] : ( in @ ( kpair @ X1176 @ V_53 ) @ V_51 ) ) )
& ( ( dsetconstr @ X1172
@ ^ [V_52: $i] : ( in @ ( kpair @ X1176 @ V_52 ) @ V_51 ) )
= ( setadjoin @ X1178 @ emptyset ) ) ) )
& ( subset @ V_51 @ ( cartprod @ X1170 @ X1172 ) ) ) ) )
=> ( ! [X1180: $i] :
( ( in @ X1180 @ X1170 )
=> ? [X1182: $i] :
( ( in @ X1182
@ ( dsetconstr @ X1172
@ ^ [V_55: $i] : ( in @ ( kpair @ X1180 @ V_55 ) @ X1174 ) ) )
& ( ( dsetconstr @ X1172
@ ^ [V_54: $i] : ( in @ ( kpair @ X1180 @ V_54 ) @ X1174 ) )
= ( setadjoin @ X1182 @ emptyset ) ) ) )
& ( subset @ X1174 @ ( cartprod @ X1170 @ X1172 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(zf_stmt_1,negated_conjecture,
~ ( ! [X4: $i,X6: $i] :
( ! [X8: $i] :
( ( in @ X8 @ X4 )
<=> ( in @ X8 @ X6 ) )
=> ( X4 = X6 ) )
=> ( ! [X10: $i] :
~ ( in @ X10 @ emptyset )
=> ( ! [X12: $i,X14: $i,X16: $i] :
( ( in @ X16 @ ( setadjoin @ X12 @ X14 ) )
<=> ( ( in @ X16 @ X14 )
| ( X16 = X12 ) ) )
=> ( ! [X18: $i,X20: $i] :
( ( in @ X20 @ ( powerset @ X18 ) )
<=> ! [X22: $i] :
( ( in @ X22 @ X20 )
=> ( in @ X22 @ X18 ) ) )
=> ( ! [X24: $i,X26: $i] :
( ( in @ X26 @ ( setunion @ X24 ) )
<=> ? [X28: $i] :
( ( in @ X26 @ X28 )
& ( in @ X28 @ X24 ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X30: $i] :
( ( in @ X30 @ omega )
=> ( in @ ( setadjoin @ X30 @ X30 ) @ omega ) )
=> ( ! [X32: $i] :
( ( ! [X34: $i] :
( ( ( in @ X34 @ X32 )
& ( in @ X34 @ omega ) )
=> ( in @ ( setadjoin @ X34 @ X34 ) @ X32 ) )
& ( in @ emptyset @ X32 ) )
=> ! [X36: $i] :
( ( in @ X36 @ omega )
=> ( in @ X36 @ X32 ) ) )
=> ( ! [X38: $i > $i > $o,X40: $i] :
( ! [X42: $i] :
( ( in @ X42 @ X40 )
=> ? [X44: $i] :
( ( X38 @ X42 @ X44 )
& ! [X46: $i] :
( ( X38 @ X42 @ X46 )
=> ( X44 = X46 ) ) ) )
=> ? [X48: $i] :
! [X50: $i] :
( ( in @ X50 @ X48 )
<=> ? [X52: $i] :
( ( in @ X52 @ X40 )
& ( X38 @ X52 @ X50 ) ) ) )
=> ( ! [X54: $i] :
( ? [X56: $i] : ( in @ X56 @ X54 )
=> ? [X58: $i] :
( ( in @ X58 @ X54 )
& ~ ? [X60: $i] :
( ( in @ X60 @ X58 )
& ( in @ X60 @ X54 ) ) ) )
=> ( ! [X62: $i] :
? [X64: $i] :
( ! [X96: $i] :
( ( in @ X96 @ X64 )
=> ! [X98: $i] :
( ( in @ X98 @ X96 )
=> ( in @ X98 @ X62 ) ) )
& ! [X90: $i,X92: $i] :
( ( ( in @ X92 @ X62 )
& ( in @ X90 @ X62 ) )
=> ( ! [X94: $i] :
( ( in @ X94 @ X64 )
=> ( ( in @ X90 @ X94 )
<=> ( in @ X92 @ X94 ) ) )
=> ( X90 = X92 ) ) )
& ! [X82: $i,X84: $i] :
( ( ( in @ X84 @ X64 )
& ( in @ X82 @ X64 ) )
=> ( ! [X88: $i] :
( ( in @ X88 @ X84 )
=> ( in @ X88 @ X82 ) )
| ! [X86: $i] :
( ( in @ X86 @ X82 )
=> ( in @ X86 @ X84 ) ) ) )
& ! [X66: $i] :
( ( ? [X70: $i] : ( in @ X70 @ X66 )
& ! [X68: $i] :
( ( in @ X68 @ X66 )
=> ( in @ X68 @ X62 ) ) )
=> ? [X72: $i,X74: $i] :
( ( in @ X72 @ X64 )
& ( in @ X74 @ X66 )
& ~ ? [X80: $i] :
( ( in @ X80 @ X72 )
& ( in @ X80 @ X66 ) )
& ! [X76: $i] :
( ( in @ X76 @ X64 )
=> ( ( in @ X74 @ X76 )
| ! [X78: $i] :
( ( in @ X78 @ X76 )
=> ( in @ X78 @ X72 ) ) ) ) ) ) )
=> ( ! [X100: $i > $o] :
( ? [X102: $i] :
( ( X100 @ X102 )
& ! [X104: $i] :
( ( X100 @ X104 )
=> ( X102 = X104 ) ) )
=> ( X100
@ ( descr
@ ^ [V_1: $i] : ( X100 @ V_1 ) ) ) )
=> ( ! [X106: $i,X108: $i > $o,X110: $i] :
( ( in @ X110 @ X106 )
=> ( ( X108 @ X110 )
=> ( in @ X110
@ ( dsetconstr @ X106
@ ^ [V_2: $i] : ( X108 @ V_2 ) ) ) ) )
=> ( ! [X112: $i,X114: $i > $o,X116: $i] :
( ( in @ X116
@ ( dsetconstr @ X112
@ ^ [V_3: $i] : ( X114 @ V_3 ) ) )
=> ( in @ X116 @ X112 ) )
=> ( ! [X118: $i,X120: $i > $o,X122: $i] :
( ( in @ X122
@ ( dsetconstr @ X118
@ ^ [V_4: $i] : ( X120 @ V_4 ) ) )
=> ( X120 @ X122 ) )
=> ( ! [X124: $i > $o] :
( ? [X126: $i] :
( ( X124 @ X126 )
& ! [X128: $i] :
( ( X124 @ X128 )
=> ( X126 = X128 ) ) )
=> ? [X130: $i] :
( ( X124 @ X130 )
& ! [X132: $i] :
( ( X124 @ X132 )
=> ( X130 = X132 ) ) ) )
=> ( ! [X134: $o,X136: $i] :
( ( in @ X136 @ ( prop2set @ X134 ) )
=> X134 )
=> ( ! [X138: $i] :
( ( in @ X138 @ emptyset )
=> ! [X140: $o] : X140 )
=> ( ! [X142: $i] :
~ ( in @ X142 @ emptyset )
=> ( ! [X144: $i] :
~ ( in @ X144 @ emptyset )
=> ( ! [X146: $i > $o] :
( ? [X148: $i] :
( ( X146 @ X148 )
& ! [X150: $i] :
( ( X146 @ X150 )
=> ( X148 = X150 ) ) )
=> ? [X152: $i] : ( X146 @ X152 ) )
=> ( ! [X154: $i,X156: $i] :
( ! [X158: $i] :
( ( in @ X158 @ X154 )
=> ( in @ X158 @ X156 ) )
=> ( ! [X160: $i] :
( ( in @ X160 @ X156 )
=> ( in @ X160 @ X154 ) )
=> ( X154 = X156 ) ) )
=> ( ! [X162: $i] :
( ! [X164: $i] :
~ ( in @ X164 @ X162 )
=> ( X162 = emptyset ) )
=> ( ! [X166: $i] :
( ! [X168: $i] :
~ ( in @ X168 @ X166 )
=> ( X166 = emptyset ) )
=> ( ! [X170: $i,X172: $i > $o,X174: $i] :
( ( in @ X174 @ X170 )
=> ( ( in @ X174
@ ( dsetconstr @ X170
@ ^ [V_5: $i] : ( X172 @ V_5 ) ) )
<=> ( X172 @ X174 ) ) )
=> ( ! [X176: $i] :
( ( X176 != emptyset )
=> ? [X178: $i] : ( in @ X178 @ X176 ) )
=> ( ! [X180: $i,X182: $i > $o,X184: $i] :
( ( in @ X184 @ X180 )
=> ( ( X182 @ X184 )
=> ( ( dsetconstr @ X180
@ ^ [V_6: $i] : ( X182 @ V_6 ) )
!= emptyset ) ) )
=> ( ! [X186: $i] :
( ? [X188: $i] : ( in @ X188 @ X186 )
=> ( X186 != emptyset ) )
=> ( ! [X190: $i,X192: $i] : ( in @ X190 @ ( setadjoin @ X190 @ X192 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X194: $i,X196: $i,X198: $i] :
( ( in @ X198 @ X196 )
=> ( in @ X198 @ ( setadjoin @ X194 @ X196 ) ) )
=> ( ! [X200: $i,X202: $i,X204: $i] :
( ( in @ X204 @ ( setadjoin @ X200 @ X202 ) )
=> ! [X206: $o] :
( ( ( X204 = X200 )
=> X206 )
=> ( ( ( in @ X204 @ X202 )
=> X206 )
=> X206 ) ) )
=> ( ! [X208: $i,X210: $i,X212: $i] :
( ( in @ X212 @ ( setadjoin @ X208 @ X210 ) )
=> ( ( in @ X212 @ X210 )
| ( X212 = X208 ) ) )
=> ( ! [X214: $i] :
( ( dsetconstr @ X214
@ ^ [V_7: $i] : $true )
= X214 )
=> ( ! [X216: $i,X218: $i] :
( ! [X220: $i] :
( ( in @ X220 @ X218 )
=> ( in @ X220 @ X216 ) )
=> ( in @ X218 @ ( powerset @ X216 ) ) )
=> ( ! [X222: $i] : ( in @ emptyset @ ( powerset @ X222 ) )
=> ( ! [X224: $i] : ( in @ emptyset @ ( powerset @ X224 ) )
=> ( ! [X226: $i,X228: $i,X230: $i] :
( ( in @ X228 @ ( powerset @ X226 ) )
=> ( ( in @ X230 @ X228 )
=> ( in @ X230 @ X226 ) ) )
=> ( ! [X232: $i,X234: $i,X236: $i] :
( ( in @ X234 @ X236 )
=> ( ( in @ X236 @ X232 )
=> ( in @ X234 @ ( setunion @ X232 ) ) ) )
=> ( ! [X238: $i,X240: $i] :
( ( in @ X240 @ ( setunion @ X238 ) )
=> ! [X242: $o] :
( ! [X244: $i] :
( ( in @ X240 @ X244 )
=> ( ( in @ X244 @ X238 )
=> X242 ) )
=> X242 ) )
=> ( ! [X246: $i,X248: $i] :
( ( in @ X248 @ X246 )
=> ( in @ X248 @ ( powerset @ ( setunion @ X246 ) ) ) )
=> ( ! [X250: $i > $o] :
( ? [X252: $i] :
( ( X250 @ X252 )
& ! [X254: $i] :
( ( X250 @ X254 )
=> ( X252 = X254 ) ) )
=> ? [X256: $i] :
! [X258: $i] :
( ( X250 @ X258 )
<=> ( X258 = X256 ) ) )
=> ( ! [X260: $i] :
( ( X260 != emptyset )
=> ? [X262: $i] : ( in @ X262 @ X260 ) )
=> ( ! [X264: $i,X266: $i] :
( ( in @ X264 @ ( setadjoin @ X266 @ emptyset ) )
=> ( X264 = X266 ) )
=> ( ! [X268: $i,X270: $i] :
( ( X268 != X270 )
=> ~ ( in @ X270 @ ( setadjoin @ X268 @ emptyset ) ) )
=> ( ! [X272: $i,X274: $i] :
( ( X272 = X274 )
=> ( in @ X272 @ ( setadjoin @ X274 @ emptyset ) ) )
=> ( ! [X276: $i,X278: $i] :
( ( in @ X276 @ ( setadjoin @ X278 @ emptyset ) )
=> ( in @ X278 @ ( setadjoin @ X276 @ emptyset ) ) )
=> ( ! [X280: $i,X282: $i,X284: $i] :
( ( in @ X284 @ ( setadjoin @ X280 @ ( setadjoin @ X282 @ emptyset ) ) )
=> ( ( X284 = X282 )
| ( X284 = X280 ) ) )
=> ( ! [X286: $i,X288: $i] : ( in @ X286 @ ( setadjoin @ X286 @ ( setadjoin @ X288 @ emptyset ) ) )
=> ( ! [X290: $i,X292: $i] : ( in @ X292 @ ( setadjoin @ X290 @ ( setadjoin @ X292 @ emptyset ) ) )
=> ( ! [X294: $i,X296: $i > $o] :
( ? [X298: $i] :
( ( in @ X298 @ X294 )
& ( X296 @ X298 ) )
=> ( ( dsetconstr @ X294
@ ^ [V_8: $i] : ( X296 @ V_8 ) )
!= emptyset ) )
=> ( ! [X300: $i > $o,X302: $i] :
( ( in @ X302 @ emptyset )
=> ( X300 @ X302 ) )
=> ( ! [X304: $i,X306: $i > $o] :
( ~ ! [X308: $i] :
( ( in @ X308 @ X304 )
=> ( X306 @ X308 ) )
=> ? [X310: $i] :
( ( in @ X310 @ X304 )
& ~ ( X306 @ X310 ) ) )
=> ( ! [X312: $i,X314: $i > $o] :
( ! [X316: $i] :
( ( in @ X316 @ X312 )
=> ~ ( X314 @ X316 ) )
=> ~ ? [X318: $i] :
( ( in @ X318 @ X312 )
& ( X314 @ X318 ) ) )
=> ( ! [X320: $i,X322: $i > $o] :
( ~ ? [X324: $i] :
( ( in @ X324 @ X320 )
& ( X322 @ X324 ) )
=> ! [X326: $i] :
( ( in @ X326 @ X320 )
=> ~ ( X322 @ X326 ) ) )
=> ( ! [X328: $i,X330: $i > $o] :
( ? [X332: $i] :
( ( in @ X332 @ X328 )
& ~ ( X330 @ X332 ) )
=> ~ ! [X334: $i] :
( ( in @ X334 @ X328 )
=> ( X330 @ X334 ) ) )
=> ( ! [X336: $o] :
( X336
=> ( in @ emptyset @ ( prop2set @ X336 ) ) )
=> ( ! [X338: $o] :
( X338
=> ( set2prop @ ( prop2set @ X338 ) ) )
=> ( ! [X340: $i,X342: $i > $o] :
( ~ ? [X344: $i] :
( ( in @ X344 @ X340 )
& ( X342 @ X344 ) )
=> ! [X346: $i] :
( ( in @ X346 @ X340 )
=> ~ ( X342 @ X346 ) ) )
=> ( ! [X348: $i,X350: $i > $o] :
( ~ ! [X352: $i] :
( ( in @ X352 @ X348 )
=> ( X350 @ X352 ) )
=> ? [X354: $i] :
( ( in @ X354 @ X348 )
& ~ ( X350 @ X354 ) ) )
=> ( ! [X356: $i > $o] :
( ? [X358: $i] :
( ( X356 @ X358 )
& ! [X360: $i] :
( ( X356 @ X360 )
=> ( X358 = X360 ) ) )
=> ? [X362: $i] :
( ( X356 @ X362 )
& ! [X364: $i] :
( ( X356 @ X364 )
=> ( X362 = X364 ) ) ) )
=> ( ! [X366: $i > $o] :
( ? [X368: $i] : ( X366 @ X368 )
=> ( ! [X370: $i,X372: $i] :
( ( X366 @ X370 )
=> ( ( X366 @ X372 )
=> ( X370 = X372 ) ) )
=> ? [X374: $i] :
( ( X366 @ X374 )
& ! [X376: $i] :
( ( X366 @ X376 )
=> ( X374 = X376 ) ) ) ) )
=> ( ! [X378: $i > $o] :
( ? [X380: $i] :
! [X382: $i] :
( ( X378 @ X382 )
<=> ( X382 = X380 ) )
=> ? [X384: $i] :
( ( X378 @ X384 )
& ! [X386: $i] :
( ( X378 @ X386 )
=> ( X384 = X386 ) ) ) )
=> ( ! [X388: $i,X390: $i] :
( ( X388 = X390 )
=> ! [X392: $i,X394: $i] :
( ( X392 = X394 )
=> ( ( in @ X392 @ X388 )
=> ( in @ X394 @ X390 ) ) ) )
=> ( ! [X396: $i,X398: $i] :
( ( X396 = X398 )
=> ! [X400: $i,X402: $i] :
( ( X400 = X402 )
=> ( ( in @ X400 @ X396 )
<=> ( in @ X402 @ X398 ) ) ) )
=> ( ! [X404: $i > $o] :
( ? [X406: $i] :
( ( X404 @ X406 )
& ! [X408: $i] :
( ( X404 @ X408 )
=> ( X406 = X408 ) ) )
=> ! [X410: $i,X412: $i] :
( ( X404 @ X410 )
=> ( ( X404 @ X412 )
=> ( X410 = X412 ) ) ) )
=> ( ! [X414: $i > $o,X416: $i > $o] :
( ! [X418: $i,X420: $i] :
( ( X418 = X420 )
=> ( ( X414 @ X418 )
<=> ( X416 @ X420 ) ) )
=> ( ? [X422: $i] :
( ( X414 @ X422 )
& ! [X424: $i] :
( ( X414 @ X424 )
=> ( X422 = X424 ) ) )
<=> ? [X426: $i] :
( ( X416 @ X426 )
& ! [X428: $i] :
( ( X416 @ X428 )
=> ( X426 = X428 ) ) ) ) )
=> ( ( emptyset = emptyset )
=> ( ! [X430: $i,X432: $i] :
( ( X430 = X432 )
=> ! [X434: $i,X436: $i] :
( ( X434 = X436 )
=> ( ( setadjoin @ X430 @ X434 )
= ( setadjoin @ X432 @ X436 ) ) ) )
=> ( ! [X438: $i,X440: $i] :
( ( X438 = X440 )
=> ( ( powerset @ X438 )
= ( powerset @ X440 ) ) )
=> ( ! [X442: $i,X444: $i] :
( ( X442 = X444 )
=> ( ( setunion @ X442 )
= ( setunion @ X444 ) ) )
=> ( ( omega = omega )
=> ( ! [X446: $i > $o] :
( ? [X448: $i] :
( ( X446 @ X448 )
& ! [X450: $i] :
( ( X446 @ X450 )
=> ( X448 = X450 ) ) )
=> ! [X452: $i,X454: $i] :
( ( X446 @ X452 )
=> ( ( X446 @ X454 )
=> ( X452 = X454 ) ) ) )
=> ( ! [X456: $i > $o,X458: $i > $o] :
( ! [X460: $i,X462: $i] :
( ( X460 = X462 )
=> ( ( X456 @ X460 )
<=> ( X458 @ X462 ) ) )
=> ( ? [X464: $i] :
( ( X456 @ X464 )
& ! [X466: $i] :
( ( X456 @ X466 )
=> ( X464 = X466 ) ) )
=> ( ? [X468: $i] :
( ( X458 @ X468 )
& ! [X470: $i] :
( ( X458 @ X470 )
=> ( X468 = X470 ) ) )
=> ( ( descr
@ ^ [V_9: $i] : ( X456 @ V_9 ) )
= ( descr
@ ^ [V_10: $i] : ( X458 @ V_10 ) ) ) ) ) )
=> ( ! [X472: $i,X474: $i] :
( ( X472 = X474 )
=> ! [X476: $i > $o,X478: $i > $o] :
( ! [X480: $i] :
( ( in @ X480 @ X472 )
=> ! [X482: $i] :
( ( in @ X482 @ X474 )
=> ( ( X480 = X482 )
=> ( ( X476 @ X480 )
<=> ( X478 @ X482 ) ) ) ) )
=> ( ( dsetconstr @ X472
@ ^ [V_11: $i] : ( X476 @ V_11 ) )
= ( dsetconstr @ X474
@ ^ [V_12: $i] : ( X478 @ V_12 ) ) ) ) )
=> ( ! [X484: $i,X486: $i] :
( ! [X488: $i] :
( ( in @ X488 @ X484 )
=> ( in @ X488 @ X486 ) )
=> ( subset @ X484 @ X486 ) )
=> ( ! [X490: $i,X492: $i] :
( ( X490 = X492 )
=> ( subset @ X492 @ X490 ) )
=> ( ! [X494: $i,X496: $i] :
( ( X494 = X496 )
=> ( subset @ X494 @ X496 ) )
=> ( ! [X498: $i,X500: $i] :
( ! [X502: $i] :
( ( in @ X502 @ X498 )
=> ( in @ X502 @ X500 ) )
=> ( subset @ X498 @ X500 ) )
=> ( ! [X504: $i] : ( subset @ emptyset @ X504 )
=> ( ! [X506: $i,X508: $i,X510: $i] :
( ( subset @ X506 @ X508 )
=> ( ( in @ X510 @ X506 )
=> ( in @ X510 @ X508 ) ) )
=> ( ! [X512: $i,X514: $i,X516: $i] :
( ( subset @ X512 @ X514 )
=> ( ~ ( in @ X516 @ X514 )
=> ~ ( in @ X516 @ X512 ) ) )
=> ( ! [X518: $i,X520: $i,X522: $i] :
( ( in @ X522 @ X518 )
=> ( ~ ( in @ X522 @ X520 )
=> ~ ( subset @ X518 @ X520 ) ) )
=> ( ! [X524: $i,X526: $i] :
( ~ ( subset @ X524 @ X526 )
=> ( X524 != X526 ) )
=> ( ! [X528: $i,X530: $i,X532: $i] :
( ( in @ X532 @ X528 )
=> ( ~ ( in @ X532 @ X530 )
=> ( X528 != X530 ) ) )
=> ( ! [X534: $i] : ( subset @ X534 @ X534 )
=> ( ! [X536: $i,X538: $i,X540: $i] :
( ( subset @ X536 @ X538 )
=> ( ( subset @ X538 @ X540 )
=> ( subset @ X536 @ X540 ) ) )
=> ( ! [X542: $i,X544: $i] : ( subset @ X544 @ ( setadjoin @ X542 @ X544 ) )
=> ( ! [X546: $i,X548: $i,X550: $i] :
( ( subset @ X546 @ X550 )
=> ( subset @ X546 @ ( setadjoin @ X548 @ X550 ) ) )
=> ( ! [X552: $i,X554: $i] :
( ( subset @ X552 @ X554 )
=> ( in @ X552 @ ( powerset @ X554 ) ) )
=> ( ! [X556: $i,X558: $i] :
( ( subset @ X556 @ X558 )
=> ( ( subset @ X558 @ X556 )
=> ( X556 = X558 ) ) )
=> ( ! [X560: $i] :
( ( subset @ X560 @ emptyset )
=> ( X560 = emptyset ) )
=> ( ! [X562: $i,X564: $i] :
( ( subset @ X564 @ X562 )
=> ( in @ X564 @ ( powerset @ X562 ) ) )
=> ( ! [X566: $i,X568: $i] :
( ( in @ X568 @ ( powerset @ X566 ) )
=> ( subset @ X568 @ X566 ) )
=> ( ! [X570: $i] : ( in @ X570 @ ( powerset @ X570 ) )
=> ( ! [X572: $i,X574: $i] :
( ( subset @ X572 @ X574 )
=> ( subset @ ( powerset @ X572 ) @ ( powerset @ X574 ) ) )
=> ( ! [X576: $i,X578: $i > $o] :
( in
@ ( dsetconstr @ X576
@ ^ [V_13: $i] : ( X578 @ V_13 ) )
@ ( powerset @ X576 ) )
=> ( ! [X580: $i,X582: $i > $o] :
( subset
@ ( dsetconstr @ X580
@ ^ [V_14: $i] : ( X582 @ V_14 ) )
@ X580 )
=> ( ! [X584: $i,X586: $i,X588: $i] :
( ( in @ X588 @ X584 )
=> ( in @ X588 @ ( binunion @ X584 @ X586 ) ) )
=> ( ! [X590: $i,X592: $i] : ( in @ X592 @ ( setadjoin @ X590 @ ( setadjoin @ X592 @ emptyset ) ) )
=> ( ! [X594: $i,X596: $i,X598: $i] :
( ( in @ X598 @ X596 )
=> ( in @ X598 @ ( binunion @ X594 @ X596 ) ) )
=> ( ! [X600: $i,X602: $i,X604: $i,X606: $o] :
( ( in @ X604 @ ( binunion @ X600 @ X602 ) )
=> ( ( ( in @ X604 @ X600 )
=> X606 )
=> ( ( ( in @ X604 @ X602 )
=> X606 )
=> X606 ) ) )
=> ( ! [X608: $i,X610: $i,X612: $i] :
( ( in @ X612 @ ( binunion @ X608 @ X610 ) )
=> ( ( in @ X612 @ X610 )
| ( in @ X612 @ X608 ) ) )
=> ( ! [X614: $i,X616: $i] : ( subset @ X614 @ ( binunion @ X614 @ X616 ) )
=> ( ! [X618: $i,X620: $i] : ( subset @ X620 @ ( binunion @ X618 @ X620 ) )
=> ( ! [X622: $i,X624: $i,X626: $i] :
( ( in @ X626 @ X622 )
=> ( ( in @ X626 @ X624 )
=> ( in @ X626 @ ( binintersect @ X622 @ X624 ) ) ) )
=> ( ! [X628: $i,X630: $i,X632: $i] :
( ( subset @ X632 @ X628 )
=> ( ( subset @ X632 @ X630 )
=> ( subset @ X632 @ ( binintersect @ X628 @ X630 ) ) ) )
=> ( ! [X634: $i,X636: $i,X638: $i] :
( ( in @ X638 @ ( binintersect @ X634 @ X636 ) )
=> ( in @ X638 @ X634 ) )
=> ( ! [X640: $i,X642: $i] : ( subset @ ( binintersect @ X640 @ X642 ) @ X640 )
=> ( ! [X644: $i,X646: $i] :
( ( subset @ X644 @ X646 )
=> ( ( binintersect @ X644 @ X646 )
= X644 ) )
=> ( ! [X648: $i,X650: $i] :
( ( ( binintersect @ X648 @ X650 )
= X650 )
=> ( subset @ X650 @ X648 ) )
=> ( ! [X652: $i,X654: $i,X656: $i] :
( ( in @ X656 @ ( binintersect @ X652 @ X654 ) )
=> ( in @ X656 @ X654 ) )
=> ( ! [X658: $i,X660: $i] :
( ~ ? [X662: $i] :
( ( in @ X662 @ X658 )
& ( in @ X662 @ X660 ) )
=> ( ( binintersect @ X658 @ X660 )
= emptyset ) )
=> ( ! [X664: $i,X666: $i] : ( subset @ ( binintersect @ X664 @ X666 ) @ X666 )
=> ( ! [X668: $i,X670: $i] :
( ( subset @ X670 @ X668 )
=> ( ( binintersect @ X668 @ X670 )
= X670 ) )
=> ( ! [X672: $i,X674: $i] :
( ( ( binintersect @ X672 @ X674 )
= X672 )
=> ( subset @ X672 @ X674 ) )
=> ( ! [X676: $i,X678: $i,X680: $i] :
( ( binintersect @ X676 @ ( binunion @ X678 @ X680 ) )
= ( binunion @ ( binintersect @ X676 @ X678 ) @ ( binintersect @ X676 @ X680 ) ) )
=> ( ! [X682: $i,X684: $i,X686: $i] :
( ( in @ X686 @ X682 )
=> ( ~ ( in @ X686 @ X684 )
=> ( in @ X686 @ ( setminus @ X682 @ X684 ) ) ) )
=> ( ! [X688: $i,X690: $i,X692: $i] :
( ( in @ X692 @ ( setminus @ X688 @ X690 ) )
=> ( in @ X692 @ X688 ) )
=> ( ! [X694: $i,X696: $i,X698: $i] :
( ( in @ X698 @ ( setminus @ X694 @ X696 ) )
=> ~ ( in @ X698 @ X696 ) )
=> ( ! [X700: $i,X702: $i] :
( ( subset @ X700 @ X702 )
=> ( ( setminus @ X700 @ X702 )
= emptyset ) )
=> ( ! [X704: $i,X706: $i,X708: $i] :
( ~ ( in @ X708 @ ( setminus @ X704 @ X706 ) )
=> ( ( in @ X708 @ X704 )
=> ( in @ X708 @ X706 ) ) )
=> ( ! [X710: $i,X712: $i,X714: $i] :
( ~ ( in @ X714 @ ( setminus @ X710 @ X712 ) )
=> ( ~ ( in @ X714 @ X712 )
=> ~ ( in @ X714 @ X710 ) ) )
=> ( ! [X716: $i,X718: $i,X720: $i] :
( ~ ( in @ X720 @ X716 )
=> ~ ( in @ X720 @ ( setminus @ X716 @ X718 ) ) )
=> ( ! [X722: $i,X724: $i,X726: $i] :
( ( in @ X726 @ X724 )
=> ~ ( in @ X726 @ ( setminus @ X722 @ X724 ) ) )
=> ( ! [X728: $i,X730: $i] : ( subset @ ( setminus @ X728 @ X730 ) @ X728 )
=> ( ! [X732: $i,X734: $i] :
( ( ( setminus @ X732 @ X734 )
= emptyset )
=> ( subset @ X732 @ X734 ) )
=> ( ! [X736: $i,X738: $i,X740: $i] :
( ( in @ X740 @ ( symdiff @ X736 @ X738 ) )
=> ! [X742: $o] :
( ( ( in @ X740 @ X736 )
=> ( ~ ( in @ X740 @ X738 )
=> X742 ) )
=> ( ( ~ ( in @ X740 @ X736 )
=> ( ( in @ X740 @ X738 )
=> X742 ) )
=> X742 ) ) )
=> ( ! [X744: $i,X746: $i,X748: $i] :
( ( in @ X748 @ X744 )
=> ( ~ ( in @ X748 @ X746 )
=> ( in @ X748 @ ( symdiff @ X744 @ X746 ) ) ) )
=> ( ! [X750: $i,X752: $i,X754: $i] :
( ~ ( in @ X754 @ X750 )
=> ( ( in @ X754 @ X752 )
=> ( in @ X754 @ ( symdiff @ X750 @ X752 ) ) ) )
=> ( ! [X756: $i,X758: $i,X760: $i] :
( ( in @ X760 @ X756 )
=> ( ( in @ X760 @ X758 )
=> ~ ( in @ X760 @ ( symdiff @ X756 @ X758 ) ) ) )
=> ( ! [X762: $i,X764: $i,X766: $i] :
( ~ ( in @ X766 @ X762 )
=> ( ~ ( in @ X766 @ X764 )
=> ~ ( in @ X766 @ ( symdiff @ X762 @ X764 ) ) ) )
=> ( ! [X768: $i,X770: $i] : ( in @ X770 @ ( setadjoin @ X768 @ ( setadjoin @ X770 @ emptyset ) ) )
=> ( ! [X772: $i,X774: $i] : ( in @ X772 @ ( setunion @ ( setadjoin @ ( setadjoin @ X772 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X772 @ ( setadjoin @ X774 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X776: $i,X778: $i] : ( in @ X778 @ ( setunion @ ( setadjoin @ ( setadjoin @ X776 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X776 @ ( setadjoin @ X778 @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [X780: $i,X782: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X780 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X780 @ ( setadjoin @ X782 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X784: $i,X786: $i] : ( iskpair @ ( kpair @ X784 @ X786 ) )
=> ( ! [X788: $i,X790: $i] :
( ( in @ X790 @ X788 )
=> ( subset @ ( setadjoin @ X790 @ emptyset ) @ X788 ) )
=> ( ! [X792: $i,X794: $i] :
( ( in @ X794 @ X792 )
=> ( in @ ( setadjoin @ X794 @ emptyset ) @ ( powerset @ X792 ) ) )
=> ( ! [X796: $i,X798: $i,X800: $i] :
( ( in @ X800 @ X796 )
=> ( in @ ( setadjoin @ X800 @ emptyset ) @ ( powerset @ ( binunion @ X796 @ X798 ) ) ) )
=> ( ! [X802: $i,X804: $i,X806: $i] :
( ( in @ X806 @ ( setadjoin @ X802 @ ( setadjoin @ X804 @ emptyset ) ) )
=> ( ( X806 = X804 )
| ( X806 = X802 ) ) )
=> ( ! [X808: $i,X810: $i,X812: $i] :
( ( in @ X812 @ X808 )
=> ! [X814: $i] :
( ( in @ X814 @ X810 )
=> ( subset @ ( setadjoin @ X812 @ ( setadjoin @ X814 @ emptyset ) ) @ ( binunion @ X808 @ X810 ) ) ) )
=> ( ! [X816: $i,X818: $i,X820: $i] :
( ( in @ X820 @ X816 )
=> ! [X822: $i] :
( ( in @ X822 @ X818 )
=> ( in @ ( setadjoin @ X820 @ ( setadjoin @ X822 @ emptyset ) ) @ ( powerset @ ( binunion @ X816 @ X818 ) ) ) ) )
=> ( ! [X824: $i,X826: $i,X828: $i] :
( ( in @ X828 @ X824 )
=> ! [X830: $i] :
( ( in @ X830 @ X826 )
=> ( subset @ ( setadjoin @ ( setadjoin @ X828 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X828 @ ( setadjoin @ X830 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ X824 @ X826 ) ) ) ) )
=> ( ! [X832: $i,X834: $i,X836: $i] :
( ( in @ X836 @ X832 )
=> ! [X838: $i] :
( ( in @ X838 @ X834 )
=> ( in @ ( setadjoin @ ( setadjoin @ X836 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X836 @ ( setadjoin @ X838 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ X832 @ X834 ) ) ) ) ) )
=> ( ! [X840: $i,X842: $i,X844: $i] :
( ( in @ X844 @ X840 )
=> ! [X846: $i] :
( ( in @ X846 @ X842 )
=> ( in @ ( kpair @ X844 @ X846 ) @ ( powerset @ ( powerset @ ( binunion @ X840 @ X842 ) ) ) ) ) )
=> ( ! [X848: $i,X850: $i,X852: $i] :
( ( in @ X852 @ X848 )
=> ! [X854: $i] :
( ( in @ X854 @ X850 )
=> ( in @ ( kpair @ X852 @ X854 ) @ ( cartprod @ X848 @ X850 ) ) ) )
=> ( ! [X856: $i,X858: $i,X860: $i] :
( ( in @ X860 @ ( cartprod @ X856 @ X858 ) )
=> ? [X862: $i] :
( ( in @ X862 @ X856 )
& ? [X864: $i] :
( ( in @ X864 @ X858 )
& ( X860
= ( kpair @ X862 @ X864 ) ) ) ) )
=> ( ! [X866: $i,X868: $i,X870: $i] :
( ( in @ X870 @ ( cartprod @ X866 @ X868 ) )
=> ( iskpair @ X870 ) )
=> ( ! [X872: $i,X874: $i] :
( ( in @ X874 @ ( setunion @ X872 ) )
=> ? [X876: $i] :
( ( in @ X876 @ X872 )
& ( in @ X874 @ X876 ) ) )
=> ( ! [X878: $i] : ( subset @ ( setunion @ ( setadjoin @ X878 @ emptyset ) ) @ X878 )
=> ( ! [X880: $i] : ( subset @ X880 @ ( setunion @ ( setadjoin @ X880 @ emptyset ) ) )
=> ( ! [X882: $i] :
( ( setunion @ ( setadjoin @ X882 @ emptyset ) )
= X882 )
=> ( ! [X884: $i,X886: $i > $o] :
( ! [X888: $i] :
( ( in @ X888 @ X884 )
=> ! [X890: $i] :
( ( in @ X890 @ X884 )
=> ( ( X886 @ X888 )
=> ( ( X886 @ X890 )
=> ( X888 = X890 ) ) ) ) )
=> ( ? [X892: $i] :
( ( in @ X892 @ X884 )
& ( X886 @ X892 ) )
=> ? [X894: $i] :
( ( in @ X894
@ ( dsetconstr @ X884
@ ^ [V_16: $i] : ( X886 @ V_16 ) ) )
& ( ( dsetconstr @ X884
@ ^ [V_15: $i] : ( X886 @ V_15 ) )
= ( setadjoin @ X894 @ emptyset ) ) ) ) )
=> ( ! [X896: $i,X898: $i > $o] :
( ? [X900: $i] :
( ( in @ X900
@ ( dsetconstr @ X896
@ ^ [V_18: $i] : ( X898 @ V_18 ) ) )
& ( ( dsetconstr @ X896
@ ^ [V_17: $i] : ( X898 @ V_17 ) )
= ( setadjoin @ X900 @ emptyset ) ) )
=> ? [X902: $i] :
( ( in @ X902 @ X896 )
& ( X898 @ X902 ) ) )
=> ( ! [X904: $i,X906: $i > $o,X908: $i] :
( ( in @ X908 @ X904 )
=> ( ( X906 @ X908 )
=> ( ! [X910: $i] :
( ( in @ X910 @ X904 )
=> ( ( X906 @ X910 )
=> ( X910 = X908 ) ) )
=> ? [X912: $i] :
( ( in @ X912
@ ( dsetconstr @ X904
@ ^ [V_20: $i] : ( X906 @ V_20 ) ) )
& ( ( dsetconstr @ X904
@ ^ [V_19: $i] : ( X906 @ V_19 ) )
= ( setadjoin @ X912 @ emptyset ) ) ) ) ) )
=> ( ! [X914: $i,X916: $i > $o] :
( ! [X918: $i] :
( ( in @ X918 @ X914 )
=> ! [X920: $i] :
( ( in @ X920 @ X914 )
=> ( ( X916 @ X918 )
=> ( ( X916 @ X920 )
=> ( X918 = X920 ) ) ) ) )
=> ( ? [X922: $i] :
( ( in @ X922 @ X914 )
& ( X916 @ X922 ) )
=> ? [X924: $i] :
( ( in @ X924
@ ( dsetconstr @ X914
@ ^ [V_22: $i] : ( X916 @ V_22 ) ) )
& ( ( dsetconstr @ X914
@ ^ [V_21: $i] : ( X916 @ V_21 ) )
= ( setadjoin @ X924 @ emptyset ) ) ) ) )
=> ( ! [X926: $i,X928: $i] :
( ( ( setadjoin @ X926 @ emptyset )
= ( setadjoin @ X928 @ emptyset ) )
=> ( X926 = X928 ) )
=> ( ! [X930: $i,X932: $i,X934: $i] :
( ( in @ ( setadjoin @ X934 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X930 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X930 @ ( setadjoin @ X932 @ emptyset ) ) @ emptyset ) ) )
=> ( X930 = X934 ) )
=> ( ! [X936: $i] :
( ( iskpair @ X936 )
=> ? [X938: $i] :
( ( in @ X938
@ ( dsetconstr @ ( setunion @ X936 )
@ ^ [V_24: $i] : ( in @ ( setadjoin @ V_24 @ emptyset ) @ X936 ) ) )
& ( ( dsetconstr @ ( setunion @ X936 )
@ ^ [V_23: $i] : ( in @ ( setadjoin @ V_23 @ emptyset ) @ X936 ) )
= ( setadjoin @ X938 @ emptyset ) ) ) )
=> ( ! [X940: $i] :
( ? [X942: $i] :
( ( in @ X942 @ X940 )
& ( X940
= ( setadjoin @ X942 @ emptyset ) ) )
=> ( in @ ( setunion @ X940 ) @ X940 ) )
=> ( ! [X944: $i,X946: $i] :
( ( kfst @ ( kpair @ X944 @ X946 ) )
= X944 )
=> ( ! [X948: $i,X950: $i,X952: $i] :
( ( in @ X952 @ ( cartprod @ X948 @ X950 ) )
=> ( in @ ( kfst @ X952 ) @ X948 ) )
=> ( ! [X954: $i,X956: $i,X958: $i,X960: $i] :
( ( ( setadjoin @ ( setadjoin @ X954 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X954 @ ( setadjoin @ X956 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X958 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X958 @ ( setadjoin @ X960 @ emptyset ) ) @ emptyset ) ) )
=> ( X954 = X958 ) )
=> ( ! [X962: $i,X964: $i,X966: $i,X968: $i] :
( ( ( kpair @ X962 @ X964 )
= ( kpair @ X966 @ X968 ) )
=> ( X962 = X966 ) )
=> ( ! [X970: $i,X972: $i] :
( ( X970 = X972 )
=> ( ( setadjoin @ X970 @ ( setadjoin @ X972 @ emptyset ) )
= ( setadjoin @ X970 @ emptyset ) ) )
=> ( ! [X974: $i,X976: $i] :
( ( X974 = X976 )
=> ( ( setadjoin @ ( setadjoin @ X974 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X974 @ ( setadjoin @ X976 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X974 @ emptyset ) @ emptyset ) ) )
=> ( ! [X978: $i,X980: $i,X982: $i,X984: $i] :
( ( ( setadjoin @ ( setadjoin @ X978 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X978 @ ( setadjoin @ X980 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X982 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X982 @ ( setadjoin @ X984 @ emptyset ) ) @ emptyset ) ) )
=> ( ( X982 = X984 )
=> ( X980 = X984 ) ) )
=> ( ! [X986: $i,X988: $i,X990: $i] :
( ( ( setadjoin @ X986 @ ( setadjoin @ X988 @ emptyset ) )
= ( setadjoin @ X990 @ emptyset ) )
=> ( X986 = X988 ) )
=> ( ! [X992: $i,X994: $i,X996: $i,X998: $i] :
( ( ( setadjoin @ ( setadjoin @ X992 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X992 @ ( setadjoin @ X994 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ X996 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X996 @ ( setadjoin @ X998 @ emptyset ) ) @ emptyset ) ) )
=> ( X994 = X998 ) )
=> ( ! [X1000: $i,X1002: $i,X1004: $i,X1006: $i] :
( ( ( kpair @ X1000 @ X1002 )
= ( kpair @ X1004 @ X1006 ) )
=> ( X1002 = X1006 ) )
=> ( ! [X1008: $i] :
( ( iskpair @ X1008 )
=> ? [X1010: $i] :
( ( in @ X1010
@ ( dsetconstr @ ( setunion @ X1008 )
@ ^ [V_26: $i] :
( X1008
= ( kpair @ ( kfst @ X1008 ) @ V_26 ) ) ) )
& ( ( dsetconstr @ ( setunion @ X1008 )
@ ^ [V_25: $i] :
( X1008
= ( kpair @ ( kfst @ X1008 ) @ V_25 ) ) )
= ( setadjoin @ X1010 @ emptyset ) ) ) )
=> ( ! [X1012: $i,X1014: $i] :
( ( ksnd @ ( kpair @ X1012 @ X1014 ) )
= X1014 )
=> ( ! [X1016: $i] :
( ( iskpair @ X1016 )
=> ( ( kpair @ ( kfst @ X1016 ) @ ( ksnd @ X1016 ) )
= X1016 ) )
=> ( ! [X1018: $i,X1020: $i,X1022: $i] :
( ( in @ X1022 @ ( cartprod @ X1018 @ X1020 ) )
=> ( in @ ( ksnd @ X1022 ) @ X1020 ) )
=> ( ! [X1024: $i,X1026: $i,X1028: $i,X1030: $i] :
( ( in @ ( kpair @ X1028 @ X1030 ) @ ( cartprod @ X1024 @ X1026 ) )
=> ( in @ X1028 @ X1024 ) )
=> ( ! [X1032: $i,X1034: $i,X1036: $i,X1038: $i] :
( ( in @ ( kpair @ X1036 @ X1038 ) @ ( cartprod @ X1032 @ X1034 ) )
=> ( in @ X1038 @ X1034 ) )
=> ( ! [X1040: $i,X1042: $i,X1044: $i] :
( ( in @ X1044 @ X1040 )
=> ! [X1046: $i] :
( ( in @ X1046 @ X1042 )
=> ( ( kpair @ X1044 @ X1046 )
= ( kpair @ X1044 @ X1046 ) ) ) )
=> ( ! [X1048: $i,X1050: $i,X1052: $i] :
( ( in @ X1052 @ X1048 )
=> ! [X1054: $i] :
( ( in @ X1054 @ X1050 )
=> ( ( kfst @ ( kpair @ X1052 @ X1054 ) )
= X1052 ) ) )
=> ( ! [X1056: $i,X1058: $i,X1060: $i] :
( ( in @ X1060 @ X1056 )
=> ! [X1062: $i] :
( ( in @ X1062 @ X1058 )
=> ( ( ksnd @ ( kpair @ X1060 @ X1062 ) )
= X1062 ) ) )
=> ( ! [X1064: $i,X1066: $i,X1068: $i] :
( ( in @ X1068 @ ( cartprod @ X1064 @ X1066 ) )
=> ( ( kpair @ ( kfst @ X1068 ) @ ( ksnd @ X1068 ) )
= X1068 ) )
=> ( ! [X1070: $i,X1072: $i,X1074: $i > $i > $o,X1076: $i] :
( ( in @ X1076 @ X1070 )
=> ! [X1078: $i] :
( ( in @ X1078 @ X1072 )
=> ( ( X1074 @ X1076 @ X1078 )
=> ( in @ ( kpair @ X1076 @ X1078 )
@ ( dpsetconstr @ X1070 @ X1072
@ ^ [V_27: $i,V_28: $i] : ( X1074 @ V_27 @ V_28 ) ) ) ) ) )
=> ( ! [X1080: $i,X1082: $i,X1084: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1080 @ X1082
@ ^ [V_29: $i,V_30: $i] : ( X1084 @ V_29 @ V_30 ) )
@ ( cartprod @ X1080 @ X1082 ) )
=> ( ! [X1086: $i,X1088: $i,X1090: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X1086 @ X1088
@ ^ [V_31: $i,V_32: $i] : ( X1090 @ V_31 @ V_32 ) )
@ ( cartprod @ X1086 @ X1088 ) )
=> ( ! [X1092: $i,X1094: $i,X1096: $i > $i > $o,X1098: $i] :
( ( in @ X1098 @ X1092 )
=> ! [X1100: $i] :
( ( in @ X1100 @ X1094 )
=> ( ( in @ ( kpair @ X1098 @ X1100 )
@ ( dpsetconstr @ X1092 @ X1094
@ ^ [V_33: $i,V_34: $i] : ( X1096 @ V_33 @ V_34 ) ) )
=> ( X1096 @ X1098 @ X1100 ) ) ) )
=> ( ! [X1102: $i,X1104: $i,X1106: $i > $i > $o,X1108: $i,X1110: $i] :
( ( in @ ( kpair @ X1108 @ X1110 )
@ ( dpsetconstr @ X1102 @ X1104
@ ^ [V_35: $i,V_36: $i] : ( X1106 @ V_35 @ V_36 ) ) )
=> ( in @ X1108 @ X1102 ) )
=> ( ! [X1112: $i,X1114: $i,X1116: $i > $i > $o,X1118: $i,X1120: $i] :
( ( in @ ( kpair @ X1118 @ X1120 )
@ ( dpsetconstr @ X1112 @ X1114
@ ^ [V_37: $i,V_38: $i] : ( X1116 @ V_37 @ V_38 ) ) )
=> ( in @ X1120 @ X1114 ) )
=> ( ! [X1122: $i,X1124: $i,X1126: $i > $i > $o,X1128: $i,X1130: $i] :
( ( in @ ( kpair @ X1128 @ X1130 )
@ ( dpsetconstr @ X1122 @ X1124
@ ^ [V_39: $i,V_40: $i] : ( X1126 @ V_39 @ V_40 ) ) )
=> ( X1126 @ X1128 @ X1130 ) )
=> ( ! [X1132: $i,X1134: $i,X1136: $i] :
( ( ! [X1138: $i] :
( ( in @ X1138 @ X1132 )
=> ? [X1140: $i] :
( ( in @ X1140
@ ( dsetconstr @ X1134
@ ^ [V_42: $i] : ( in @ ( kpair @ X1138 @ V_42 ) @ X1136 ) ) )
& ( ( dsetconstr @ X1134
@ ^ [V_41: $i] : ( in @ ( kpair @ X1138 @ V_41 ) @ X1136 ) )
= ( setadjoin @ X1140 @ emptyset ) ) ) )
& ( subset @ X1136 @ ( cartprod @ X1132 @ X1134 ) ) )
=> ! [X1142: $i] :
( ( in @ X1142 @ X1132 )
=> ? [X1144: $i] :
( ( in @ X1144
@ ( dsetconstr @ X1134
@ ^ [V_44: $i] : ( in @ ( kpair @ X1142 @ V_44 ) @ X1136 ) ) )
& ( ( dsetconstr @ X1134
@ ^ [V_43: $i] : ( in @ ( kpair @ X1142 @ V_43 ) @ X1136 ) )
= ( setadjoin @ X1144 @ emptyset ) ) ) ) )
=> ( ! [X1146: $i,X1148: $i,X1150: $i] :
( ( ! [X1152: $i] :
( ( in @ X1152 @ X1146 )
=> ? [X1154: $i] :
( ( in @ X1154
@ ( dsetconstr @ X1148
@ ^ [V_46: $i] : ( in @ ( kpair @ X1152 @ V_46 ) @ X1150 ) ) )
& ( ( dsetconstr @ X1148
@ ^ [V_45: $i] : ( in @ ( kpair @ X1152 @ V_45 ) @ X1150 ) )
= ( setadjoin @ X1154 @ emptyset ) ) ) )
& ( subset @ X1150 @ ( cartprod @ X1146 @ X1148 ) ) )
=> ! [X1156: $i] :
( ( in @ X1156 @ X1146 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X1148
@ ^ [V_47: $i] : ( in @ ( kpair @ X1156 @ V_47 ) @ X1150 ) ) )
@ X1148 ) ) )
=> ( ! [X1158: $i,X1160: $i,X1162: $i] :
( ( ! [X1164: $i] :
( ( in @ X1164 @ X1158 )
=> ? [X1166: $i] :
( ( in @ X1166
@ ( dsetconstr @ X1160
@ ^ [V_49: $i] : ( in @ ( kpair @ X1164 @ V_49 ) @ X1162 ) ) )
& ( ( dsetconstr @ X1160
@ ^ [V_48: $i] : ( in @ ( kpair @ X1164 @ V_48 ) @ X1162 ) )
= ( setadjoin @ X1166 @ emptyset ) ) ) )
& ( subset @ X1162 @ ( cartprod @ X1158 @ X1160 ) ) )
=> ! [X1168: $i] :
( ( in @ X1168 @ X1158 )
=> ( in
@ ( setunion
@ ( dsetconstr @ X1160
@ ^ [V_50: $i] : ( in @ ( kpair @ X1168 @ V_50 ) @ X1162 ) ) )
@ X1160 ) ) )
=> ! [X1170: $i,X1172: $i,X1174: $i] :
( ( in @ X1174
@ ( dsetconstr @ ( powerset @ ( cartprod @ X1170 @ X1172 ) )
@ ^ [V_51: $i] :
( ! [X1176: $i] :
( ( in @ X1176 @ X1170 )
=> ? [X1178: $i] :
( ( in @ X1178
@ ( dsetconstr @ X1172
@ ^ [V_53: $i] : ( in @ ( kpair @ X1176 @ V_53 ) @ V_51 ) ) )
& ( ( dsetconstr @ X1172
@ ^ [V_52: $i] : ( in @ ( kpair @ X1176 @ V_52 ) @ V_51 ) )
= ( setadjoin @ X1178 @ emptyset ) ) ) )
& ( subset @ V_51 @ ( cartprod @ X1170 @ X1172 ) ) ) ) )
=> ( ! [X1180: $i] :
( ( in @ X1180 @ X1170 )
=> ? [X1182: $i] :
( ( in @ X1182
@ ( dsetconstr @ X1172
@ ^ [V_55: $i] : ( in @ ( kpair @ X1180 @ V_55 ) @ X1174 ) ) )
& ( ( dsetconstr @ X1172
@ ^ [V_54: $i] : ( in @ ( kpair @ X1180 @ V_54 ) @ X1174 ) )
= ( setadjoin @ X1182 @ emptyset ) ) ) )
& ( subset @ X1174 @ ( cartprod @ X1170 @ X1172 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference('cnf.neg',[status(esa)],[zf_stmt_0]) ).
thf(zip_derived_cl1943,plain,
~ ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
<=> ( in @ Y2 @ Y1 ) ) )
=> ( Y0 = Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] : ( (~) @ ( in @ Y0 @ emptyset ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( setadjoin @ Y0 @ Y1 ) )
<=> ( ( in @ Y2 @ Y1 )
| ( Y2 = Y0 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( powerset @ Y0 ) )
<=> ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( in @ Y2 @ Y0 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( setunion @ Y0 ) )
<=> ( ??
@ ^ [Y2: $i] :
( ( in @ Y1 @ Y2 )
& ( in @ Y2 @ Y0 ) ) ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ( !!
@ ^ [Y0: $i] :
( ( in @ Y0 @ omega )
=> ( in @ ( setadjoin @ Y0 @ Y0 ) @ omega ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( ( !!
@ ^ [Y1: $i] :
( ( ( in @ Y1 @ Y0 )
& ( in @ Y1 @ omega ) )
=> ( in @ ( setadjoin @ Y1 @ Y1 ) @ Y0 ) ) )
& ( in @ emptyset @ Y0 ) )
=> ( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ omega )
=> ( in @ Y1 @ Y0 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $i > $o] :
( !!
@ ^ [Y1: $i] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( ??
@ ^ [Y3: $i] :
( ( Y0 @ Y2 @ Y3 )
& ( !!
@ ^ [Y4: $i] :
( ( Y0 @ Y2 @ Y4 )
=> ( Y3 = Y4 ) ) ) ) ) ) )
=> ( ??
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
<=> ( ??
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( Y0 @ Y4 @ Y3 ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( ??
@ ^ [Y1: $i] : ( in @ Y1 @ Y0 ) )
=> ( ??
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
& ( (~)
@ ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
& ( in @ Y2 @ Y0 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ??
@ ^ [Y1: $i] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ Y3 @ Y0 ) ) ) ) )
& ( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( ( in @ Y3 @ Y0 )
& ( in @ Y2 @ Y0 ) )
=> ( ( !!
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ Y2 @ Y4 )
<=> ( in @ Y3 @ Y4 ) ) ) )
=> ( Y2 = Y3 ) ) ) ) )
& ( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( ( in @ Y3 @ Y1 )
& ( in @ Y2 @ Y1 ) )
=> ( ( !!
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y3 )
=> ( in @ Y4 @ Y2 ) ) )
| ( !!
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y2 )
=> ( in @ Y4 @ Y3 ) ) ) ) ) ) )
& ( !!
@ ^ [Y2: $i] :
( ( ( ??
@ ^ [Y3: $i] : ( in @ Y3 @ Y2 ) )
& ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y2 )
=> ( in @ Y3 @ Y0 ) ) ) )
=> ( ??
@ ^ [Y3: $i] :
( ??
@ ^ [Y4: $i] :
( ( in @ Y3 @ Y1 )
& ( in @ Y4 @ Y2 )
& ( (~)
@ ( ??
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y3 )
& ( in @ Y5 @ Y2 ) ) ) )
& ( !!
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( in @ Y4 @ Y5 )
| ( !!
@ ^ [Y6: $i] :
( ( in @ Y6 @ Y5 )
=> ( in @ Y6 @ Y3 ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
=> ( Y0 @ ( descr @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( Y1 @ Y2 )
=> ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
=> ( in @ Y2 @ Y0 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
=> ( Y1 @ Y2 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
=> ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $o] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( prop2set @ Y0 ) )
=> Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( in @ Y0 @ emptyset )
=> ( !!
@ ^ [Y1: $o] : Y1 ) ) )
=> ( ( !!
@ ^ [Y0: $i] : ( (~) @ ( in @ Y0 @ emptyset ) ) )
=> ( ( !!
@ ^ [Y0: $i] : ( (~) @ ( in @ Y0 @ emptyset ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
=> ( ??
@ ^ [Y1: $i] : ( Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) )
=> ( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( in @ Y2 @ Y0 ) ) )
=> ( Y0 = Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( !!
@ ^ [Y1: $i] : ( (~) @ ( in @ Y1 @ Y0 ) ) )
=> ( Y0 = emptyset ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( !!
@ ^ [Y1: $i] : ( (~) @ ( in @ Y1 @ Y0 ) ) )
=> ( Y0 = emptyset ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
<=> ( Y1 @ Y2 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( Y0 != emptyset )
=> ( ??
@ ^ [Y1: $i] : ( in @ Y1 @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( Y1 @ Y2 )
=> ( ( dsetconstr @ Y0 @ Y1 )
!= emptyset ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( ??
@ ^ [Y1: $i] : ( in @ Y1 @ Y0 ) )
=> ( Y0 != emptyset ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ Y1 ) ) ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( in @ Y2 @ ( setadjoin @ Y0 @ Y1 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( setadjoin @ Y0 @ Y1 ) )
=> ( !!
@ ^ [Y3: $o] :
( ( ( Y2 = Y0 )
=> Y3 )
=> ( ( ( in @ Y2 @ Y1 )
=> Y3 )
=> Y3 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( setadjoin @ Y0 @ Y1 ) )
=> ( ( in @ Y2 @ Y1 )
| ( Y2 = Y0 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( dsetconstr @ Y0
@ ^ [Y1: $i] : $true )
= Y0 ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( in @ Y2 @ Y0 ) ) )
=> ( in @ Y1 @ ( powerset @ Y0 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] : ( in @ emptyset @ ( powerset @ Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] : ( in @ emptyset @ ( powerset @ Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y1 @ ( powerset @ Y0 ) )
=> ( ( in @ Y2 @ Y1 )
=> ( in @ Y2 @ Y0 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y1 @ Y2 )
=> ( ( in @ Y2 @ Y0 )
=> ( in @ Y1 @ ( setunion @ Y0 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( setunion @ Y0 ) )
=> ( !!
@ ^ [Y2: $o] :
( ( !!
@ ^ [Y3: $i] :
( ( in @ Y1 @ Y3 )
=> ( ( in @ Y3 @ Y0 )
=> Y2 ) ) )
=> Y2 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( in @ Y1 @ ( powerset @ ( setunion @ Y0 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
=> ( ??
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
<=> ( Y2 = Y1 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( Y0 != emptyset )
=> ( ??
@ ^ [Y1: $i] : ( in @ Y1 @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y0 @ ( setadjoin @ Y1 @ emptyset ) )
=> ( Y0 = Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 != Y1 )
=> ( (~) @ ( in @ Y1 @ ( setadjoin @ Y0 @ emptyset ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( in @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y0 @ ( setadjoin @ Y1 @ emptyset ) )
=> ( in @ Y1 @ ( setadjoin @ Y0 @ emptyset ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) )
=> ( ( Y2 = Y1 )
| ( Y2 = Y0 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( in @ Y0 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( Y1 @ Y2 ) ) )
=> ( ( dsetconstr @ Y0 @ Y1 )
!= emptyset ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ emptyset )
=> ( Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( (~)
@ ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( Y1 @ Y2 ) ) ) )
=> ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( (~) @ ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( (~) @ ( Y1 @ Y2 ) ) ) )
=> ( (~)
@ ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( (~)
@ ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( Y1 @ Y2 ) ) ) )
=> ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( (~) @ ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( (~) @ ( Y1 @ Y2 ) ) ) )
=> ( (~)
@ ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $o] :
( Y0
=> ( in @ emptyset @ ( prop2set @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $o] :
( Y0
=> ( set2prop @ ( prop2set @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( (~)
@ ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( Y1 @ Y2 ) ) ) )
=> ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( (~) @ ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( (~)
@ ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( Y1 @ Y2 ) ) ) )
=> ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( (~) @ ( Y1 @ Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
=> ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( ( ??
@ ^ [Y1: $i] : ( Y0 @ Y1 ) )
=> ( ( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y1 )
=> ( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
=> ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( ( ??
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
<=> ( Y2 = Y1 ) ) ) )
=> ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( Y2 = Y3 )
=> ( ( in @ Y2 @ Y0 )
=> ( in @ Y3 @ Y1 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( Y2 = Y3 )
=> ( ( in @ Y2 @ Y0 )
<=> ( in @ Y3 @ Y1 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
=> ( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y1 )
=> ( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( !!
@ ^ [Y1: $i > $o] :
( ( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( Y2 = Y3 )
=> ( ( Y0 @ Y2 )
<=> ( Y1 @ Y3 ) ) ) ) )
=> ( ( ??
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( !!
@ ^ [Y3: $i] :
( ( Y0 @ Y3 )
=> ( Y2 = Y3 ) ) ) ) )
<=> ( ??
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( !!
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
=> ( Y2 = Y3 ) ) ) ) ) ) ) ) )
=> ( ( emptyset = emptyset )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( Y2 = Y3 )
=> ( ( setadjoin @ Y0 @ Y2 )
= ( setadjoin @ Y1 @ Y3 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( ( powerset @ Y0 )
= ( powerset @ Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( ( setunion @ Y0 )
= ( setunion @ Y1 ) ) ) ) )
=> ( ( omega = omega )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( ( ??
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
=> ( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( Y0 @ Y1 )
=> ( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i > $o] :
( !!
@ ^ [Y1: $i > $o] :
( ( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( Y2 = Y3 )
=> ( ( Y0 @ Y2 )
<=> ( Y1 @ Y3 ) ) ) ) )
=> ( ( ??
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
& ( !!
@ ^ [Y3: $i] :
( ( Y0 @ Y3 )
=> ( Y2 = Y3 ) ) ) ) )
=> ( ( ??
@ ^ [Y2: $i] :
( ( Y1 @ Y2 )
& ( !!
@ ^ [Y3: $i] :
( ( Y1 @ Y3 )
=> ( Y2 = Y3 ) ) ) ) )
=> ( ( descr @ Y0 )
= ( descr @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( !!
@ ^ [Y2: $i > $o] :
( !!
@ ^ [Y3: $i > $o] :
( ( !!
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( !!
@ ^ [Y5: $i] :
( ( in @ Y5 @ Y1 )
=> ( ( Y4 = Y5 )
=> ( ( Y2 @ Y4 )
<=> ( Y3 @ Y5 ) ) ) ) ) ) )
=> ( ( dsetconstr @ Y0 @ Y2 )
= ( dsetconstr @ Y1 @ Y3 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) )
=> ( subset @ Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( subset @ Y1 @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( subset @ Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) )
=> ( subset @ Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] : ( subset @ emptyset @ Y0 ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( subset @ Y0 @ Y1 )
=> ( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( subset @ Y0 @ Y1 )
=> ( ( (~) @ ( in @ Y2 @ Y1 ) )
=> ( (~) @ ( in @ Y2 @ Y0 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( (~) @ ( in @ Y2 @ Y1 ) )
=> ( (~) @ ( subset @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( (~) @ ( subset @ Y0 @ Y1 ) )
=> ( Y0 != Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( (~) @ ( in @ Y2 @ Y1 ) )
=> ( Y0 != Y1 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] : ( subset @ Y0 @ Y0 ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( subset @ Y0 @ Y1 )
=> ( ( subset @ Y1 @ Y2 )
=> ( subset @ Y0 @ Y2 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( subset @ Y1 @ ( setadjoin @ Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( subset @ Y0 @ Y2 )
=> ( subset @ Y0 @ ( setadjoin @ Y1 @ Y2 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( subset @ Y0 @ Y1 )
=> ( in @ Y0 @ ( powerset @ Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( subset @ Y0 @ Y1 )
=> ( ( subset @ Y1 @ Y0 )
=> ( Y0 = Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( subset @ Y0 @ emptyset )
=> ( Y0 = emptyset ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( subset @ Y1 @ Y0 )
=> ( in @ Y1 @ ( powerset @ Y0 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( powerset @ Y0 ) )
=> ( subset @ Y1 @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] : ( in @ Y0 @ ( powerset @ Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( subset @ Y0 @ Y1 )
=> ( subset @ ( powerset @ Y0 ) @ ( powerset @ Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] : ( in @ ( dsetconstr @ Y0 @ Y1 ) @ ( powerset @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] : ( subset @ ( dsetconstr @ Y0 @ Y1 ) @ Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ ( binunion @ Y0 @ Y1 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( in @ Y2 @ ( binunion @ Y0 @ Y1 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $o] :
( ( in @ Y2 @ ( binunion @ Y0 @ Y1 ) )
=> ( ( ( in @ Y2 @ Y0 )
=> Y3 )
=> ( ( ( in @ Y2 @ Y1 )
=> Y3 )
=> Y3 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( binunion @ Y0 @ Y1 ) )
=> ( ( in @ Y2 @ Y1 )
| ( in @ Y2 @ Y0 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( subset @ Y0 @ ( binunion @ Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( subset @ Y1 @ ( binunion @ Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( in @ Y2 @ Y1 )
=> ( in @ Y2 @ ( binintersect @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( subset @ Y2 @ Y0 )
=> ( ( subset @ Y2 @ Y1 )
=> ( subset @ Y2 @ ( binintersect @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( binintersect @ Y0 @ Y1 ) )
=> ( in @ Y2 @ Y0 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( subset @ ( binintersect @ Y0 @ Y1 ) @ Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( subset @ Y0 @ Y1 )
=> ( ( binintersect @ Y0 @ Y1 )
= Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( ( binintersect @ Y0 @ Y1 )
= Y1 )
=> ( subset @ Y1 @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( binintersect @ Y0 @ Y1 ) )
=> ( in @ Y2 @ Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( (~)
@ ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( in @ Y2 @ Y1 ) ) ) )
=> ( ( binintersect @ Y0 @ Y1 )
= emptyset ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( subset @ ( binintersect @ Y0 @ Y1 ) @ Y1 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( subset @ Y1 @ Y0 )
=> ( ( binintersect @ Y0 @ Y1 )
= Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( ( binintersect @ Y0 @ Y1 )
= Y0 )
=> ( subset @ Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( binintersect @ Y0 @ ( binunion @ Y1 @ Y2 ) )
= ( binunion @ ( binintersect @ Y0 @ Y1 ) @ ( binintersect @ Y0 @ Y2 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( (~) @ ( in @ Y2 @ Y1 ) )
=> ( in @ Y2 @ ( setminus @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( setminus @ Y0 @ Y1 ) )
=> ( in @ Y2 @ Y0 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( setminus @ Y0 @ Y1 ) )
=> ( (~) @ ( in @ Y2 @ Y1 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( subset @ Y0 @ Y1 )
=> ( ( setminus @ Y0 @ Y1 )
= emptyset ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( (~) @ ( in @ Y2 @ ( setminus @ Y0 @ Y1 ) ) )
=> ( ( in @ Y2 @ Y0 )
=> ( in @ Y2 @ Y1 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( (~) @ ( in @ Y2 @ ( setminus @ Y0 @ Y1 ) ) )
=> ( ( (~) @ ( in @ Y2 @ Y1 ) )
=> ( (~) @ ( in @ Y2 @ Y0 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( (~) @ ( in @ Y2 @ Y0 ) )
=> ( (~) @ ( in @ Y2 @ ( setminus @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y1 )
=> ( (~) @ ( in @ Y2 @ ( setminus @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( subset @ ( setminus @ Y0 @ Y1 ) @ Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( ( setminus @ Y0 @ Y1 )
= emptyset )
=> ( subset @ Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( symdiff @ Y0 @ Y1 ) )
=> ( !!
@ ^ [Y3: $o] :
( ( ( in @ Y2 @ Y0 )
=> ( ( (~) @ ( in @ Y2 @ Y1 ) )
=> Y3 ) )
=> ( ( ( (~) @ ( in @ Y2 @ Y0 ) )
=> ( ( in @ Y2 @ Y1 )
=> Y3 ) )
=> Y3 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( (~) @ ( in @ Y2 @ Y1 ) )
=> ( in @ Y2 @ ( symdiff @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( (~) @ ( in @ Y2 @ Y0 ) )
=> ( ( in @ Y2 @ Y1 )
=> ( in @ Y2 @ ( symdiff @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( in @ Y2 @ Y1 )
=> ( (~) @ ( in @ Y2 @ ( symdiff @ Y0 @ Y1 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( (~) @ ( in @ Y2 @ Y0 ) )
=> ( ( (~) @ ( in @ Y2 @ Y1 ) )
=> ( (~) @ ( in @ Y2 @ ( symdiff @ Y0 @ Y1 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( in @ Y1 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( in @ Y0 @ ( setunion @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( in @ Y1 @ ( setunion @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] : ( iskpair @ ( kpair @ Y0 @ Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( subset @ ( setadjoin @ Y1 @ emptyset ) @ Y0 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
=> ( in @ ( setadjoin @ Y1 @ emptyset ) @ ( powerset @ Y0 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( in @ ( setadjoin @ Y2 @ emptyset ) @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) )
=> ( ( Y2 = Y1 )
| ( Y2 = Y0 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( subset @ ( setadjoin @ Y2 @ ( setadjoin @ Y3 @ emptyset ) ) @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( in @ ( setadjoin @ Y2 @ ( setadjoin @ Y3 @ emptyset ) ) @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( subset @ ( setadjoin @ ( setadjoin @ Y2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y2 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( in @ ( setadjoin @ ( setadjoin @ Y2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y2 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( in @ ( kpair @ Y2 @ Y3 ) @ ( powerset @ ( powerset @ ( binunion @ Y0 @ Y1 ) ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( in @ ( kpair @ Y2 @ Y3 ) @ ( cartprod @ Y0 @ Y1 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
=> ( ??
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
& ( ??
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
& ( Y2
= ( kpair @ Y3 @ Y4 ) ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
=> ( iskpair @ Y2 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( in @ Y1 @ ( setunion @ Y0 ) )
=> ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( in @ Y1 @ Y2 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] : ( subset @ ( setunion @ ( setadjoin @ Y0 @ emptyset ) ) @ Y0 ) )
=> ( ( !!
@ ^ [Y0: $i] : ( subset @ Y0 @ ( setunion @ ( setadjoin @ Y0 @ emptyset ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( setunion @ ( setadjoin @ Y0 @ emptyset ) )
= Y0 ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ( Y1 @ Y2 )
=> ( ( Y1 @ Y3 )
=> ( Y2 = Y3 ) ) ) ) ) ) )
=> ( ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( Y1 @ Y2 ) ) )
=> ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
& ( ( dsetconstr @ Y0 @ Y1 )
= ( setadjoin @ Y2 @ emptyset ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
& ( ( dsetconstr @ Y0 @ Y1 )
= ( setadjoin @ Y2 @ emptyset ) ) ) )
=> ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( Y1 @ Y2 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( ( Y1 @ Y2 )
=> ( ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ( Y1 @ Y3 )
=> ( Y3 = Y2 ) ) ) )
=> ( ??
@ ^ [Y3: $i] :
( ( in @ Y3 @ ( dsetconstr @ Y0 @ Y1 ) )
& ( ( dsetconstr @ Y0 @ Y1 )
= ( setadjoin @ Y3 @ emptyset ) ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i > $o] :
( ( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ( Y1 @ Y2 )
=> ( ( Y1 @ Y3 )
=> ( Y2 = Y3 ) ) ) ) ) ) )
=> ( ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
& ( Y1 @ Y2 ) ) )
=> ( ??
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( dsetconstr @ Y0 @ Y1 ) )
& ( ( dsetconstr @ Y0 @ Y1 )
= ( setadjoin @ Y2 @ emptyset ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( ( setadjoin @ Y0 @ emptyset )
= ( setadjoin @ Y1 @ emptyset ) )
=> ( Y0 = Y1 ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ ( setadjoin @ Y2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) ) )
=> ( Y0 = Y2 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( iskpair @ Y0 )
=> ( ??
@ ^ [Y1: $i] :
( ( in @ Y1
@ ( dsetconstr @ ( setunion @ Y0 )
@ ^ [Y2: $i] : ( in @ ( setadjoin @ Y2 @ emptyset ) @ Y0 ) ) )
& ( ( dsetconstr @ ( setunion @ Y0 )
@ ^ [Y2: $i] : ( in @ ( setadjoin @ Y2 @ emptyset ) @ Y0 ) )
= ( setadjoin @ Y1 @ emptyset ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( ??
@ ^ [Y1: $i] :
( ( in @ Y1 @ Y0 )
& ( Y0
= ( setadjoin @ Y1 @ emptyset ) ) ) )
=> ( in @ ( setunion @ Y0 ) @ Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( kfst @ ( kpair @ Y0 @ Y1 ) )
= Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
=> ( in @ ( kfst @ Y2 ) @ Y0 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Y2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y2 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) )
=> ( Y0 = Y2 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( ( kpair @ Y0 @ Y1 )
= ( kpair @ Y2 @ Y3 ) )
=> ( Y0 = Y2 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) )
= ( setadjoin @ Y0 @ emptyset ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( Y0 = Y1 )
=> ( ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ emptyset ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Y2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y2 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) )
=> ( ( Y2 = Y3 )
=> ( Y1 = Y3 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) )
= ( setadjoin @ Y2 @ emptyset ) )
=> ( Y0 = Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( ( setadjoin @ ( setadjoin @ Y0 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y0 @ ( setadjoin @ Y1 @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ Y2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ Y2 @ ( setadjoin @ Y3 @ emptyset ) ) @ emptyset ) ) )
=> ( Y1 = Y3 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( ( kpair @ Y0 @ Y1 )
= ( kpair @ Y2 @ Y3 ) )
=> ( Y1 = Y3 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( iskpair @ Y0 )
=> ( ??
@ ^ [Y1: $i] :
( ( in @ Y1
@ ( dsetconstr @ ( setunion @ Y0 )
@ ^ [Y2: $i] :
( Y0
= ( kpair @ ( kfst @ Y0 ) @ Y2 ) ) ) )
& ( ( dsetconstr @ ( setunion @ Y0 )
@ ^ [Y2: $i] :
( Y0
= ( kpair @ ( kfst @ Y0 ) @ Y2 ) ) )
= ( setadjoin @ Y1 @ emptyset ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( ( ksnd @ ( kpair @ Y0 @ Y1 ) )
= Y1 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( ( iskpair @ Y0 )
=> ( ( kpair @ ( kfst @ Y0 ) @ ( ksnd @ Y0 ) )
= Y0 ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
=> ( in @ ( ksnd @ Y2 ) @ Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( in @ ( kpair @ Y2 @ Y3 ) @ ( cartprod @ Y0 @ Y1 ) )
=> ( in @ Y2 @ Y0 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( !!
@ ^ [Y3: $i] :
( ( in @ ( kpair @ Y2 @ Y3 ) @ ( cartprod @ Y0 @ Y1 ) )
=> ( in @ Y3 @ Y1 ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( kpair @ Y2 @ Y3 )
= ( kpair @ Y2 @ Y3 ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( kfst @ ( kpair @ Y2 @ Y3 ) )
= Y2 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ Y0 )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y1 )
=> ( ( ksnd @ ( kpair @ Y2 @ Y3 ) )
= Y3 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2 @ ( cartprod @ Y0 @ Y1 ) )
=> ( ( kpair @ ( kfst @ Y2 ) @ ( ksnd @ Y2 ) )
= Y2 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i > $i > $o] :
( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !!
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( Y2 @ Y3 @ Y4 )
=> ( in @ ( kpair @ Y3 @ Y4 ) @ ( dpsetconstr @ Y0 @ Y1 @ Y2 ) ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i > $i > $o] : ( subset @ ( dpsetconstr @ Y0 @ Y1 @ Y2 ) @ ( cartprod @ Y0 @ Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i > $i > $o] : ( subset @ ( dpsetconstr @ Y0 @ Y1 @ Y2 ) @ ( cartprod @ Y0 @ Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i > $i > $o] :
( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( !!
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y1 )
=> ( ( in @ ( kpair @ Y3 @ Y4 ) @ ( dpsetconstr @ Y0 @ Y1 @ Y2 ) )
=> ( Y2 @ Y3 @ Y4 ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i > $i > $o] :
( !!
@ ^ [Y3: $i] :
( !!
@ ^ [Y4: $i] :
( ( in @ ( kpair @ Y3 @ Y4 ) @ ( dpsetconstr @ Y0 @ Y1 @ Y2 ) )
=> ( in @ Y3 @ Y0 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i > $i > $o] :
( !!
@ ^ [Y3: $i] :
( !!
@ ^ [Y4: $i] :
( ( in @ ( kpair @ Y3 @ Y4 ) @ ( dpsetconstr @ Y0 @ Y1 @ Y2 ) )
=> ( in @ Y4 @ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i > $i > $o] :
( !!
@ ^ [Y3: $i] :
( !!
@ ^ [Y4: $i] :
( ( in @ ( kpair @ Y3 @ Y4 ) @ ( dpsetconstr @ Y0 @ Y1 @ Y2 ) )
=> ( Y2 @ Y3 @ Y4 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ??
@ ^ [Y4: $i] :
( ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) )
= ( setadjoin @ Y4 @ emptyset ) ) ) ) ) )
& ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ??
@ ^ [Y4: $i] :
( ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) )
= ( setadjoin @ Y4 @ emptyset ) ) ) ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ??
@ ^ [Y4: $i] :
( ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) )
= ( setadjoin @ Y4 @ emptyset ) ) ) ) ) )
& ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( in
@ ( setunion
@ ( dsetconstr @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) )
@ Y1 ) ) ) ) ) ) )
=> ( ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ??
@ ^ [Y4: $i] :
( ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) )
= ( setadjoin @ Y4 @ emptyset ) ) ) ) ) )
& ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) )
=> ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( in
@ ( setunion
@ ( dsetconstr @ Y1
@ ^ [Y4: $i] : ( in @ ( kpair @ Y3 @ Y4 ) @ Y2 ) ) )
@ Y1 ) ) ) ) ) ) )
=> ( !!
@ ^ [Y0: $i] :
( !!
@ ^ [Y1: $i] :
( !!
@ ^ [Y2: $i] :
( ( in @ Y2
@ ( dsetconstr @ ( powerset @ ( cartprod @ Y0 @ Y1 ) )
@ ^ [Y3: $i] :
( ( !!
@ ^ [Y4: $i] :
( ( in @ Y4 @ Y0 )
=> ( ??
@ ^ [Y5: $i] :
( ( in @ Y5
@ ( dsetconstr @ Y1
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) ) )
& ( ( dsetconstr @ Y1
@ ^ [Y6: $i] : ( in @ ( kpair @ Y4 @ Y6 ) @ Y3 ) )
= ( setadjoin @ Y5 @ emptyset ) ) ) ) ) )
& ( subset @ Y3 @ ( cartprod @ Y0 @ Y1 ) ) ) ) )
=> ( ( !!
@ ^ [Y3: $i] :
( ( in @ Y3 @ Y0 )
=> ( ??
@ ^ [Y4: $i] :
( ( in @ Y4
@ ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) ) )
& ( ( dsetconstr @ Y1
@ ^ [Y5: $i] : ( in @ ( kpair @ Y3 @ Y5 ) @ Y2 ) )
= ( setadjoin @ Y4 @ emptyset ) ) ) ) ) )
& ( subset @ Y2 @ ( cartprod @ Y0 @ Y1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(cnf,[status(esa)],[zf_stmt_1]) ).
thf(zip_derived_cl1944,plain,
$false,
inference(eprover,[status(thm)],[zip_derived_cl1943]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.17/0.17 % Problem : SEU675^1 : TPTP v8.1.2. Released v3.7.0.
% 0.17/0.18 % Command : python3 /export/starexec/sandbox2/solver/bin/portfolio.lams.parallel.py %s %d /export/starexec/sandbox2/tmp/tmp.0eGY29e6zI true
% 0.19/0.39 % Computer : n031.cluster.edu
% 0.19/0.39 % Model : x86_64 x86_64
% 0.19/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.19/0.39 % Memory : 8042.1875MB
% 0.19/0.39 % OS : Linux 3.10.0-693.el7.x86_64
% 0.19/0.39 % CPULimit : 300
% 0.19/0.39 % WCLimit : 300
% 0.19/0.39 % DateTime : Wed Aug 23 13:36:05 EDT 2023
% 0.19/0.39 % CPUTime :
% 0.19/0.39 % Running portfolio for 300 s
% 0.19/0.39 % File : /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.19/0.40 % Number of cores: 8
% 0.19/0.40 % Python version: Python 3.6.8
% 0.19/0.40 % Running in HO mode
% 0.25/0.69 % Total configuration time : 828
% 0.25/0.69 % Estimated wc time : 1656
% 0.25/0.69 % Estimated cpu time (8 cpus) : 207.0
% 0.25/0.78 % /export/starexec/sandbox2/solver/bin/lams/40_c.s.sh running for 80s
% 0.25/0.78 % /export/starexec/sandbox2/solver/bin/lams/35_full_unif4.sh running for 80s
% 0.25/0.79 % /export/starexec/sandbox2/solver/bin/lams/40_c_ic.sh running for 80s
% 0.25/0.79 % /export/starexec/sandbox2/solver/bin/lams/15_e_short1.sh running for 30s
% 0.25/0.79 % /export/starexec/sandbox2/solver/bin/lams/40_noforms.sh running for 90s
% 0.25/0.79 % /export/starexec/sandbox2/solver/bin/lams/40_b.comb.sh running for 70s
% 0.25/0.79 % /export/starexec/sandbox2/solver/bin/lams/20_acsne_simpl.sh running for 40s
% 1.41/0.85 % /export/starexec/sandbox2/solver/bin/lams/30_sp5.sh running for 60s
% 1.41/0.91 % /export/starexec/sandbox2/solver/bin/lams/30_b.l.sh running for 90s
% 59.31/8.29 % Solved by lams/15_e_short1.sh.
% 59.31/8.29 % done 1 iterations in 7.331s
% 59.31/8.29 % SZS status Theorem for '/export/starexec/sandbox2/benchmark/theBenchmark.p'
% 59.31/8.29 % SZS output start Refutation
% See solution above
% 59.31/8.33
% 59.31/8.33
% 59.31/8.33 % Terminating...
% 60.06/8.41 % Runner terminated.
% 60.06/8.42 % Zipperpin 1.5 exiting
%------------------------------------------------------------------------------