%------------------------------------------------------------------------------
% File : Leo-III---1.7.12
% Problem : SEU709^1 : TPTP v8.2.0. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n004.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 : Tue May 21 03:39:13 EDT 2024
% Result : Theorem 17.48s 7.08s
% Output : Refutation 18.59s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 478
% Syntax : Number of formulae : 481 ( 226 unt; 252 typ; 225 def)
% Number of atoms : 2099 ( 505 equ; 2 cnn)
% Maximal formula atoms : 560 ( 9 avg)
% Number of connectives : 5016 ( 116 ~; 24 |; 95 &;3360 @)
% ( 0 <=>;1421 =>; 0 <=; 0 <~>)
% Maximal formula depth : 239 ( 4 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 168 ( 168 >; 0 *; 0 +; 0 <<)
% Number of symbols : 260 ( 256 usr; 234 con; 0-4 aty)
% Number of variables : 1420 ( 22 ^1320 !; 78 ?;1420 :)
% Comments :
%------------------------------------------------------------------------------
thf(in_type,type,
in: $i > $i > $o ).
thf(setextAx_type,type,
setextAx: $o ).
thf(setextAx_def,definition,
( setextAx
= ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
= ( in @ C @ B ) )
=> ( A = B ) ) ) ) ).
thf(emptyset_type,type,
emptyset: $i ).
thf(emptysetAx_type,type,
emptysetAx: $o ).
thf(emptysetAx_def,definition,
( emptysetAx
= ( ! [A: $i] :
~ ( in @ A @ emptyset ) ) ) ).
thf(setadjoin_type,type,
setadjoin: $i > $i > $i ).
thf(setadjoinAx_type,type,
setadjoinAx: $o ).
thf(setadjoinAx_def,definition,
( setadjoinAx
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ B ) )
= ( ( C = A )
| ( in @ C @ B ) ) ) ) ) ).
thf(powerset_type,type,
powerset: $i > $i ).
thf(powersetAx_type,type,
powersetAx: $o ).
thf(powersetAx_def,definition,
( powersetAx
= ( ! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
= ( ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) ) ) ) ) ) ).
thf(setunion_type,type,
setunion: $i > $i ).
thf(setunionAx_type,type,
setunionAx: $o ).
thf(setunionAx_def,definition,
( setunionAx
= ( ! [A: $i,B: $i] :
( ( in @ B @ ( setunion @ A ) )
= ( ? [C: $i] :
( ( in @ B @ C )
& ( in @ C @ A ) ) ) ) ) ) ).
thf(omega_type,type,
omega: $i ).
thf(omega0Ax_type,type,
omega0Ax: $o ).
thf(omega0Ax_def,definition,
( omega0Ax
= ( in @ emptyset @ omega ) ) ).
thf(omegaSAx_type,type,
omegaSAx: $o ).
thf(omegaSAx_def,definition,
( omegaSAx
= ( ! [A: $i] :
( ( in @ A @ omega )
=> ( in @ ( setadjoin @ A @ A ) @ omega ) ) ) ) ).
thf(omegaIndAx_type,type,
omegaIndAx: $o ).
thf(omegaIndAx_def,definition,
( omegaIndAx
= ( ! [A: $i] :
( ( ( in @ emptyset @ A )
& ! [B: $i] :
( ( ( in @ B @ omega )
& ( in @ B @ A ) )
=> ( in @ ( setadjoin @ B @ B ) @ A ) ) )
=> ! [B: $i] :
( ( in @ B @ omega )
=> ( in @ B @ A ) ) ) ) ) ).
thf(replAx_type,type,
replAx: $o ).
thf(replAx_def,definition,
( replAx
= ( ! [A: $i > $i > $o,B: $i] :
( ! [C: $i] :
( ( in @ C @ B )
=> ( exu @ ( A @ C ) ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
= ( ? [E: $i] :
( ( in @ E @ B )
& ( A @ E @ D ) ) ) ) ) ) ) ).
thf(foundationAx_type,type,
foundationAx: $o ).
thf(foundationAx_def,definition,
( foundationAx
= ( ! [A: $i] :
( ? [B: $i] : ( in @ B @ A )
=> ? [B: $i] :
( ( in @ B @ A )
& ~ ? [C: $i] :
( ( in @ C @ B )
& ( in @ C @ A ) ) ) ) ) ) ).
thf(wellorderingAx_type,type,
wellorderingAx: $o ).
thf(wellorderingAx_def,definition,
( wellorderingAx
= ( ! [A: $i] :
? [B: $i] :
( ! [C: $i] :
( ( in @ C @ B )
=> ! [D: $i] :
( ( in @ D @ C )
=> ( in @ D @ A ) ) )
& ! [C: $i,D: $i] :
( ( ( in @ C @ A )
& ( in @ D @ A ) )
=> ( ! [E: $i] :
( ( in @ E @ B )
=> ( ( in @ C @ E )
= ( in @ D @ E ) ) )
=> ( C = D ) ) )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B ) )
=> ( ! [E: $i] :
( ( in @ E @ C )
=> ( in @ E @ D ) )
| ! [E: $i] :
( ( in @ E @ D )
=> ( in @ E @ C ) ) ) )
& ! [C: $i] :
( ( ! [D: $i] :
( ( in @ D @ C )
=> ( in @ D @ A ) )
& ? [D: $i] : ( in @ D @ C ) )
=> ? [D: $i,E: $i] :
( ( in @ D @ B )
& ( in @ E @ C )
& ~ ? [F: $i] :
( ( in @ F @ D )
& ( in @ F @ C ) )
& ! [F: $i] :
( ( in @ F @ B )
=> ( ! [G: $i] :
( ( in @ G @ F )
=> ( in @ G @ D ) )
| ( in @ E @ F ) ) ) ) ) ) ) ) ).
thf(descr_type,type,
descr: ( $i > $o ) > $i ).
thf(descrp_type,type,
descrp: $o ).
thf(descrp_def,definition,
( descrp
= ( ! [A: $i > $o] :
( ( exu @ A )
=> ( A @ ( descr @ A ) ) ) ) ) ).
thf(dsetconstr_type,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(dsetconstrI_type,type,
dsetconstrI: $o ).
thf(dsetconstrI_def,definition,
( dsetconstrI
= ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ A )
=> ( ( B @ C )
=> ( in @ C @ ( dsetconstr @ A @ B ) ) ) ) ) ) ).
thf(dsetconstrEL_type,type,
dsetconstrEL: $o ).
thf(dsetconstrEL_def,definition,
( dsetconstrEL
= ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ ( dsetconstr @ A @ B ) )
=> ( in @ C @ A ) ) ) ) ).
thf(dsetconstrER_type,type,
dsetconstrER: $o ).
thf(dsetconstrER_def,definition,
( dsetconstrER
= ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ ( dsetconstr @ A @ B ) )
=> ( B @ C ) ) ) ) ).
thf(exuE1_type,type,
exuE1: $o ).
thf(exuE1_def,definition,
( exuE1
= ( ! [A: $i > $o] :
( ( exu @ A )
=> ? [B: $i] :
( ( A @ B )
& ! [C: $i] :
( ( A @ C )
=> ( B = C ) ) ) ) ) ) ).
thf(prop2set_type,type,
prop2set: $o > $i ).
thf(prop2setE_type,type,
prop2setE: $o ).
thf(prop2setE_def,definition,
( prop2setE
= ( ! [A: $o,B: $i] :
( ( in @ B @ ( prop2set @ A ) )
=> A ) ) ) ).
thf(emptysetE_type,type,
emptysetE: $o ).
thf(emptysetE_def,definition,
( emptysetE
= ( ! [A: $i] :
( ( in @ A @ emptyset )
=> ! [B: $o] : B ) ) ) ).
thf(emptysetimpfalse_type,type,
emptysetimpfalse: $o ).
thf(emptysetimpfalse_def,definition,
( emptysetimpfalse
= ( ! [A: $i] :
( ( in @ A @ emptyset )
=> $false ) ) ) ).
thf(notinemptyset_type,type,
notinemptyset: $o ).
thf(notinemptyset_def,definition,
( notinemptyset
= ( ! [A: $i] :
~ ( in @ A @ emptyset ) ) ) ).
thf(exuE3e_type,type,
exuE3e: $o ).
thf(exuE3e_def,definition,
( exuE3e
= ( ! [A: $i > $o] :
( ( exu @ A )
=> ? [B: $i] : ( A @ B ) ) ) ) ).
thf(setext_type,type,
setext: $o ).
thf(setext_def,definition,
( setext
= ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) )
=> ( A = B ) ) ) ) ) ).
thf(emptyI_type,type,
emptyI: $o ).
thf(emptyI_def,definition,
( emptyI
= ( ! [A: $i] :
( ! [B: $i] :
~ ( in @ B @ A )
=> ( A = emptyset ) ) ) ) ).
thf(noeltsimpempty_type,type,
noeltsimpempty: $o ).
thf(noeltsimpempty_def,definition,
( noeltsimpempty
= ( ! [A: $i] :
( ! [B: $i] :
~ ( in @ B @ A )
=> ( A = emptyset ) ) ) ) ).
thf(setbeta_type,type,
setbeta: $o ).
thf(setbeta_def,definition,
( setbeta
= ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ A )
=> ( ( in @ C @ ( dsetconstr @ A @ B ) )
= ( B @ C ) ) ) ) ) ).
thf(nonemptyE1_type,type,
nonemptyE1: $o ).
thf(nonemptyE1_def,definition,
( nonemptyE1
= ( ! [A: $i] :
( ( nonempty @ A )
=> ? [B: $i] : ( in @ B @ A ) ) ) ) ).
thf(nonemptyI_type,type,
nonemptyI: $o ).
thf(nonemptyI_def,definition,
( nonemptyI
= ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ A )
=> ( ( B @ C )
=> ( nonempty @ ( dsetconstr @ A @ B ) ) ) ) ) ) ).
thf(nonemptyI1_type,type,
nonemptyI1: $o ).
thf(nonemptyI1_def,definition,
( nonemptyI1
= ( ! [A: $i] :
( ? [B: $i] : ( in @ B @ A )
=> ( nonempty @ A ) ) ) ) ).
thf(setadjoinIL_type,type,
setadjoinIL: $o ).
thf(setadjoinIL_def,definition,
( setadjoinIL
= ( ! [A: $i,B: $i] : ( in @ A @ ( setadjoin @ A @ B ) ) ) ) ).
thf(emptyinunitempty_type,type,
emptyinunitempty: $o ).
thf(emptyinunitempty_def,definition,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
thf(setadjoinIR_type,type,
setadjoinIR: $o ).
thf(setadjoinIR_def,definition,
( setadjoinIR
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ B )
=> ( in @ C @ ( setadjoin @ A @ B ) ) ) ) ) ).
thf(setadjoinE_type,type,
setadjoinE: $o ).
thf(setadjoinE_def,definition,
( setadjoinE
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ B ) )
=> ! [D: $o] :
( ( ( C = A )
=> D )
=> ( ( ( in @ C @ B )
=> D )
=> D ) ) ) ) ) ).
thf(setadjoinOr_type,type,
setadjoinOr: $o ).
thf(setadjoinOr_def,definition,
( setadjoinOr
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ B ) )
=> ( ( C = A )
| ( in @ C @ B ) ) ) ) ) ).
thf(setoftrueEq_type,type,
setoftrueEq: $o ).
thf(setoftrueEq_def,definition,
( setoftrueEq
= ( ! [A: $i] :
( ( dsetconstr @ A
@ ^ [B: $i] : $true )
= A ) ) ) ).
thf(powersetI_type,type,
powersetI: $o ).
thf(powersetI_def,definition,
( powersetI
= ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) )
=> ( in @ B @ ( powerset @ A ) ) ) ) ) ).
thf(emptyinPowerset_type,type,
emptyinPowerset: $o ).
thf(emptyinPowerset_def,definition,
( emptyinPowerset
= ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
thf(emptyInPowerset_type,type,
emptyInPowerset: $o ).
thf(emptyInPowerset_def,definition,
( emptyInPowerset
= ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
thf(powersetE_type,type,
powersetE: $o ).
thf(powersetE_def,definition,
( powersetE
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( ( in @ C @ B )
=> ( in @ C @ A ) ) ) ) ) ).
thf(setunionI_type,type,
setunionI: $o ).
thf(setunionI_def,definition,
( setunionI
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ B @ C )
=> ( ( in @ C @ A )
=> ( in @ B @ ( setunion @ A ) ) ) ) ) ) ).
thf(setunionE_type,type,
setunionE: $o ).
thf(setunionE_def,definition,
( setunionE
= ( ! [A: $i,B: $i] :
( ( in @ B @ ( setunion @ A ) )
=> ! [C: $o] :
( ! [D: $i] :
( ( in @ B @ D )
=> ( ( in @ D @ A )
=> C ) )
=> C ) ) ) ) ).
thf(subPowSU_type,type,
subPowSU: $o ).
thf(subPowSU_def,definition,
( subPowSU
= ( ! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( in @ B @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
thf(exuE2_type,type,
exuE2: $o ).
thf(exuE2_def,definition,
( exuE2
= ( ! [A: $i > $o] :
( ( exu @ A )
=> ? [B: $i] :
! [C: $i] :
( ( A @ C )
= ( C = B ) ) ) ) ) ).
thf(nonemptyImpWitness_type,type,
nonemptyImpWitness: $o ).
thf(nonemptyImpWitness_def,definition,
( nonemptyImpWitness
= ( ! [A: $i] :
( ( nonempty @ A )
=> ? [B: $i] :
( ( in @ B @ A )
& $true ) ) ) ) ).
thf(uniqinunit_type,type,
uniqinunit: $o ).
thf(uniqinunit_def,definition,
( uniqinunit
= ( ! [A: $i,B: $i] :
( ( in @ A @ ( setadjoin @ B @ emptyset ) )
=> ( A = B ) ) ) ) ).
thf(notinsingleton_type,type,
notinsingleton: $o ).
thf(notinsingleton_def,definition,
( notinsingleton
= ( ! [A: $i,B: $i] :
( ( A != B )
=> ~ ( in @ B @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
thf(eqinunit_type,type,
eqinunit: $o ).
thf(eqinunit_def,definition,
( eqinunit
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( in @ A @ ( setadjoin @ B @ emptyset ) ) ) ) ) ).
thf(singletonsswitch_type,type,
singletonsswitch: $o ).
thf(singletonsswitch_def,definition,
( singletonsswitch
= ( ! [A: $i,B: $i] :
( ( in @ A @ ( setadjoin @ B @ emptyset ) )
=> ( in @ B @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
thf(upairsetE_type,type,
upairsetE: $o ).
thf(upairsetE_def,definition,
( upairsetE
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) )
=> ( ( C = A )
| ( C = B ) ) ) ) ) ).
thf(upairsetIL_type,type,
upairsetIL: $o ).
thf(upairsetIL_def,definition,
( upairsetIL
= ( ! [A: $i,B: $i] : ( in @ A @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) ) ) ) ).
thf(upairsetIR_type,type,
upairsetIR: $o ).
thf(upairsetIR_def,definition,
( upairsetIR
= ( ! [A: $i,B: $i] : ( in @ B @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) ) ) ) ).
thf(emptyE1_type,type,
emptyE1: $o ).
thf(emptyE1_def,definition,
( emptyE1
= ( ! [A: $i,B: $i > $o] :
( ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ( ( ( dsetconstr @ A @ B )
= emptyset )
=> $false ) ) ) ) ).
thf(vacuousDall_type,type,
vacuousDall: $o ).
thf(vacuousDall_def,definition,
( vacuousDall
= ( ! [A: $i > $o,B: $i] :
( ( in @ B @ emptyset )
=> ( A @ B ) ) ) ) ).
thf(quantDeMorgan1_type,type,
quantDeMorgan1: $o ).
thf(quantDeMorgan1_def,definition,
( quantDeMorgan1
= ( ! [A: $i,B: $i > $o] :
( ~ ! [C: $i] :
( ( in @ C @ A )
=> ( B @ C ) )
=> ? [C: $i] :
( ( in @ C @ A )
& ~ ( B @ C ) ) ) ) ) ).
thf(quantDeMorgan2_type,type,
quantDeMorgan2: $o ).
thf(quantDeMorgan2_def,definition,
( quantDeMorgan2
= ( ! [A: $i,B: $i > $o] :
( ! [C: $i] :
( ( in @ C @ A )
=> ~ ( B @ C ) )
=> ~ ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) ) ) ) ) ).
thf(quantDeMorgan3_type,type,
quantDeMorgan3: $o ).
thf(quantDeMorgan3_def,definition,
( quantDeMorgan3
= ( ! [A: $i,B: $i > $o] :
( ~ ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ! [C: $i] :
( ( in @ C @ A )
=> ~ ( B @ C ) ) ) ) ) ).
thf(quantDeMorgan4_type,type,
quantDeMorgan4: $o ).
thf(quantDeMorgan4_def,definition,
( quantDeMorgan4
= ( ! [A: $i,B: $i > $o] :
( ? [C: $i] :
( ( in @ C @ A )
& ~ ( B @ C ) )
=> ~ ! [C: $i] :
( ( in @ C @ A )
=> ( B @ C ) ) ) ) ) ).
thf(prop2setI_type,type,
prop2setI: $o ).
thf(prop2setI_def,definition,
( prop2setI
= ( ! [A: $o] :
( A
=> ( in @ emptyset @ ( prop2set @ A ) ) ) ) ) ).
thf(set2prop_type,type,
set2prop: $i > $o ).
thf(prop2set2propI_type,type,
prop2set2propI: $o ).
thf(prop2set2propI_def,definition,
( prop2set2propI
= ( ! [A: $o] :
( A
=> ( set2prop @ ( prop2set @ A ) ) ) ) ) ).
thf(notdexE_type,type,
notdexE: $o ).
thf(notdexE_def,definition,
( notdexE
= ( ! [A: $i,B: $i > $o] :
( ~ ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ! [C: $i] :
( ( in @ C @ A )
=> ~ ( B @ C ) ) ) ) ) ).
thf(notdallE_type,type,
notdallE: $o ).
thf(notdallE_def,definition,
( notdallE
= ( ! [A: $i,B: $i > $o] :
( ~ ! [C: $i] :
( ( in @ C @ A )
=> ( B @ C ) )
=> ? [C: $i] :
( ( in @ C @ A )
& ~ ( B @ C ) ) ) ) ) ).
thf(exuI1_type,type,
exuI1: $o ).
thf(exuI1_def,definition,
( exuI1
= ( ! [A: $i > $o] :
( ? [B: $i] :
( ( A @ B )
& ! [C: $i] :
( ( A @ C )
=> ( B = C ) ) )
=> ( exu @ A ) ) ) ) ).
thf(exuI3_type,type,
exuI3: $o ).
thf(exuI3_def,definition,
( exuI3
= ( ! [A: $i > $o] :
( ? [B: $i] : ( A @ B )
=> ( ! [B: $i,C: $i] :
( ( A @ B )
=> ( ( A @ C )
=> ( B = C ) ) )
=> ( exu @ A ) ) ) ) ) ).
thf(exuI2_type,type,
exuI2: $o ).
thf(exuI2_def,definition,
( exuI2
= ( ! [A: $i > $o] :
( ? [B: $i] :
! [C: $i] :
( ( A @ C )
= ( C = B ) )
=> ( exu @ A ) ) ) ) ).
thf(inCongP_type,type,
inCongP: $o ).
thf(inCongP_def,definition,
( inCongP
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [C: $i,D: $i] :
( ( C = D )
=> ( ( in @ C @ A )
=> ( in @ D @ B ) ) ) ) ) ) ).
thf(in__Cong_type,type,
in__Cong: $o ).
thf(in__Cong_def,definition,
( in__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [C: $i,D: $i] :
( ( C = D )
=> ( ( in @ C @ A )
= ( in @ D @ B ) ) ) ) ) ) ).
thf(exuE3u_type,type,
exuE3u: $o ).
thf(exuE3u_def,definition,
( exuE3u
= ( ! [A: $i > $o] :
( ( exu @ A )
=> ! [B: $i,C: $i] :
( ( A @ B )
=> ( ( A @ C )
=> ( B = C ) ) ) ) ) ) ).
thf(exu__Cong_type,type,
exu__Cong: $o ).
thf(exu__Cong_def,definition,
( exu__Cong
= ( ! [A: $i > $o,B: $i > $o] :
( ! [C: $i,D: $i] :
( ( C = D )
=> ( ( A @ C )
= ( B @ D ) ) )
=> ( ( exu @ A )
= ( exu @ B ) ) ) ) ) ).
thf(emptyset__Cong_type,type,
emptyset__Cong: $o ).
thf(emptyset__Cong_def,definition,
( emptyset__Cong
= ( emptyset = emptyset ) ) ).
thf(setadjoin__Cong_type,type,
setadjoin__Cong: $o ).
thf(setadjoin__Cong_def,definition,
( setadjoin__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [C: $i,D: $i] :
( ( C = D )
=> ( ( setadjoin @ A @ C )
= ( setadjoin @ B @ D ) ) ) ) ) ) ).
thf(powerset__Cong_type,type,
powerset__Cong: $o ).
thf(powerset__Cong_def,definition,
( powerset__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( powerset @ A )
= ( powerset @ B ) ) ) ) ) ).
thf(setunion__Cong_type,type,
setunion__Cong: $o ).
thf(setunion__Cong_def,definition,
( setunion__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( setunion @ A )
= ( setunion @ B ) ) ) ) ) ).
thf(omega__Cong_type,type,
omega__Cong: $o ).
thf(omega__Cong_def,definition,
( omega__Cong
= ( omega = omega ) ) ).
thf(exuEu_type,type,
exuEu: $o ).
thf(exuEu_def,definition,
( exuEu
= ( ! [A: $i > $o] :
( ( exu @ A )
=> ! [B: $i,C: $i] :
( ( A @ B )
=> ( ( A @ C )
=> ( B = C ) ) ) ) ) ) ).
thf(descr__Cong_type,type,
descr__Cong: $o ).
thf(descr__Cong_def,definition,
( descr__Cong
= ( ! [A: $i > $o,B: $i > $o] :
( ! [C: $i,D: $i] :
( ( C = D )
=> ( ( A @ C )
= ( B @ D ) ) )
=> ( ( exu @ A )
=> ( ( exu @ B )
=> ( ( descr @ A )
= ( descr @ B ) ) ) ) ) ) ) ).
thf(dsetconstr__Cong_type,type,
dsetconstr__Cong: $o ).
thf(dsetconstr__Cong_def,definition,
( dsetconstr__Cong
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [C: $i > $o,D: $i > $o] :
( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( E = F )
=> ( ( C @ E )
= ( D @ F ) ) ) ) )
=> ( ( dsetconstr @ A @ C )
= ( dsetconstr @ B @ D ) ) ) ) ) ) ).
thf(subset_type,type,
subset: $i > $i > $o ).
thf(subsetI1_type,type,
subsetI1: $o ).
thf(subsetI1_def,definition,
( subsetI1
= ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( subset @ A @ B ) ) ) ) ).
thf(eqimpsubset2_type,type,
eqimpsubset2: $o ).
thf(eqimpsubset2_def,definition,
( eqimpsubset2
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ B @ A ) ) ) ) ).
thf(eqimpsubset1_type,type,
eqimpsubset1: $o ).
thf(eqimpsubset1_def,definition,
( eqimpsubset1
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ A @ B ) ) ) ) ).
thf(subsetI2_type,type,
subsetI2: $o ).
thf(subsetI2_def,definition,
( subsetI2
= ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( subset @ A @ B ) ) ) ) ).
thf(emptysetsubset_type,type,
emptysetsubset: $o ).
thf(emptysetsubset_def,definition,
( emptysetsubset
= ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
thf(subsetE_type,type,
subsetE: $o ).
thf(subsetE_def,definition,
( subsetE
= ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( in @ C @ A )
=> ( in @ C @ B ) ) ) ) ) ).
thf(subsetE2_type,type,
subsetE2: $o ).
thf(subsetE2_def,definition,
( subsetE2
= ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ~ ( in @ C @ B )
=> ~ ( in @ C @ A ) ) ) ) ) ).
thf(notsubsetI_type,type,
notsubsetI: $o ).
thf(notsubsetI_def,definition,
( notsubsetI
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ~ ( subset @ A @ B ) ) ) ) ) ).
thf(notequalI1_type,type,
notequalI1: $o ).
thf(notequalI1_def,definition,
( notequalI1
= ( ! [A: $i,B: $i] :
( ~ ( subset @ A @ B )
=> ( A != B ) ) ) ) ).
thf(notequalI2_type,type,
notequalI2: $o ).
thf(notequalI2_def,definition,
( notequalI2
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( A != B ) ) ) ) ) ).
thf(subsetRefl_type,type,
subsetRefl: $o ).
thf(subsetRefl_def,definition,
( subsetRefl
= ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
thf(subsetTrans_type,type,
subsetTrans: $o ).
thf(subsetTrans_def,definition,
( subsetTrans
= ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ C )
=> ( subset @ A @ C ) ) ) ) ) ).
thf(setadjoinSub_type,type,
setadjoinSub: $o ).
thf(setadjoinSub_def,definition,
( setadjoinSub
= ( ! [A: $i,B: $i] : ( subset @ B @ ( setadjoin @ A @ B ) ) ) ) ).
thf(setadjoinSub2_type,type,
setadjoinSub2: $o ).
thf(setadjoinSub2_def,definition,
( setadjoinSub2
= ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ C )
=> ( subset @ A @ ( setadjoin @ B @ C ) ) ) ) ) ).
thf(subset2powerset_type,type,
subset2powerset: $o ).
thf(subset2powerset_def,definition,
( subset2powerset
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( in @ A @ ( powerset @ B ) ) ) ) ) ).
thf(setextsub_type,type,
setextsub: $o ).
thf(setextsub_def,definition,
( setextsub
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ A )
=> ( A = B ) ) ) ) ) ).
thf(subsetemptysetimpeq_type,type,
subsetemptysetimpeq: $o ).
thf(subsetemptysetimpeq_def,definition,
( subsetemptysetimpeq
= ( ! [A: $i] :
( ( subset @ A @ emptyset )
=> ( A = emptyset ) ) ) ) ).
thf(powersetI1_type,type,
powersetI1: $o ).
thf(powersetI1_def,definition,
( powersetI1
= ( ! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( in @ B @ ( powerset @ A ) ) ) ) ) ).
thf(powersetE1_type,type,
powersetE1: $o ).
thf(powersetE1_def,definition,
( powersetE1
= ( ! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( subset @ B @ A ) ) ) ) ).
thf(inPowerset_type,type,
inPowerset: $o ).
thf(inPowerset_def,definition,
( inPowerset
= ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
thf(powersetsubset_type,type,
powersetsubset: $o ).
thf(powersetsubset_def,definition,
( powersetsubset
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
thf(sepInPowerset_type,type,
sepInPowerset: $o ).
thf(sepInPowerset_def,definition,
( sepInPowerset
= ( ! [A: $i,B: $i > $o] : ( in @ ( dsetconstr @ A @ B ) @ ( powerset @ A ) ) ) ) ).
thf(sepSubset_type,type,
sepSubset: $o ).
thf(sepSubset_def,definition,
( sepSubset
= ( ! [A: $i,B: $i > $o] : ( subset @ ( dsetconstr @ A @ B ) @ A ) ) ) ).
thf(binunion_type,type,
binunion: $i > $i > $i ).
thf(binunionIL_type,type,
binunionIL: $o ).
thf(binunionIL_def,definition,
( binunionIL
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( in @ C @ ( binunion @ A @ B ) ) ) ) ) ).
thf(upairset2IR_type,type,
upairset2IR: $o ).
thf(upairset2IR_def,definition,
( upairset2IR
= ( ! [A: $i,B: $i] : ( in @ B @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) ) ) ) ).
thf(binunionIR_type,type,
binunionIR: $o ).
thf(binunionIR_def,definition,
( binunionIR
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ B )
=> ( in @ C @ ( binunion @ A @ B ) ) ) ) ) ).
thf(binunionEcases_type,type,
binunionEcases: $o ).
thf(binunionEcases_def,definition,
( binunionEcases
= ( ! [A: $i,B: $i,C: $i,D: $o] :
( ( in @ C @ ( binunion @ A @ B ) )
=> ( ( ( in @ C @ A )
=> D )
=> ( ( ( in @ C @ B )
=> D )
=> D ) ) ) ) ) ).
thf(binunionE_type,type,
binunionE: $o ).
thf(binunionE_def,definition,
( binunionE
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( binunion @ A @ B ) )
=> ( ( in @ C @ A )
| ( in @ C @ B ) ) ) ) ) ).
thf(binunionLsub_type,type,
binunionLsub: $o ).
thf(binunionLsub_def,definition,
( binunionLsub
= ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
thf(binunionRsub_type,type,
binunionRsub: $o ).
thf(binunionRsub_def,definition,
( binunionRsub
= ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
thf(binintersect_type,type,
binintersect: $i > $i > $i ).
thf(binintersectI_type,type,
binintersectI: $o ).
thf(binintersectI_def,definition,
( binintersectI
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ( in @ C @ B )
=> ( in @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
thf(binintersectSubset5_type,type,
binintersectSubset5: $o ).
thf(binintersectSubset5_def,definition,
( binintersectSubset5
= ( ! [A: $i,B: $i,C: $i] :
( ( subset @ C @ A )
=> ( ( subset @ C @ B )
=> ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
thf(binintersectEL_type,type,
binintersectEL: $o ).
thf(binintersectEL_def,definition,
( binintersectEL
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( binintersect @ A @ B ) )
=> ( in @ C @ A ) ) ) ) ).
thf(binintersectLsub_type,type,
binintersectLsub: $o ).
thf(binintersectLsub_def,definition,
( binintersectLsub
= ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
thf(binintersectSubset2_type,type,
binintersectSubset2: $o ).
thf(binintersectSubset2_def,definition,
( binintersectSubset2
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( binintersect @ A @ B )
= A ) ) ) ) ).
thf(binintersectSubset3_type,type,
binintersectSubset3: $o ).
thf(binintersectSubset3_def,definition,
( binintersectSubset3
= ( ! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= B )
=> ( subset @ B @ A ) ) ) ) ).
thf(binintersectER_type,type,
binintersectER: $o ).
thf(binintersectER_def,definition,
( binintersectER
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( binintersect @ A @ B ) )
=> ( in @ C @ B ) ) ) ) ).
thf(disjointsetsI1_type,type,
disjointsetsI1: $o ).
thf(disjointsetsI1_def,definition,
( disjointsetsI1
= ( ! [A: $i,B: $i] :
( ~ ? [C: $i] :
( ( in @ C @ A )
& ( in @ C @ B ) )
=> ( ( binintersect @ A @ B )
= emptyset ) ) ) ) ).
thf(binintersectRsub_type,type,
binintersectRsub: $o ).
thf(binintersectRsub_def,definition,
( binintersectRsub
= ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
thf(binintersectSubset4_type,type,
binintersectSubset4: $o ).
thf(binintersectSubset4_def,definition,
( binintersectSubset4
= ( ! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( ( binintersect @ A @ B )
= B ) ) ) ) ).
thf(binintersectSubset1_type,type,
binintersectSubset1: $o ).
thf(binintersectSubset1_def,definition,
( binintersectSubset1
= ( ! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= A )
=> ( subset @ A @ B ) ) ) ) ).
thf(bs114d_type,type,
bs114d: $o ).
thf(bs114d_def,definition,
( bs114d
= ( ! [A: $i,B: $i,C: $i] :
( ( binintersect @ A @ ( binunion @ B @ C ) )
= ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
thf(setminus_type,type,
setminus: $i > $i > $i ).
thf(setminusI_type,type,
setminusI: $o ).
thf(setminusI_def,definition,
( setminusI
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( in @ C @ ( setminus @ A @ B ) ) ) ) ) ) ).
thf(setminusEL_type,type,
setminusEL: $o ).
thf(setminusEL_def,definition,
( setminusEL
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setminus @ A @ B ) )
=> ( in @ C @ A ) ) ) ) ).
thf(setminusER_type,type,
setminusER: $o ).
thf(setminusER_def,definition,
( setminusER
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setminus @ A @ B ) )
=> ~ ( in @ C @ B ) ) ) ) ).
thf(setminusSubset2_type,type,
setminusSubset2: $o ).
thf(setminusSubset2_def,definition,
( setminusSubset2
= ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( setminus @ A @ B )
= emptyset ) ) ) ) ).
thf(setminusERneg_type,type,
setminusERneg: $o ).
thf(setminusERneg_def,definition,
( setminusERneg
= ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ ( setminus @ A @ B ) )
=> ( ( in @ C @ A )
=> ( in @ C @ B ) ) ) ) ) ).
thf(setminusELneg_type,type,
setminusELneg: $o ).
thf(setminusELneg_def,definition,
( setminusELneg
= ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ ( setminus @ A @ B ) )
=> ( ~ ( in @ C @ B )
=> ~ ( in @ C @ A ) ) ) ) ) ).
thf(setminusILneg_type,type,
setminusILneg: $o ).
thf(setminusILneg_def,definition,
( setminusILneg
= ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ A )
=> ~ ( in @ C @ ( setminus @ A @ B ) ) ) ) ) ).
thf(setminusIRneg_type,type,
setminusIRneg: $o ).
thf(setminusIRneg_def,definition,
( setminusIRneg
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ B )
=> ~ ( in @ C @ ( setminus @ A @ B ) ) ) ) ) ).
thf(setminusLsub_type,type,
setminusLsub: $o ).
thf(setminusLsub_def,definition,
( setminusLsub
= ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
thf(setminusSubset1_type,type,
setminusSubset1: $o ).
thf(setminusSubset1_def,definition,
( setminusSubset1
= ( ! [A: $i,B: $i] :
( ( ( setminus @ A @ B )
= emptyset )
=> ( subset @ A @ B ) ) ) ) ).
thf(symdiff_type,type,
symdiff: $i > $i > $i ).
thf(symdiffE_type,type,
symdiffE: $o ).
thf(symdiffE_def,definition,
( symdiffE
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( symdiff @ A @ B ) )
=> ! [D: $o] :
( ( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> D ) )
=> ( ( ~ ( in @ C @ A )
=> ( ( in @ C @ B )
=> D ) )
=> D ) ) ) ) ) ).
thf(symdiffI1_type,type,
symdiffI1: $o ).
thf(symdiffI1_def,definition,
( symdiffI1
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( in @ C @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf(symdiffI2_type,type,
symdiffI2: $o ).
thf(symdiffI2_def,definition,
( symdiffI2
= ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ A )
=> ( ( in @ C @ B )
=> ( in @ C @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf(symdiffIneg1_type,type,
symdiffIneg1: $o ).
thf(symdiffIneg1_def,definition,
( symdiffIneg1
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ( in @ C @ B )
=> ~ ( in @ C @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf(symdiffIneg2_type,type,
symdiffIneg2: $o ).
thf(symdiffIneg2_def,definition,
( symdiffIneg2
= ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ~ ( in @ C @ ( symdiff @ A @ B ) ) ) ) ) ) ).
thf(iskpair_type,type,
iskpair: $i > $o ).
thf(secondinupair_type,type,
secondinupair: $o ).
thf(secondinupair_def,definition,
( secondinupair
= ( ! [A: $i,B: $i] : ( in @ B @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) ) ) ) ).
thf(setukpairIL_type,type,
setukpairIL: $o ).
thf(setukpairIL_def,definition,
( setukpairIL
= ( ! [A: $i,B: $i] : ( in @ A @ ( setunion @ ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf(setukpairIR_type,type,
setukpairIR: $o ).
thf(setukpairIR_def,definition,
( setukpairIR
= ( ! [A: $i,B: $i] : ( in @ B @ ( setunion @ ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
thf(kpairiskpair_type,type,
kpairiskpair: $o ).
thf(kpairiskpair_def,definition,
( kpairiskpair
= ( ! [A: $i,B: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) ) ) ) ) ).
thf(kpair_type,type,
kpair: $i > $i > $i ).
thf(kpairp_type,type,
kpairp: $o ).
thf(kpairp_def,definition,
( kpairp
= ( ! [A: $i,B: $i] : ( iskpair @ ( kpair @ A @ B ) ) ) ) ).
thf(cartprod_type,type,
cartprod: $i > $i > $i ).
thf(singletonsubset_type,type,
singletonsubset: $o ).
thf(singletonsubset_def,definition,
( singletonsubset
= ( ! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( subset @ ( setadjoin @ B @ emptyset ) @ A ) ) ) ) ).
thf(singletoninpowerset_type,type,
singletoninpowerset: $o ).
thf(singletoninpowerset_def,definition,
( singletoninpowerset
= ( ! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( in @ ( setadjoin @ B @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
thf(singletoninpowunion_type,type,
singletoninpowunion: $o ).
thf(singletoninpowunion_def,definition,
( singletoninpowunion
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( in @ ( setadjoin @ C @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
thf(upairset2E_type,type,
upairset2E: $o ).
thf(upairset2E_def,definition,
( upairset2E
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) )
=> ( ( C = A )
| ( C = B ) ) ) ) ) ).
thf(upairsubunion_type,type,
upairsubunion: $o ).
thf(upairsubunion_def,definition,
( upairsubunion
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( subset @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
thf(upairinpowunion_type,type,
upairinpowunion: $o ).
thf(upairinpowunion_def,definition,
( upairinpowunion
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( in @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
thf(ubforcartprodlem1_type,type,
ubforcartprodlem1: $o ).
thf(ubforcartprodlem1_def,definition,
( ubforcartprodlem1
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( subset @ ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
thf(ubforcartprodlem2_type,type,
ubforcartprodlem2: $o ).
thf(ubforcartprodlem2_def,definition,
( ubforcartprodlem2
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( in @ ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
thf(ubforcartprodlem3_type,type,
ubforcartprodlem3: $o ).
thf(ubforcartprodlem3_def,definition,
( ubforcartprodlem3
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( in @ ( kpair @ C @ D ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
thf(cartprodpairin_type,type,
cartprodpairin: $o ).
thf(cartprodpairin_def,definition,
( cartprodpairin
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( in @ ( kpair @ C @ D ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
thf(cartprodmempair1_type,type,
cartprodmempair1: $o ).
thf(cartprodmempair1_def,definition,
( cartprodmempair1
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ? [D: $i] :
( ( in @ D @ A )
& ? [E: $i] :
( ( in @ E @ B )
& ( C
= ( kpair @ D @ E ) ) ) ) ) ) ) ).
thf(cartprodmempair_type,type,
cartprodmempair: $o ).
thf(cartprodmempair_def,definition,
( cartprodmempair
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ( iskpair @ C ) ) ) ) ).
thf(setunionE2_type,type,
setunionE2: $o ).
thf(setunionE2_def,definition,
( setunionE2
= ( ! [A: $i,B: $i] :
( ( in @ B @ ( setunion @ A ) )
=> ? [C: $i] :
( ( in @ C @ A )
& ( in @ B @ C ) ) ) ) ) ).
thf(setunionsingleton1_type,type,
setunionsingleton1: $o ).
thf(setunionsingleton1_def,definition,
( setunionsingleton1
= ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
thf(setunionsingleton2_type,type,
setunionsingleton2: $o ).
thf(setunionsingleton2_def,definition,
( setunionsingleton2
= ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
thf(setunionsingleton_type,type,
setunionsingleton: $o ).
thf(setunionsingleton_def,definition,
( setunionsingleton
= ( ! [A: $i] :
( ( setunion @ ( setadjoin @ A @ emptyset ) )
= A ) ) ) ).
thf(singleton_type,type,
singleton: $i > $o ).
thf(singletonprop_type,type,
singletonprop: $o ).
thf(singletonprop_def,definition,
( singletonprop
= ( ! [A: $i,B: $i > $o] :
( ! [C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( B @ C )
=> ( ( B @ D )
=> ( C = D ) ) ) ) )
=> ( ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ( singleton @ ( dsetconstr @ A @ B ) ) ) ) ) ) ).
thf(ex1E1_type,type,
ex1E1: $o ).
thf(ex1E1_def,definition,
( ex1E1
= ( ! [A: $i,B: $i > $o] :
( ( ex1 @ A @ B )
=> ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) ) ) ) ) ).
thf(ex1I_type,type,
ex1I: $o ).
thf(ex1I_def,definition,
( ex1I
= ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ A )
=> ( ( B @ C )
=> ( ! [D: $i] :
( ( in @ D @ A )
=> ( ( B @ D )
=> ( D = C ) ) )
=> ( ex1 @ A @ B ) ) ) ) ) ) ).
thf(ex1I2_type,type,
ex1I2: $o ).
thf(ex1I2_def,definition,
( ex1I2
= ( ! [A: $i,B: $i > $o] :
( ! [C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( B @ C )
=> ( ( B @ D )
=> ( C = D ) ) ) ) )
=> ( ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ( ex1 @ A @ B ) ) ) ) ) ).
thf(singletonsuniq_type,type,
singletonsuniq: $o ).
thf(singletonsuniq_def,definition,
( singletonsuniq
= ( ! [A: $i,B: $i] :
( ( ( setadjoin @ A @ emptyset )
= ( setadjoin @ B @ emptyset ) )
=> ( A = B ) ) ) ) ).
thf(setukpairinjL1_type,type,
setukpairinjL1: $o ).
thf(setukpairinjL1_def,definition,
( setukpairinjL1
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) ) )
=> ( A = C ) ) ) ) ).
thf(kfstsingleton_type,type,
kfstsingleton: $o ).
thf(kfstsingleton_def,definition,
( kfstsingleton
= ( ! [A: $i] :
( ( iskpair @ A )
=> ( singleton
@ ( dsetconstr @ ( setunion @ A )
@ ^ [B: $i] : ( in @ ( setadjoin @ B @ emptyset ) @ A ) ) ) ) ) ) ).
thf(theprop_type,type,
theprop: $o ).
thf(theprop_def,definition,
( theprop
= ( ! [A: $i] :
( ( singleton @ A )
=> ( in @ ( setunion @ A ) @ A ) ) ) ) ).
thf(kfst_type,type,
kfst: $i > $i ).
thf(kfstpairEq_type,type,
kfstpairEq: $o ).
thf(kfstpairEq_def,definition,
( kfstpairEq
= ( ! [A: $i,B: $i] :
( ( kfst @ ( kpair @ A @ B ) )
= A ) ) ) ).
thf(cartprodfstin_type,type,
cartprodfstin: $o ).
thf(cartprodfstin_def,definition,
( cartprodfstin
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ( in @ ( kfst @ C ) @ A ) ) ) ) ).
thf(setukpairinjL2_type,type,
setukpairinjL2: $o ).
thf(setukpairinjL2_def,definition,
( setukpairinjL2
= ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) )
=> ( A = C ) ) ) ) ).
thf(setukpairinjL_type,type,
setukpairinjL: $o ).
thf(setukpairinjL_def,definition,
( setukpairinjL
= ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( kpair @ A @ B )
= ( kpair @ C @ D ) )
=> ( A = C ) ) ) ) ).
thf(setukpairinjR11_type,type,
setukpairinjR11: $o ).
thf(setukpairinjR11_def,definition,
( setukpairinjR11
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) )
= ( setadjoin @ A @ emptyset ) ) ) ) ) ).
thf(setukpairinjR12_type,type,
setukpairinjR12: $o ).
thf(setukpairinjR12_def,definition,
( setukpairinjR12
= ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ A @ emptyset ) @ emptyset ) ) ) ) ) ).
thf(setukpairinjR1_type,type,
setukpairinjR1: $o ).
thf(setukpairinjR1_def,definition,
( setukpairinjR1
= ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) )
=> ( ( C = D )
=> ( B = D ) ) ) ) ) ).
thf(upairequniteq_type,type,
upairequniteq: $o ).
thf(upairequniteq_def,definition,
( upairequniteq
= ( ! [A: $i,B: $i,C: $i] :
( ( ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) )
= ( setadjoin @ C @ emptyset ) )
=> ( A = B ) ) ) ) ).
thf(setukpairinjR2_type,type,
setukpairinjR2: $o ).
thf(setukpairinjR2_def,definition,
( setukpairinjR2
= ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) )
=> ( B = D ) ) ) ) ).
thf(setukpairinjR_type,type,
setukpairinjR: $o ).
thf(setukpairinjR_def,definition,
( setukpairinjR
= ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( kpair @ A @ B )
= ( kpair @ C @ D ) )
=> ( B = D ) ) ) ) ).
thf(ksndsingleton_type,type,
ksndsingleton: $o ).
thf(ksndsingleton_def,definition,
( ksndsingleton
= ( ! [A: $i] :
( ( iskpair @ A )
=> ( singleton
@ ( dsetconstr @ ( setunion @ A )
@ ^ [B: $i] :
( A
= ( kpair @ ( kfst @ A ) @ B ) ) ) ) ) ) ) ).
thf(ksnd_type,type,
ksnd: $i > $i ).
thf(ksndpairEq_type,type,
ksndpairEq: $o ).
thf(ksndpairEq_def,definition,
( ksndpairEq
= ( ! [A: $i,B: $i] :
( ( ksnd @ ( kpair @ A @ B ) )
= B ) ) ) ).
thf(kpairsurjEq_type,type,
kpairsurjEq: $o ).
thf(kpairsurjEq_def,definition,
( kpairsurjEq
= ( ! [A: $i] :
( ( iskpair @ A )
=> ( ( kpair @ ( kfst @ A ) @ ( ksnd @ A ) )
= A ) ) ) ) ).
thf(cartprodsndin_type,type,
cartprodsndin: $o ).
thf(cartprodsndin_def,definition,
( cartprodsndin
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ( in @ ( ksnd @ C ) @ B ) ) ) ) ).
thf(cartprodpairmemEL_type,type,
cartprodpairmemEL: $o ).
thf(cartprodpairmemEL_def,definition,
( cartprodpairmemEL
= ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( in @ ( kpair @ C @ D ) @ ( cartprod @ A @ B ) )
=> ( in @ C @ A ) ) ) ) ).
thf(cartprodpairmemER_type,type,
cartprodpairmemER: $o ).
thf(cartprodpairmemER_def,definition,
( cartprodpairmemER
= ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( in @ ( kpair @ C @ D ) @ ( cartprod @ A @ B ) )
=> ( in @ D @ B ) ) ) ) ).
thf(cartprodmempaircEq_type,type,
cartprodmempaircEq: $o ).
thf(cartprodmempaircEq_def,definition,
( cartprodmempaircEq
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( ( kpair @ C @ D )
= ( kpair @ C @ D ) ) ) ) ) ) ).
thf(cartprodfstpairEq_type,type,
cartprodfstpairEq: $o ).
thf(cartprodfstpairEq_def,definition,
( cartprodfstpairEq
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( ( kfst @ ( kpair @ C @ D ) )
= C ) ) ) ) ) ).
thf(cartprodsndpairEq_type,type,
cartprodsndpairEq: $o ).
thf(cartprodsndpairEq_def,definition,
( cartprodsndpairEq
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( ( ksnd @ ( kpair @ C @ D ) )
= D ) ) ) ) ) ).
thf(cartprodpairsurjEq_type,type,
cartprodpairsurjEq: $o ).
thf(cartprodpairsurjEq_def,definition,
( cartprodpairsurjEq
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ( ( kpair @ ( kfst @ C ) @ ( ksnd @ C ) )
= C ) ) ) ) ).
thf(dpsetconstr_type,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(dpsetconstrI_type,type,
dpsetconstrI: $o ).
thf(dpsetconstrI_def,definition,
( dpsetconstrI
= ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i] :
( ( in @ D @ A )
=> ! [E: $i] :
( ( in @ E @ B )
=> ( ( C @ D @ E )
=> ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) ) ) ) ) ) ) ).
thf(dpsetconstrSub_type,type,
dpsetconstrSub: $o ).
thf(dpsetconstrSub_def,definition,
( dpsetconstrSub
= ( ! [A: $i,B: $i,C: $i > $i > $o] : ( subset @ ( dpsetconstr @ A @ B @ C ) @ ( cartprod @ A @ B ) ) ) ) ).
thf(setOfPairsIsBReln_type,type,
setOfPairsIsBReln: $o ).
thf(setOfPairsIsBReln_def,definition,
( setOfPairsIsBReln
= ( ! [A: $i,B: $i,C: $i > $i > $o] : ( breln @ A @ B @ ( dpsetconstr @ A @ B @ C ) ) ) ) ).
thf(dpsetconstrERa_type,type,
dpsetconstrERa: $o ).
thf(dpsetconstrERa_def,definition,
( dpsetconstrERa
= ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i] :
( ( in @ D @ A )
=> ! [E: $i] :
( ( in @ E @ B )
=> ( ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) )
=> ( C @ D @ E ) ) ) ) ) ) ).
thf(dpsetconstrEL1_type,type,
dpsetconstrEL1: $o ).
thf(dpsetconstrEL1_def,definition,
( dpsetconstrEL1
= ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i,E: $i] :
( ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) )
=> ( in @ D @ A ) ) ) ) ).
thf(dpsetconstrEL2_type,type,
dpsetconstrEL2: $o ).
thf(dpsetconstrEL2_def,definition,
( dpsetconstrEL2
= ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i,E: $i] :
( ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) )
=> ( in @ E @ B ) ) ) ) ).
thf(dpsetconstrER_type,type,
dpsetconstrER: $o ).
thf(dpsetconstrER_def,definition,
( dpsetconstrER
= ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i,E: $i] :
( ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) )
=> ( C @ D @ E ) ) ) ) ).
thf(func_type,type,
func: $i > $i > $i > $o ).
thf(funcSet_type,type,
funcSet: $i > $i > $i ).
thf(funcImageSingleton_type,type,
funcImageSingleton: $o ).
thf(funcImageSingleton_def,definition,
( funcImageSingleton
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( singleton
@ ( dsetconstr @ B
@ ^ [E: $i] : ( in @ ( kpair @ D @ E ) @ C ) ) ) ) ) ) ) ).
thf(apProp_type,type,
apProp: $o ).
thf(apProp_def,definition,
( apProp
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in
@ ( setunion
@ ( dsetconstr @ B
@ ^ [E: $i] : ( in @ ( kpair @ D @ E ) @ C ) ) )
@ B ) ) ) ) ) ).
thf(ap_type,type,
ap: $i > $i > $i > $i > $i ).
thf(app_type,type,
app: $o ).
thf(app_def,definition,
( app
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( ap @ A @ B @ C @ D ) @ B ) ) ) ) ) ).
thf(infuncsetfunc_type,type,
infuncsetfunc: $o ).
thf(infuncsetfunc_def,definition,
( infuncsetfunc
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ( func @ A @ B @ C ) ) ) ) ).
thf(ap2p_type,type,
ap2p: $o ).
thf(ap2p_def,definition,
( ap2p
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( ap @ A @ B @ C @ D ) @ B ) ) ) ) ) ).
thf(funcinfuncset_type,type,
funcinfuncset: $o ).
thf(funcinfuncset_def,definition,
( funcinfuncset
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ( in @ C @ ( funcSet @ A @ B ) ) ) ) ) ).
thf(lamProp_type,type,
lamProp: $o ).
thf(lamProp_def,definition,
( lamProp
= ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ( func @ A @ B
@ ( dpsetconstr @ A @ B
@ ^ [D: $i] : ( (=) @ $i @ ( C @ D ) ) ) ) ) ) ) ).
thf(lam_type,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(lamp_type,type,
lamp: $o ).
thf(lamp_def,definition,
( lamp
= ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ( func @ A @ B @ ( lam @ A @ B @ C ) ) ) ) ) ).
thf(lam2p_type,type,
lam2p: $o ).
thf(lam2p_def,definition,
( lam2p
= ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ( in @ ( lam @ A @ B @ C ) @ ( funcSet @ A @ B ) ) ) ) ) ).
thf(brelnall1_type,type,
brelnall1: $o ).
thf(brelnall1_def,definition,
( brelnall1
= ( ! [A: $i,B: $i,C: $i] :
( ( breln @ A @ B @ C )
=> ! [D: $i > $o] :
( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ C )
=> ( D @ ( kpair @ E @ F ) ) ) ) )
=> ! [E: $i] :
( ( in @ E @ C )
=> ( D @ E ) ) ) ) ) ) ).
thf(brelnall2_type,type,
brelnall2: $o ).
thf(brelnall2_def,definition,
( brelnall2
= ( ! [A: $i,B: $i,C: $i] :
( ( breln @ A @ B @ C )
=> ! [D: $i > $o] :
( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ C )
=> ( D @ ( kpair @ E @ F ) ) ) ) )
=> ! [E: $i] :
( ( in @ E @ C )
=> ( D @ E ) ) ) ) ) ) ).
thf(ex1E2_type,type,
ex1E2: $o ).
thf(ex1E2_def,definition,
( ex1E2
= ( ! [A: $i,B: $i > $o] :
( ( ex1 @ A @ B )
=> ! [C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( B @ C )
=> ( ( B @ D )
=> ( C = D ) ) ) ) ) ) ) ) ).
thf(funcGraphProp1_type,type,
funcGraphProp1: $o ).
thf(funcGraphProp1_def,definition,
( funcGraphProp1
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( kpair @ D @ ( ap @ A @ B @ C @ D ) ) @ C ) ) ) ) ) ).
thf(funcGraphProp3_type,type,
funcGraphProp3: $o ).
thf(funcGraphProp3_def,definition,
( funcGraphProp3
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( kpair @ D @ ( ap @ A @ B @ C @ D ) ) @ C ) ) ) ) ) ).
thf(funcGraphProp2_type,type,
funcGraphProp2: $o ).
thf(funcGraphProp2_def,definition,
( funcGraphProp2
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ! [E: $i] :
( ( in @ E @ B )
=> ( ( in @ ( kpair @ D @ E ) @ C )
=> ( ( ap @ A @ B @ C @ D )
= E ) ) ) ) ) ) ) ).
thf(funcextLem_type,type,
funcextLem: $o ).
thf(funcextLem_def,definition,
( funcextLem
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( func @ A @ B @ D )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ( ( ap @ A @ B @ C @ E )
= ( ap @ A @ B @ D @ E ) ) )
=> ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ D )
=> ( in @ ( kpair @ E @ F ) @ C ) ) ) ) ) ) ) ) ) ).
thf(funcGraphProp4_type,type,
funcGraphProp4: $o ).
thf(funcGraphProp4_def,definition,
( funcGraphProp4
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ! [E: $i] :
( ( in @ E @ B )
=> ( ( in @ ( kpair @ D @ E ) @ C )
=> ( ( ap @ A @ B @ C @ D )
= E ) ) ) ) ) ) ) ).
thf(subbreln_type,type,
subbreln: $o ).
thf(subbreln_def,definition,
( subbreln
= ( ! [A: $i,B: $i,C: $i] :
( ( breln @ A @ B @ C )
=> ! [D: $i] :
( ( breln @ A @ B @ D )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ C )
=> ( in @ ( kpair @ E @ F ) @ D ) ) ) )
=> ( subset @ C @ D ) ) ) ) ) ) ).
thf(eqbreln_type,type,
eqbreln: $o ).
thf(eqbreln_def,definition,
( eqbreln
= ( ! [A: $i,B: $i,C: $i] :
( ( breln @ A @ B @ C )
=> ! [D: $i] :
( ( breln @ A @ B @ D )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ C )
=> ( in @ ( kpair @ E @ F ) @ D ) ) ) )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ D )
=> ( in @ ( kpair @ E @ F ) @ C ) ) ) )
=> ( C = D ) ) ) ) ) ) ) ).
thf(funcext_type,type,
funcext: $o ).
thf(funcext_def,definition,
( funcext
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( func @ A @ B @ D )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ( ( ap @ A @ B @ C @ E )
= ( ap @ A @ B @ D @ E ) ) )
=> ( C = D ) ) ) ) ) ) ).
thf(funcext2_type,type,
funcext2: $o ).
thf(funcext2_def,definition,
( funcext2
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ ( funcSet @ A @ B ) )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ( ( ap @ A @ B @ C @ E )
= ( ap @ A @ B @ D @ E ) ) )
=> ( C = D ) ) ) ) ) ) ).
thf(ap2apEq1_type,type,
ap2apEq1: $o ).
thf(ap2apEq1_def,definition,
( ap2apEq1
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( ap @ A @ B @ C @ D )
= ( ap @ A @ B @ C @ D ) ) ) ) ) ) ).
thf(ap2apEq2_type,type,
ap2apEq2: $o ).
thf(ap2apEq2_def,definition,
( ap2apEq2
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( ap @ A @ B @ C @ D )
= ( ap @ A @ B @ C @ D ) ) ) ) ) ) ).
thf(beta1_type,type,
beta1: $o ).
thf(beta1_def,definition,
( beta1
= ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( ap @ A @ B @ ( lam @ A @ B @ C ) @ D )
= ( C @ D ) ) ) ) ) ) ).
thf(eta1_type,type,
eta1: $o ).
thf(eta1_def,definition,
( eta1
= ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ( ( lam @ A @ B @ ( ap @ A @ B @ C ) )
= C ) ) ) ) ).
thf(lam2lamEq_type,type,
lam2lamEq: $o ).
thf(lam2lamEq_def,definition,
( lam2lamEq
= ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ( ( lam @ A @ B @ C )
= ( lam @ A @ B @ C ) ) ) ) ) ).
thf(beta2_type,type,
beta2: $o ).
thf(beta2_def,definition,
( beta2
= ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( ap @ A @ B @ ( lam @ A @ B @ C ) @ D )
= ( C @ D ) ) ) ) ) ) ).
thf(eta2_type,type,
eta2: $o ).
thf(eta2_def,definition,
( eta2
= ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ( ( lam @ A @ B @ ( ap @ A @ B @ C ) )
= C ) ) ) ) ).
thf(iffalseProp1_type,type,
iffalseProp1: $o ).
thf(iffalseProp1_def,definition,
( iffalseProp1
= ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ~ B
=> ( in @ D
@ ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) ) ) ) ) ) ) ) ).
thf(iffalseProp2_type,type,
iffalseProp2: $o ).
thf(iffalseProp2_def,definition,
( iffalseProp2
= ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ~ B
=> ( ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) )
= ( setadjoin @ D @ emptyset ) ) ) ) ) ) ) ).
thf(iftrueProp1_type,type,
iftrueProp1: $o ).
thf(iftrueProp1_def,definition,
( iftrueProp1
= ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( B
=> ( in @ C
@ ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) ) ) ) ) ) ) ) ).
thf(iftrueProp2_type,type,
iftrueProp2: $o ).
thf(iftrueProp2_def,definition,
( iftrueProp2
= ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( B
=> ( ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) )
= ( setadjoin @ C @ emptyset ) ) ) ) ) ) ) ).
thf(ifSingleton_type,type,
ifSingleton: $o ).
thf(ifSingleton_def,definition,
( ifSingleton
= ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( singleton
@ ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) ) ) ) ) ) ) ).
thf(if_type,type,
if: $i > $o > $i > $i > $i ).
thf(ifp_type,type,
ifp: $o ).
thf(ifp_def,definition,
( ifp
= ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( if @ A @ B @ C @ D ) @ A ) ) ) ) ) ).
thf(theeq_type,type,
theeq: $o ).
thf(theeq_def,definition,
( theeq
= ( ! [A: $i] :
( ( singleton @ A )
=> ! [B: $i] :
( ( in @ B @ A )
=> ( ( setunion @ A )
= B ) ) ) ) ) ).
thf(iftrue_type,type,
iftrue: $o ).
thf(iftrue_def,definition,
( iftrue
= ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( B
=> ( ( if @ A @ B @ C @ D )
= C ) ) ) ) ) ) ).
thf(iffalse_type,type,
iffalse: $o ).
thf(iffalse_def,definition,
( iffalse
= ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ~ B
=> ( ( if @ A @ B @ C @ D )
= D ) ) ) ) ) ) ).
thf(1,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( if @ A @ B @ C @ D ) @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',iftrueorfalse) ).
thf(2,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( if @ A @ B @ C @ D ) @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(3,plain,
~ ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
= ( in @ C @ B ) )
=> ( A = B ) )
=> ( ! [A: $i] :
~ ( in @ A @ emptyset )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ B ) )
= ( ( C = A )
| ( in @ C @ B ) ) )
=> ( ! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
= ( ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) ) ) )
=> ( ! [A: $i,B: $i] :
( ( in @ B @ ( setunion @ A ) )
= ( ? [C: $i] :
( ( in @ B @ C )
& ( in @ C @ A ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [A: $i] :
( ( in @ A @ omega )
=> ( in @ ( setadjoin @ A @ A ) @ omega ) )
=> ( ! [A: $i] :
( ( ( in @ emptyset @ A )
& ! [B: $i] :
( ( ( in @ B @ omega )
& ( in @ B @ A ) )
=> ( in @ ( setadjoin @ B @ B ) @ A ) ) )
=> ! [B: $i] :
( ( in @ B @ omega )
=> ( in @ B @ A ) ) )
=> ( ! [A: $i > $i > $o,B: $i] :
( ! [C: $i] :
( ( in @ C @ B )
=> ? [D: $i] :
( ( A @ C @ D )
& ! [E: $i] :
( ( A @ C @ E )
=> ( D = E ) ) ) )
=> ? [C: $i] :
! [D: $i] :
( ( in @ D @ C )
= ( ? [E: $i] :
( ( in @ E @ B )
& ( A @ E @ D ) ) ) ) )
=> ( ! [A: $i] :
( ? [B: $i] : ( in @ B @ A )
=> ? [B: $i] :
( ( in @ B @ A )
& ~ ? [C: $i] :
( ( in @ C @ B )
& ( in @ C @ A ) ) ) )
=> ( ! [A: $i] :
? [B: $i] :
( ! [C: $i] :
( ( in @ C @ B )
=> ! [D: $i] :
( ( in @ D @ C )
=> ( in @ D @ A ) ) )
& ! [C: $i,D: $i] :
( ( ( in @ C @ A )
& ( in @ D @ A ) )
=> ( ! [E: $i] :
( ( in @ E @ B )
=> ( ( in @ C @ E )
= ( in @ D @ E ) ) )
=> ( C = D ) ) )
& ! [C: $i,D: $i] :
( ( ( in @ C @ B )
& ( in @ D @ B ) )
=> ( ! [E: $i] :
( ( in @ E @ C )
=> ( in @ E @ D ) )
| ! [E: $i] :
( ( in @ E @ D )
=> ( in @ E @ C ) ) ) )
& ! [C: $i] :
( ( ! [D: $i] :
( ( in @ D @ C )
=> ( in @ D @ A ) )
& ? [D: $i] : ( in @ D @ C ) )
=> ? [D: $i,E: $i] :
( ( in @ D @ B )
& ( in @ E @ C )
& ~ ? [F: $i] :
( ( in @ F @ D )
& ( in @ F @ C ) )
& ! [F: $i] :
( ( in @ F @ B )
=> ( ! [G: $i] :
( ( in @ G @ F )
=> ( in @ G @ D ) )
| ( in @ E @ F ) ) ) ) ) )
=> ( ! [A: $i > $o] :
( ? [B: $i] :
( ( A @ B )
& ! [C: $i] :
( ( A @ C )
=> ( B = C ) ) )
=> ( A @ ( descr @ A ) ) )
=> ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ A )
=> ( ( B @ C )
=> ( in @ C @ ( dsetconstr @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ ( dsetconstr @ A @ B ) )
=> ( in @ C @ A ) )
=> ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ ( dsetconstr @ A @ B ) )
=> ( B @ C ) )
=> ( ! [A: $o,B: $i] :
( ( in @ B @ ( prop2set @ A ) )
=> A )
=> ( ! [A: $i] :
( ( in @ A @ emptyset )
=> ! [B: $o] : B )
=> ( ! [A: $i] :
~ ( in @ A @ emptyset )
=> ( ! [A: $i] :
~ ( in @ A @ emptyset )
=> ( ! [A: $i > $o] :
( ? [B: $i] :
( ( A @ B )
& ! [C: $i] :
( ( A @ C )
=> ( B = C ) ) )
=> ? [B: $i] : ( A @ B ) )
=> ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) )
=> ( A = B ) ) )
=> ( ! [A: $i] :
( ! [B: $i] :
~ ( in @ B @ A )
=> ( A = emptyset ) )
=> ( ! [A: $i] :
( ! [B: $i] :
~ ( in @ B @ A )
=> ( A = emptyset ) )
=> ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ A )
=> ( ( in @ C @ ( dsetconstr @ A @ B ) )
= ( B @ C ) ) )
=> ( ! [A: $i] :
( ( A != emptyset )
=> ? [B: $i] : ( in @ B @ A ) )
=> ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ A )
=> ( ( B @ C )
=> ( ( dsetconstr @ A @ B )
!= emptyset ) ) )
=> ( ! [A: $i] :
( ? [B: $i] : ( in @ B @ A )
=> ( A != emptyset ) )
=> ( ! [A: $i,B: $i] : ( in @ A @ ( setadjoin @ A @ B ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ B )
=> ( in @ C @ ( setadjoin @ A @ B ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ B ) )
=> ! [D: $o] :
( ( ( C = A )
=> D )
=> ( ( ( in @ C @ B )
=> D )
=> D ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ B ) )
=> ( ( C = A )
| ( in @ C @ B ) ) )
=> ( ! [A: $i] :
( ( dsetconstr @ A
@ ^ [B: $i] : $true )
= A )
=> ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ B )
=> ( in @ C @ A ) )
=> ( in @ B @ ( powerset @ A ) ) )
=> ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) )
=> ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( ( in @ C @ B )
=> ( in @ C @ A ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ B @ C )
=> ( ( in @ C @ A )
=> ( in @ B @ ( setunion @ A ) ) ) )
=> ( ! [A: $i,B: $i] :
( ( in @ B @ ( setunion @ A ) )
=> ! [C: $o] :
( ! [D: $i] :
( ( in @ B @ D )
=> ( ( in @ D @ A )
=> C ) )
=> C ) )
=> ( ! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( in @ B @ ( powerset @ ( setunion @ A ) ) ) )
=> ( ! [A: $i > $o] :
( ? [B: $i] :
( ( A @ B )
& ! [C: $i] :
( ( A @ C )
=> ( B = C ) ) )
=> ? [B: $i] :
! [C: $i] :
( ( A @ C )
= ( C = B ) ) )
=> ( ! [A: $i] :
( ( A != emptyset )
=> ? [B: $i] : ( in @ B @ A ) )
=> ( ! [A: $i,B: $i] :
( ( in @ A @ ( setadjoin @ B @ emptyset ) )
=> ( A = B ) )
=> ( ! [A: $i,B: $i] :
( ( A != B )
=> ~ ( in @ B @ ( setadjoin @ A @ emptyset ) ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ( in @ A @ ( setadjoin @ B @ emptyset ) ) )
=> ( ! [A: $i,B: $i] :
( ( in @ A @ ( setadjoin @ B @ emptyset ) )
=> ( in @ B @ ( setadjoin @ A @ emptyset ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) )
=> ( ( C = A )
| ( C = B ) ) )
=> ( ! [A: $i,B: $i] : ( in @ A @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) )
=> ( ! [A: $i,B: $i] : ( in @ B @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) )
=> ( ! [A: $i,B: $i > $o] :
( ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ( ( dsetconstr @ A @ B )
!= emptyset ) )
=> ( ! [A: $i > $o,B: $i] :
( ( in @ B @ emptyset )
=> ( A @ B ) )
=> ( ! [A: $i,B: $i > $o] :
( ~ ! [C: $i] :
( ( in @ C @ A )
=> ( B @ C ) )
=> ? [C: $i] :
( ( in @ C @ A )
& ~ ( B @ C ) ) )
=> ( ! [A: $i,B: $i > $o] :
( ! [C: $i] :
( ( in @ C @ A )
=> ~ ( B @ C ) )
=> ~ ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) ) )
=> ( ! [A: $i,B: $i > $o] :
( ~ ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ! [C: $i] :
( ( in @ C @ A )
=> ~ ( B @ C ) ) )
=> ( ! [A: $i,B: $i > $o] :
( ? [C: $i] :
( ( in @ C @ A )
& ~ ( B @ C ) )
=> ~ ! [C: $i] :
( ( in @ C @ A )
=> ( B @ C ) ) )
=> ( ! [A: $o] :
( A
=> ( in @ emptyset @ ( prop2set @ A ) ) )
=> ( ! [A: $o] :
( A
=> ( set2prop @ ( prop2set @ A ) ) )
=> ( ! [A: $i,B: $i > $o] :
( ~ ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ! [C: $i] :
( ( in @ C @ A )
=> ~ ( B @ C ) ) )
=> ( ! [A: $i,B: $i > $o] :
( ~ ! [C: $i] :
( ( in @ C @ A )
=> ( B @ C ) )
=> ? [C: $i] :
( ( in @ C @ A )
& ~ ( B @ C ) ) )
=> ( ! [A: $i > $o] :
( ? [B: $i] : ( A @ B )
=> ( ! [B: $i,C: $i] :
( ( A @ B )
=> ( ( A @ C )
=> ( B = C ) ) )
=> ? [B: $i] :
( ( A @ B )
& ! [C: $i] :
( ( A @ C )
=> ( B = C ) ) ) ) )
=> ( ! [A: $i > $o] :
( ? [B: $i] :
! [C: $i] :
( ( A @ C )
= ( C = B ) )
=> ? [B: $i] :
( ( A @ B )
& ! [C: $i] :
( ( A @ C )
=> ( B = C ) ) ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [C: $i,D: $i] :
( ( C = D )
=> ( ( in @ C @ A )
=> ( in @ D @ B ) ) ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [C: $i,D: $i] :
( ( C = D )
=> ( ( in @ C @ A )
= ( in @ D @ B ) ) ) )
=> ( ! [A: $i > $o] :
( ? [B: $i] :
( ( A @ B )
& ! [C: $i] :
( ( A @ C )
=> ( B = C ) ) )
=> ! [B: $i,C: $i] :
( ( A @ B )
=> ( ( A @ C )
=> ( B = C ) ) ) )
=> ( ! [A: $i > $o,B: $i > $o] :
( ! [C: $i,D: $i] :
( ( C = D )
=> ( ( A @ C )
= ( B @ D ) ) )
=> ( ( ? [C: $i] :
( ( A @ C )
& ! [D: $i] :
( ( A @ D )
=> ( C = D ) ) ) )
= ( ? [C: $i] :
( ( B @ C )
& ! [D: $i] :
( ( B @ D )
=> ( C = D ) ) ) ) ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [C: $i,D: $i] :
( ( C = D )
=> ( ( setadjoin @ A @ C )
= ( setadjoin @ B @ D ) ) ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( powerset @ A )
= ( powerset @ B ) ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( setunion @ A )
= ( setunion @ B ) ) )
=> ( ! [A: $i > $o] :
( ? [B: $i] :
( ( A @ B )
& ! [C: $i] :
( ( A @ C )
=> ( B = C ) ) )
=> ! [B: $i,C: $i] :
( ( A @ B )
=> ( ( A @ C )
=> ( B = C ) ) ) )
=> ( ! [A: $i > $o,B: $i > $o] :
( ! [C: $i,D: $i] :
( ( C = D )
=> ( ( A @ C )
= ( B @ D ) ) )
=> ( ? [C: $i] :
( ( A @ C )
& ! [D: $i] :
( ( A @ D )
=> ( C = D ) ) )
=> ( ? [C: $i] :
( ( B @ C )
& ! [D: $i] :
( ( B @ D )
=> ( C = D ) ) )
=> ( ( descr @ A )
= ( descr @ B ) ) ) ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ! [C: $i > $o,D: $i > $o] :
( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( E = F )
=> ( ( C @ E )
= ( D @ F ) ) ) ) )
=> ( ( dsetconstr @ A @ C )
= ( dsetconstr @ B @ D ) ) ) )
=> ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( subset @ A @ B ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ B @ A ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ( subset @ A @ B ) )
=> ( ! [A: $i,B: $i] :
( ! [C: $i] :
( ( in @ C @ A )
=> ( in @ C @ B ) )
=> ( subset @ A @ B ) )
=> ( ! [A: $i] : ( subset @ emptyset @ A )
=> ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( in @ C @ A )
=> ( in @ C @ B ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ~ ( in @ C @ B )
=> ~ ( in @ C @ A ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ~ ( subset @ A @ B ) ) )
=> ( ! [A: $i,B: $i] :
( ~ ( subset @ A @ B )
=> ( A != B ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( A != B ) ) )
=> ( ! [A: $i] : ( subset @ A @ A )
=> ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ C )
=> ( subset @ A @ C ) ) )
=> ( ! [A: $i,B: $i] : ( subset @ B @ ( setadjoin @ A @ B ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( subset @ A @ C )
=> ( subset @ A @ ( setadjoin @ B @ C ) ) )
=> ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( in @ A @ ( powerset @ B ) ) )
=> ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( subset @ B @ A )
=> ( A = B ) ) )
=> ( ! [A: $i] :
( ( subset @ A @ emptyset )
=> ( A = emptyset ) )
=> ( ! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( in @ B @ ( powerset @ A ) ) )
=> ( ! [A: $i,B: $i] :
( ( in @ B @ ( powerset @ A ) )
=> ( subset @ B @ A ) )
=> ( ! [A: $i] : ( in @ A @ ( powerset @ A ) )
=> ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) )
=> ( ! [A: $i,B: $i > $o] : ( in @ ( dsetconstr @ A @ B ) @ ( powerset @ A ) )
=> ( ! [A: $i,B: $i > $o] : ( subset @ ( dsetconstr @ A @ B ) @ A )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( in @ C @ ( binunion @ A @ B ) ) )
=> ( ! [A: $i,B: $i] : ( in @ B @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ B )
=> ( in @ C @ ( binunion @ A @ B ) ) )
=> ( ! [A: $i,B: $i,C: $i,D: $o] :
( ( in @ C @ ( binunion @ A @ B ) )
=> ( ( ( in @ C @ A )
=> D )
=> ( ( ( in @ C @ B )
=> D )
=> D ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( binunion @ A @ B ) )
=> ( ( in @ C @ A )
| ( in @ C @ B ) ) )
=> ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) )
=> ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ( in @ C @ B )
=> ( in @ C @ ( binintersect @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( subset @ C @ A )
=> ( ( subset @ C @ B )
=> ( subset @ C @ ( binintersect @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( binintersect @ A @ B ) )
=> ( in @ C @ A ) )
=> ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A )
=> ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( binintersect @ A @ B )
= A ) )
=> ( ! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= B )
=> ( subset @ B @ A ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( binintersect @ A @ B ) )
=> ( in @ C @ B ) )
=> ( ! [A: $i,B: $i] :
( ~ ? [C: $i] :
( ( in @ C @ A )
& ( in @ C @ B ) )
=> ( ( binintersect @ A @ B )
= emptyset ) )
=> ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B )
=> ( ! [A: $i,B: $i] :
( ( subset @ B @ A )
=> ( ( binintersect @ A @ B )
= B ) )
=> ( ! [A: $i,B: $i] :
( ( ( binintersect @ A @ B )
= A )
=> ( subset @ A @ B ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( binintersect @ A @ ( binunion @ B @ C ) )
= ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( in @ C @ ( setminus @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setminus @ A @ B ) )
=> ( in @ C @ A ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setminus @ A @ B ) )
=> ~ ( in @ C @ B ) )
=> ( ! [A: $i,B: $i] :
( ( subset @ A @ B )
=> ( ( setminus @ A @ B )
= emptyset ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ ( setminus @ A @ B ) )
=> ( ( in @ C @ A )
=> ( in @ C @ B ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ ( setminus @ A @ B ) )
=> ( ~ ( in @ C @ B )
=> ~ ( in @ C @ A ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ A )
=> ~ ( in @ C @ ( setminus @ A @ B ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ B )
=> ~ ( in @ C @ ( setminus @ A @ B ) ) )
=> ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A )
=> ( ! [A: $i,B: $i] :
( ( ( setminus @ A @ B )
= emptyset )
=> ( subset @ A @ B ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( symdiff @ A @ B ) )
=> ! [D: $o] :
( ( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> D ) )
=> ( ( ~ ( in @ C @ A )
=> ( ( in @ C @ B )
=> D ) )
=> D ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ( in @ C @ ( symdiff @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ A )
=> ( ( in @ C @ B )
=> ( in @ C @ ( symdiff @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( ( in @ C @ B )
=> ~ ( in @ C @ ( symdiff @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ~ ( in @ C @ A )
=> ( ~ ( in @ C @ B )
=> ~ ( in @ C @ ( symdiff @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i] : ( in @ B @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) )
=> ( ! [A: $i,B: $i] : ( in @ A @ ( setunion @ ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [A: $i,B: $i] : ( in @ B @ ( setunion @ ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) ) ) )
=> ( ! [A: $i,B: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) ) )
=> ( ! [A: $i,B: $i] : ( iskpair @ ( kpair @ A @ B ) )
=> ( ! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( subset @ ( setadjoin @ B @ emptyset ) @ A ) )
=> ( ! [A: $i,B: $i] :
( ( in @ B @ A )
=> ( in @ ( setadjoin @ B @ emptyset ) @ ( powerset @ A ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ( in @ ( setadjoin @ C @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) )
=> ( ( C = A )
| ( C = B ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( subset @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ ( binunion @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( in @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( subset @ ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( in @ ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( in @ ( kpair @ C @ D ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( in @ ( kpair @ C @ D ) @ ( cartprod @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ? [D: $i] :
( ( in @ D @ A )
& ? [E: $i] :
( ( in @ E @ B )
& ( C
= ( kpair @ D @ E ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ( iskpair @ C ) )
=> ( ! [A: $i,B: $i] :
( ( in @ B @ ( setunion @ A ) )
=> ? [C: $i] :
( ( in @ C @ A )
& ( in @ B @ C ) ) )
=> ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A )
=> ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) )
=> ( ! [A: $i] :
( ( setunion @ ( setadjoin @ A @ emptyset ) )
= A )
=> ( ! [A: $i,B: $i > $o] :
( ! [C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( B @ C )
=> ( ( B @ D )
=> ( C = D ) ) ) ) )
=> ( ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ( singleton @ ( dsetconstr @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i > $o] :
( ( singleton @ ( dsetconstr @ A @ B ) )
=> ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) ) )
=> ( ! [A: $i,B: $i > $o,C: $i] :
( ( in @ C @ A )
=> ( ( B @ C )
=> ( ! [D: $i] :
( ( in @ D @ A )
=> ( ( B @ D )
=> ( D = C ) ) )
=> ( singleton @ ( dsetconstr @ A @ B ) ) ) ) )
=> ( ! [A: $i,B: $i > $o] :
( ! [C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( B @ C )
=> ( ( B @ D )
=> ( C = D ) ) ) ) )
=> ( ? [C: $i] :
( ( in @ C @ A )
& ( B @ C ) )
=> ( singleton @ ( dsetconstr @ A @ B ) ) ) )
=> ( ! [A: $i,B: $i] :
( ( ( setadjoin @ A @ emptyset )
= ( setadjoin @ B @ emptyset ) )
=> ( A = B ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) ) )
=> ( A = C ) )
=> ( ! [A: $i] :
( ( iskpair @ A )
=> ( singleton
@ ( dsetconstr @ ( setunion @ A )
@ ^ [B: $i] : ( in @ ( setadjoin @ B @ emptyset ) @ A ) ) ) )
=> ( ! [A: $i] :
( ( singleton @ A )
=> ( in @ ( setunion @ A ) @ A ) )
=> ( ! [A: $i,B: $i] :
( ( kfst @ ( kpair @ A @ B ) )
= A )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ( in @ ( kfst @ C ) @ A ) )
=> ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) )
=> ( A = C ) )
=> ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( kpair @ A @ B )
= ( kpair @ C @ D ) )
=> ( A = C ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) )
= ( setadjoin @ A @ emptyset ) ) )
=> ( ! [A: $i,B: $i] :
( ( A = B )
=> ( ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ A @ emptyset ) @ emptyset ) ) )
=> ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) )
=> ( ( C = D )
=> ( B = D ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) )
= ( setadjoin @ C @ emptyset ) )
=> ( A = B ) )
=> ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( setadjoin @ ( setadjoin @ A @ emptyset ) @ ( setadjoin @ ( setadjoin @ A @ ( setadjoin @ B @ emptyset ) ) @ emptyset ) )
= ( setadjoin @ ( setadjoin @ C @ emptyset ) @ ( setadjoin @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) @ emptyset ) ) )
=> ( B = D ) )
=> ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( ( kpair @ A @ B )
= ( kpair @ C @ D ) )
=> ( B = D ) )
=> ( ! [A: $i] :
( ( iskpair @ A )
=> ( singleton
@ ( dsetconstr @ ( setunion @ A )
@ ^ [B: $i] :
( A
= ( kpair @ ( kfst @ A ) @ B ) ) ) ) )
=> ( ! [A: $i,B: $i] :
( ( ksnd @ ( kpair @ A @ B ) )
= B )
=> ( ! [A: $i] :
( ( iskpair @ A )
=> ( ( kpair @ ( kfst @ A ) @ ( ksnd @ A ) )
= A ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ( in @ ( ksnd @ C ) @ B ) )
=> ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( in @ ( kpair @ C @ D ) @ ( cartprod @ A @ B ) )
=> ( in @ C @ A ) )
=> ( ! [A: $i,B: $i,C: $i,D: $i] :
( ( in @ ( kpair @ C @ D ) @ ( cartprod @ A @ B ) )
=> ( in @ D @ B ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( ( kfst @ ( kpair @ C @ D ) )
= C ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ B )
=> ( ( ksnd @ ( kpair @ C @ D ) )
= D ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( cartprod @ A @ B ) )
=> ( ( kpair @ ( kfst @ C ) @ ( ksnd @ C ) )
= C ) )
=> ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i] :
( ( in @ D @ A )
=> ! [E: $i] :
( ( in @ E @ B )
=> ( ( C @ D @ E )
=> ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i > $i > $o] : ( subset @ ( dpsetconstr @ A @ B @ C ) @ ( cartprod @ A @ B ) )
=> ( ! [A: $i,B: $i,C: $i > $i > $o] : ( subset @ ( dpsetconstr @ A @ B @ C ) @ ( cartprod @ A @ B ) )
=> ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i] :
( ( in @ D @ A )
=> ! [E: $i] :
( ( in @ E @ B )
=> ( ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) )
=> ( C @ D @ E ) ) ) )
=> ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i,E: $i] :
( ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) )
=> ( in @ D @ A ) )
=> ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i,E: $i] :
( ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) )
=> ( in @ E @ B ) )
=> ( ! [A: $i,B: $i,C: $i > $i > $o,D: $i,E: $i] :
( ( in @ ( kpair @ D @ E ) @ ( dpsetconstr @ A @ B @ C ) )
=> ( C @ D @ E ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( singleton
@ ( dsetconstr @ B
@ ^ [E: $i] : ( in @ ( kpair @ D @ E ) @ C ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in
@ ( setunion
@ ( dsetconstr @ B
@ ^ [E: $i] : ( in @ ( kpair @ D @ E ) @ C ) ) )
@ B ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( ap @ A @ B @ C @ D ) @ B ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ( func @ A @ B @ C ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( ap @ A @ B @ C @ D ) @ B ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ( in @ C @ ( funcSet @ A @ B ) ) )
=> ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ( func @ A @ B
@ ( dpsetconstr @ A @ B
@ ^ [D: $i] : ( (=) @ $i @ ( C @ D ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ( func @ A @ B @ ( lam @ A @ B @ C ) ) )
=> ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ( in @ ( lam @ A @ B @ C ) @ ( funcSet @ A @ B ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( subset @ C @ ( cartprod @ A @ B ) )
=> ! [D: $i > $o] :
( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ C )
=> ( D @ ( kpair @ E @ F ) ) ) ) )
=> ! [E: $i] :
( ( in @ E @ C )
=> ( D @ E ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( subset @ C @ ( cartprod @ A @ B ) )
=> ! [D: $i > $o] :
( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ C )
=> ( D @ ( kpair @ E @ F ) ) ) ) )
=> ! [E: $i] :
( ( in @ E @ C )
=> ( D @ E ) ) ) )
=> ( ! [A: $i,B: $i > $o] :
( ( singleton @ ( dsetconstr @ A @ B ) )
=> ! [C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( B @ C )
=> ( ( B @ D )
=> ( C = D ) ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( kpair @ D @ ( ap @ A @ B @ C @ D ) ) @ C ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( kpair @ D @ ( ap @ A @ B @ C @ D ) ) @ C ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( in @ D @ A )
=> ! [E: $i] :
( ( in @ E @ B )
=> ( ( in @ ( kpair @ D @ E ) @ C )
=> ( ( ap @ A @ B @ C @ D )
= E ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( func @ A @ B @ D )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ( ( ap @ A @ B @ C @ E )
= ( ap @ A @ B @ D @ E ) ) )
=> ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ D )
=> ( in @ ( kpair @ E @ F ) @ C ) ) ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ! [E: $i] :
( ( in @ E @ B )
=> ( ( in @ ( kpair @ D @ E ) @ C )
=> ( ( ap @ A @ B @ C @ D )
= E ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( subset @ C @ ( cartprod @ A @ B ) )
=> ! [D: $i] :
( ( subset @ D @ ( cartprod @ A @ B ) )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ C )
=> ( in @ ( kpair @ E @ F ) @ D ) ) ) )
=> ( subset @ C @ D ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( subset @ C @ ( cartprod @ A @ B ) )
=> ! [D: $i] :
( ( subset @ D @ ( cartprod @ A @ B ) )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ C )
=> ( in @ ( kpair @ E @ F ) @ D ) ) ) )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ! [F: $i] :
( ( in @ F @ B )
=> ( ( in @ ( kpair @ E @ F ) @ D )
=> ( in @ ( kpair @ E @ F ) @ C ) ) ) )
=> ( C = D ) ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ! [D: $i] :
( ( func @ A @ B @ D )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ( ( ap @ A @ B @ C @ E )
= ( ap @ A @ B @ D @ E ) ) )
=> ( C = D ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ! [D: $i] :
( ( in @ D @ ( funcSet @ A @ B ) )
=> ( ! [E: $i] :
( ( in @ E @ A )
=> ( ( ap @ A @ B @ C @ E )
= ( ap @ A @ B @ D @ E ) ) )
=> ( C = D ) ) ) )
=> ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( ap @ A @ B @ ( lam @ A @ B @ C ) @ D )
= ( C @ D ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( func @ A @ B @ C )
=> ( ( lam @ A @ B @ ( ap @ A @ B @ C ) )
= C ) )
=> ( ! [A: $i,B: $i,C: $i > $i] :
( ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( C @ D ) @ B ) )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ( ap @ A @ B @ ( lam @ A @ B @ C ) @ D )
= ( C @ D ) ) ) )
=> ( ! [A: $i,B: $i,C: $i] :
( ( in @ C @ ( funcSet @ A @ B ) )
=> ( ( lam @ A @ B @ ( ap @ A @ B @ C ) )
= C ) )
=> ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ~ B
=> ( in @ D
@ ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) ) ) ) ) )
=> ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ~ B
=> ( ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) )
= ( setadjoin @ D @ emptyset ) ) ) ) )
=> ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( B
=> ( in @ C
@ ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) ) ) ) ) )
=> ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( B
=> ( ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) )
= ( setadjoin @ C @ emptyset ) ) ) ) )
=> ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( singleton
@ ( dsetconstr @ A
@ ^ [E: $i] :
( ( B
& ( E = C ) )
| ( ~ B
& ( E = D ) ) ) ) ) ) )
=> ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( if @ A @ B @ C @ D ) @ A ) ) )
=> ( ! [A: $i] :
( ( singleton @ A )
=> ! [B: $i] :
( ( in @ B @ A )
=> ( ( setunion @ A )
= B ) ) )
=> ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( B
=> ( ( if @ A @ B @ C @ D )
= C ) ) ) )
=> ( ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( ~ B
=> ( ( if @ A @ B @ C @ D )
= D ) ) ) )
=> ! [A: $i,B: $o,C: $i] :
( ( in @ C @ A )
=> ! [D: $i] :
( ( in @ D @ A )
=> ( in @ ( if @ A @ B @ C @ D ) @ ( setadjoin @ C @ ( setadjoin @ D @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(849,plain,
$false,
inference(e,[status(thm)],[3]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SEU709^1 : TPTP v8.2.0. Released v3.7.0.
% 0.07/0.15 % Command : run_Leo-III %s %d
% 0.13/0.36 % Computer : n004.cluster.edu
% 0.13/0.36 % Model : x86_64 x86_64
% 0.13/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.36 % Memory : 8042.1875MB
% 0.13/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.36 % CPULimit : 300
% 0.13/0.36 % WCLimit : 300
% 0.13/0.36 % DateTime : Sun May 19 16:24:39 EDT 2024
% 0.13/0.36 % CPUTime :
% 1.03/1.01 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 2.09/1.40 % [INFO] Parsing done (396ms).
% 2.09/1.42 % [INFO] Running in sequential loop mode.
% 3.02/1.73 % [INFO] eprover registered as external prover.
% 3.02/1.73 % [INFO] cvc4 registered as external prover.
% 3.02/1.74 % [INFO] Scanning for conjecture ...
% 4.55/2.24 % [INFO] Found a conjecture (or negated_conjecture) and 0 axioms. Running axiom selection ...
% 5.05/2.34 % [INFO] Axiom selection finished. Selected 0 axioms (removed 0 axioms).
% 5.05/2.34 % [INFO] Problem is higher-order (TPTP THF).
% 5.30/2.34 % [INFO] Type checking passed.
% 5.30/2.35 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 17.48/7.08 % External prover 'e' found a proof!
% 17.48/7.08 % [INFO] Killing All external provers ...
% 17.48/7.08 % Time passed: 6565ms (effective reasoning time: 5659ms)
% 17.48/7.08 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 17.48/7.08 % Axioms used in derivation (0):
% 17.48/7.08 % No. of inferences in proof: 4
% 17.48/7.08 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 6565 ms resp. 5659 ms w/o parsing
% 18.59/7.21 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 18.59/7.21 % [INFO] Killing All external provers ...
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