%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : SEU546^1 : TPTP v8.1.0. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n006.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 : 600s
% DateTime : Tue Jul 19 13:53:07 EDT 2022
% Result : Theorem 187.84s 187.82s
% Output : Proof 187.84s
% Verified :
% SZS Type : Refutation
% Derivation depth : 68
% Number of leaves : 226
% Syntax : Number of formulae : 296 ( 157 unt; 13 typ; 65 def)
% Number of atoms : 5260 (1514 equ; 0 cnn)
% Maximal formula atoms : 144 ( 18 avg)
% Number of connectives : 21733 (3349 ~; 16 |; 0 &;11546 @)
% ( 15 <=>;6807 =>; 0 <=; 0 <~>)
% Maximal formula depth : 74 ( 12 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 926 ( 926 >; 0 *; 0 +; 0 <<)
% Number of symbols : 95 ( 93 usr; 85 con; 0-2 aty)
% Number of variables : 6824 ( 41 ^6783 !; 0 ?;6824 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_setunion,type,
setunion: $i > $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i > $o ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_descr,type,
descr: ( $i > $o ) > $i ).
thf(ty_dsetconstr,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(ty_powerset,type,
powerset: $i > $i ).
thf(ty_omega,type,
omega: $i ).
thf(ty_prop2set,type,
prop2set: $o > $i ).
thf(ty_set2prop,type,
set2prop: $i > $o ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_setadjoin,type,
setadjoin: $i > $i > $i ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: $i] :
~ ( ( eigen__0 @ X1 )
=> ( eigen__1 = X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: $i] :
( ( eigen__0 @ X1 )
= ( X1 = eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( eigen__0 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( X2 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i] :
( ( eigen__2 = X1 )
=> ( X1 = eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ( eigen__0 @ eigen__2 )
=> ( eigen__1 = eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( eigen__1 = eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( ( eigen__0 @ eigen__2 )
= ( eigen__2 = eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( eigen__1 = eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( sP2
=> ~ ! [X1: $i] :
( ( eigen__0 @ X1 )
=> ( eigen__1 = X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( ( eigen__2 = eigen__1 )
=> sP8 ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( eigen__2 = eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: $i] :
( ( eigen__0 @ X1 )
=> ( eigen__1 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ! [X1: $i] :
( ( eigen__0 @ X1 )
=> ~ ! [X2: $i] :
( ( eigen__0 @ X2 )
=> ( X1 = X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( eigen__0 @ eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( sP2 = sP6 ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(def_exu,definition,
( exu
= ( ^ [X1: $i > $o] :
~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ).
thf(def_setextAx,definition,
( setextAx
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_emptysetAx,definition,
( emptysetAx
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_setadjoinAx,definition,
( setadjoinAx
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_powersetAx,definition,
( powersetAx
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_setunionAx,definition,
( setunionAx
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_omega0Ax,definition,
( omega0Ax
= ( in @ emptyset @ omega ) ) ).
thf(def_omegaSAx,definition,
( omegaSAx
= ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ) ) ) ).
thf(def_omegaIndAx,definition,
( omegaIndAx
= ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) ) ) ) ).
thf(def_replAx,definition,
( replAx
= ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( exu @ ( X1 @ X3 ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) ) ) ) ).
thf(def_foundationAx,definition,
( foundationAx
= ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_wellorderingAx,definition,
( wellorderingAx
= ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) ) ) ) ).
thf(def_descrp,definition,
( descrp
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ( X1 @ ( descr @ X1 ) ) ) ) ) ).
thf(def_dsetconstrI,definition,
( dsetconstrI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_dsetconstrEL,definition,
( dsetconstrEL
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ) ).
thf(def_dsetconstrER,definition,
( dsetconstrER
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) ) ) ) ).
thf(def_exuE1,definition,
( exuE1
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ).
thf(def_prop2setE,definition,
( prop2setE
= ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 ) ) ) ).
thf(def_emptysetE,definition,
( emptysetE
= ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 ) ) ) ).
thf(def_emptysetimpfalse,definition,
( emptysetimpfalse
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_notinemptyset,definition,
( notinemptyset
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_exuE3e,definition,
( exuE3e
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) ) ).
thf(def_setext,definition,
( setext
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) ) ) ) ).
thf(def_emptyI,definition,
( emptyI
= ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_noeltsimpempty,definition,
( noeltsimpempty
= ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_setbeta,definition,
( setbeta
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) ) ) ) ).
thf(def_nonempty,definition,
( nonempty
= ( ^ [X1: $i] : ( X1 != emptyset ) ) ) ).
thf(def_nonemptyE1,definition,
( nonemptyE1
= ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ) ) ) ).
thf(def_nonemptyI,definition,
( nonemptyI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( nonempty @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_nonemptyI1,definition,
( nonemptyI1
= ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( nonempty @ X1 ) ) ) ) ).
thf(def_setadjoinIL,definition,
( setadjoinIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ) ).
thf(def_emptyinunitempty,definition,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
thf(def_setadjoinIR,definition,
( setadjoinIR
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) ).
thf(def_setadjoinE,definition,
( setadjoinE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ) ) ) ).
thf(def_setadjoinOr,definition,
( setadjoinOr
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_setoftrueEq,definition,
( setoftrueEq
= ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 ) ) ) ).
thf(def_powersetI,definition,
( powersetI
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) ) ) ) ).
thf(def_emptyinPowerset,definition,
( emptyinPowerset
= ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ).
thf(def_emptyInPowerset,definition,
( emptyInPowerset
= ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ).
thf(def_powersetE,definition,
( powersetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_setunionI,definition,
( setunionI
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) ) ) ) ).
thf(def_setunionE,definition,
( setunionE
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) ) ) ) ).
thf(def_subPowSU,definition,
( subPowSU
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ) ).
thf(def_exuE2,definition,
( exuE2
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) ) ) ) ).
thf(def_nonemptyImpWitness,definition,
( nonemptyImpWitness
= ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ) ) ) ).
thf(def_uniqinunit,definition,
( uniqinunit
= ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_notinsingleton,definition,
( notinsingleton
= ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_eqinunit,definition,
( eqinunit
= ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_singletonsswitch,definition,
( singletonsswitch
= ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(def_upairsetE,definition,
( upairsetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) ) ) ) ).
thf(def_upairsetIL,definition,
( upairsetIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_upairsetIR,definition,
( upairsetIR
= ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ).
thf(def_emptyE1,definition,
( emptyE1
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) ) ) ).
thf(def_vacuousDall,definition,
( vacuousDall
= ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) ) ) ) ).
thf(def_quantDeMorgan1,definition,
( quantDeMorgan1
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_quantDeMorgan2,definition,
( quantDeMorgan2
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_quantDeMorgan3,definition,
( quantDeMorgan3
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_quantDeMorgan4,definition,
( quantDeMorgan4
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_prop2setI,definition,
( prop2setI
= ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) ) ) ) ).
thf(def_prop2set2propI,definition,
( prop2set2propI
= ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) ) ) ) ).
thf(def_notdexE,definition,
( notdexE
= ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_notdallE,definition,
( notdallE
= ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_exuI1,definition,
( exuI1
= ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) ) ) ).
thf(def_exuI3,definition,
( exuI3
= ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( exu @ X1 ) ) ) ) ) ).
thf(exuI2,conjecture,
( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(h1,negated_conjecture,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[exuI2]) ).
thf(h2,assumption,
! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
! [X1: $i] :
~ ( in @ X1 @ emptyset ),
introduced(assumption,[]) ).
thf(h5,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h7,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h8,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h9,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h10,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h11,assumption,
~ ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h12,assumption,
in @ emptyset @ omega,
introduced(assumption,[]) ).
thf(h13,assumption,
~ ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h14,assumption,
! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ),
introduced(assumption,[]) ).
thf(h15,assumption,
~ ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h16,assumption,
! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h17,assumption,
~ ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h18,assumption,
! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h19,assumption,
~ ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h20,assumption,
! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h21,assumption,
~ ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h22,assumption,
! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h23,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h24,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h25,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h26,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h27,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h28,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ),
introduced(assumption,[]) ).
thf(h29,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h30,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) ),
introduced(assumption,[]) ).
thf(h31,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h32,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ),
introduced(assumption,[]) ).
thf(h33,assumption,
~ ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h34,assumption,
! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 ),
introduced(assumption,[]) ).
thf(h35,assumption,
~ ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h36,assumption,
! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 ),
introduced(assumption,[]) ).
thf(h37,assumption,
~ ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h38,assumption,
~ ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h39,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h40,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h41,assumption,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h42,assumption,
! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) ),
introduced(assumption,[]) ).
thf(h43,assumption,
~ ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h44,assumption,
! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ),
introduced(assumption,[]) ).
thf(h45,assumption,
~ ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h46,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h47,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h48,assumption,
~ ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h49,assumption,
! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ),
introduced(assumption,[]) ).
thf(h50,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h51,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) ),
introduced(assumption,[]) ).
thf(h52,assumption,
~ ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h53,assumption,
! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) ),
introduced(assumption,[]) ).
thf(h54,assumption,
~ ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h55,assumption,
! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h56,assumption,
~ ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h57,assumption,
in @ emptyset @ ( setadjoin @ emptyset @ emptyset ),
introduced(assumption,[]) ).
thf(h58,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h59,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h60,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h61,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ),
introduced(assumption,[]) ).
thf(h62,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h63,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h64,assumption,
~ ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h65,assumption,
! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 ),
introduced(assumption,[]) ).
thf(h66,assumption,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h67,assumption,
! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h68,assumption,
~ ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h69,assumption,
! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ),
introduced(assumption,[]) ).
thf(h70,assumption,
~ ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h71,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h72,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h73,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h74,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h75,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h76,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) ),
introduced(assumption,[]) ).
thf(h77,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h78,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h79,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h80,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) ),
introduced(assumption,[]) ).
thf(h81,assumption,
~ ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h82,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h83,assumption,
! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ),
introduced(assumption,[]) ).
thf(h84,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h85,assumption,
! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h86,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h87,assumption,
! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h88,assumption,
~ ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h89,assumption,
! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h90,assumption,
~ ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h91,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) ),
introduced(assumption,[]) ).
thf(h92,assumption,
~ ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h93,assumption,
! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h94,assumption,
~ ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h95,assumption,
! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ),
introduced(assumption,[]) ).
thf(h96,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h97,assumption,
! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ),
introduced(assumption,[]) ).
thf(h98,assumption,
~ ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h99,assumption,
! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h100,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h101,assumption,
! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h102,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h103,assumption,
! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h104,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h105,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h106,assumption,
~ ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h107,assumption,
! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h108,assumption,
~ ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h109,assumption,
! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h110,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h111,assumption,
~ ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h112,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h113,assumption,
~ ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h114,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h115,assumption,
~ ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ),
introduced(assumption,[]) ).
thf(h116,assumption,
~ ( ~ ! [X1: $i] :
~ ! [X2: $i] :
( ( eigen__0 @ X2 )
= ( X2 = X1 ) )
=> ~ sP13 ),
introduced(assumption,[]) ).
thf(h117,assumption,
~ ! [X1: $i] :
~ ! [X2: $i] :
( ( eigen__0 @ X2 )
= ( X2 = X1 ) ),
introduced(assumption,[]) ).
thf(h118,assumption,
sP13,
introduced(assumption,[]) ).
thf(h119,assumption,
sP1,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP15
| sP2
| ~ sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP7
| ~ sP14
| sP11 ),
inference(prop_rule,[status(thm)],]) ).
thf(3,plain,
sP6,
inference(prop_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP1
| sP7 ),
inference(all_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP10
| ~ sP11
| sP8 ),
inference(prop_rule,[status(thm)],]) ).
thf(6,plain,
( ~ sP4
| sP10 ),
inference(all_rule,[status(thm)],]) ).
thf(7,plain,
( ~ sP3
| sP4 ),
inference(all_rule,[status(thm)],]) ).
thf(8,plain,
sP3,
inference(eq_sym,[status(thm)],]) ).
thf(9,plain,
( sP5
| ~ sP8 ),
inference(prop_rule,[status(thm)],]) ).
thf(10,plain,
( sP5
| sP14 ),
inference(prop_rule,[status(thm)],]) ).
thf(11,plain,
( sP12
| ~ sP5 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(12,plain,
( ~ sP9
| ~ sP2
| ~ sP12 ),
inference(prop_rule,[status(thm)],]) ).
thf(13,plain,
( ~ sP13
| sP9 ),
inference(all_rule,[status(thm)],]) ).
thf(14,plain,
( ~ sP1
| sP15 ),
inference(all_rule,[status(thm)],]) ).
thf(15,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h119,h117,h118,h116,h114,h115,h32,h113,h101,h112,h103,h111,h109,h110,h107,h108,h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,h119,h118]) ).
thf(16,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h117,h118,h116,h114,h115,h32,h113,h101,h112,h103,h111,h109,h110,h107,h108,h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negall(discharge,[h119]),tab_negall(eigenvar,eigen__1)],[h117,15,h119]) ).
thf(17,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h116,h114,h115,h32,h113,h101,h112,h103,h111,h109,h110,h107,h108,h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h117,h118])],[h116,16,h117,h118]) ).
thf(18,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h114,h115,h32,h113,h101,h112,h103,h111,h109,h110,h107,h108,h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negall(discharge,[h116]),tab_negall(eigenvar,eigen__0)],[h115,17,h116]) ).
thf(19,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h32,h113,h101,h112,h103,h111,h109,h110,h107,h108,h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h114,h115])],[h113,18,h114,h115]) ).
thf(20,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h101,h112,h103,h111,h109,h110,h107,h108,h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h32,h113])],[h112,19,h32,h113]) ).
thf(21,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h103,h111,h109,h110,h107,h108,h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h101,h112])],[h111,20,h101,h112]) ).
thf(22,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h109,h110,h107,h108,h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h103,h111])],[h110,21,h103,h111]) ).
thf(23,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h107,h108,h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h109,h110])],[h108,22,h109,h110]) ).
thf(24,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h101,h106,h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h107,h108])],[h106,23,h107,h108]) ).
thf(25,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h103,h105,h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h101,h106])],[h105,24,h101,h106]) ).
thf(26,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h103,h104,h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h103,h105])],[h104,25,h103,h105]) ).
thf(27,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h101,h102,h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h103,h104])],[h102,26,h103,h104]) ).
thf(28,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h99,h100,h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h101,h102])],[h100,27,h101,h102]) ).
thf(29,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h97,h98,h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h99,h100])],[h98,28,h99,h100]) ).
thf(30,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h95,h96,h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h97,h98])],[h96,29,h97,h98]) ).
thf(31,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h93,h94,h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h95,h96])],[h94,30,h95,h96]) ).
thf(32,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h91,h92,h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h93,h94])],[h92,31,h93,h94]) ).
thf(33,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h89,h90,h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h91,h92])],[h90,32,h91,h92]) ).
thf(34,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h87,h88,h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h89,h90])],[h88,33,h89,h90]) ).
thf(35,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h85,h86,h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h87,h88])],[h86,34,h87,h88]) ).
thf(36,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h83,h84,h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h85,h86])],[h84,35,h85,h86]) ).
thf(37,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h49,h82,h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h83,h84])],[h82,36,h83,h84]) ).
thf(38,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h80,h81,h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h49,h82])],[h81,37,h49,h82]) ).
thf(39,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h78,h79,h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h80,h81])],[h79,38,h80,h81]) ).
thf(40,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h76,h77,h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h78,h79])],[h77,39,h78,h79]) ).
thf(41,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h74,h75,h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h76,h77])],[h75,40,h76,h77]) ).
thf(42,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h72,h73,h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h74,h75])],[h73,41,h74,h75]) ).
thf(43,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h69,h71,h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h72,h73])],[h71,42,h72,h73]) ).
thf(44,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h69,h70,h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h69,h71])],[h70,43,h69,h71]) ).
thf(45,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h67,h68,h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h69,h70])],[h68,44,h69,h70]) ).
thf(46,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h65,h66,h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h67,h68])],[h66,45,h67,h68]) ).
thf(47,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h63,h64,h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h65,h66])],[h64,46,h65,h66]) ).
thf(48,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h61,h62,h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h63,h64])],[h62,47,h63,h64]) ).
thf(49,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h59,h60,h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h61,h62])],[h60,48,h61,h62]) ).
thf(50,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h57,h58,h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h59,h60])],[h58,49,h59,h60]) ).
thf(51,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h55,h56,h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h57,h58])],[h56,50,h57,h58]) ).
thf(52,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h53,h54,h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h55,h56])],[h54,51,h55,h56]) ).
thf(53,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h51,h52,h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h53,h54])],[h52,52,h53,h54]) ).
thf(54,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h49,h50,h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h51,h52])],[h50,53,h51,h52]) ).
thf(55,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h47,h48,h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h49,h50])],[h48,54,h49,h50]) ).
thf(56,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h44,h46,h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h47,h48])],[h46,55,h47,h48]) ).
thf(57,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h44,h45,h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h44,h46])],[h45,56,h44,h46]) ).
thf(58,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h42,h43,h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h44,h45])],[h43,57,h44,h45]) ).
thf(59,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h40,h41,h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h42,h43])],[h41,58,h42,h43]) ).
thf(60,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h39,h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h40,h41])],[h39,59,h40,h41]) ).
thf(61,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h38,h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h4,h39])],[h38,60,h4,h39]) ).
thf(62,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h36,h37,h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h4,h38])],[h37,61,h4,h38]) ).
thf(63,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h34,h35,h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h36,h37])],[h35,62,h36,h37]) ).
thf(64,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h32,h33,h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h34,h35])],[h33,63,h34,h35]) ).
thf(65,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h30,h31,h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h32,h33])],[h31,64,h32,h33]) ).
thf(66,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h28,h29,h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h30,h31])],[h29,65,h30,h31]) ).
thf(67,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h26,h27,h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h28,h29])],[h27,66,h28,h29]) ).
thf(68,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h24,h25,h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h26,h27])],[h25,67,h26,h27]) ).
thf(69,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h22,h23,h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h24,h25])],[h23,68,h24,h25]) ).
thf(70,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h20,h21,h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h22,h23])],[h21,69,h22,h23]) ).
thf(71,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h18,h19,h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h20,h21])],[h19,70,h20,h21]) ).
thf(72,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h16,h17,h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h18,h19])],[h17,71,h18,h19]) ).
thf(73,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h14,h15,h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h16,h17])],[h15,72,h16,h17]) ).
thf(74,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h12,h13,h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h14,h15])],[h13,73,h14,h15]) ).
thf(75,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h10,h11,h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h12,h13])],[h11,74,h12,h13]) ).
thf(76,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h8,h9,h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h10,h11])],[h9,75,h10,h11]) ).
thf(77,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h6,h7,h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h8,h9])],[h7,76,h8,h9]) ).
thf(78,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h5,h2,h3,h1,h0]),tab_negimp(discharge,[h6,h7])],[h5,77,h6,h7]) ).
thf(79,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h2,h3,h1,h0]),tab_negimp(discharge,[h4,h5])],[h3,78,h4,h5]) ).
thf(80,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,79,h2,h3]) ).
thf(81,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[80,h0]) ).
thf(0,theorem,
( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ( X3 != X1 )
=> ( X3 = X2 ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) )
=> ( ! [X1: $i > $o,X2: $i] :
( ( in @ X2 @ emptyset )
=> ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( in @ emptyset @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $o] :
( X1
=> ( set2prop @ ( prop2set @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) )
=> ( ! [X1: $i,X2: $i > $o] :
( ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( X2 @ X3 ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ( X1 @ X2 )
=> ( ! [X2: $i,X3: $i] :
( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ! [X1: $i > $o] :
( ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[80,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : SEU546^1 : TPTP v8.1.0. Released v3.7.0.
% 0.14/0.14 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.14/0.35 % Computer : n006.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 600
% 0.14/0.35 % DateTime : Sun Jun 19 19:43:40 EDT 2022
% 0.14/0.35 % CPUTime :
% 187.84/187.82 % SZS status Theorem
% 187.84/187.82 % Mode: mode86:USE_SINE=true:SINE_TOLERANCE=2.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=3.:SINE_DEPTH=0
% 187.84/187.82 % Inferences: 573
% 187.84/187.82 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------