%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : SEU516^1 : TPTP v8.1.2. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n027.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 17:18:07 EDT 2023
% Result : Theorem 0.19s 0.51s
% Output : Proof 0.19s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_setunion,type,
setunion: $i > $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_setadjoin,type,
setadjoin: $i > $i > $i ).
thf(ty_descr,type,
descr: ( $i > $o ) > $i ).
thf(ty_dsetconstr,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_powerset,type,
powerset: $i > $i ).
thf(ty_omega,type,
omega: $i ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(sP1,plain,
( sP1
<=> ( eigen__2 = eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ( in @ eigen__2 @ ( setadjoin @ eigen__0 @ eigen__1 ) )
= ( ~ sP1
=> ( in @ eigen__2 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i] :
( ( in @ X1 @ ( setadjoin @ eigen__0 @ eigen__1 ) )
= ( ( X1 != eigen__0 )
=> ( in @ X1 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( in @ eigen__2 @ ( setadjoin @ eigen__0 @ eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( in @ eigen__2 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( ~ sP1
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setadjoin @ eigen__0 @ X1 ) )
= ( ( X2 != eigen__0 )
=> ( in @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(def_exu,definition,
( exu
= ( ^ [X1: $i > $o] :
? [X2: $i] :
( ( X1 @ X2 )
& ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 = X3 ) ) ) ) ) ).
thf(def_setextAx,definition,
( setextAx
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [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] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( 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] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( in @ X1 @ omega )
@ ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ) ) ) ).
thf(def_omegaIndAx,definition,
( omegaIndAx
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( ( in @ emptyset @ X1 )
& ! [X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( ( in @ X2 @ omega )
& ( in @ X2 @ X1 ) )
@ ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
@ ! [X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ omega )
@ ( in @ X2 @ X1 ) ) ) ) ) ).
thf(def_replAx,definition,
( replAx
= ( ! [X1: $i > $i > $o,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( exu
@ ^ [X4: $i] : ( X1 @ X3 @ X4 ) ) )
@ ? [X3: $i] :
! [X4: $i] :
( ( in @ X4 @ X3 )
<=> ? [X5: $i] :
( ( in @ X5 @ X2 )
& ( X1 @ X5 @ X4 ) ) ) ) ) ) ).
thf(def_foundationAx,definition,
( foundationAx
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ? [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] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X3 )
@ ( in @ X4 @ X1 ) ) )
& ! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( in @ X3 @ X1 )
& ( in @ X4 @ X1 ) )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X2 )
@ ( ( in @ X3 @ X5 )
<=> ( in @ X4 @ X5 ) ) )
@ ( X3 = X4 ) ) )
& ! [X3: $i,X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ( in @ X3 @ X2 )
& ( in @ X4 @ X2 ) )
@ ( ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X3 )
@ ( in @ X5 @ X4 ) )
| ! [X5: $i] :
( ^ [X6: $o,X7: $o] :
( X6
=> X7 )
@ ( in @ X5 @ X4 )
@ ( in @ X5 @ X3 ) ) ) )
& ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( ! [X4: $i] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( 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] :
( ^ [X7: $o,X8: $o] :
( X7
=> X8 )
@ ( in @ X6 @ X2 )
@ ( ! [X7: $i] :
( ^ [X8: $o,X9: $o] :
( X8
=> X9 )
@ ( in @ X7 @ X6 )
@ ( in @ X7 @ X4 ) )
| ( in @ X5 @ X6 ) ) ) ) ) ) ) ) ).
thf(def_descrp,definition,
( descrp
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ( X1
@ ( descr
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ) ).
thf(def_dsetconstrI,definition,
( dsetconstrI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X2 @ X3 )
@ ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) ) ) ) ) ) ).
thf(def_dsetconstrEL,definition,
( dsetconstrEL
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) )
@ ( in @ X3 @ X1 ) ) ) ) ).
thf(def_dsetconstrER,definition,
( dsetconstrER
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) )
@ ( X2 @ X3 ) ) ) ) ).
thf(def_exuE1,definition,
( exuE1
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ? [X2: $i] :
( ( X1 @ X2 )
& ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X1 @ X3 )
@ ( X2 = X3 ) ) ) ) ) ) ).
thf(def_prop2set,definition,
( prop2set
= ( ^ [X1: $o] :
( dsetconstr @ ( powerset @ emptyset )
@ ^ [X2: $i] : X1 ) ) ) ).
thf(def_prop2setE,definition,
( prop2setE
= ( ! [X1: $o,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ( in @ X2 @ ( prop2set @ X1 ) )
@ X1 ) ) ) ).
thf(def_emptysetE,definition,
( emptysetE
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( in @ X1 @ emptyset )
@ ! [X2: $o] : X2 ) ) ) ).
thf(def_emptysetimpfalse,definition,
( emptysetimpfalse
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( in @ X1 @ emptyset )
@ $false ) ) ) ).
thf(def_notinemptyset,definition,
( notinemptyset
= ( ! [X1: $i] : ( (~) @ ( in @ X1 @ emptyset ) ) ) ) ).
thf(def_exuE3e,definition,
( exuE3e
= ( ! [X1: $i > $o] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( exu
@ ^ [X2: $i] : ( X1 @ X2 ) )
@ ? [X2: $i] : ( X1 @ X2 ) ) ) ) ).
thf(def_setext,definition,
( setext
= ( ! [X1: $i,X2: $i] :
( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( in @ X3 @ X2 ) )
@ ( ^ [X3: $o,X4: $o] :
( X3
=> X4 )
@ ! [X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ X1 ) )
@ ( X1 = X2 ) ) ) ) ) ).
thf(def_emptyI,definition,
( emptyI
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: $i] : ( (~) @ ( in @ X2 @ X1 ) )
@ ( X1 = emptyset ) ) ) ) ).
thf(def_noeltsimpempty,definition,
( noeltsimpempty
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ! [X2: $i] : ( (~) @ ( in @ X2 @ X1 ) )
@ ( X1 = emptyset ) ) ) ) ).
thf(def_setbeta,definition,
( setbeta
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ( in @ X3
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) )
<=> ( X2 @ X3 ) ) ) ) ) ).
thf(def_nonempty,definition,
( nonempty
= ( ^ [X1: $i] : ( (~) @ ( X1 = emptyset ) ) ) ) ).
thf(def_nonemptyE1,definition,
( nonemptyE1
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ( nonempty @ X1 )
@ ? [X2: $i] : ( in @ X2 @ X1 ) ) ) ) ).
thf(def_nonemptyI,definition,
( nonemptyI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X1 )
@ ( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( X2 @ X3 )
@ ( nonempty
@ ( dsetconstr @ X1
@ ^ [X4: $i] : ( X2 @ X4 ) ) ) ) ) ) ) ).
thf(def_nonemptyI1,definition,
( nonemptyI1
= ( ! [X1: $i] :
( ^ [X2: $o,X3: $o] :
( X2
=> X3 )
@ ? [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] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ X2 )
@ ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) ).
thf(def_setadjoinE,definition,
( setadjoinE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
@ ! [X4: $o] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( X3 = X1 )
@ X4 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X3 @ X2 )
@ X4 )
@ X4 ) ) ) ) ) ).
thf(setadjoinOr,conjecture,
( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( sP7
=> ( ! [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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(h0,negated_conjecture,
~ ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( sP7
=> ( ! [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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[setadjoinOr]) ).
thf(h1,assumption,
! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) ),
introduced(assumption,[]) ).
thf(h2,assumption,
~ ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( sP7
=> ( ! [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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
! [X1: $i] :
~ ( in @ X1 @ emptyset ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( sP7
=> ( ! [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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h5,assumption,
sP7,
introduced(assumption,[]) ).
thf(h6,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h7,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h8,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h9,assumption,
! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) ),
introduced(assumption,[]) ).
thf(h10,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h11,assumption,
in @ emptyset @ omega,
introduced(assumption,[]) ).
thf(h12,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h13,assumption,
! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ),
introduced(assumption,[]) ).
thf(h14,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
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 ) ) ),
introduced(assumption,[]) ).
thf(h16,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
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 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h18,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
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 ) ) ) ),
introduced(assumption,[]) ).
thf(h20,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
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 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h22,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h23,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h24,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h25,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) ),
introduced(assumption,[]) ).
thf(h26,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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h27,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ),
introduced(assumption,[]) ).
thf(h28,assumption,
~ ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h29,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) ),
introduced(assumption,[]) ).
thf(h30,assumption,
~ ( ! [X1: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h31,assumption,
! [X1: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : X1 ) )
=> X1 ),
introduced(assumption,[]) ).
thf(h32,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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h33,assumption,
! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 ),
introduced(assumption,[]) ).
thf(h34,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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h35,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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h36,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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h37,assumption,
! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h38,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 ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h39,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(h40,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 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h41,assumption,
! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ),
introduced(assumption,[]) ).
thf(h42,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 ) ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h43,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 ) ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h44,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) ),
introduced(assumption,[]) ).
thf(h45,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 ) ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h46,assumption,
! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ),
introduced(assumption,[]) ).
thf(h47,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 ) ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h48,assumption,
! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) ),
introduced(assumption,[]) ).
thf(h49,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 ) ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h50,assumption,
! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) ),
introduced(assumption,[]) ).
thf(h51,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 ) ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h52,assumption,
! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ),
introduced(assumption,[]) ).
thf(h53,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 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h54,assumption,
in @ emptyset @ ( setadjoin @ emptyset @ emptyset ),
introduced(assumption,[]) ).
thf(h55,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 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h56,assumption,
! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h57,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 ) ) ) ),
introduced(assumption,[]) ).
thf(h58,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(h59,assumption,
~ ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ),
introduced(assumption,[]) ).
thf(h60,assumption,
~ ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setadjoin @ eigen__0 @ X1 ) )
=> ( ( X2 != eigen__0 )
=> ( in @ X2 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(h61,assumption,
~ ! [X1: $i] :
( ( in @ X1 @ ( setadjoin @ eigen__0 @ eigen__1 ) )
=> ( ( X1 != eigen__0 )
=> ( in @ X1 @ eigen__1 ) ) ),
introduced(assumption,[]) ).
thf(h62,assumption,
~ ( sP4
=> sP6 ),
introduced(assumption,[]) ).
thf(h63,assumption,
sP4,
introduced(assumption,[]) ).
thf(h64,assumption,
~ sP6,
introduced(assumption,[]) ).
thf(h65,assumption,
~ sP1,
introduced(assumption,[]) ).
thf(h66,assumption,
~ sP5,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP6
| sP1
| sP5 ),
inference(prop_rule,[status(thm)],]) ).
thf(2,plain,
( ~ sP2
| ~ sP4
| sP6 ),
inference(prop_rule,[status(thm)],]) ).
thf(3,plain,
( ~ sP3
| sP2 ),
inference(all_rule,[status(thm)],]) ).
thf(4,plain,
( ~ sP8
| sP3 ),
inference(all_rule,[status(thm)],]) ).
thf(5,plain,
( ~ sP7
| sP8 ),
inference(all_rule,[status(thm)],]) ).
thf(6,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h65,h66,h63,h64,h62,h61,h60,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0])],[1,2,3,4,5,h5,h63,h65,h66]) ).
thf(7,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h63,h64,h62,h61,h60,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h65,h66])],[h64,6,h65,h66]) ).
thf(8,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h62,h61,h60,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h63,h64])],[h62,7,h63,h64]) ).
thf(9,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h61,h60,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negall(discharge,[h62]),tab_negall(eigenvar,eigen__2)],[h61,8,h62]) ).
thf(10,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h60,h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negall(discharge,[h61]),tab_negall(eigenvar,eigen__1)],[h60,9,h61]) ).
thf(11,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h58,h59,h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negall(discharge,[h60]),tab_negall(eigenvar,eigen__0)],[h59,10,h60]) ).
thf(12,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h56,h57,h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h58,h59])],[h57,11,h58,h59]) ).
thf(13,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h54,h55,h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h56,h57])],[h55,12,h56,h57]) ).
thf(14,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h52,h53,h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h54,h55])],[h53,13,h54,h55]) ).
thf(15,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h50,h51,h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h52,h53])],[h51,14,h52,h53]) ).
thf(16,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h48,h49,h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h50,h51])],[h49,15,h50,h51]) ).
thf(17,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h46,h47,h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h48,h49])],[h47,16,h48,h49]) ).
thf(18,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h44,h45,h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h46,h47])],[h45,17,h46,h47]) ).
thf(19,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h41,h43,h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h44,h45])],[h43,18,h44,h45]) ).
thf(20,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h41,h42,h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h41,h43])],[h42,19,h41,h43]) ).
thf(21,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h39,h40,h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h41,h42])],[h40,20,h41,h42]) ).
thf(22,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h37,h38,h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h39,h40])],[h38,21,h39,h40]) ).
thf(23,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h36,h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h37,h38])],[h36,22,h37,h38]) ).
thf(24,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h35,h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h3,h36])],[h35,23,h3,h36]) ).
thf(25,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h33,h34,h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h3,h35])],[h34,24,h3,h35]) ).
thf(26,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h31,h32,h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h33,h34])],[h32,25,h33,h34]) ).
thf(27,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h29,h30,h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h31,h32])],[h30,26,h31,h32]) ).
thf(28,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h27,h28,h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h29,h30])],[h28,27,h29,h30]) ).
thf(29,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h25,h26,h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h27,h28])],[h26,28,h27,h28]) ).
thf(30,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h23,h24,h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h25,h26])],[h24,29,h25,h26]) ).
thf(31,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h21,h22,h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h23,h24])],[h22,30,h23,h24]) ).
thf(32,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h19,h20,h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h21,h22])],[h20,31,h21,h22]) ).
thf(33,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h17,h18,h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h19,h20])],[h18,32,h19,h20]) ).
thf(34,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h15,h16,h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h17,h18])],[h16,33,h17,h18]) ).
thf(35,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h13,h14,h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h15,h16])],[h14,34,h15,h16]) ).
thf(36,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h11,h12,h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h13,h14])],[h12,35,h13,h14]) ).
thf(37,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h9,h10,h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h11,h12])],[h10,36,h11,h12]) ).
thf(38,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h7,h8,h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h9,h10])],[h8,37,h9,h10]) ).
thf(39,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h6,h3,h4,h1,h2,h0]),tab_negimp(discharge,[h7,h8])],[h6,38,h7,h8]) ).
thf(40,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h3,h4,h1,h2,h0]),tab_negimp(discharge,[h5,h6])],[h4,39,h5,h6]) ).
thf(41,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h1,h2,h0]),tab_negimp(discharge,[h3,h4])],[h2,40,h3,h4]) ).
thf(42,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h0]),tab_negimp(discharge,[h1,h2])],[h0,41,h1,h2]) ).
thf(0,theorem,
( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( sP7
=> ( ! [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: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h0])],[42,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU516^1 : TPTP v8.1.2. Released v3.7.0.
% 0.00/0.13 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.12/0.34 % Computer : n027.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Wed Aug 23 14:12:17 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.51 % SZS status Theorem
% 0.19/0.51 % Mode: cade22grackle2xfee4
% 0.19/0.51 % Steps: 2299
% 0.19/0.51 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------