TPTP Problem File: SEU523^1.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : SEU523^1 : TPTP v7.0.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Preliminary Notions - Power Sets and Unions
% Version  : Especial.
% English  : (! A:i.! x:i.in x (setunion A) -> (! phi:o.(! B:i.in x B ->
%            in B A -> phi) -> phi))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC025g [Bro08]

% Status   : Theorem
% Rating   : 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.60 v5.2.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% Syntax   : Number of formulae    :   93 (   0 unit;  50 type;  42 defn)
%            Number of atoms       :  539 (  54 equality; 255 variable)
%            Maximal formula depth :   50 (   6 average)
%            Number of connectives :  395 (   7   ~;   4   |;  18   &; 259   @)
%                                         (   7 <=>; 100  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   26 (  26   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   54 (  50   :;   0   =)
%            Number of variables   :  130 (   2 sgn;  99   !;  16   ?;  15   ^)
%                                         ( 130   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=88
%          : 
%------------------------------------------------------------------------------
thf(in_type,type,(
    in: $i > $i > $o )).

thf(exu_type,type,(
    exu: ( $i > $o ) > $o )).

thf(exu,definition,
    ( exu
    = ( ^ [Xphi: $i > $o] :
        ? [Xx: $i] :
          ( ( Xphi @ Xx )
          & ! [Xy: $i] :
              ( ( Xphi @ Xy )
             => ( Xx = Xy ) ) ) ) )).

thf(setextAx_type,type,(
    setextAx: $o )).

thf(setextAx,definition,
    ( setextAx
    = ( ! [A: $i,B: $i] :
          ( ! [Xx: $i] :
              ( ( in @ Xx @ A )
            <=> ( in @ Xx @ B ) )
         => ( A = B ) ) ) )).

thf(emptyset_type,type,(
    emptyset: $i )).

thf(emptysetAx_type,type,(
    emptysetAx: $o )).

thf(emptysetAx,definition,
    ( emptysetAx
    = ( ! [Xx: $i] :
          ~ ( in @ Xx @ emptyset ) ) )).

thf(setadjoin_type,type,(
    setadjoin: $i > $i > $i )).

thf(setadjoinAx_type,type,(
    setadjoinAx: $o )).

thf(setadjoinAx,definition,
    ( setadjoinAx
    = ( ! [Xx: $i,A: $i,Xy: $i] :
          ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
        <=> ( ( Xy = Xx )
            | ( in @ Xy @ A ) ) ) ) )).

thf(powerset_type,type,(
    powerset: $i > $i )).

thf(powersetAx_type,type,(
    powersetAx: $o )).

thf(powersetAx,definition,
    ( powersetAx
    = ( ! [A: $i,B: $i] :
          ( ( in @ B @ ( powerset @ A ) )
        <=> ! [Xx: $i] :
              ( ( in @ Xx @ B )
             => ( in @ Xx @ A ) ) ) ) )).

thf(setunion_type,type,(
    setunion: $i > $i )).

thf(setunionAx_type,type,(
    setunionAx: $o )).

thf(setunionAx,definition,
    ( setunionAx
    = ( ! [A: $i,Xx: $i] :
          ( ( in @ Xx @ ( setunion @ A ) )
        <=> ? [B: $i] :
              ( ( in @ Xx @ B )
              & ( in @ B @ A ) ) ) ) )).

thf(omega_type,type,(
    omega: $i )).

thf(omega0Ax_type,type,(
    omega0Ax: $o )).

thf(omega0Ax,definition,
    ( omega0Ax
    = ( in @ emptyset @ omega ) )).

thf(omegaSAx_type,type,(
    omegaSAx: $o )).

thf(omegaSAx,definition,
    ( omegaSAx
    = ( ! [Xx: $i] :
          ( ( in @ Xx @ omega )
         => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) )).

thf(omegaIndAx_type,type,(
    omegaIndAx: $o )).

thf(omegaIndAx,definition,
    ( omegaIndAx
    = ( ! [A: $i] :
          ( ( ( in @ emptyset @ A )
            & ! [Xx: $i] :
                ( ( ( in @ Xx @ omega )
                  & ( in @ Xx @ A ) )
               => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
         => ! [Xx: $i] :
              ( ( in @ Xx @ omega )
             => ( in @ Xx @ A ) ) ) ) )).

thf(replAx_type,type,(
    replAx: $o )).

thf(replAx,definition,
    ( replAx
    = ( ! [Xphi: $i > $i > $o,A: $i] :
          ( ! [Xx: $i] :
              ( ( in @ Xx @ A )
             => ( exu
                @ ^ [Xy: $i] :
                    ( Xphi @ Xx @ Xy ) ) )
         => ? [B: $i] :
            ! [Xx: $i] :
              ( ( in @ Xx @ B )
            <=> ? [Xy: $i] :
                  ( ( in @ Xy @ A )
                  & ( Xphi @ Xy @ Xx ) ) ) ) ) )).

thf(foundationAx_type,type,(
    foundationAx: $o )).

thf(foundationAx,definition,
    ( foundationAx
    = ( ! [A: $i] :
          ( ? [Xx: $i] :
              ( in @ Xx @ A )
         => ? [B: $i] :
              ( ( in @ B @ A )
              & ~ ( ? [Xx: $i] :
                      ( ( in @ Xx @ B )
                      & ( in @ Xx @ A ) ) ) ) ) ) )).

thf(wellorderingAx_type,type,(
    wellorderingAx: $o )).

thf(wellorderingAx,definition,
    ( wellorderingAx
    = ( ! [A: $i] :
        ? [B: $i] :
          ( ! [C: $i] :
              ( ( in @ C @ B )
             => ! [Xx: $i] :
                  ( ( in @ Xx @ C )
                 => ( in @ Xx @ A ) ) )
          & ! [Xx: $i,Xy: $i] :
              ( ( ( in @ Xx @ A )
                & ( in @ Xy @ A ) )
             => ( ! [C: $i] :
                    ( ( in @ C @ B )
                   => ( ( in @ Xx @ C )
                    <=> ( in @ Xy @ C ) ) )
               => ( Xx = Xy ) ) )
          & ! [C: $i,D: $i] :
              ( ( ( in @ C @ B )
                & ( in @ D @ B ) )
             => ( ! [Xx: $i] :
                    ( ( in @ Xx @ C )
                   => ( in @ Xx @ D ) )
                | ! [Xx: $i] :
                    ( ( in @ Xx @ D )
                   => ( in @ Xx @ C ) ) ) )
          & ! [C: $i] :
              ( ( ! [Xx: $i] :
                    ( ( in @ Xx @ C )
                   => ( in @ Xx @ A ) )
                & ? [Xx: $i] :
                    ( in @ Xx @ C ) )
             => ? [D: $i,Xx: $i] :
                  ( ( in @ D @ B )
                  & ( in @ Xx @ C )
                  & ~ ( ? [Xy: $i] :
                          ( ( in @ Xy @ D )
                          & ( in @ Xy @ C ) ) )
                  & ! [E: $i] :
                      ( ( in @ E @ B )
                     => ( ! [Xy: $i] :
                            ( ( in @ Xy @ E )
                           => ( in @ Xy @ D ) )
                        | ( in @ Xx @ E ) ) ) ) ) ) ) )).

thf(descr_type,type,(
    descr: ( $i > $o ) > $i )).

thf(descrp_type,type,(
    descrp: $o )).

thf(descrp,definition,
    ( descrp
    = ( ! [Xphi: $i > $o] :
          ( ( exu
            @ ^ [Xx: $i] :
                ( Xphi @ Xx ) )
         => ( Xphi
            @ ( descr
              @ ^ [Xx: $i] :
                  ( Xphi @ Xx ) ) ) ) ) )).

thf(dsetconstr_type,type,(
    dsetconstr: $i > ( $i > $o ) > $i )).

thf(dsetconstrI_type,type,(
    dsetconstrI: $o )).

thf(dsetconstrI,definition,
    ( dsetconstrI
    = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
          ( ( in @ Xx @ A )
         => ( ( Xphi @ Xx )
           => ( in @ Xx
              @ ( dsetconstr @ A
                @ ^ [Xy: $i] :
                    ( Xphi @ Xy ) ) ) ) ) ) )).

thf(dsetconstrEL_type,type,(
    dsetconstrEL: $o )).

thf(dsetconstrEL,definition,
    ( dsetconstrEL
    = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
          ( ( in @ Xx
            @ ( dsetconstr @ A
              @ ^ [Xy: $i] :
                  ( Xphi @ Xy ) ) )
         => ( in @ Xx @ A ) ) ) )).

thf(dsetconstrER_type,type,(
    dsetconstrER: $o )).

thf(dsetconstrER,definition,
    ( dsetconstrER
    = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
          ( ( in @ Xx
            @ ( dsetconstr @ A
              @ ^ [Xy: $i] :
                  ( Xphi @ Xy ) ) )
         => ( Xphi @ Xx ) ) ) )).

thf(exuE1_type,type,(
    exuE1: $o )).

thf(exuE1,definition,
    ( exuE1
    = ( ! [Xphi: $i > $o] :
          ( ( exu
            @ ^ [Xx: $i] :
                ( Xphi @ Xx ) )
         => ? [Xx: $i] :
              ( ( Xphi @ Xx )
              & ! [Xy: $i] :
                  ( ( Xphi @ Xy )
                 => ( Xx = Xy ) ) ) ) ) )).

thf(prop2set_type,type,(
    prop2set: $o > $i )).

thf(prop2set,definition,
    ( prop2set
    = ( ^ [Xphi: $o] :
          ( dsetconstr @ ( powerset @ emptyset )
          @ ^ [Xx: $i] : Xphi ) ) )).

thf(prop2setE_type,type,(
    prop2setE: $o )).

thf(prop2setE,definition,
    ( prop2setE
    = ( ! [Xphi: $o,Xx: $i] :
          ( ( in @ Xx @ ( prop2set @ Xphi ) )
         => Xphi ) ) )).

thf(emptysetE_type,type,(
    emptysetE: $o )).

thf(emptysetE,definition,
    ( emptysetE
    = ( ! [Xx: $i] :
          ( ( in @ Xx @ emptyset )
         => ! [Xphi: $o] : Xphi ) ) )).

thf(emptysetimpfalse_type,type,(
    emptysetimpfalse: $o )).

thf(emptysetimpfalse,definition,
    ( emptysetimpfalse
    = ( ! [Xx: $i] :
          ( ( in @ Xx @ emptyset )
         => $false ) ) )).

thf(notinemptyset_type,type,(
    notinemptyset: $o )).

thf(notinemptyset,definition,
    ( notinemptyset
    = ( ! [Xx: $i] :
          ~ ( in @ Xx @ emptyset ) ) )).

thf(exuE3e_type,type,(
    exuE3e: $o )).

thf(exuE3e,definition,
    ( exuE3e
    = ( ! [Xphi: $i > $o] :
          ( ( exu
            @ ^ [Xx: $i] :
                ( Xphi @ Xx ) )
         => ? [Xx: $i] :
              ( Xphi @ Xx ) ) ) )).

thf(setext_type,type,(
    setext: $o )).

thf(setext,definition,
    ( setext
    = ( ! [A: $i,B: $i] :
          ( ! [Xx: $i] :
              ( ( in @ Xx @ A )
             => ( in @ Xx @ B ) )
         => ( ! [Xx: $i] :
                ( ( in @ Xx @ B )
               => ( in @ Xx @ A ) )
           => ( A = B ) ) ) ) )).

thf(emptyI_type,type,(
    emptyI: $o )).

thf(emptyI,definition,
    ( emptyI
    = ( ! [A: $i] :
          ( ! [Xx: $i] :
              ~ ( in @ Xx @ A )
         => ( A = emptyset ) ) ) )).

thf(noeltsimpempty_type,type,(
    noeltsimpempty: $o )).

thf(noeltsimpempty,definition,
    ( noeltsimpempty
    = ( ! [A: $i] :
          ( ! [Xx: $i] :
              ~ ( in @ Xx @ A )
         => ( A = emptyset ) ) ) )).

thf(setbeta_type,type,(
    setbeta: $o )).

thf(setbeta,definition,
    ( setbeta
    = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
          ( ( in @ Xx @ A )
         => ( ( in @ Xx
              @ ( dsetconstr @ A
                @ ^ [Xy: $i] :
                    ( Xphi @ Xy ) ) )
          <=> ( Xphi @ Xx ) ) ) ) )).

thf(nonempty_type,type,(
    nonempty: $i > $o )).

thf(nonempty,definition,
    ( nonempty
    = ( ^ [Xx: $i] : ( Xx != emptyset ) ) )).

thf(nonemptyE1_type,type,(
    nonemptyE1: $o )).

thf(nonemptyE1,definition,
    ( nonemptyE1
    = ( ! [A: $i] :
          ( ( nonempty @ A )
         => ? [Xx: $i] :
              ( in @ Xx @ A ) ) ) )).

thf(nonemptyI_type,type,(
    nonemptyI: $o )).

thf(nonemptyI,definition,
    ( nonemptyI
    = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
          ( ( in @ Xx @ A )
         => ( ( Xphi @ Xx )
           => ( nonempty
              @ ( dsetconstr @ A
                @ ^ [Xy: $i] :
                    ( Xphi @ Xy ) ) ) ) ) ) )).

thf(nonemptyI1_type,type,(
    nonemptyI1: $o )).

thf(nonemptyI1,definition,
    ( nonemptyI1
    = ( ! [A: $i] :
          ( ? [Xx: $i] :
              ( in @ Xx @ A )
         => ( nonempty @ A ) ) ) )).

thf(setadjoinIL_type,type,(
    setadjoinIL: $o )).

thf(setadjoinIL,definition,
    ( setadjoinIL
    = ( ! [Xx: $i,Xy: $i] :
          ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) )).

thf(emptyinunitempty_type,type,(
    emptyinunitempty: $o )).

thf(emptyinunitempty,definition,
    ( emptyinunitempty
    = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) )).

thf(setadjoinIR_type,type,(
    setadjoinIR: $o )).

thf(setadjoinIR,definition,
    ( setadjoinIR
    = ( ! [Xx: $i,A: $i,Xy: $i] :
          ( ( in @ Xy @ A )
         => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) )).

thf(setadjoinE_type,type,(
    setadjoinE: $o )).

thf(setadjoinE,definition,
    ( setadjoinE
    = ( ! [Xx: $i,A: $i,Xy: $i] :
          ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
         => ! [Xphi: $o] :
              ( ( ( Xy = Xx )
               => Xphi )
             => ( ( ( in @ Xy @ A )
                 => Xphi )
               => Xphi ) ) ) ) )).

thf(setadjoinOr_type,type,(
    setadjoinOr: $o )).

thf(setadjoinOr,definition,
    ( setadjoinOr
    = ( ! [Xx: $i,A: $i,Xy: $i] :
          ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
         => ( ( Xy = Xx )
            | ( in @ Xy @ A ) ) ) ) )).

thf(setoftrueEq_type,type,(
    setoftrueEq: $o )).

thf(setoftrueEq,definition,
    ( setoftrueEq
    = ( ! [A: $i] :
          ( ( dsetconstr @ A
            @ ^ [Xx: $i] : $true )
          = A ) ) )).

thf(powersetI_type,type,(
    powersetI: $o )).

thf(powersetI,definition,
    ( powersetI
    = ( ! [A: $i,B: $i] :
          ( ! [Xx: $i] :
              ( ( in @ Xx @ B )
             => ( in @ Xx @ A ) )
         => ( in @ B @ ( powerset @ A ) ) ) ) )).

thf(emptyinPowerset_type,type,(
    emptyinPowerset: $o )).

thf(emptyinPowerset,definition,
    ( emptyinPowerset
    = ( ! [A: $i] :
          ( in @ emptyset @ ( powerset @ A ) ) ) )).

thf(emptyInPowerset_type,type,(
    emptyInPowerset: $o )).

thf(emptyInPowerset,definition,
    ( emptyInPowerset
    = ( ! [A: $i] :
          ( in @ emptyset @ ( powerset @ A ) ) ) )).

thf(powersetE_type,type,(
    powersetE: $o )).

thf(powersetE,definition,
    ( powersetE
    = ( ! [A: $i,B: $i,Xx: $i] :
          ( ( in @ B @ ( powerset @ A ) )
         => ( ( in @ Xx @ B )
           => ( in @ Xx @ A ) ) ) ) )).

thf(setunionI_type,type,(
    setunionI: $o )).

thf(setunionI,definition,
    ( setunionI
    = ( ! [A: $i,Xx: $i,B: $i] :
          ( ( in @ Xx @ B )
         => ( ( in @ B @ A )
           => ( in @ Xx @ ( setunion @ A ) ) ) ) ) )).

thf(setunionE,conjecture,
    ( setextAx
   => ( emptysetAx
     => ( setadjoinAx
       => ( powersetAx
         => ( setunionAx
           => ( omega0Ax
             => ( omegaSAx
               => ( omegaIndAx
                 => ( replAx
                   => ( foundationAx
                     => ( wellorderingAx
                       => ( descrp
                         => ( dsetconstrI
                           => ( dsetconstrEL
                             => ( dsetconstrER
                               => ( exuE1
                                 => ( prop2setE
                                   => ( emptysetE
                                     => ( emptysetimpfalse
                                       => ( notinemptyset
                                         => ( exuE3e
                                           => ( setext
                                             => ( emptyI
                                               => ( noeltsimpempty
                                                 => ( setbeta
                                                   => ( nonemptyE1
                                                     => ( nonemptyI
                                                       => ( nonemptyI1
                                                         => ( setadjoinIL
                                                           => ( emptyinunitempty
                                                             => ( setadjoinIR
                                                               => ( setadjoinE
                                                                 => ( setadjoinOr
                                                                   => ( setoftrueEq
                                                                     => ( powersetI
                                                                       => ( emptyinPowerset
                                                                         => ( emptyInPowerset
                                                                           => ( powersetE
                                                                             => ( setunionI
                                                                               => ! [A: $i,Xx: $i] :
                                                                                    ( ( in @ Xx @ ( setunion @ A ) )
                                                                                   => ! [Xphi: $o] :
                                                                                        ( ! [B: $i] :
                                                                                            ( ( in @ Xx @ B )
                                                                                           => ( ( in @ B @ A )
                                                                                             => Xphi ) )
                                                                                       => Xphi ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

%------------------------------------------------------------------------------