TPTP Problem File: SEU298+1.p

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%------------------------------------------------------------------------------
% File     : SEU298+1 : TPTP v7.4.0. Released v3.3.0.
% Domain   : Set theory
% Problem  : MPTP bushy problem s1_xboole_0__e4_27_3_1__finset_1
% Version  : [Urb07] axioms : Especial.
% English  :

% Refs     : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
%          : [Urb07] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb07]
% Names    : bushy-s1_xboole_0__e4_27_3_1__finset_1 [Urb07]

% Status   : Theorem
% Rating   : 0.44 v7.4.0, 0.30 v7.3.0, 0.34 v7.2.0, 0.31 v7.1.0, 0.43 v7.0.0, 0.37 v6.4.0, 0.38 v6.3.0, 0.46 v6.2.0, 0.48 v6.1.0, 0.57 v6.0.0, 0.52 v5.5.0, 0.56 v5.4.0, 0.57 v5.3.0, 0.59 v5.2.0, 0.40 v5.1.0, 0.43 v5.0.0, 0.54 v4.1.0, 0.52 v4.0.1, 0.57 v4.0.0, 0.58 v3.7.0, 0.60 v3.5.0, 0.68 v3.4.0, 0.63 v3.3.0
% Syntax   : Number of formulae    :   42 (  10 unit)
%            Number of atoms       :  139 (   8 equality)
%            Maximal formula depth :   14 (   5 average)
%            Number of connectives :  113 (  16 ~  ;   0  |;  72  &)
%                                         (   2 <=>;  23 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   1 propositional; 0-2 arity)
%            Number of functors    :    4 (   0 constant; 1-2 arity)
%            Number of variables   :   60 (   0 singleton;  37 !;  23 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%------------------------------------------------------------------------------
fof(s1_xboole_0__e4_27_3_1__finset_1,conjecture,(
    ! [A,B] :
      ( ( ordinal(A)
        & element(B,powerset(powerset(succ(A)))) )
     => ? [C] :
        ! [D] :
          ( in(D,C)
        <=> ( in(D,powerset(A))
            & ? [E] :
                ( in(E,B)
                & D = set_difference(E,singleton(A)) ) ) ) ) )).

fof(rc2_finset_1,axiom,(
    ! [A] :
    ? [B] :
      ( element(B,powerset(A))
      & empty(B)
      & relation(B)
      & function(B)
      & one_to_one(B)
      & epsilon_transitive(B)
      & epsilon_connected(B)
      & ordinal(B)
      & natural(B)
      & finite(B) ) )).

fof(rc3_funct_1,axiom,(
    ? [A] :
      ( relation(A)
      & function(A)
      & one_to_one(A) ) )).

fof(rc2_ordinal1,axiom,(
    ? [A] :
      ( relation(A)
      & function(A)
      & one_to_one(A)
      & empty(A)
      & epsilon_transitive(A)
      & epsilon_connected(A)
      & ordinal(A) ) )).

fof(rc1_funct_1,axiom,(
    ? [A] :
      ( relation(A)
      & function(A) ) )).

fof(rc2_funct_1,axiom,(
    ? [A] :
      ( relation(A)
      & empty(A)
      & function(A) ) )).

fof(cc2_funct_1,axiom,(
    ! [A] :
      ( ( relation(A)
        & empty(A)
        & function(A) )
     => ( relation(A)
        & function(A)
        & one_to_one(A) ) ) )).

fof(rc1_relat_1,axiom,(
    ? [A] :
      ( empty(A)
      & relation(A) ) )).

fof(fc3_relat_1,axiom,(
    ! [A,B] :
      ( ( relation(A)
        & relation(B) )
     => relation(set_difference(A,B)) ) )).

fof(rc2_relat_1,axiom,(
    ? [A] :
      ( ~ empty(A)
      & relation(A) ) )).

fof(rc1_arytm_3,axiom,(
    ? [A] :
      ( ~ empty(A)
      & epsilon_transitive(A)
      & epsilon_connected(A)
      & ordinal(A)
      & natural(A) ) )).

fof(fc2_arytm_3,axiom,(
    ! [A] :
      ( ( ordinal(A)
        & natural(A) )
     => ( ~ empty(succ(A))
        & epsilon_transitive(succ(A))
        & epsilon_connected(succ(A))
        & ordinal(succ(A))
        & natural(succ(A)) ) ) )).

fof(rc1_finset_1,axiom,(
    ? [A] :
      ( ~ empty(A)
      & finite(A) ) )).

fof(cc1_finset_1,axiom,(
    ! [A] :
      ( empty(A)
     => finite(A) ) )).

fof(rc3_finset_1,axiom,(
    ! [A] :
      ( ~ empty(A)
     => ? [B] :
          ( element(B,powerset(A))
          & ~ empty(B)
          & finite(B) ) ) )).

fof(cc2_finset_1,axiom,(
    ! [A] :
      ( finite(A)
     => ! [B] :
          ( element(B,powerset(A))
         => finite(B) ) ) )).

fof(fc12_finset_1,axiom,(
    ! [A,B] :
      ( finite(A)
     => finite(set_difference(A,B)) ) )).

fof(cc1_funct_1,axiom,(
    ! [A] :
      ( empty(A)
     => function(A) ) )).

fof(cc1_relat_1,axiom,(
    ! [A] :
      ( empty(A)
     => relation(A) ) )).

fof(cc2_arytm_3,axiom,(
    ! [A] :
      ( ( empty(A)
        & ordinal(A) )
     => ( epsilon_transitive(A)
        & epsilon_connected(A)
        & ordinal(A)
        & natural(A) ) ) )).

fof(cc2_ordinal1,axiom,(
    ! [A] :
      ( ( epsilon_transitive(A)
        & epsilon_connected(A) )
     => ordinal(A) ) )).

fof(rc1_ordinal1,axiom,(
    ? [A] :
      ( epsilon_transitive(A)
      & epsilon_connected(A)
      & ordinal(A) ) )).

fof(cc3_ordinal1,axiom,(
    ! [A] :
      ( empty(A)
     => ( epsilon_transitive(A)
        & epsilon_connected(A)
        & ordinal(A) ) ) )).

fof(rc3_ordinal1,axiom,(
    ? [A] :
      ( ~ empty(A)
      & epsilon_transitive(A)
      & epsilon_connected(A)
      & ordinal(A) ) )).

fof(rc1_subset_1,axiom,(
    ! [A] :
      ( ~ empty(A)
     => ? [B] :
          ( element(B,powerset(A))
          & ~ empty(B) ) ) )).

fof(rc2_subset_1,axiom,(
    ! [A] :
    ? [B] :
      ( element(B,powerset(A))
      & empty(B) ) )).

fof(rc1_xboole_0,axiom,(
    ? [A] : empty(A) )).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ empty(A) )).

fof(antisymmetry_r2_hidden,axiom,(
    ! [A,B] :
      ( in(A,B)
     => ~ in(B,A) ) )).

fof(dt_k1_ordinal1,axiom,(
    $true )).

fof(dt_k1_tarski,axiom,(
    $true )).

fof(dt_k1_zfmisc_1,axiom,(
    $true )).

fof(dt_k4_xboole_0,axiom,(
    $true )).

fof(dt_m1_subset_1,axiom,(
    $true )).

fof(fc1_finset_1,axiom,(
    ! [A] :
      ( ~ empty(singleton(A))
      & finite(singleton(A)) ) )).

fof(cc1_arytm_3,axiom,(
    ! [A] :
      ( ordinal(A)
     => ! [B] :
          ( element(B,A)
         => ( epsilon_transitive(B)
            & epsilon_connected(B)
            & ordinal(B) ) ) ) )).

fof(fc1_ordinal1,axiom,(
    ! [A] : ~ empty(succ(A)) )).

fof(cc1_ordinal1,axiom,(
    ! [A] :
      ( ordinal(A)
     => ( epsilon_transitive(A)
        & epsilon_connected(A) ) ) )).

fof(fc3_ordinal1,axiom,(
    ! [A] :
      ( ordinal(A)
     => ( ~ empty(succ(A))
        & epsilon_transitive(succ(A))
        & epsilon_connected(succ(A))
        & ordinal(succ(A)) ) ) )).

fof(fc1_subset_1,axiom,(
    ! [A] : ~ empty(powerset(A)) )).

fof(fc2_subset_1,axiom,(
    ! [A] : ~ empty(singleton(A)) )).

fof(s1_tarski__e4_27_3_1__finset_1__1,axiom,(
    ! [A,B] :
      ( ( ordinal(A)
        & element(B,powerset(powerset(succ(A)))) )
     => ( ! [C,D,E] :
            ( ( C = D
              & ? [F] :
                  ( in(F,B)
                  & D = set_difference(F,singleton(A)) )
              & C = E
              & ? [G] :
                  ( in(G,B)
                  & E = set_difference(G,singleton(A)) ) )
           => D = E )
       => ? [C] :
          ! [D] :
            ( in(D,C)
          <=> ? [E] :
                ( in(E,powerset(A))
                & E = D
                & ? [H] :
                    ( in(H,B)
                    & D = set_difference(H,singleton(A)) ) ) ) ) ) )).

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