TPTP Problem File: SEU099+1.p

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%------------------------------------------------------------------------------
% File     : SEU099+1 : TPTP v9.0.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : Finite sets, theorem 30
% Version  : [Urb06] axioms : Especial.
% English  :

% Refs     : [Dar90] Darmochwal (1990), Finite Sets
%          : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb06]
% Names    : finset_1__t30_finset_1 [Urb06]

% Status   : Theorem
% Rating   : 1.00 v7.4.0, 0.97 v7.1.0, 0.96 v7.0.0, 1.00 v3.7.0, 0.95 v3.5.0, 1.00 v3.2.0
% Syntax   : Number of formulae    :   72 (  12 unt;   0 def)
%            Number of atoms       :  234 (  13 equ)
%            Maximal formula atoms :   19 (   3 avg)
%            Number of connectives :  202 (  40   ~;   3   |; 116   &)
%                                         (   7 <=>;  36  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :   21 (  20 usr;   0 prp; 1-2 aty)
%            Number of functors    :    5 (   5 usr;   2 con; 0-2 aty)
%            Number of variables   :  116 (  89   !;  27   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
    ! [A,B] :
      ( in(A,B)
     => ~ in(B,A) ) ).

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

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

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

fof(cc1_ordinal1,axiom,
    ! [A] :
      ( ordinal(A)
     => ( epsilon_transitive(A)
        & epsilon_connected(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_finset_1,axiom,
    ! [A] :
      ( finite(A)
     => ! [B] :
          ( element(B,powerset(A))
         => finite(B) ) ) ).

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

fof(cc2_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(cc4_arytm_3,axiom,
    ! [A] :
      ( element(A,positive_rationals)
     => ( ordinal(A)
       => ( epsilon_transitive(A)
          & epsilon_connected(A)
          & ordinal(A)
          & natural(A) ) ) ) ).

fof(commutativity_k2_xboole_0,axiom,
    ! [A,B] : set_union2(A,B) = set_union2(B,A) ).

fof(d1_tarski,axiom,
    ! [A,B] :
      ( B = singleton(A)
    <=> ! [C] :
          ( in(C,B)
        <=> C = A ) ) ).

fof(d2_xboole_0,axiom,
    ! [A,B,C] :
      ( C = set_union2(A,B)
    <=> ! [D] :
          ( in(D,C)
        <=> ( in(D,A)
            | in(D,B) ) ) ) ).

fof(d9_ordinal1,axiom,
    ! [A] :
      ( inclusion_linear(A)
    <=> ! [B,C] :
          ( ( in(B,A)
            & in(C,A) )
         => inclusion_comparable(B,C) ) ) ).

fof(d9_xboole_0,axiom,
    ! [A,B] :
      ( inclusion_comparable(A,B)
    <=> ( subset(A,B)
        | subset(B,A) ) ) ).

fof(existence_m1_subset_1,axiom,
    ! [A] :
    ? [B] : element(B,A) ).

fof(fc12_relat_1,axiom,
    ( empty(empty_set)
    & relation(empty_set)
    & relation_empty_yielding(empty_set) ) ).

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

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

fof(fc1_xboole_0,axiom,
    empty(empty_set) ).

fof(fc2_ordinal1,axiom,
    ( relation(empty_set)
    & relation_empty_yielding(empty_set)
    & function(empty_set)
    & one_to_one(empty_set)
    & empty(empty_set)
    & epsilon_transitive(empty_set)
    & epsilon_connected(empty_set)
    & ordinal(empty_set) ) ).

fof(fc2_relat_1,axiom,
    ! [A,B] :
      ( ( relation(A)
        & relation(B) )
     => relation(set_union2(A,B)) ) ).

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

fof(fc2_xboole_0,axiom,
    ! [A,B] :
      ( ~ empty(A)
     => ~ empty(set_union2(A,B)) ) ).

fof(fc3_xboole_0,axiom,
    ! [A,B] :
      ( ~ empty(A)
     => ~ empty(set_union2(B,A)) ) ).

fof(fc4_relat_1,axiom,
    ( empty(empty_set)
    & relation(empty_set) ) ).

fof(fc8_arytm_3,axiom,
    ~ empty(positive_rationals) ).

fof(fc9_finset_1,axiom,
    ! [A,B] :
      ( ( finite(A)
        & finite(B) )
     => finite(set_union2(A,B)) ) ).

fof(idempotence_k2_xboole_0,axiom,
    ! [A,B] : set_union2(A,A) = A ).

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

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

fof(rc1_funcop_1,axiom,
    ? [A] :
      ( relation(A)
      & function(A)
      & function_yielding(A) ) ).

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

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

fof(rc1_ordinal2,axiom,
    ? [A] :
      ( epsilon_transitive(A)
      & epsilon_connected(A)
      & ordinal(A)
      & being_limit_ordinal(A) ) ).

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

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

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

fof(rc2_arytm_3,axiom,
    ? [A] :
      ( element(A,positive_rationals)
      & ~ empty(A)
      & epsilon_transitive(A)
      & epsilon_connected(A)
      & ordinal(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(rc2_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & empty(A)
      & function(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(rc2_ordinal2,axiom,
    ? [A] :
      ( relation(A)
      & function(A)
      & transfinite_sequence(A)
      & ordinal_yielding(A) ) ).

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

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

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

fof(rc3_arytm_3,axiom,
    ? [A] :
      ( element(A,positive_rationals)
      & empty(A)
      & epsilon_transitive(A)
      & epsilon_connected(A)
      & ordinal(A)
      & natural(A) ) ).

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

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

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

fof(rc3_relat_1,axiom,
    ? [A] :
      ( relation(A)
      & relation_empty_yielding(A) ) ).

fof(rc4_finset_1,axiom,
    ! [A] :
      ( ~ empty(A)
     => ? [B] :
          ( element(B,powerset(A))
          & ~ empty(B)
          & finite(B) ) ) ).

fof(rc4_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & relation_empty_yielding(A)
      & function(A) ) ).

fof(rc4_ordinal1,axiom,
    ? [A] :
      ( relation(A)
      & function(A)
      & transfinite_sequence(A) ) ).

fof(rc5_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & relation_non_empty(A)
      & function(A) ) ).

fof(reflexivity_r1_tarski,axiom,
    ! [A,B] : subset(A,A) ).

fof(reflexivity_r3_xboole_0,axiom,
    ! [A,B] : inclusion_comparable(A,A) ).

fof(s2_finset_1__e6_46__finset_1,axiom,
    ! [A] :
      ( ( finite(A)
        & ~ ( empty_set != empty_set
            & ! [B] :
                ~ ( in(B,empty_set)
                  & ! [C] :
                      ( in(C,empty_set)
                     => subset(B,C) ) ) )
        & ! [D,E] :
            ( ( in(D,A)
              & subset(E,A)
              & ~ ( E != empty_set
                  & ! [F] :
                      ~ ( in(F,E)
                        & ! [G] :
                            ( in(G,E)
                           => subset(F,G) ) ) ) )
           => ~ ( set_union2(E,singleton(D)) != empty_set
                & ! [H] :
                    ~ ( in(H,set_union2(E,singleton(D)))
                      & ! [I] :
                          ( in(I,set_union2(E,singleton(D)))
                         => subset(H,I) ) ) ) ) )
     => ~ ( A != empty_set
          & ! [J] :
              ~ ( in(J,A)
                & ! [K] :
                    ( in(K,A)
                   => subset(J,K) ) ) ) ) ).

fof(symmetry_r3_xboole_0,axiom,
    ! [A,B] :
      ( inclusion_comparable(A,B)
     => inclusion_comparable(B,A) ) ).

fof(t1_boole,axiom,
    ! [A] : set_union2(A,empty_set) = A ).

fof(t1_subset,axiom,
    ! [A,B] :
      ( in(A,B)
     => element(A,B) ) ).

fof(t1_xboole_1,axiom,
    ! [A,B,C] :
      ( ( subset(A,B)
        & subset(B,C) )
     => subset(A,C) ) ).

fof(t2_subset,axiom,
    ! [A,B] :
      ( element(A,B)
     => ( empty(B)
        | in(A,B) ) ) ).

fof(t30_finset_1,conjecture,
    ! [A] :
      ~ ( finite(A)
        & A != empty_set
        & inclusion_linear(A)
        & ! [B] :
            ~ ( in(B,A)
              & ! [C] :
                  ( in(C,A)
                 => subset(B,C) ) ) ) ).

fof(t3_subset,axiom,
    ! [A,B] :
      ( element(A,powerset(B))
    <=> subset(A,B) ) ).

fof(t4_subset,axiom,
    ! [A,B,C] :
      ( ( in(A,B)
        & element(B,powerset(C)) )
     => element(A,C) ) ).

fof(t5_subset,axiom,
    ! [A,B,C] :
      ~ ( in(A,B)
        & element(B,powerset(C))
        & empty(C) ) ).

fof(t6_boole,axiom,
    ! [A] :
      ( empty(A)
     => A = empty_set ) ).

fof(t7_boole,axiom,
    ! [A,B] :
      ~ ( in(A,B)
        & empty(B) ) ).

fof(t8_boole,axiom,
    ! [A,B] :
      ~ ( empty(A)
        & A != B
        & empty(B) ) ).

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