%------------------------------------------------------------------------------
% File     : SET981+1 : TPTP v8.2.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : Basic properties of sets, theorem 135
% Version  : [Urb06] axioms : Especial.
% English  :

% Refs     : [Byl90] Bylinski (1990), Some Basic Properties of Sets
%          : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb06]
% Names    : zfmisc_1__t135_zfmisc_1 [Urb06]

% Status   : Theorem
% Rating   : 1.00 v5.4.0, 0.96 v5.3.0, 1.00 v3.2.0
% Syntax   : Number of formulae    :   25 (   9 unt;   0 def)
%            Number of atoms       :   58 (  19 equ)
%            Maximal formula atoms :    5 (   2 avg)
%            Number of connectives :   46 (  13   ~;   3   |;  10   &)
%                                         (  13 <=>;   7  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :    7 (   7 usr;   1 con; 0-2 aty)
%            Number of variables   :   66 (  58   !;   8   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%          : Infinox says this has no finite (counter-) models.
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
    ! [A,B] :
      ( in(A,B)
     => ~ in(B,A) ) ).

fof(commutativity_k2_tarski,axiom,
    ! [A,B] : unordered_pair(A,B) = unordered_pair(B,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(d1_xboole_0,axiom,
    ! [A] :
      ( A = empty_set
    <=> ! [B] : ~ in(B,A) ) ).

fof(d2_tarski,axiom,
    ! [A,B,C] :
      ( C = unordered_pair(A,B)
    <=> ! [D] :
          ( in(D,C)
        <=> ( D = A
            | D = B ) ) ) ).

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

fof(d2_zfmisc_1,axiom,
    ! [A,B,C] :
      ( C = cartesian_product2(A,B)
    <=> ! [D] :
          ( in(D,C)
        <=> ? [E,F] :
              ( in(E,A)
              & in(F,B)
              & D = ordered_pair(E,F) ) ) ) ).

fof(d3_tarski,axiom,
    ! [A,B] :
      ( subset(A,B)
    <=> ! [C] :
          ( in(C,A)
         => in(C,B) ) ) ).

fof(d4_tarski,axiom,
    ! [A,B] :
      ( B = union(A)
    <=> ! [C] :
          ( in(C,B)
        <=> ? [D] :
              ( in(C,D)
              & in(D,A) ) ) ) ).

fof(d5_tarski,axiom,
    ! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).

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

fof(fc1_zfmisc_1,axiom,
    ! [A,B] : ~ empty(ordered_pair(A,B)) ).

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

fof(l50_zfmisc_1,axiom,
    ! [A,B] :
      ( in(A,B)
     => subset(A,union(B)) ) ).

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

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

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

fof(s1_xboole_0__e2_121_2__zfmisc_1,axiom,
    ! [A] :
    ? [B] :
    ! [C] :
      ( in(C,B)
    <=> ( in(C,union(A))
        & ? [D] :
            ( C = D
            & ? [E] :
                ( in(E,D)
                & in(E,A) ) ) ) ) ).

fof(t102_zfmisc_1,axiom,
    ! [A,B,C] :
      ~ ( in(A,cartesian_product2(B,C))
        & ! [D,E] : ordered_pair(D,E) != A ) ).

fof(t135_zfmisc_1,conjecture,
    ! [A,B] :
      ( ( subset(A,cartesian_product2(A,B))
        | subset(A,cartesian_product2(B,A)) )
     => A = empty_set ) ).

fof(t15_xboole_1,axiom,
    ! [A,B] :
      ( set_union2(A,B) = empty_set
     => A = empty_set ) ).

fof(t7_tarski,axiom,
    ! [A,B] :
      ~ ( in(A,B)
        & ! [C] :
            ~ ( in(C,B)
              & ! [D] :
                  ~ ( in(D,B)
                    & in(D,C) ) ) ) ).

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