%------------------------------------------------------------------------------
% File     : SET959+1 : TPTP v8.2.0. Bugfixed v4.0.0.
% Domain   : Set theory
% Problem  : Basic properties of sets, theorem 112
% 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__t112_zfmisc_1 [Urb06]

% Status   : Theorem
% Rating   : 0.33 v8.2.0, 0.31 v8.1.0, 0.33 v7.5.0, 0.38 v7.4.0, 0.20 v7.3.0, 0.24 v7.2.0, 0.21 v7.1.0, 0.22 v7.0.0, 0.20 v6.4.0, 0.23 v6.3.0, 0.25 v6.2.0, 0.36 v6.1.0, 0.33 v6.0.0, 0.26 v5.4.0, 0.32 v5.3.0, 0.41 v5.2.0, 0.25 v5.1.0, 0.29 v4.1.0, 0.30 v4.0.1, 0.35 v4.0.0
% Syntax   : Number of formulae    :    8 (   5 unt;   0 def)
%            Number of atoms       :   17 (   6 equ)
%            Maximal formula atoms :    7 (   2 avg)
%            Number of connectives :   16 (   7   ~;   0   |;   4   &)
%                                         (   2 <=>;   3  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 1-2 aty)
%            Number of functors    :    3 (   3 usr;   0 con; 1-2 aty)
%            Number of variables   :   23 (  21   !;   2   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
% Bugfixes : v4.0.0 - Removed duplicate formula t2_tarski
%------------------------------------------------------------------------------
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(d5_tarski,axiom,
    ! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).

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

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

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

fof(t112_zfmisc_1,conjecture,
    ! [A,B] :
      ( ( ! [C] :
            ~ ( in(C,A)
              & ! [D,E] : C != ordered_pair(D,E) )
        & ! [C] :
            ~ ( in(C,B)
              & ! [D,E] : C != ordered_pair(D,E) )
        & ! [C,D] :
            ( in(ordered_pair(C,D),A)
          <=> in(ordered_pair(C,D),B) ) )
     => A = B ) ).

fof(t2_tarski,axiom,
    ! [A,B] :
      ( ! [C] :
          ( in(C,A)
        <=> in(C,B) )
     => A = B ) ).

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