%------------------------------------------------------------------------------
% File     : SEU189+1 : TPTP v8.2.0. Released v3.3.0.
% Domain   : Set theory
% Problem  : MPTP bushy problem t65_relat_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-t65_relat_1 [Urb07]

% Status   : Theorem
% Rating   : 0.14 v8.2.0, 0.08 v8.1.0, 0.06 v7.4.0, 0.07 v7.2.0, 0.03 v7.1.0, 0.04 v7.0.0, 0.03 v6.4.0, 0.08 v6.3.0, 0.04 v6.2.0, 0.08 v6.1.0, 0.10 v6.0.0, 0.04 v5.5.0, 0.07 v5.4.0, 0.11 v5.3.0, 0.15 v5.2.0, 0.00 v4.1.0, 0.04 v4.0.1, 0.09 v4.0.0, 0.12 v3.7.0, 0.05 v3.4.0, 0.11 v3.3.0
% Syntax   : Number of formulae    :   25 (   8 unt;   0 def)
%            Number of atoms       :   51 (   9 equ)
%            Maximal formula atoms :    4 (   2 avg)
%            Number of connectives :   36 (  10   ~;   2   |;  11   &)
%                                         (   1 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   3 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    6 (   4 usr;   1 prp; 0-2 aty)
%            Number of functors    :    3 (   3 usr;   1 con; 0-1 aty)
%            Number of variables   :   24 (  19   !;   5   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%------------------------------------------------------------------------------
fof(existence_m1_subset_1,axiom,
    ! [A] :
    ? [B] : element(B,A) ).

fof(dt_m1_subset_1,axiom,
    $true ).

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

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

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

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

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

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

fof(fc5_relat_1,axiom,
    ! [A] :
      ( ( ~ empty(A)
        & relation(A) )
     => ~ empty(relation_dom(A)) ) ).

fof(fc6_relat_1,axiom,
    ! [A] :
      ( ( ~ empty(A)
        & relation(A) )
     => ~ empty(relation_rng(A)) ) ).

fof(fc7_relat_1,axiom,
    ! [A] :
      ( empty(A)
     => ( empty(relation_dom(A))
        & relation(relation_dom(A)) ) ) ).

fof(fc8_relat_1,axiom,
    ! [A] :
      ( empty(A)
     => ( empty(relation_rng(A))
        & relation(relation_rng(A)) ) ) ).

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

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

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

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

fof(dt_k1_relat_1,axiom,
    $true ).

fof(dt_k1_xboole_0,axiom,
    $true ).

fof(dt_k2_relat_1,axiom,
    $true ).

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

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

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

fof(t65_relat_1,conjecture,
    ! [A] :
      ( relation(A)
     => ( relation_dom(A) = empty_set
      <=> relation_rng(A) = empty_set ) ) ).

fof(t60_relat_1,axiom,
    ( relation_dom(empty_set) = empty_set
    & relation_rng(empty_set) = empty_set ) ).

fof(t64_relat_1,axiom,
    ! [A] :
      ( relation(A)
     => ( ( relation_dom(A) = empty_set
          | relation_rng(A) = empty_set )
       => A = empty_set ) ) ).

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