%------------------------------------------------------------------------------
% File     : SEU242+1 : TPTP v9.0.0. Released v3.3.0.
% Domain   : Set theory
% Problem  : MPTP bushy problem l4_wellord1
% 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-l4_wellord1 [Urb07]

% Status   : Theorem
% Rating   : 0.21 v9.0.0, 0.22 v8.2.0, 0.19 v8.1.0, 0.14 v7.5.0, 0.16 v7.4.0, 0.13 v7.3.0, 0.14 v7.2.0, 0.10 v7.1.0, 0.17 v7.0.0, 0.20 v6.4.0, 0.23 v6.3.0, 0.25 v6.2.0, 0.20 v6.1.0, 0.27 v6.0.0, 0.26 v5.5.0, 0.37 v5.4.0, 0.43 v5.3.0, 0.44 v5.2.0, 0.30 v5.1.0, 0.33 v5.0.0, 0.42 v4.1.0, 0.43 v4.0.1, 0.39 v4.0.0, 0.42 v3.7.0, 0.40 v3.5.0, 0.42 v3.4.0, 0.47 v3.3.0
% Syntax   : Number of formulae    :   36 (  19 unt;   0 def)
%            Number of atoms       :   72 (  10 equ)
%            Maximal formula atoms :    7 (   2 avg)
%            Number of connectives :   54 (  18   ~;   1   |;  20   &)
%                                         (   3 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   13 (   4 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :   10 (   8 usr;   1 prp; 0-2 aty)
%            Number of functors    :    8 (   8 usr;   1 con; 0-2 aty)
%            Number of variables   :   44 (  38   !;   6   ?)
% 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_funct_1,axiom,
    ! [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(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(d14_relat_2,axiom,
    ! [A] :
      ( relation(A)
     => ( connected(A)
      <=> is_connected_in(A,relation_field(A)) ) ) ).

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

fof(d6_relat_1,axiom,
    ! [A] :
      ( relation(A)
     => relation_field(A) = set_union2(relation_dom(A),relation_rng(A)) ) ).

fof(d6_relat_2,axiom,
    ! [A] :
      ( relation(A)
     => ! [B] :
          ( is_connected_in(A,B)
        <=> ! [C,D] :
              ~ ( in(C,B)
                & in(D,B)
                & C != D
                & ~ in(ordered_pair(C,D),A)
                & ~ in(ordered_pair(D,C),A) ) ) ) ).

fof(dt_k1_relat_1,axiom,
    $true ).

fof(dt_k1_tarski,axiom,
    $true ).

fof(dt_k1_xboole_0,axiom,
    $true ).

fof(dt_k2_relat_1,axiom,
    $true ).

fof(dt_k2_tarski,axiom,
    $true ).

fof(dt_k2_xboole_0,axiom,
    $true ).

fof(dt_k3_relat_1,axiom,
    $true ).

fof(dt_k4_tarski,axiom,
    $true ).

fof(dt_m1_subset_1,axiom,
    $true ).

fof(existence_m1_subset_1,axiom,
    ! [A] :
    ? [B] : element(B,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(l4_wellord1,conjecture,
    ! [A] :
      ( relation(A)
     => ( connected(A)
      <=> ! [B,C] :
            ~ ( in(B,relation_field(A))
              & in(C,relation_field(A))
              & B != C
              & ~ in(ordered_pair(B,C),A)
              & ~ in(ordered_pair(C,B),A) ) ) ) ).

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

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

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

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

fof(rc3_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & function(A)
      & one_to_one(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(t2_subset,axiom,
    ! [A,B] :
      ( element(A,B)
     => ( empty(B)
        | in(A,B) ) ) ).

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) ) ).

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