%------------------------------------------------------------------------------
% File     : SEU419+1 : TPTP v8.2.0. Released v3.4.0.
% Domain   : Set Theory
% Problem  : First and Second Order Cutting of Binary Relations T12
% Version  : [Urb08] axioms : Especial.
% English  :

% Refs     : [Ret05] Retel (2005), Properties of First and Second Order Cut
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : t12_relset_2 [Urb08]

% Status   : Theorem
% Rating   : 0.86 v8.2.0, 0.89 v7.5.0, 0.88 v7.4.0, 0.80 v7.3.0, 0.76 v7.2.0, 0.72 v7.1.0, 0.83 v7.0.0, 0.80 v6.4.0, 0.77 v6.3.0, 0.88 v6.2.0, 0.92 v6.1.0, 0.93 v6.0.0, 0.91 v5.5.0, 0.89 v5.4.0, 0.93 v5.2.0, 0.90 v5.0.0, 0.88 v4.1.0, 0.87 v4.0.0, 0.88 v3.7.0, 0.85 v3.5.0, 0.89 v3.4.0
% Syntax   : Number of formulae    :   40 (  17 unt;   0 def)
%            Number of atoms       :   76 (   9 equ)
%            Maximal formula atoms :    4 (   1 avg)
%            Number of connectives :   48 (  12   ~;   1   |;  17   &)
%                                         (   6 <=>;  12  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   4 avg)
%            Maximal term depth    :    4 (   1 avg)
%            Number of predicates  :    8 (   6 usr;   1 prp; 0-2 aty)
%            Number of functors    :    7 (   7 usr;   1 con; 0-2 aty)
%            Number of variables   :   58 (  52   !;   6   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Normal version: includes the axioms (which may be theorems from
%            other articles) and background that are possibly necessary.
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
fof(t12_relset_2,conjecture,
    ! [A,B] :
      ( v1_relat_1(B)
     => ! [C] :
          ( v1_relat_1(C)
         => k9_relat_1(k4_xboole_0(B,C),k1_tarski(A)) = k4_xboole_0(k9_relat_1(B,k1_tarski(A)),k9_relat_1(C,k1_tarski(A))) ) ) ).

fof(antisymmetry_r2_hidden,axiom,
    ! [A,B] :
      ( r2_hidden(A,B)
     => ~ r2_hidden(B,A) ) ).

fof(cc1_relat_1,axiom,
    ! [A] :
      ( v1_xboole_0(A)
     => v1_relat_1(A) ) ).

fof(commutativity_k2_tarski,axiom,
    ! [A,B] : k2_tarski(A,B) = k2_tarski(B,A) ).

fof(d10_xboole_0,axiom,
    ! [A,B] :
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ).

fof(d3_tarski,axiom,
    ! [A,B] :
      ( r1_tarski(A,B)
    <=> ! [C] :
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ).

fof(d4_xboole_0,axiom,
    ! [A,B,C] :
      ( C = k4_xboole_0(A,B)
    <=> ! [D] :
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ).

fof(d5_tarski,axiom,
    ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ).

fof(dt_k1_tarski,axiom,
    $true ).

fof(dt_k1_xboole_0,axiom,
    $true ).

fof(dt_k1_zfmisc_1,axiom,
    $true ).

fof(dt_k2_tarski,axiom,
    $true ).

fof(dt_k4_tarski,axiom,
    $true ).

fof(dt_k4_xboole_0,axiom,
    $true ).

fof(dt_k9_relat_1,axiom,
    $true ).

fof(dt_m1_subset_1,axiom,
    $true ).

fof(existence_m1_subset_1,axiom,
    ! [A] :
    ? [B] : m1_subset_1(B,A) ).

fof(fc12_relat_1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0)
    & v3_relat_1(k1_xboole_0) ) ).

fof(fc1_subset_1,axiom,
    ! [A] : ~ v1_xboole_0(k1_zfmisc_1(A)) ).

fof(fc2_subset_1,axiom,
    ! [A] : ~ v1_xboole_0(k1_tarski(A)) ).

fof(fc3_relat_1,axiom,
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_relat_1(B) )
     => v1_relat_1(k4_xboole_0(A,B)) ) ).

fof(fc3_subset_1,axiom,
    ! [A,B] : ~ v1_xboole_0(k2_tarski(A,B)) ).

fof(fc4_relat_1,axiom,
    ( v1_xboole_0(k1_xboole_0)
    & v1_relat_1(k1_xboole_0) ) ).

fof(rc1_relat_1,axiom,
    ? [A] :
      ( v1_xboole_0(A)
      & v1_relat_1(A) ) ).

fof(rc1_subset_1,axiom,
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B) ) ) ).

fof(rc2_relat_1,axiom,
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_relat_1(A) ) ).

fof(rc2_subset_1,axiom,
    ! [A] :
    ? [B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
      & v1_xboole_0(B) ) ).

fof(rc3_relat_1,axiom,
    ? [A] :
      ( v1_relat_1(A)
      & v3_relat_1(A) ) ).

fof(reflexivity_r1_tarski,axiom,
    ! [A,B] : r1_tarski(A,A) ).

fof(t1_subset,axiom,
    ! [A,B] :
      ( r2_hidden(A,B)
     => m1_subset_1(A,B) ) ).

fof(t2_subset,axiom,
    ! [A,B] :
      ( m1_subset_1(A,B)
     => ( v1_xboole_0(B)
        | r2_hidden(A,B) ) ) ).

fof(t3_boole,axiom,
    ! [A] : k4_xboole_0(A,k1_xboole_0) = A ).

fof(t3_subset,axiom,
    ! [A,B] :
      ( m1_subset_1(A,k1_zfmisc_1(B))
    <=> r1_tarski(A,B) ) ).

fof(t4_boole,axiom,
    ! [A] : k4_xboole_0(k1_xboole_0,A) = k1_xboole_0 ).

fof(t4_subset,axiom,
    ! [A,B,C] :
      ( ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C)) )
     => m1_subset_1(A,C) ) ).

fof(t5_subset,axiom,
    ! [A,B,C] :
      ~ ( r2_hidden(A,B)
        & m1_subset_1(B,k1_zfmisc_1(C))
        & v1_xboole_0(C) ) ).

fof(t6_boole,axiom,
    ! [A] :
      ( v1_xboole_0(A)
     => A = k1_xboole_0 ) ).

fof(t7_boole,axiom,
    ! [A,B] :
      ~ ( r2_hidden(A,B)
        & v1_xboole_0(B) ) ).

fof(t8_boole,axiom,
    ! [A,B] :
      ~ ( v1_xboole_0(A)
        & A != B
        & v1_xboole_0(B) ) ).

fof(t9_relset_2,axiom,
    ! [A,B,C] :
      ( v1_relat_1(C)
     => ( r2_hidden(A,k9_relat_1(C,k1_tarski(B)))
      <=> r2_hidden(k4_tarski(B,A),C) ) ) ).

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