SET007 Axioms: SET007+13.ax


%------------------------------------------------------------------------------
% File     : SET007+13 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of Binary Relations
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : relat_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   63 (  17 unit)
%            Number of atoms       :  200 (   6 equality)
%            Maximal formula depth :   13 (   4 average)
%            Number of connectives :  146 (   9 ~  ;   0  |;  37  &)
%                                         (  24 <=>;  76 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   1 propositional; 0-2 arity)
%            Number of functors    :   11 (   0 constant; 1-2 arity)
%            Number of variables   :   79 (   0 singleton;  79 !;   0 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(d1_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r1_relat_2(A,B)
        <=> ! [C] :
              ( r2_hidden(C,B)
             => r2_hidden(k4_tarski(C,C),A) ) ) ) )).

fof(d2_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r2_relat_2(A,B)
        <=> ! [C] :
              ~ ( r2_hidden(C,B)
                & r2_hidden(k4_tarski(C,C),A) ) ) ) )).

fof(d3_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r3_relat_2(A,B)
        <=> ! [C,D] :
              ( ( r2_hidden(C,B)
                & r2_hidden(D,B)
                & r2_hidden(k4_tarski(C,D),A) )
             => r2_hidden(k4_tarski(D,C),A) ) ) ) )).

fof(d4_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r4_relat_2(A,B)
        <=> ! [C,D] :
              ( ( r2_hidden(C,B)
                & r2_hidden(D,B)
                & r2_hidden(k4_tarski(C,D),A)
                & r2_hidden(k4_tarski(D,C),A) )
             => C = D ) ) ) )).

fof(d5_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r5_relat_2(A,B)
        <=> ! [C,D] :
              ~ ( r2_hidden(C,B)
                & r2_hidden(D,B)
                & r2_hidden(k4_tarski(C,D),A)
                & r2_hidden(k4_tarski(D,C),A) ) ) ) )).

fof(d6_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r6_relat_2(A,B)
        <=> ! [C,D] :
              ~ ( r2_hidden(C,B)
                & r2_hidden(D,B)
                & C != D
                & ~ r2_hidden(k4_tarski(C,D),A)
                & ~ r2_hidden(k4_tarski(D,C),A) ) ) ) )).

fof(d7_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r7_relat_2(A,B)
        <=> ! [C,D] :
              ~ ( r2_hidden(C,B)
                & r2_hidden(D,B)
                & ~ r2_hidden(k4_tarski(C,D),A)
                & ~ r2_hidden(k4_tarski(D,C),A) ) ) ) )).

fof(d8_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r8_relat_2(A,B)
        <=> ! [C,D,E] :
              ( ( r2_hidden(C,B)
                & r2_hidden(D,B)
                & r2_hidden(E,B)
                & r2_hidden(k4_tarski(C,D),A)
                & r2_hidden(k4_tarski(D,E),A) )
             => r2_hidden(k4_tarski(C,E),A) ) ) ) )).

fof(d9_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v1_relat_2(A)
      <=> r1_relat_2(A,k3_relat_1(A)) ) ) )).

fof(d10_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v2_relat_2(A)
      <=> r2_relat_2(A,k3_relat_1(A)) ) ) )).

fof(d11_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v3_relat_2(A)
      <=> r3_relat_2(A,k3_relat_1(A)) ) ) )).

fof(d12_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v4_relat_2(A)
      <=> r4_relat_2(A,k3_relat_1(A)) ) ) )).

fof(d13_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v5_relat_2(A)
      <=> r5_relat_2(A,k3_relat_1(A)) ) ) )).

fof(d14_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v6_relat_2(A)
      <=> r6_relat_2(A,k3_relat_1(A)) ) ) )).

fof(d15_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v7_relat_2(A)
      <=> r7_relat_2(A,k3_relat_1(A)) ) ) )).

fof(d16_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v8_relat_2(A)
      <=> r8_relat_2(A,k3_relat_1(A)) ) ) )).

fof(t1_relat_2,axiom,(
    $true )).

fof(t2_relat_2,axiom,(
    $true )).

fof(t3_relat_2,axiom,(
    $true )).

fof(t4_relat_2,axiom,(
    $true )).

fof(t5_relat_2,axiom,(
    $true )).

fof(t6_relat_2,axiom,(
    $true )).

fof(t7_relat_2,axiom,(
    $true )).

fof(t8_relat_2,axiom,(
    $true )).

fof(t9_relat_2,axiom,(
    $true )).

fof(t10_relat_2,axiom,(
    $true )).

fof(t11_relat_2,axiom,(
    $true )).

fof(t12_relat_2,axiom,(
    $true )).

fof(t13_relat_2,axiom,(
    $true )).

fof(t14_relat_2,axiom,(
    $true )).

fof(t15_relat_2,axiom,(
    $true )).

fof(t16_relat_2,axiom,(
    $true )).

fof(t17_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v1_relat_2(A)
      <=> r1_tarski(k6_relat_1(k3_relat_1(A)),A) ) ) )).

fof(t18_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v2_relat_2(A)
      <=> r1_xboole_0(k6_relat_1(k3_relat_1(A)),A) ) ) )).

fof(t19_relat_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( r4_relat_2(B,A)
      <=> r5_relat_2(k4_xboole_0(B,k6_relat_1(A)),A) ) ) )).

fof(t20_relat_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( r5_relat_2(B,A)
       => r4_relat_2(k2_xboole_0(B,k6_relat_1(A)),A) ) ) )).

fof(t21_relat_2,axiom,(
    $true )).

fof(t22_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( ( v3_relat_2(A)
          & v8_relat_2(A) )
       => v1_relat_2(A) ) ) )).

fof(t23_relat_2,axiom,(
    ! [A] :
      ( v3_relat_2(k6_relat_1(A))
      & v8_relat_2(k6_relat_1(A)) ) )).

fof(t24_relat_2,axiom,(
    ! [A] :
      ( v4_relat_2(k6_relat_1(A))
      & v1_relat_2(k6_relat_1(A)) ) )).

fof(t25_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( ( v2_relat_2(A)
          & v8_relat_2(A) )
       => v5_relat_2(A) ) ) )).

fof(t26_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v5_relat_2(A)
       => ( v2_relat_2(A)
          & v4_relat_2(A) ) ) ) )).

fof(t27_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v1_relat_2(A)
       => v1_relat_2(k4_relat_1(A)) ) ) )).

fof(t28_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v2_relat_2(A)
       => v2_relat_2(k4_relat_1(A)) ) ) )).

fof(t29_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v1_relat_2(A)
       => ( k1_relat_1(A) = k1_relat_1(k4_relat_1(A))
          & k2_relat_1(A) = k2_relat_1(k4_relat_1(A)) ) ) ) )).

fof(t30_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v3_relat_2(A)
      <=> A = k4_relat_1(A) ) ) )).

fof(t31_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( ( v1_relat_2(A)
              & v1_relat_2(B) )
           => ( v1_relat_2(k2_xboole_0(A,B))
              & v1_relat_2(k3_xboole_0(A,B)) ) ) ) ) )).

fof(t32_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( ( v2_relat_2(A)
              & v2_relat_2(B) )
           => ( v2_relat_2(k2_xboole_0(A,B))
              & v2_relat_2(k3_xboole_0(A,B)) ) ) ) ) )).

fof(t33_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( v2_relat_2(A)
           => v2_relat_2(k4_xboole_0(A,B)) ) ) ) )).

fof(t34_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v3_relat_2(A)
       => v3_relat_2(k4_relat_1(A)) ) ) )).

fof(t35_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( ( v3_relat_2(A)
              & v3_relat_2(B) )
           => ( v3_relat_2(k2_xboole_0(A,B))
              & v3_relat_2(k3_xboole_0(A,B))
              & v3_relat_2(k4_xboole_0(A,B)) ) ) ) ) )).

fof(t36_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v5_relat_2(A)
       => v5_relat_2(k4_relat_1(A)) ) ) )).

fof(t37_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( ( v5_relat_2(A)
              & v5_relat_2(B) )
           => v5_relat_2(k3_xboole_0(A,B)) ) ) ) )).

fof(t38_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( v5_relat_2(A)
           => v5_relat_2(k4_xboole_0(A,B)) ) ) ) )).

fof(t39_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v4_relat_2(A)
      <=> r1_tarski(k3_xboole_0(A,k4_relat_1(A)),k6_relat_1(k1_relat_1(A))) ) ) )).

fof(t40_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v4_relat_2(A)
       => v4_relat_2(k4_relat_1(A)) ) ) )).

fof(t41_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( v4_relat_2(A)
           => ( v4_relat_2(k3_xboole_0(A,B))
              & v4_relat_2(k4_xboole_0(A,B)) ) ) ) ) )).

fof(t42_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v8_relat_2(A)
       => v8_relat_2(k4_relat_1(A)) ) ) )).

fof(t43_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( v1_relat_1(B)
         => ( ( v8_relat_2(A)
              & v8_relat_2(B) )
           => v8_relat_2(k3_xboole_0(A,B)) ) ) ) )).

fof(t44_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v8_relat_2(A)
      <=> r1_tarski(k5_relat_1(A,A),A) ) ) )).

fof(t45_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v6_relat_2(A)
      <=> r1_tarski(k4_xboole_0(k2_zfmisc_1(k3_relat_1(A),k3_relat_1(A)),k6_relat_1(k3_relat_1(A))),k2_xboole_0(A,k4_relat_1(A))) ) ) )).

fof(t46_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v7_relat_2(A)
       => ( v6_relat_2(A)
          & v1_relat_2(A) ) ) ) )).

fof(t47_relat_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v7_relat_2(A)
      <=> k2_zfmisc_1(k3_relat_1(A),k3_relat_1(A)) = k2_xboole_0(A,k4_relat_1(A)) ) ) )).
%------------------------------------------------------------------------------