SET007 Axioms: SET007+16.ax


%------------------------------------------------------------------------------
% File     : SET007+16 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Relations Defined on Sets
% 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    : relset_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   89 (  20 unit)
%            Number of atoms       :  253 (  34 equality)
%            Maximal formula depth :   19 (   6 average)
%            Number of connectives :  193 (  29 ~  ;   0  |;  42  &)
%                                         (  11 <=>; 111 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   1 propositional; 0-3 arity)
%            Number of functors    :   33 (   5 constant; 0-6 arity)
%            Number of variables   :  275 (   0 singleton; 266 !;   9 ?)
%            Maximal term depth    :    3 (   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(cc1_relset_1,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B)))
     => v1_relat_1(C) ) )).

fof(d1_relset_1,axiom,(
    ! [A,B,C] :
      ( m1_relset_1(C,A,B)
    <=> r1_tarski(C,k2_zfmisc_1(A,B)) ) )).

fof(t1_relset_1,axiom,(
    $true )).

fof(t2_relset_1,axiom,(
    $true )).

fof(t3_relset_1,axiom,(
    $true )).

fof(t4_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,B,C)
     => ( r1_tarski(A,D)
       => m2_relset_1(A,B,C) ) ) )).

fof(t5_relset_1,axiom,(
    $true )).

fof(t6_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,B)
     => ~ ( r2_hidden(C,D)
          & ! [E,F] :
              ~ ( C = k4_tarski(E,F)
                & r2_hidden(E,A)
                & r2_hidden(F,B) ) ) ) )).

fof(t7_relset_1,axiom,(
    $true )).

fof(t8_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( r2_hidden(C,A)
        & r2_hidden(D,B) )
     => m2_relset_1(k1_tarski(k4_tarski(C,D)),A,B) ) )).

fof(t9_relset_1,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( r1_tarski(k1_relat_1(B),A)
       => m2_relset_1(B,A,k2_relat_1(B)) ) ) )).

fof(t10_relset_1,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( r1_tarski(k2_relat_1(B),A)
       => m2_relset_1(B,k1_relat_1(B),A) ) ) )).

fof(t11_relset_1,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(C)
     => ( ( r1_tarski(k1_relat_1(C),A)
          & r1_tarski(k2_relat_1(C),B) )
       => m2_relset_1(C,A,B) ) ) )).

fof(t12_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => ( r1_tarski(k1_relat_1(C),A)
        & r1_tarski(k2_relat_1(C),B) ) ) )).

fof(t13_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,C)
     => ( r1_tarski(k1_relat_1(D),B)
       => m2_relset_1(D,B,C) ) ) )).

fof(t14_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,C,A)
     => ( r1_tarski(k2_relat_1(D),B)
       => m2_relset_1(D,C,B) ) ) )).

fof(t15_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,C)
     => ( r1_tarski(A,B)
       => m2_relset_1(D,B,C) ) ) )).

fof(t16_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,C,A)
     => ( r1_tarski(A,B)
       => m2_relset_1(D,C,B) ) ) )).

fof(t17_relset_1,axiom,(
    ! [A,B,C,D,E] :
      ( m2_relset_1(E,A,C)
     => ( ( r1_tarski(A,B)
          & r1_tarski(C,D) )
       => m2_relset_1(E,B,D) ) ) )).

fof(t18_relset_1,axiom,(
    $true )).

fof(t19_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => r1_tarski(k3_relat_1(C),k2_xboole_0(A,B)) ) )).

fof(t20_relset_1,axiom,(
    $true )).

fof(t21_relset_1,axiom,(
    $true )).

fof(t22_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,B,A)
     => ( ! [D] :
            ~ ( r2_hidden(D,B)
              & ! [E] : ~ r2_hidden(k4_tarski(D,E),C) )
      <=> k4_relset_1(B,A,C) = B ) ) )).

fof(t23_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => ( ! [D] :
            ~ ( r2_hidden(D,B)
              & ! [E] : ~ r2_hidden(k4_tarski(E,D),C) )
      <=> k5_relset_1(A,B,C) = B ) ) )).

fof(t24_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => ( k4_relset_1(B,A,k6_relset_1(A,B,C)) = k5_relset_1(A,B,C)
        & k5_relset_1(B,A,k6_relset_1(A,B,C)) = k4_relset_1(A,B,C) ) ) )).

fof(t25_relset_1,axiom,(
    ! [A,B] : m2_relset_1(k1_xboole_0,A,B) )).

fof(t26_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => ( m2_relset_1(C,k1_xboole_0,B)
       => C = k1_xboole_0 ) ) )).

fof(t27_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,B,A)
     => ( m2_relset_1(C,B,k1_xboole_0)
       => C = k1_xboole_0 ) ) )).

fof(t28_relset_1,axiom,(
    ! [A] : r1_tarski(k6_relat_1(A),k2_zfmisc_1(A,A)) )).

fof(t29_relset_1,axiom,(
    ! [A] : m2_relset_1(k6_relat_1(A),A,A) )).

fof(t30_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,B)
     => ( r1_tarski(k6_relat_1(C),D)
       => ( r1_tarski(C,k4_relset_1(A,B,D))
          & r1_tarski(C,k5_relset_1(A,B,D)) ) ) ) )).

fof(t31_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,B,A)
     => ( r1_tarski(k6_relat_1(B),C)
       => ( B = k4_relset_1(B,A,C)
          & r1_tarski(B,k5_relset_1(B,A,C)) ) ) ) )).

fof(t32_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => ( r1_tarski(k6_relat_1(B),C)
       => ( r1_tarski(B,k4_relset_1(A,B,C))
          & B = k5_relset_1(A,B,C) ) ) ) )).

fof(t33_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,C)
     => m2_relset_1(k8_relset_1(A,C,D,B),B,C) ) )).

fof(t34_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,B,A)
     => ( r1_tarski(B,C)
       => k8_relset_1(B,A,D,C) = D ) ) )).

fof(t35_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,C,A)
     => m2_relset_1(k9_relset_1(C,A,B,D),C,B) ) )).

fof(t36_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m2_relset_1(D,A,B)
     => ( r1_tarski(B,C)
       => k9_relset_1(A,B,C,D) = D ) ) )).

fof(t37_relset_1,axiom,(
    $true )).

fof(t38_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => ( k10_relset_1(A,B,C,A) = k5_relset_1(A,B,C)
        & k11_relset_1(A,B,C,B) = k4_relset_1(A,B,C) ) ) )).

fof(t39_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,B,A)
     => ( k10_relset_1(B,A,C,k11_relset_1(B,A,C,A)) = k5_relset_1(B,A,C)
        & k11_relset_1(B,A,C,k10_relset_1(B,A,C,B)) = k4_relset_1(B,A,C) ) ) )).

fof(t40_relset_1,axiom,(
    $true )).

fof(t41_relset_1,axiom,(
    $true )).

fof(t42_relset_1,axiom,(
    $true )).

fof(t43_relset_1,axiom,(
    $true )).

fof(t44_relset_1,axiom,(
    $true )).

fof(t45_relset_1,axiom,(
    ! [A,B] :
      ( m2_relset_1(B,A,A)
     => ( k5_relat_1(B,k6_relat_1(A)) = B
        & k5_relat_1(k6_relat_1(A),B) = B ) ) )).

fof(t46_relset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k6_relat_1(A) != k1_xboole_0 ) )).

fof(t47_relset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_relset_1(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ( r2_hidden(D,k4_relset_1(A,B,C))
                  <=> ? [E] :
                        ( m1_subset_1(E,B)
                        & r2_hidden(k4_tarski(D,E),C) ) ) ) ) ) ) )).

fof(t48_relset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_relset_1(C,B,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ( r2_hidden(D,k5_relset_1(B,A,C))
                  <=> ? [E] :
                        ( m1_subset_1(E,B)
                        & r2_hidden(k4_tarski(E,D),C) ) ) ) ) ) ) )).

fof(t49_relset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_relset_1(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ~ ( r2_hidden(D,k4_relset_1(A,B,C))
                      & ! [E] :
                          ( m1_subset_1(E,B)
                         => ~ r2_hidden(E,k5_relset_1(A,B,C)) ) ) ) ) ) ) )).

fof(t50_relset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_relset_1(C,B,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ~ ( r2_hidden(D,k5_relset_1(B,A,C))
                      & ! [E] :
                          ( m1_subset_1(E,B)
                         => ~ r2_hidden(E,k4_relset_1(B,A,C)) ) ) ) ) ) ) )).

fof(t51_relset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m2_relset_1(D,A,B)
                 => ! [E] :
                      ( m2_relset_1(E,B,C)
                     => ! [F] :
                          ( m1_subset_1(F,A)
                         => ! [G] :
                              ( m1_subset_1(G,C)
                             => ( r2_hidden(k4_tarski(F,G),k7_relset_1(A,B,B,C,D,E))
                              <=> ? [H] :
                                    ( m1_subset_1(H,B)
                                    & r2_hidden(k4_tarski(F,H),D)
                                    & r2_hidden(k4_tarski(H,G),E) ) ) ) ) ) ) ) ) ) )).

fof(t52_relset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m2_relset_1(D,C,A)
                 => ! [E] :
                      ( m1_subset_1(E,A)
                     => ( r2_hidden(E,k10_relset_1(C,A,D,B))
                      <=> ? [F] :
                            ( m1_subset_1(F,C)
                            & r2_hidden(k4_tarski(F,E),D)
                            & r2_hidden(F,B) ) ) ) ) ) ) ) )).

fof(t53_relset_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m2_relset_1(D,A,C)
                 => ! [E] :
                      ( m1_subset_1(E,A)
                     => ( r2_hidden(E,k11_relset_1(A,C,D,B))
                      <=> ? [F] :
                            ( m1_subset_1(F,C)
                            & r2_hidden(k4_tarski(E,F),D)
                            & r2_hidden(F,B) ) ) ) ) ) ) ) )).

fof(s1_relset_1,axiom,(
    ? [A] :
      ( m2_relset_1(A,f1_s1_relset_1,f2_s1_relset_1)
      & ! [B,C] :
          ( r2_hidden(k4_tarski(B,C),A)
        <=> ( r2_hidden(B,f1_s1_relset_1)
            & r2_hidden(C,f2_s1_relset_1)
            & p1_s1_relset_1(B,C) ) ) ) )).

fof(s2_relset_1,axiom,(
    ? [A] :
      ( m2_relset_1(A,f1_s2_relset_1,f2_s2_relset_1)
      & ! [B] :
          ( m1_subset_1(B,f1_s2_relset_1)
         => ! [C] :
              ( m1_subset_1(C,f2_s2_relset_1)
             => ( r2_hidden(k4_tarski(B,C),A)
              <=> p1_s2_relset_1(B,C) ) ) ) ) )).

fof(dt_m1_relset_1,axiom,(
    $true )).

fof(existence_m1_relset_1,axiom,(
    ! [A,B] :
    ? [C] : m1_relset_1(C,A,B) )).

fof(dt_m2_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
     => m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,B))) ) )).

fof(existence_m2_relset_1,axiom,(
    ! [A,B] :
    ? [C] : m2_relset_1(C,A,B) )).

fof(redefinition_m2_relset_1,axiom,(
    ! [A,B,C] :
      ( m2_relset_1(C,A,B)
    <=> m1_relset_1(C,A,B) ) )).

fof(dt_k1_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => m2_relset_1(k1_relset_1(A,B,C,D),A,B) ) )).

fof(commutativity_k1_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => k1_relset_1(A,B,C,D) = k1_relset_1(A,B,D,C) ) )).

fof(idempotence_k1_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => k1_relset_1(A,B,C,C) = C ) )).

fof(redefinition_k1_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => k1_relset_1(A,B,C,D) = k2_xboole_0(C,D) ) )).

fof(dt_k2_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => m2_relset_1(k2_relset_1(A,B,C,D),A,B) ) )).

fof(commutativity_k2_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => k2_relset_1(A,B,C,D) = k2_relset_1(A,B,D,C) ) )).

fof(idempotence_k2_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => k2_relset_1(A,B,C,C) = C ) )).

fof(redefinition_k2_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => k2_relset_1(A,B,C,D) = k3_xboole_0(C,D) ) )).

fof(dt_k3_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => m2_relset_1(k3_relset_1(A,B,C,D),A,B) ) )).

fof(redefinition_k3_relset_1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_relset_1(C,A,B)
        & m1_relset_1(D,A,B) )
     => k3_relset_1(A,B,C,D) = k4_xboole_0(C,D) ) )).

fof(dt_k4_relset_1,axiom,(
    ! [A,B,C] :
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k4_relset_1(A,B,C),k1_zfmisc_1(A)) ) )).

fof(redefinition_k4_relset_1,axiom,(
    ! [A,B,C] :
      ( m1_relset_1(C,A,B)
     => k4_relset_1(A,B,C) = k1_relat_1(C) ) )).

fof(dt_k5_relset_1,axiom,(
    ! [A,B,C] :
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k5_relset_1(A,B,C),k1_zfmisc_1(B)) ) )).

fof(redefinition_k5_relset_1,axiom,(
    ! [A,B,C] :
      ( m1_relset_1(C,A,B)
     => k5_relset_1(A,B,C) = k2_relat_1(C) ) )).

fof(dt_k6_relset_1,axiom,(
    ! [A,B,C] :
      ( m1_relset_1(C,A,B)
     => m2_relset_1(k6_relset_1(A,B,C),B,A) ) )).

fof(involutiveness_k6_relset_1,axiom,(
    ! [A,B,C] :
      ( m1_relset_1(C,A,B)
     => k6_relset_1(A,B,k6_relset_1(A,B,C)) = C ) )).

fof(redefinition_k6_relset_1,axiom,(
    ! [A,B,C] :
      ( m1_relset_1(C,A,B)
     => k6_relset_1(A,B,C) = k4_relat_1(C) ) )).

fof(dt_k7_relset_1,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( m1_relset_1(E,A,B)
        & m1_relset_1(F,C,D) )
     => m2_relset_1(k7_relset_1(A,B,C,D,E,F),A,D) ) )).

fof(redefinition_k7_relset_1,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( m1_relset_1(E,A,B)
        & m1_relset_1(F,C,D) )
     => k7_relset_1(A,B,C,D,E,F) = k5_relat_1(E,F) ) )).

fof(dt_k8_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m1_relset_1(C,A,B)
     => m2_relset_1(k8_relset_1(A,B,C,D),A,B) ) )).

fof(redefinition_k8_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m1_relset_1(C,A,B)
     => k8_relset_1(A,B,C,D) = k7_relat_1(C,D) ) )).

fof(dt_k9_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m1_relset_1(D,A,B)
     => m2_relset_1(k9_relset_1(A,B,C,D),A,B) ) )).

fof(redefinition_k9_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m1_relset_1(D,A,B)
     => k9_relset_1(A,B,C,D) = k8_relat_1(C,D) ) )).

fof(dt_k10_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k10_relset_1(A,B,C,D),k1_zfmisc_1(B)) ) )).

fof(redefinition_k10_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m1_relset_1(C,A,B)
     => k10_relset_1(A,B,C,D) = k9_relat_1(C,D) ) )).

fof(dt_k11_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m1_relset_1(C,A,B)
     => m1_subset_1(k11_relset_1(A,B,C,D),k1_zfmisc_1(A)) ) )).

fof(redefinition_k11_relset_1,axiom,(
    ! [A,B,C,D] :
      ( m1_relset_1(C,A,B)
     => k11_relset_1(A,B,C,D) = k10_relat_1(C,D) ) )).
%------------------------------------------------------------------------------