SET007 Axioms: SET007+8.ax


%------------------------------------------------------------------------------
% File     : SET007+8 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Families of 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    : setfam_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   94 (  36 unit)
%            Number of atoms       :  224 (  55 equality)
%            Maximal formula depth :   11 (   4 average)
%            Number of connectives :  171 (  41 ~  ;   3  |;  32  &)
%                                         (  19 <=>;  76 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   1 propositional; 0-2 arity)
%            Number of functors    :   20 (   1 constant; 0-3 arity)
%            Number of variables   :  173 (   2 singleton; 164 !;   9 ?)
%            Maximal term depth    :    4 (   2 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(fc1_setfam_1,axiom,(
    ! [A,B] :
      ( ( v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => v1_xboole_0(k7_setfam_1(A,B)) ) )).

fof(rc1_setfam_1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_setfam_1(A) ) )).

fof(fc2_setfam_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v1_xboole_0(k1_tarski(A))
        & v1_setfam_1(k1_tarski(A)) ) ) )).

fof(fc3_setfam_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B) )
     => ( ~ v1_xboole_0(k2_tarski(A,B))
        & v1_setfam_1(k2_tarski(A,B)) ) ) )).

fof(fc4_setfam_1,axiom,(
    ! [A,B] :
      ( ( v1_setfam_1(A)
        & v1_setfam_1(B) )
     => v1_setfam_1(k2_xboole_0(A,B)) ) )).

fof(cc1_setfam_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_setfam_1(A) )
     => ! [B] :
          ( m1_subset_1(B,A)
         => ~ v1_xboole_0(B) ) ) )).

fof(rc2_setfam_1,axiom,(
    ? [A] : ~ v2_setfam_1(A) )).

fof(cc2_setfam_1,axiom,(
    ! [A] :
      ( ~ v2_setfam_1(A)
     => ~ v1_xboole_0(A) ) )).

fof(rc3_setfam_1,axiom,(
    ! [A] :
      ( ~ v2_setfam_1(A)
     => ? [B] :
          ( m1_subset_1(B,A)
          & ~ v1_xboole_0(B) ) ) )).

fof(d1_setfam_1,axiom,(
    ! [A,B] :
      ( ( A != k1_xboole_0
       => ( B = k1_setfam_1(A)
        <=> ! [C] :
              ( r2_hidden(C,B)
            <=> ! [D] :
                  ( r2_hidden(D,A)
                 => r2_hidden(C,D) ) ) ) )
      & ( A = k1_xboole_0
       => ( B = k1_setfam_1(A)
        <=> B = k1_xboole_0 ) ) ) )).

fof(t1_setfam_1,axiom,(
    $true )).

fof(t2_setfam_1,axiom,(
    k1_setfam_1(k1_xboole_0) = k1_xboole_0 )).

fof(t3_setfam_1,axiom,(
    ! [A] : r1_tarski(k1_setfam_1(A),k3_tarski(A)) )).

fof(t4_setfam_1,axiom,(
    ! [A,B] :
      ( r2_hidden(A,B)
     => r1_tarski(k1_setfam_1(B),A) ) )).

fof(t5_setfam_1,axiom,(
    ! [A] :
      ( r2_hidden(k1_xboole_0,A)
     => k1_setfam_1(A) = k1_xboole_0 ) )).

fof(t6_setfam_1,axiom,(
    ! [A,B] :
      ( ! [C] :
          ( r2_hidden(C,A)
         => r1_tarski(B,C) )
     => ( A = k1_xboole_0
        | r1_tarski(B,k1_setfam_1(A)) ) ) )).

fof(t7_setfam_1,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => ( A = k1_xboole_0
        | r1_tarski(k1_setfam_1(B),k1_setfam_1(A)) ) ) )).

fof(t8_setfam_1,axiom,(
    ! [A,B,C] :
      ( ( r2_hidden(A,B)
        & r1_tarski(A,C) )
     => r1_tarski(k1_setfam_1(B),C) ) )).

fof(t9_setfam_1,axiom,(
    ! [A,B,C] :
      ( ( r2_hidden(A,B)
        & r1_xboole_0(A,C) )
     => r1_xboole_0(k1_setfam_1(B),C) ) )).

fof(t10_setfam_1,axiom,(
    ! [A,B] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & k1_setfam_1(k2_xboole_0(A,B)) != k3_xboole_0(k1_setfam_1(A),k1_setfam_1(B)) ) )).

fof(t11_setfam_1,axiom,(
    ! [A] : k1_setfam_1(k1_tarski(A)) = A )).

fof(t12_setfam_1,axiom,(
    ! [A,B] : k1_setfam_1(k2_tarski(A,B)) = k3_xboole_0(A,B) )).

fof(d2_setfam_1,axiom,(
    ! [A,B] :
      ( r1_setfam_1(A,B)
    <=> ! [C] :
          ~ ( r2_hidden(C,A)
            & ! [D] :
                ~ ( r2_hidden(D,B)
                  & r1_tarski(C,D) ) ) ) )).

fof(d3_setfam_1,axiom,(
    ! [A,B] :
      ( r2_setfam_1(A,B)
    <=> ! [C] :
          ~ ( r2_hidden(C,B)
            & ! [D] :
                ~ ( r2_hidden(D,A)
                  & r1_tarski(D,C) ) ) ) )).

fof(t13_setfam_1,axiom,(
    $true )).

fof(t14_setfam_1,axiom,(
    $true )).

fof(t15_setfam_1,axiom,(
    $true )).

fof(t16_setfam_1,axiom,(
    $true )).

fof(t17_setfam_1,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r1_setfam_1(A,B) ) )).

fof(t18_setfam_1,axiom,(
    ! [A,B] :
      ( r1_setfam_1(A,B)
     => r1_tarski(k3_tarski(A),k3_tarski(B)) ) )).

fof(t19_setfam_1,axiom,(
    ! [A,B] :
      ( r2_setfam_1(B,A)
     => ( A = k1_xboole_0
        | r1_tarski(k1_setfam_1(B),k1_setfam_1(A)) ) ) )).

fof(t20_setfam_1,axiom,(
    ! [A] : r1_setfam_1(k1_xboole_0,A) )).

fof(t21_setfam_1,axiom,(
    ! [A] :
      ( r1_setfam_1(A,k1_xboole_0)
     => A = k1_xboole_0 ) )).

fof(t22_setfam_1,axiom,(
    $true )).

fof(t23_setfam_1,axiom,(
    ! [A,B,C] :
      ( ( r1_setfam_1(A,B)
        & r1_setfam_1(B,C) )
     => r1_setfam_1(A,C) ) )).

fof(t24_setfam_1,axiom,(
    ! [A,B] :
      ( r1_setfam_1(B,k1_tarski(A))
     => ! [C] :
          ( r2_hidden(C,B)
         => r1_tarski(C,A) ) ) )).

fof(t25_setfam_1,axiom,(
    ! [A,B,C] :
      ( r1_setfam_1(C,k2_tarski(A,B))
     => ! [D] :
          ~ ( r2_hidden(D,C)
            & ~ r1_tarski(D,A)
            & ~ r1_tarski(D,B) ) ) )).

fof(d4_setfam_1,axiom,(
    ! [A,B,C] :
      ( C = k2_setfam_1(A,B)
    <=> ! [D] :
          ( r2_hidden(D,C)
        <=> ? [E,F] :
              ( r2_hidden(E,A)
              & r2_hidden(F,B)
              & D = k2_xboole_0(E,F) ) ) ) )).

fof(d5_setfam_1,axiom,(
    ! [A,B,C] :
      ( C = k3_setfam_1(A,B)
    <=> ! [D] :
          ( r2_hidden(D,C)
        <=> ? [E,F] :
              ( r2_hidden(E,A)
              & r2_hidden(F,B)
              & D = k3_xboole_0(E,F) ) ) ) )).

fof(d6_setfam_1,axiom,(
    ! [A,B,C] :
      ( C = k4_setfam_1(A,B)
    <=> ! [D] :
          ( r2_hidden(D,C)
        <=> ? [E,F] :
              ( r2_hidden(E,A)
              & r2_hidden(F,B)
              & D = k4_xboole_0(E,F) ) ) ) )).

fof(t26_setfam_1,axiom,(
    $true )).

fof(t27_setfam_1,axiom,(
    $true )).

fof(t28_setfam_1,axiom,(
    $true )).

fof(t29_setfam_1,axiom,(
    ! [A] : r1_setfam_1(A,k2_setfam_1(A,A)) )).

fof(t30_setfam_1,axiom,(
    ! [A] : r1_setfam_1(k3_setfam_1(A,A),A) )).

fof(t31_setfam_1,axiom,(
    ! [A] : r1_setfam_1(k4_setfam_1(A,A),A) )).

fof(t32_setfam_1,axiom,(
    $true )).

fof(t33_setfam_1,axiom,(
    $true )).

fof(t34_setfam_1,axiom,(
    ! [A,B] :
      ( ~ r1_xboole_0(A,B)
     => k3_xboole_0(k1_setfam_1(A),k1_setfam_1(B)) = k1_setfam_1(k3_setfam_1(A,B)) ) )).

fof(t35_setfam_1,axiom,(
    ! [A,B] :
      ( B != k1_xboole_0
     => k2_xboole_0(A,k1_setfam_1(B)) = k1_setfam_1(k2_setfam_1(k1_tarski(A),B)) ) )).

fof(t36_setfam_1,axiom,(
    ! [A,B] : k3_xboole_0(A,k3_tarski(B)) = k3_tarski(k3_setfam_1(k1_tarski(A),B)) )).

fof(t37_setfam_1,axiom,(
    ! [A,B] :
      ( B != k1_xboole_0
     => k4_xboole_0(A,k3_tarski(B)) = k1_setfam_1(k4_setfam_1(k1_tarski(A),B)) ) )).

fof(t38_setfam_1,axiom,(
    ! [A,B] :
      ( B != k1_xboole_0
     => k4_xboole_0(A,k1_setfam_1(B)) = k3_tarski(k4_setfam_1(k1_tarski(A),B)) ) )).

fof(t39_setfam_1,axiom,(
    ! [A,B] : k3_tarski(k3_setfam_1(A,B)) = k3_xboole_0(k3_tarski(A),k3_tarski(B)) )).

fof(t40_setfam_1,axiom,(
    ! [A,B] :
      ~ ( A != k1_xboole_0
        & B != k1_xboole_0
        & ~ r1_tarski(k2_xboole_0(k1_setfam_1(A),k1_setfam_1(B)),k1_setfam_1(k2_setfam_1(A,B))) ) )).

fof(t41_setfam_1,axiom,(
    ! [A,B] : r1_tarski(k1_setfam_1(k4_setfam_1(A,B)),k4_xboole_0(k1_setfam_1(A),k1_setfam_1(B))) )).

fof(t42_setfam_1,axiom,(
    $true )).

fof(t43_setfam_1,axiom,(
    $true )).

fof(t44_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,B)
                <=> r2_hidden(D,C) ) )
           => B = C ) ) ) )).

fof(d7_setfam_1,axiom,(
    $true )).

fof(d8_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( C = k7_setfam_1(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,C)
                <=> r2_hidden(k3_subset_1(A,D),B) ) ) ) ) ) )).

fof(t45_setfam_1,axiom,(
    $true )).

fof(t46_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ~ ( B != k1_xboole_0
          & k7_setfam_1(A,B) = k1_xboole_0 ) ) )).

fof(t47_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( B != k1_xboole_0
       => k6_subset_1(A,k2_subset_1(A),k5_setfam_1(A,B)) = k6_setfam_1(A,k7_setfam_1(A,B)) ) ) )).

fof(t48_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( B != k1_xboole_0
       => k5_setfam_1(A,k7_setfam_1(A,B)) = k6_subset_1(A,k2_subset_1(A),k6_setfam_1(A,B)) ) ) )).

fof(t49_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r2_hidden(k3_subset_1(A,C),k7_setfam_1(A,B))
          <=> r2_hidden(C,B) ) ) ) )).

fof(t50_setfam_1,axiom,(
    $true )).

fof(t51_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( r1_tarski(k7_setfam_1(A,B),k7_setfam_1(A,C))
           => r1_tarski(B,C) ) ) ) )).

fof(t52_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( r1_tarski(k7_setfam_1(A,B),C)
          <=> r1_tarski(B,k7_setfam_1(A,C)) ) ) ) )).

fof(t53_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( k7_setfam_1(A,B) = k7_setfam_1(A,C)
           => B = C ) ) ) )).

fof(t54_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => k7_setfam_1(A,k4_subset_1(k1_zfmisc_1(A),B,C)) = k4_subset_1(k1_zfmisc_1(A),k7_setfam_1(A,B),k7_setfam_1(A,C)) ) ) )).

fof(t55_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( B = k1_tarski(A)
       => k7_setfam_1(A,B) = k1_tarski(k1_xboole_0) ) ) )).

fof(d9_setfam_1,axiom,(
    ! [A] :
      ( v1_setfam_1(A)
    <=> ~ r2_hidden(k1_xboole_0,A) ) )).

fof(t56_setfam_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(k3_tarski(A),B)
        & r2_hidden(C,A) )
     => r1_tarski(C,B) ) )).

fof(t57_setfam_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(C,k3_tarski(k2_xboole_0(A,B)))
        & ! [D] :
            ( r2_hidden(D,B)
           => r1_xboole_0(D,C) ) )
     => r1_tarski(C,k3_tarski(A)) ) )).

fof(d10_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ( B != k1_xboole_0
         => k8_setfam_1(A,B) = k6_setfam_1(A,B) )
        & ( B = k1_xboole_0
         => k8_setfam_1(A,B) = A ) ) ) )).

fof(t58_setfam_1,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( r2_hidden(B,A)
       => ( r2_hidden(B,k8_setfam_1(A,C))
        <=> ! [D] :
              ( r2_hidden(D,C)
             => r2_hidden(B,D) ) ) ) ) )).

fof(t59_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( r1_tarski(B,C)
           => r1_tarski(k8_setfam_1(A,C),k8_setfam_1(A,B)) ) ) ) )).

fof(d11_setfam_1,axiom,(
    ! [A] :
      ( v2_setfam_1(A)
    <=> ! [B] :
          ( ~ v1_xboole_0(B)
         => ~ r2_hidden(B,A) ) ) )).

fof(reflexivity_r1_setfam_1,axiom,(
    ! [A,B] : r1_setfam_1(A,A) )).

fof(reflexivity_r2_setfam_1,axiom,(
    ! [A,B] : r2_setfam_1(B,B) )).

fof(dt_k1_setfam_1,axiom,(
    $true )).

fof(dt_k2_setfam_1,axiom,(
    $true )).

fof(commutativity_k2_setfam_1,axiom,(
    ! [A,B] : k2_setfam_1(A,B) = k2_setfam_1(B,A) )).

fof(dt_k3_setfam_1,axiom,(
    $true )).

fof(commutativity_k3_setfam_1,axiom,(
    ! [A,B] : k3_setfam_1(A,B) = k3_setfam_1(B,A) )).

fof(dt_k4_setfam_1,axiom,(
    $true )).

fof(dt_k5_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k5_setfam_1(A,B),k1_zfmisc_1(A)) ) )).

fof(redefinition_k5_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => k5_setfam_1(A,B) = k3_tarski(B) ) )).

fof(dt_k6_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k6_setfam_1(A,B),k1_zfmisc_1(A)) ) )).

fof(redefinition_k6_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => k6_setfam_1(A,B) = k1_setfam_1(B) ) )).

fof(dt_k7_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k7_setfam_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(involutiveness_k7_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => k7_setfam_1(A,k7_setfam_1(A,B)) = B ) )).

fof(dt_k8_setfam_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k8_setfam_1(A,B),k1_zfmisc_1(A)) ) )).
%------------------------------------------------------------------------------