%------------------------------------------------------------------------------
% File     : SET007+517 : TPTP v9.0.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Algebraic Operation on Subsets of Many Sorted 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    : closure3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   48 (   2 unt;   0 def)
%            Number of atoms       :  283 (  26 equ)
%            Maximal formula atoms :   12 (   5 avg)
%            Number of connectives :  274 (  39   ~;   0   |; 118   &)
%                                         (  12 <=>; 105  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   9 avg)
%            Maximal term depth    :    4 (   1 avg)
%            Number of predicates  :   44 (  42 usr;   1 prp; 0-4 aty)
%            Number of functors    :   32 (  32 usr;   1 con; 0-4 aty)
%            Number of variables   :  177 ( 171   !;   6   ?)
% 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_closure3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ( v1_struct_0(g1_struct_0(u1_struct_0(A)))
        & ~ v3_struct_0(g1_struct_0(u1_struct_0(A))) ) ) ).

fof(fc2_closure3,axiom,
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => ( v1_relat_1(k2_closure3(A,B,C))
        & v3_relat_1(k2_closure3(A,B,C))
        & v1_funct_1(k2_closure3(A,B,C))
        & v1_pre_circ(k2_closure3(A,B,C),A) ) ) ).

fof(rc1_closure3,axiom,
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_pboole(B,A) )
     => ? [C] :
          ( m1_relset_1(C,k6_closure2(A,B),k6_closure2(A,B))
          & ~ v1_xboole_0(C)
          & v1_relat_1(C)
          & v1_funct_1(C)
          & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
          & v1_partfun1(C,k6_closure2(A,B),k6_closure2(A,B))
          & v7_closure2(C,A,B)
          & v8_closure2(C,A,B)
          & v9_closure2(C,A,B)
          & v1_closure3(C,A,B) ) ) ).

fof(fc3_closure3,axiom,
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v4_msualg_1(B,A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ( v11_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
        & v14_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
        & v15_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
        & v16_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A))) ) ) ).

fof(fc4_closure3,axiom,
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v4_msualg_1(B,A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ( v11_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
        & v14_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
        & v15_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
        & v16_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
        & v2_closure3(k5_closure3(A,B),g1_struct_0(u1_struct_0(A))) ) ) ).

fof(t1_closure3,axiom,
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_pboole(C,A)
             => k1_funct_4(B,C) = C ) ) ) ).

fof(t2_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_pboole(C,A)
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => ( r2_hidden(C,D)
               => r2_pboole(A,k6_mssubfam(A,B,k5_closure2(A,B,D)),C) ) ) ) ) ).

fof(t3_closure3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v4_msualg_1(B,A)
            & v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B))))
             => ( r1_tarski(C,k6_msualg_2(A,B))
               => ! [D] :
                    ( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
                   => ( r6_pboole(u1_struct_0(A),D,k6_mssubfam(u1_struct_0(A),u4_msualg_1(A,B),k5_closure2(u1_struct_0(A),u4_msualg_1(A,B),C)))
                     => v3_msualg_2(D,A,B) ) ) ) ) ) ) ).

fof(d1_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => ( r1_closure3(A,B,C,D)
              <=> ! [E] :
                    ~ ( r2_hidden(E,D)
                      & ! [F] :
                          ~ ( r2_hidden(F,C)
                            & r1_tarski(E,F) ) ) ) ) ) ) ).

fof(d2_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => ( r2_closure3(A,B,C,D)
              <=> ! [E] :
                    ~ ( r2_hidden(E,C)
                      & ! [F] :
                          ~ ( r2_hidden(F,D)
                            & r1_tarski(F,E) ) ) ) ) ) ) ).

fof(t4_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
                 => ( ( r1_closure3(A,B,D,C)
                      & r1_closure3(A,B,E,D) )
                   => r1_closure3(A,B,E,C) ) ) ) ) ) ).

fof(t5_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
                 => ( ( r2_closure3(A,B,C,D)
                      & r2_closure3(A,B,D,E) )
                   => r2_closure3(A,B,C,E) ) ) ) ) ) ).

fof(t6_closure3,axiom,
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,A) )
         => B = k1_funct_4(k1_pboole(A),k7_relat_1(B,k1_closure3(A,B))) ) ) ).

fof(t7_closure3,axiom,
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v2_relat_1(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( ( v2_relat_1(C)
                & m1_pboole(C,A) )
             => ( ( k1_closure3(A,B) = k1_closure3(A,C)
                  & k7_relat_1(B,k1_closure3(A,B)) = k7_relat_1(C,k1_closure3(A,C)) )
               => r6_pboole(A,B,C) ) ) ) ) ).

fof(t8_closure3,axiom,
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( ~ r2_hidden(C,k1_closure3(A,B))
               => k1_funct_1(B,C) = k1_xboole_0 ) ) ) ) ).

fof(t9_closure3,axiom,
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_closure2(C,A,B,k6_closure2(A,B))
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ~ ( r2_hidden(E,k1_funct_1(C,D))
                        & ! [F] :
                            ( m1_closure2(F,A,B,k6_closure2(A,B))
                           => ~ ( r2_hidden(E,k1_funct_1(F,D))
                                & v1_pre_circ(F,A)
                                & v1_finset_1(k1_closure3(A,F))
                                & r2_pboole(A,F,C) ) ) ) ) ) ) ) ).

fof(t10_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => r6_pboole(A,k2_closure3(A,B,C),k2_mboolean(A,k5_closure2(A,B,C))) ) ) ).

fof(t11_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => r6_pboole(A,k2_closure3(A,B,k3_closure3(A,B,C,D)),k2_pboole(A,k2_closure3(A,B,C),k2_closure3(A,B,D))) ) ) ) ).

fof(t12_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => ( r1_tarski(C,D)
               => r2_pboole(A,k2_closure3(A,B,C),k2_closure3(A,B,D)) ) ) ) ) ).

fof(t13_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => r2_pboole(A,k2_closure3(A,B,k4_closure3(A,B,C,D)),k3_pboole(A,k2_closure3(A,B,C),k2_closure3(A,B,D))) ) ) ) ).

fof(d6_closure3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_struct_0(A) )
     => ! [B] :
          ( ( v15_closure2(B,A)
            & l1_closure2(B,A) )
         => ( v2_closure3(B,A)
          <=> v1_closure3(k12_closure2(A,B),u1_struct_0(A),u4_msualg_1(A,B)) ) ) ) ).

fof(d7_closure3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ! [C] :
              ( ( v11_closure2(C,g1_struct_0(u1_struct_0(A)))
                & l1_closure2(C,g1_struct_0(u1_struct_0(A))) )
             => ( C = k5_closure3(A,B)
              <=> ( u4_msualg_1(g1_struct_0(u1_struct_0(A)),C) = u4_msualg_1(A,B)
                  & u1_closure2(g1_struct_0(u1_struct_0(A)),C) = k6_msualg_2(A,B) ) ) ) ) ) ).

fof(t15_closure3,axiom,
    $true ).

fof(t16_closure3,axiom,
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v4_msualg_1(B,A)
            & v5_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ( v4_closure2(k6_msualg_2(A,B),u1_struct_0(A),u4_msualg_1(A,B))
            & m1_subset_1(k6_msualg_2(A,B),k1_zfmisc_1(k1_closure2(u1_struct_0(A),u4_msualg_1(A,B)))) ) ) ) ).

fof(dt_m1_closure3,axiom,
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => ! [D] :
          ( m1_closure3(D,A,B,C)
         => m1_pboole(D,A) ) ) ).

fof(existence_m1_closure3,axiom,
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => ? [D] : m1_closure3(D,A,B,C) ) ).

fof(redefinition_m1_closure3,axiom,
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => ! [D] :
          ( m1_closure3(D,A,B,C)
        <=> m1_subset_1(D,C) ) ) ).

fof(reflexivity_r1_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => r1_closure3(A,B,D,D) ) ).

fof(reflexivity_r2_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => r2_closure3(A,B,C,C) ) ).

fof(dt_k1_closure3,axiom,
    $true ).

fof(dt_k2_closure3,axiom,
    ! [A,B,C] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B))) )
     => m4_pboole(k2_closure3(A,B,C),A,B) ) ).

fof(dt_k3_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => m1_subset_1(k3_closure3(A,B,C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ).

fof(commutativity_k3_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => k3_closure3(A,B,C,D) = k3_closure3(A,B,D,C) ) ).

fof(idempotence_k3_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => k3_closure3(A,B,C,C) = C ) ).

fof(redefinition_k3_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => k3_closure3(A,B,C,D) = k2_xboole_0(C,D) ) ).

fof(dt_k4_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => m1_subset_1(k4_closure3(A,B,C,D),k1_zfmisc_1(k1_closure2(A,B))) ) ).

fof(commutativity_k4_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => k4_closure3(A,B,C,D) = k4_closure3(A,B,D,C) ) ).

fof(idempotence_k4_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => k4_closure3(A,B,C,C) = C ) ).

fof(redefinition_k4_closure3,axiom,
    ! [A,B,C,D] :
      ( ( m1_pboole(B,A)
        & m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
     => k4_closure3(A,B,C,D) = k3_xboole_0(C,D) ) ).

fof(dt_k5_closure3,axiom,
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v5_msualg_1(B,A)
        & l3_msualg_1(B,A) )
     => ( v11_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A)))
        & l1_closure2(k5_closure3(A,B),g1_struct_0(u1_struct_0(A))) ) ) ).

fof(d3_closure3,axiom,
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( C = k1_closure3(A,B)
            <=> C = a_2_0_closure3(A,B) ) ) ) ).

fof(d4_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,B)))
         => ! [D] :
              ( m4_pboole(D,A,B)
             => ( D = k2_closure3(A,B,C)
              <=> ! [E] :
                    ( r2_hidden(E,A)
                   => k1_funct_1(D,E) = k3_tarski(a_4_0_closure3(A,B,C,E)) ) ) ) ) ) ).

fof(t14_closure3,axiom,
    ! [A,B] :
      ( m1_pboole(B,A)
     => ! [C] :
          ( ! [D] :
              ( r2_hidden(D,C)
             => m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B))) )
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,B)))
             => ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
                 => ( ( E = a_3_0_closure3(A,B,C)
                      & D = k3_tarski(C) )
                   => r6_pboole(A,k2_closure3(A,B,E),k2_closure3(A,B,D)) ) ) ) ) ) ).

fof(d5_closure3,axiom,
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k6_closure2(A,B),k6_closure2(A,B))
                & m2_relset_1(C,k6_closure2(A,B),k6_closure2(A,B)) )
             => ( v1_closure3(C,A,B)
              <=> ! [D] :
                    ( m1_closure3(D,A,B,k6_closure2(A,B))
                   => ~ ( r6_pboole(A,D,k7_closure2(A,B,C,D))
                        & ! [E] :
                            ( m1_subset_1(E,k1_zfmisc_1(k1_closure2(A,B)))
                           => ~ ( E = a_4_1_closure3(A,B,C,D)
                                & r6_pboole(A,D,k2_closure3(A,B,E)) ) ) ) ) ) ) ) ) ).

fof(fraenkel_a_2_0_closure3,axiom,
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_pboole(C,B) )
     => ( r2_hidden(A,a_2_0_closure3(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,B)
            & A = D
            & k1_funct_1(C,D) != k1_xboole_0 ) ) ) ).

fof(fraenkel_a_4_0_closure3,axiom,
    ! [A,B,C,D,E] :
      ( ( m1_pboole(C,B)
        & m1_subset_1(D,k1_zfmisc_1(k1_closure2(B,C))) )
     => ( r2_hidden(A,a_4_0_closure3(B,C,D,E))
      <=> ? [F] :
            ( m1_closure2(F,B,C,k6_closure2(B,C))
            & A = k1_funct_1(F,E)
            & r2_hidden(F,D) ) ) ) ).

fof(fraenkel_a_3_0_closure3,axiom,
    ! [A,B,C,D] :
      ( m1_pboole(C,B)
     => ( r2_hidden(A,a_3_0_closure3(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(k1_closure2(B,C)))
            & A = k2_closure3(B,C,E)
            & r2_hidden(E,D) ) ) ) ).

fof(fraenkel_a_4_1_closure3,axiom,
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & m1_pboole(C,B)
        & v1_funct_1(D)
        & v1_funct_2(D,k6_closure2(B,C),k6_closure2(B,C))
        & m2_relset_1(D,k6_closure2(B,C),k6_closure2(B,C))
        & m1_closure3(E,B,C,k6_closure2(B,C)) )
     => ( r2_hidden(A,a_4_1_closure3(B,C,D,E))
      <=> ? [F] :
            ( m1_closure3(F,B,C,k6_closure2(B,C))
            & A = k7_closure2(B,C,D,F)
            & v1_pre_circ(F,B)
            & v1_finset_1(k1_closure3(B,F))
            & r2_pboole(B,F,E) ) ) ) ).

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