SET007 Axioms: SET007+457.ax


%------------------------------------------------------------------------------
% File     : SET007+457 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : More on the Lattice of Many Sorted Equivalence 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    : msualg_7 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   0 unit)
%            Number of atoms       :  274 (  24 equality)
%            Maximal formula depth :   17 (   9 average)
%            Number of connectives :  277 (  38 ~  ;   2  |; 116  &)
%                                         (   8 <=>; 113 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   35 (   0 propositional; 1-4 arity)
%            Number of functors    :   25 (   5 constant; 0-3 arity)
%            Number of variables   :  104 (   0 singleton;  96 !;   8 ?)
%            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(fc1_msualg_7,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_pboole(B,A) )
     => ( ~ v3_struct_0(k5_msualg_5(A,B))
        & v3_lattices(k5_msualg_5(A,B))
        & v4_lattices(k5_msualg_5(A,B))
        & v5_lattices(k5_msualg_5(A,B))
        & v6_lattices(k5_msualg_5(A,B))
        & v7_lattices(k5_msualg_5(A,B))
        & v8_lattices(k5_msualg_5(A,B))
        & v9_lattices(k5_msualg_5(A,B))
        & v10_lattices(k5_msualg_5(A,B))
        & v13_lattices(k5_msualg_5(A,B))
        & v14_lattices(k5_msualg_5(A,B))
        & v15_lattices(k5_msualg_5(A,B)) ) ) )).

fof(fc2_msualg_7,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_pboole(B,A) )
     => ( ~ v3_struct_0(k5_msualg_5(A,B))
        & v3_lattices(k5_msualg_5(A,B))
        & v4_lattices(k5_msualg_5(A,B))
        & v5_lattices(k5_msualg_5(A,B))
        & v6_lattices(k5_msualg_5(A,B))
        & v7_lattices(k5_msualg_5(A,B))
        & v8_lattices(k5_msualg_5(A,B))
        & v9_lattices(k5_msualg_5(A,B))
        & v10_lattices(k5_msualg_5(A,B))
        & v13_lattices(k5_msualg_5(A,B))
        & v14_lattices(k5_msualg_5(A,B))
        & v15_lattices(k5_msualg_5(A,B))
        & v4_lattice3(k5_msualg_5(A,B)) ) ) )).

fof(rc1_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ? [B] :
          ( m2_nat_lat(B,A)
          & ~ v3_struct_0(B)
          & v4_lattices(B)
          & v5_lattices(B)
          & v6_lattices(B)
          & v7_lattices(B)
          & v8_lattices(B)
          & v9_lattices(B)
          & v10_lattices(B)
          & v4_lattice3(B) ) ) )).

fof(cc1_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v1_msualg_7(B,A)
           => v4_lattice3(B) ) ) ) )).

fof(cc2_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v2_msualg_7(B,A)
           => v4_lattice3(B) ) ) ) )).

fof(t1_msualg_7,axiom,(
    ! [A,B] :
      ( r2_hidden(A,u1_struct_0(k2_msualg_5(B)))
    <=> ( v3_relat_2(A)
        & v8_relat_2(A)
        & v1_partfun1(A,B,B)
        & m2_relset_1(A,B,B) ) ) )).

fof(t2_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ( v2_msualg_4(k2_msualg_3(A,B),A,B)
            & m1_msualg_4(k2_msualg_3(A,B),A,B,B) ) ) ) )).

fof(t3_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ( v2_msualg_4(k11_pboole(A,B,B),A,B)
            & m1_msualg_4(k11_pboole(A,B,B),A,B,B) ) ) ) )).

fof(t4_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => k5_lattices(k5_msualg_5(A,B)) = k2_msualg_3(A,B) ) ) )).

fof(t5_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => k6_lattices(k5_msualg_5(A,B)) = k11_pboole(A,B,B) ) ) )).

fof(t6_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B))))
             => m1_subset_1(C,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B)))) ) ) ) )).

fof(t7_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k5_msualg_5(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k5_msualg_5(A,B)))
                 => ! [E] :
                      ( ( v2_msualg_4(E,A,B)
                        & m1_msualg_4(E,A,B,B) )
                     => ! [F] :
                          ( ( v2_msualg_4(F,A,B)
                            & m1_msualg_4(F,A,B,B) )
                         => ( ( C = E
                              & D = F )
                           => ( r3_lattices(k5_msualg_5(A,B),C,D)
                            <=> r2_pboole(A,E,F) ) ) ) ) ) ) ) ) )).

fof(t8_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B))))
                 => ( D = C
                   => ! [E] :
                        ( ( v2_msualg_4(E,A,B)
                          & m1_msualg_4(E,A,B,B) )
                       => ! [F] :
                            ( ( v2_msualg_4(F,A,B)
                              & m1_msualg_4(F,A,B,B) )
                           => ( ( r6_pboole(A,E,k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)))
                                & r2_hidden(F,C) )
                             => r2_pboole(A,E,F) ) ) ) ) ) ) ) ) )).

fof(t9_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B))))
                 => ( D = C
                   => ( v1_xboole_0(C)
                      | ( v2_msualg_4(k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)),A,B)
                        & m1_msualg_4(k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)),A,B,B) ) ) ) ) ) ) ) )).

fof(d1_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l3_lattices(A) )
     => ( v4_lattice3(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & r4_lattice3(A,C,B)
                & ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r4_lattice3(A,D,B)
                     => r1_lattices(A,C,D) ) ) ) ) ) ) )).

fof(t10_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => v4_lattice3(k5_msualg_5(A,B)) ) ) )).

fof(t11_msualg_7,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_pboole(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_msualg_5(A,B))))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k1_closure2(A,k11_pboole(A,B,B))))
                 => ( D = C
                   => ( v1_xboole_0(C)
                      | ! [E] :
                          ( ( v2_msualg_4(E,A,B)
                            & m1_msualg_4(E,A,B,B) )
                         => ! [F] :
                              ( ( v2_msualg_4(F,A,B)
                                & m1_msualg_4(F,A,B,B) )
                             => ( ( r6_pboole(A,E,k6_mssubfam(A,k11_pboole(A,B,B),k5_closure2(A,k11_pboole(A,B,B),D)))
                                  & F = k16_lattice3(k5_msualg_5(A,B),C) )
                               => r6_pboole(A,E,F) ) ) ) ) ) ) ) ) ) )).

fof(d2_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v1_msualg_7(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
               => r2_hidden(k16_lattice3(A,C),u1_struct_0(B)) ) ) ) ) )).

fof(d3_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v2_msualg_7(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
               => r2_hidden(k15_lattice3(A,C),u1_struct_0(B)) ) ) ) ) )).

fof(t12_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                         => ( ( C = E
                              & D = F )
                           => ( k3_lattices(A,C,D) = k3_lattices(B,E,F)
                              & k4_lattices(A,C,D) = k4_lattices(B,E,F) ) ) ) ) ) ) ) ) )).

fof(t13_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ( D = E
                       => ( r3_lattice3(A,D,C)
                        <=> r3_lattice3(B,E,C) ) ) ) ) ) ) ) )).

fof(t14_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ( D = E
                       => ( r4_lattice3(A,D,C)
                        <=> r4_lattice3(B,E,C) ) ) ) ) ) ) ) )).

fof(t15_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v1_msualg_7(B,A)
           => v4_lattice3(B) ) ) ) )).

fof(t16_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v2_msualg_7(B,A)
           => v4_lattice3(B) ) ) ) )).

fof(t17_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v1_msualg_7(B,A)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
               => k16_lattice3(A,C) = k16_lattice3(B,C) ) ) ) ) )).

fof(t18_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v2_msualg_7(B,A)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
               => k15_lattice3(A,C) = k15_lattice3(B,C) ) ) ) ) )).

fof(t19_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v1_msualg_7(B,A)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                   => ( C = D
                     => k16_lattice3(A,C) = k16_lattice3(B,D) ) ) ) ) ) ) )).

fof(t20_msualg_7,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v10_lattices(A)
        & v4_lattice3(A)
        & l3_lattices(A) )
     => ! [B] :
          ( m2_nat_lat(B,A)
         => ( v2_msualg_7(B,A)
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                   => ( C = D
                     => k15_lattice3(A,C) = k15_lattice3(B,D) ) ) ) ) ) ) )).

fof(d4_msualg_7,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r1_xreal_0(A,B)
           => ! [C] :
                ( ( ~ v3_struct_0(C)
                  & v3_lattices(C)
                  & v10_lattices(C)
                  & l3_lattices(C) )
               => ( C = k1_msualg_7(A,B)
                <=> ( u1_struct_0(C) = k1_rcomp_1(A,B)
                    & u2_lattices(C) = k1_realset1(k2_real_lat,k1_rcomp_1(A,B))
                    & u1_lattices(C) = k1_realset1(k1_real_lat,k1_rcomp_1(A,B)) ) ) ) ) ) ) )).

fof(t21_msualg_7,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r1_xreal_0(A,B)
           => v4_lattice3(k1_msualg_7(A,B)) ) ) ) )).

fof(t22_msualg_7,axiom,(
    ? [A] :
      ( m2_nat_lat(A,k1_msualg_7(np__0,np__1))
      & v2_msualg_7(A,k1_msualg_7(np__0,np__1))
      & ~ v1_msualg_7(A,k1_msualg_7(np__0,np__1)) ) )).

fof(t23_msualg_7,axiom,(
    ? [A] :
      ( ~ v3_struct_0(A)
      & v10_lattices(A)
      & v4_lattice3(A)
      & l3_lattices(A)
      & ? [B] :
          ( m2_nat_lat(B,A)
          & v2_msualg_7(B,A)
          & ~ v1_msualg_7(B,A) ) ) )).

fof(t24_msualg_7,axiom,(
    ? [A] :
      ( m2_nat_lat(A,k1_msualg_7(np__0,np__1))
      & v1_msualg_7(A,k1_msualg_7(np__0,np__1))
      & ~ v2_msualg_7(A,k1_msualg_7(np__0,np__1)) ) )).

fof(t25_msualg_7,axiom,(
    ? [A] :
      ( ~ v3_struct_0(A)
      & v10_lattices(A)
      & v4_lattice3(A)
      & l3_lattices(A)
      & ? [B] :
          ( m2_nat_lat(B,A)
          & v1_msualg_7(B,A)
          & ~ v2_msualg_7(B,A) ) ) )).

fof(dt_k1_msualg_7,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => ( ~ v3_struct_0(k1_msualg_7(A,B))
        & v3_lattices(k1_msualg_7(A,B))
        & v10_lattices(k1_msualg_7(A,B))
        & l3_lattices(k1_msualg_7(A,B)) ) ) )).
%------------------------------------------------------------------------------