SET007 Axioms: SET007+844.ax


%------------------------------------------------------------------------------
% File     : SET007+844 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Operation of Addition of Relational Structures
% 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    : latsum_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   30 (   0 unit)
%            Number of atoms       :  304 (  12 equality)
%            Maximal formula depth :   21 (  11 average)
%            Number of connectives :  311 (  37 ~  ;   1  |; 162  &)
%                                         (   7 <=>; 104 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   18 (   0 propositional; 1-3 arity)
%            Number of functors    :   11 (   0 constant; 1-6 arity)
%            Number of variables   :  107 (   0 singleton; 106 !;   1 ?)
%            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_latsum_1,axiom,(
    ! [A,B] :
      ( ( l1_orders_2(A)
        & ~ v3_struct_0(B)
        & l1_orders_2(B) )
     => ( ~ v3_struct_0(k1_latsum_1(A,B))
        & v1_orders_2(k1_latsum_1(A,B)) ) ) )).

fof(fc2_latsum_1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A)
        & l1_orders_2(B) )
     => ( ~ v3_struct_0(k1_latsum_1(A,B))
        & v1_orders_2(k1_latsum_1(A,B)) ) ) )).

fof(t1_latsum_1,axiom,(
    ! [A,B,C,D] :
      ~ ( r2_hidden(A,k2_xboole_0(C,D))
        & r2_hidden(B,k2_xboole_0(C,D))
        & ~ ( r2_hidden(A,k4_xboole_0(C,D))
            & r2_hidden(B,k4_xboole_0(C,D)) )
        & ~ ( r2_hidden(A,D)
            & r2_hidden(B,D) )
        & ~ ( r2_hidden(A,k4_xboole_0(C,D))
            & r2_hidden(B,D) )
        & ~ ( r2_hidden(A,D)
            & r2_hidden(B,k4_xboole_0(C,D)) ) ) )).

fof(d1_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ( r1_latsum_1(A,B)
          <=> ! [C,D] :
                ( ( r2_hidden(C,k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
                  & r2_hidden(D,k3_xboole_0(u1_struct_0(A),u1_struct_0(B))) )
               => ( r2_hidden(k4_tarski(C,D),u1_orders_2(A))
                <=> r2_hidden(k4_tarski(C,D),u1_orders_2(B)) ) ) ) ) ) )).

fof(d2_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C] :
              ( ( v1_orders_2(C)
                & l1_orders_2(C) )
             => ( C = k1_latsum_1(A,B)
              <=> ( u1_struct_0(C) = k2_xboole_0(u1_struct_0(A),u1_struct_0(B))
                  & u1_orders_2(C) = k2_xboole_0(k2_xboole_0(u1_orders_2(A),u1_orders_2(B)),k7_relset_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(B),u1_struct_0(B),u1_orders_2(A),u1_orders_2(B))) ) ) ) ) ) )).

fof(t2_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ( r1_tarski(u1_orders_2(A),u1_orders_2(k1_latsum_1(A,B)))
            & r1_tarski(u1_orders_2(B),u1_orders_2(k1_latsum_1(A,B))) ) ) ) )).

fof(t3_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ( ( v2_orders_2(A)
              & v2_orders_2(B) )
           => v2_orders_2(k1_latsum_1(A,B)) ) ) ) )).

fof(t4_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C,D] :
              ( ( r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B)))
                & r2_hidden(C,u1_struct_0(A))
                & r2_hidden(D,u1_struct_0(A))
                & r1_latsum_1(A,B)
                & v3_orders_2(A) )
             => r2_hidden(k4_tarski(C,D),u1_orders_2(A)) ) ) ) )).

fof(t5_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C,D] :
              ( ( r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B)))
                & r2_hidden(C,u1_struct_0(B))
                & r2_hidden(D,u1_struct_0(B))
                & r1_latsum_1(A,B)
                & v3_orders_2(B) )
             => r2_hidden(k4_tarski(C,D),u1_orders_2(B)) ) ) ) )).

fof(t6_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C,D] :
              ( ( r2_hidden(k4_tarski(C,D),u1_orders_2(A))
               => r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B))) )
              & ( r2_hidden(k4_tarski(C,D),u1_orders_2(B))
               => r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B))) ) ) ) ) )).

fof(t7_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
             => ( r2_hidden(C,u1_struct_0(A))
                | r2_hidden(C,k4_xboole_0(u1_struct_0(B),u1_struct_0(A))) ) ) ) ) )).

fof(t8_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ! [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(k1_latsum_1(A,B)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k1_latsum_1(A,B)))
                         => ( ( C = E
                              & D = F
                              & r1_latsum_1(A,B)
                              & v3_orders_2(A) )
                           => ( r1_orders_2(A,C,D)
                            <=> r1_orders_2(k1_latsum_1(A,B),E,F) ) ) ) ) ) ) ) ) )).

fof(t9_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k1_latsum_1(A,B)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(B))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                         => ( ( C = E
                              & D = F
                              & r1_latsum_1(A,B)
                              & v3_orders_2(B) )
                           => ( r1_orders_2(k1_latsum_1(A,B),C,D)
                            <=> r1_orders_2(B,E,F) ) ) ) ) ) ) ) ) )).

fof(t10_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & v1_lattice3(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( r2_hidden(C,u1_struct_0(A))
             => m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B))) ) ) ) )).

fof(t11_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & v1_lattice3(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( r2_hidden(C,u1_struct_0(B))
             => m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B))) ) ) ) )).

fof(t12_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( r2_hidden(C,k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
             => m1_subset_1(C,u1_struct_0(A)) ) ) ) )).

fof(t13_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( r2_hidden(C,k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
             => m1_subset_1(C,u1_struct_0(B)) ) ) ) )).

fof(t14_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & v1_lattice3(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k1_latsum_1(A,B)))
                 => ( ( r2_hidden(C,u1_struct_0(A))
                      & r2_hidden(D,u1_struct_0(B))
                      & r1_latsum_1(A,B) )
                   => ( r1_orders_2(k1_latsum_1(A,B),C,D)
                    <=> ? [E] :
                          ( m1_subset_1(E,u1_struct_0(k1_latsum_1(A,B)))
                          & r2_hidden(E,k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
                          & r1_orders_2(k1_latsum_1(A,B),C,E)
                          & r1_orders_2(k1_latsum_1(A,B),E,D) ) ) ) ) ) ) ) )).

fof(t15_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ! [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
                              & r1_latsum_1(A,B)
                              & v3_orders_2(A)
                              & v3_orders_2(B) )
                           => ( r1_orders_2(A,C,D)
                            <=> r1_orders_2(B,E,F) ) ) ) ) ) ) ) ) )).

fof(t16_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( v1_waybel_0(B,A)
            & v12_waybel_0(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ( ( r2_hidden(C,B)
                      & r2_hidden(D,B) )
                   => r2_hidden(k13_lattice3(A,C,D),B) ) ) ) ) ) )).

fof(t17_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C,D] :
              ( ( v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
                & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
                & r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B)))
                & r2_hidden(C,u1_struct_0(B)) )
             => r2_hidden(D,u1_struct_0(B)) ) ) ) )).

fof(t18_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k1_latsum_1(A,B)))
                 => ( ( v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
                      & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
                      & r1_orders_2(k1_latsum_1(A,B),C,D)
                      & r2_hidden(C,u1_struct_0(B)) )
                   => r2_hidden(D,u1_struct_0(B)) ) ) ) ) ) )).

fof(t19_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & v1_lattice3(B)
            & l1_orders_2(B) )
         => ! [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))
                         => ( ( v1_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
                              & v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
                              & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
                              & v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
                              & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
                              & r1_latsum_1(A,B)
                              & C = E
                              & D = F )
                           => k13_lattice3(A,C,D) = k13_lattice3(B,E,F) ) ) ) ) ) ) ) )).

fof(t20_latsum_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_yellow_0(A)
        & v1_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & v1_yellow_0(B)
            & v1_lattice3(B)
            & l1_orders_2(B) )
         => ( ( ~ v1_xboole_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)))
              & v1_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
              & v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
              & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B))) )
           => r2_hidden(k3_yellow_0(B),u1_struct_0(A)) ) ) ) )).

fof(t21_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C,D] :
              ( ( v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
                & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
                & r2_hidden(k4_tarski(C,D),u1_orders_2(k1_latsum_1(A,B)))
                & r2_hidden(D,u1_struct_0(A)) )
             => r2_hidden(C,u1_struct_0(A)) ) ) ) )).

fof(t22_latsum_1,axiom,(
    ! [A,B,C] :
      ( l1_orders_2(C)
     => ! [D] :
          ( l1_orders_2(D)
         => ~ ( r2_hidden(k4_tarski(A,B),u1_orders_2(k1_latsum_1(C,D)))
              & v13_waybel_0(k3_xboole_0(u1_struct_0(C),u1_struct_0(D)),C)
              & m1_subset_1(k3_xboole_0(u1_struct_0(C),u1_struct_0(D)),k1_zfmisc_1(u1_struct_0(C)))
              & ~ ( r2_hidden(A,u1_struct_0(C))
                  & r2_hidden(B,u1_struct_0(C)) )
              & ~ ( r2_hidden(A,u1_struct_0(D))
                  & r2_hidden(B,u1_struct_0(D)) )
              & ~ ( r2_hidden(A,k4_xboole_0(u1_struct_0(C),u1_struct_0(D)))
                  & r2_hidden(B,k4_xboole_0(u1_struct_0(D),u1_struct_0(C))) ) ) ) ) )).

fof(t23_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k1_latsum_1(A,B)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k1_latsum_1(A,B)))
                 => ( ( v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
                      & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
                      & r1_orders_2(k1_latsum_1(A,B),C,D)
                      & r2_hidden(D,u1_struct_0(A)) )
                   => r2_hidden(C,u1_struct_0(A)) ) ) ) ) ) )).

fof(t24_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ( ( r1_latsum_1(A,B)
              & v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
              & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
              & v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
              & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
              & v3_orders_2(A)
              & v4_orders_2(A)
              & v3_orders_2(B)
              & v4_orders_2(B) )
           => v4_orders_2(k1_latsum_1(A,B)) ) ) ) )).

fof(t25_latsum_1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ( ( v13_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),A)
              & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(A)))
              & v12_waybel_0(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),B)
              & m1_subset_1(k3_xboole_0(u1_struct_0(A),u1_struct_0(B)),k1_zfmisc_1(u1_struct_0(B)))
              & r1_latsum_1(A,B)
              & v3_orders_2(A)
              & v3_orders_2(B) )
           => v3_orders_2(k1_latsum_1(A,B)) ) ) ) )).

fof(dt_k1_latsum_1,axiom,(
    ! [A,B] :
      ( ( l1_orders_2(A)
        & l1_orders_2(B) )
     => ( v1_orders_2(k1_latsum_1(A,B))
        & l1_orders_2(k1_latsum_1(A,B)) ) ) )).
%------------------------------------------------------------------------------