SET007 Axioms: SET007+662.ax


%------------------------------------------------------------------------------
% File     : SET007+662 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Isomorphism between Finite Chains
% 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    : orders_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   19 (   1 unit)
%            Number of atoms       :  137 (  13 equality)
%            Maximal formula depth :   26 (   8 average)
%            Number of connectives :  136 (  18 ~  ;   1  |;  70  &)
%                                         (   3 <=>;  44 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   36 (   0 propositional; 1-3 arity)
%            Number of functors    :   13 (   3 constant; 0-2 arity)
%            Number of variables   :   39 (   0 singleton;  35 !;   4 ?)
%            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(cc1_orders_4,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v3_struct_0(A)
       => ( v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A) ) ) ) )).

fof(cc2_orders_4,axiom,(
    ! [A] :
      ( m1_orders_4(A)
     => ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A) ) ) )).

fof(rc1_orders_4,axiom,(
    ? [A] :
      ( m1_orders_4(A)
      & ~ v3_struct_0(A)
      & v2_orders_2(A)
      & v3_orders_2(A)
      & v4_orders_2(A) ) )).

fof(cc3_orders_4,axiom,(
    ! [A] :
      ( m1_orders_4(A)
     => ( ~ v3_struct_0(A)
       => ( ~ v3_struct_0(A)
          & v2_orders_2(A)
          & v3_orders_2(A)
          & v4_orders_2(A)
          & v16_waybel_0(A)
          & v1_lattice3(A)
          & v2_lattice3(A) ) ) ) )).

fof(rc2_orders_4,axiom,(
    ? [A] :
      ( m1_orders_4(A)
      & ~ v3_struct_0(A)
      & v2_orders_2(A)
      & v3_orders_2(A)
      & v4_orders_2(A)
      & v2_yellow_0(A)
      & v16_waybel_0(A)
      & v24_waybel_0(A)
      & v1_waybel_8(A)
      & v2_waybel_8(A)
      & v3_waybel_8(A)
      & ~ v1_yellow_3(A)
      & v1_lattice3(A)
      & v2_lattice3(A)
      & v2_waybel_3(A)
      & v3_waybel_3(A)
      & v6_group_1(A) ) )).

fof(rc3_orders_4,axiom,(
    ? [A] :
      ( m1_orders_4(A)
      & v2_orders_2(A)
      & v3_orders_2(A)
      & v4_orders_2(A)
      & v1_orders_4(A) ) )).

fof(cc4_orders_4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v16_waybel_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_yellow_0(B,A)
         => ( ( ~ v3_struct_0(B)
              & v4_yellow_0(B,A) )
           => ( ~ v3_struct_0(B)
              & v16_waybel_0(B) ) ) ) ) )).

fof(cc5_orders_4,axiom,(
    ! [A] :
      ( ( v6_group_1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_yellow_0(B,A)
         => v6_group_1(B) ) ) )).

fof(d1_orders_4,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( m1_orders_4(A)
      <=> ( ( ~ v3_struct_0(A)
            & v2_orders_2(A)
            & v3_orders_2(A)
            & v4_orders_2(A)
            & v16_waybel_0(A)
            & l1_orders_2(A) )
          | v3_struct_0(A) ) ) ) )).

fof(d2_orders_4,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ( v1_orders_4(A)
      <=> v1_card_4(u1_struct_0(A)) ) ) )).

fof(t1_orders_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
           => ( v4_yellow_0(k2_yellow_1(A),k2_yellow_1(B))
              & m1_yellow_0(k2_yellow_1(A),k2_yellow_1(B)) ) ) ) ) )).

fof(d3_orders_4,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B,C] :
          ( r1_orders_4(A,B,C)
        <=> ( k2_xboole_0(B,C) = u1_struct_0(A)
            & ! [D] :
                ( m1_subset_1(D,u1_struct_0(A))
               => ! [E] :
                    ( m1_subset_1(E,u1_struct_0(A))
                   => ( ( r2_hidden(D,B)
                        & r2_hidden(E,C) )
                     => r2_orders_2(A,D,E) ) ) ) ) ) ) )).

fof(t2_orders_4,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B,C] :
          ( r1_orders_4(A,B,C)
         => r1_xboole_0(B,C) ) ) )).

fof(t3_orders_4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v4_orders_2(A)
        & v2_yellow_0(A)
        & l1_orders_2(A) )
     => r1_orders_4(A,k4_xboole_0(u1_struct_0(A),k1_struct_0(A,k4_yellow_0(A))),k1_struct_0(A,k4_yellow_0(A))) ) )).

fof(t4_orders_4,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ( v23_waybel_0(C,A,B)
               => ( ~ ( u1_struct_0(A) != k1_xboole_0
                      & u1_struct_0(B) = k1_xboole_0 )
                  & ~ ( u1_struct_0(B) != k1_xboole_0
                      & u1_struct_0(A) = k1_xboole_0 )
                  & ~ ( u1_struct_0(B) = k1_xboole_0
                      & u1_struct_0(A) != k1_xboole_0 )
                  & ( u1_struct_0(A) = k1_xboole_0
                   => u1_struct_0(B) = k1_xboole_0 )
                  & ( u1_struct_0(B) = k1_xboole_0
                   => u1_struct_0(A) = k1_xboole_0 ) ) ) ) ) ) )).

fof(t5_orders_4,axiom,(
    ! [A] :
      ( ( v4_orders_2(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( v4_orders_2(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( r1_orders_4(A,C,D)
                   => ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
                       => ! [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B)))
                           => ( r1_orders_4(B,E,F)
                             => ! [G] :
                                  ( ( v1_funct_1(G)
                                    & v1_funct_2(G,u1_struct_0(k5_yellow_0(A,C)),u1_struct_0(k5_yellow_0(B,E)))
                                    & m2_relset_1(G,u1_struct_0(k5_yellow_0(A,C)),u1_struct_0(k5_yellow_0(B,E))) )
                                 => ( v23_waybel_0(G,k5_yellow_0(A,C),k5_yellow_0(B,E))
                                   => ! [H] :
                                        ( ( v1_funct_1(H)
                                          & v1_funct_2(H,u1_struct_0(k5_yellow_0(A,D)),u1_struct_0(k5_yellow_0(B,F)))
                                          & m2_relset_1(H,u1_struct_0(k5_yellow_0(A,D)),u1_struct_0(k5_yellow_0(B,F))) )
                                       => ~ ( v23_waybel_0(H,k5_yellow_0(A,D),k5_yellow_0(B,F))
                                            & ! [I] :
                                                ( ( v1_funct_1(I)
                                                  & v1_funct_2(I,u1_struct_0(A),u1_struct_0(B))
                                                  & m2_relset_1(I,u1_struct_0(A),u1_struct_0(B)) )
                                               => ~ ( I = k1_funct_4(G,H)
                                                    & v23_waybel_0(I,A,B) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t6_orders_4,axiom,(
    ! [A] :
      ( ( v6_group_1(A)
        & m1_orders_4(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k1_card_1(u1_struct_0(A)) = B
           => r5_waybel_1(A,k2_yellow_1(B)) ) ) ) )).

fof(dt_m1_orders_4,axiom,(
    ! [A] :
      ( m1_orders_4(A)
     => l1_orders_2(A) ) )).

fof(existence_m1_orders_4,axiom,(
    ? [A] : m1_orders_4(A) )).
%------------------------------------------------------------------------------