SET007 Axioms: SET007+449.ax


%------------------------------------------------------------------------------
% File     : SET007+449 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Category of Posets
% 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   43 (   2 unit)
%            Number of atoms       :  308 (  40 equality)
%            Maximal formula depth :   25 (   8 average)
%            Number of connectives :  303 (  38 ~  ;   0  |; 141  &)
%                                         (  13 <=>; 111 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   40 (   1 propositional; 0-3 arity)
%            Number of functors    :   36 (   3 constant; 0-4 arity)
%            Number of variables   :  122 (   0 singleton; 108 !;  14 ?)
%            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(rc1_orders_3,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & ~ v3_struct_0(A)
      & v1_orders_2(A)
      & v2_orders_2(A)
      & v3_orders_2(A)
      & v4_orders_2(A)
      & v1_orders_3(A) ) )).

fof(rc2_orders_3,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & v3_struct_0(A)
      & v1_orders_2(A)
      & v2_orders_2(A)
      & v3_orders_2(A)
      & v4_orders_2(A)
      & v1_orders_3(A) ) )).

fof(fc1_orders_3,axiom,
    ( v3_struct_0(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0)))
    & v1_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0)))
    & v2_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0)))
    & v3_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0)))
    & v4_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0))) )).

fof(fc2_orders_3,axiom,(
    ! [A] :
      ( ( v3_struct_0(A)
        & l1_orders_2(A) )
     => ( v1_relat_1(u1_orders_2(A))
        & v1_funct_1(u1_orders_2(A))
        & v2_funct_1(u1_orders_2(A))
        & v1_xboole_0(u1_orders_2(A)) ) ) )).

fof(cc1_orders_3,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v3_struct_0(A)
       => v1_orders_3(A) ) ) )).

fof(rc3_orders_3,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & ~ v3_struct_0(A)
      & v1_orders_2(A)
      & v2_orders_2(A)
      & v3_orders_2(A)
      & v4_orders_2(A)
      & ~ v3_orders_3(A) ) )).

fof(rc4_orders_3,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & ~ v3_struct_0(A)
      & v1_orders_2(A)
      & v2_orders_2(A)
      & v3_orders_2(A)
      & v4_orders_2(A)
      & v1_orders_3(A)
      & v3_orders_3(A) ) )).

fof(rc5_orders_3,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v4_orders_3(A) ) )).

fof(fc3_orders_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ~ v1_xboole_0(k1_orders_3(A,A)) ) )).

fof(fc4_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ~ v1_xboole_0(k2_orders_3(A)) ) )).

fof(fc5_orders_3,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & l1_orders_2(A)
        & ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & l1_orders_2(B) )
     => v1_fraenkel(k1_orders_3(A,B)) ) )).

fof(fc6_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ( ~ v3_struct_0(k4_orders_3(A))
        & v2_altcat_1(k4_orders_3(A))
        & v6_altcat_1(k4_orders_3(A)) ) ) )).

fof(fc7_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ( ~ v3_struct_0(k4_orders_3(A))
        & v2_altcat_1(k4_orders_3(A))
        & v6_altcat_1(k4_orders_3(A))
        & v11_altcat_1(k4_orders_3(A))
        & v12_altcat_1(k4_orders_3(A)) ) ) )).

fof(d1_orders_3,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_orders_3(A)
      <=> u1_orders_2(A) = k6_partfun1(u1_struct_0(A)) ) ) )).

fof(d2_orders_3,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_orders_3(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(A)))
                    & C != k1_xboole_0
                    & D != k1_xboole_0
                    & B = k4_subset_1(u1_struct_0(A),C,D)
                    & r1_xboole_0(C,D)
                    & u1_orders_2(A) = k2_xboole_0(k2_wellord1(u1_orders_2(A),C),k2_wellord1(u1_orders_2(A),D)) ) ) ) ) ) )).

fof(d3_orders_3,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v3_orders_3(A)
      <=> v2_orders_3(k2_pre_topc(A),A) ) ) )).

fof(t1_orders_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_orders_3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_orders_2(A,B,C)
              <=> B = C ) ) ) ) )).

fof(t2_orders_3,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( ( v1_relat_2(A)
            & v4_relat_2(A)
            & v8_relat_2(A)
            & v1_partfun1(A,k1_tarski(B),k1_tarski(B))
            & m2_relset_1(A,k1_tarski(B),k1_tarski(B)) )
         => A = k6_partfun1(k1_tarski(B)) ) ) )).

fof(t3_orders_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( ( v2_orders_2(A)
              & k2_pre_topc(A) = k1_struct_0(A,B) )
           => v1_orders_3(A) ) ) ) )).

fof(t4_orders_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ~ ( k2_pre_topc(A) = k1_tarski(B)
            & v3_orders_3(A) ) ) )).

fof(t5_orders_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_orders_3(A)
        & l1_orders_2(A) )
     => ( ~ ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => B = C ) )
       => v3_orders_3(A) ) ) )).

fof(d4_orders_3,axiom,(
    ! [A] :
      ( v4_orders_3(A)
    <=> ! [B] :
          ( r2_hidden(B,A)
         => ( ~ v3_struct_0(B)
            & v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & l1_orders_2(B) ) ) ) )).

fof(d5_orders_3,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)) )
             => ( v5_orders_3(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ( r1_orders_2(A,D,E)
                         => ! [F] :
                              ( m1_subset_1(F,u1_struct_0(B))
                             => ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(B))
                                 => ( ( F = k1_funct_1(C,D)
                                      & G = k1_funct_1(C,E) )
                                   => r1_orders_2(B,F,G) ) ) ) ) ) ) ) ) ) ) )).

fof(d6_orders_3,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C] :
              ( C = k1_orders_3(A,B)
            <=> ! [D] :
                  ( r2_hidden(D,C)
                <=> ? [E] :
                      ( v1_funct_1(E)
                      & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
                      & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B))
                      & D = E
                      & r2_hidden(E,k1_funct_2(u1_struct_0(A),u1_struct_0(B)))
                      & v5_orders_3(E,A,B) ) ) ) ) ) )).

fof(t6_orders_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & l1_orders_2(C) )
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D) )
                 => ! [E] :
                      ( ( v1_relat_1(E)
                        & v1_funct_1(E) )
                     => ( ( r2_hidden(D,k1_orders_3(A,B))
                          & r2_hidden(E,k1_orders_3(B,C)) )
                       => r2_hidden(k5_relat_1(D,E),k1_orders_3(A,C)) ) ) ) ) ) ) )).

fof(t7_orders_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => r2_hidden(k6_partfun1(u1_struct_0(A)),k1_orders_3(A,A)) ) )).

fof(d7_orders_3,axiom,(
    ! [A,B] :
      ( B = k2_orders_3(A)
    <=> ! [C] :
          ( r2_hidden(C,B)
        <=> ? [D] :
              ( l1_struct_0(D)
              & r2_hidden(D,A)
              & C = u1_struct_0(D) ) ) ) )).

fof(t8_orders_3,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => k2_orders_3(k1_tarski(A)) = k1_tarski(u1_struct_0(A)) ) )).

fof(t9_orders_3,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( l1_struct_0(B)
         => k2_orders_3(k2_tarski(A,B)) = k2_tarski(u1_struct_0(A),u1_struct_0(B)) ) ) )).

fof(t10_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ! [B] :
          ( m1_orders_3(B,A)
         => ! [C] :
              ( m1_orders_3(C,A)
             => r1_tarski(k1_orders_3(B,C),k1_ens_1(k2_orders_3(A))) ) ) ) )).

fof(t11_orders_3,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => r1_tarski(k1_orders_3(A,B),k1_funct_2(u1_struct_0(A),u1_struct_0(B))) ) ) )).

fof(d8_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ! [B] :
          ( ( v1_cat_1(B)
            & v2_cat_1(B)
            & v1_cat_5(B)
            & l1_cat_1(B) )
         => ( B = k3_orders_3(A)
          <=> ( u1_cat_1(B) = A
              & ! [C] :
                  ( m1_orders_3(C,A)
                 => ! [D] :
                      ( m1_orders_3(D,A)
                     => ! [E] :
                          ( m1_subset_1(E,k1_ens_1(k2_orders_3(A)))
                         => ( r2_hidden(E,k1_orders_3(C,D))
                           => m1_subset_1(k4_tarski(k4_tarski(C,D),E),u2_cat_1(B)) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,u2_cat_1(B))
                 => ? [D] :
                      ( m1_orders_3(D,A)
                      & ? [E] :
                          ( m1_orders_3(E,A)
                          & ? [F] :
                              ( m1_subset_1(F,k1_ens_1(k2_orders_3(A)))
                              & C = k4_tarski(k4_tarski(D,E),F)
                              & r2_hidden(F,k1_orders_3(D,E)) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,u2_cat_1(B))
                 => ! [D] :
                      ( m1_subset_1(D,u2_cat_1(B))
                     => ! [E] :
                          ( m1_orders_3(E,A)
                         => ! [F] :
                              ( m1_orders_3(F,A)
                             => ! [G] :
                                  ( m1_orders_3(G,A)
                                 => ! [H] :
                                      ( m1_subset_1(H,k1_ens_1(k2_orders_3(A)))
                                     => ! [I] :
                                          ( m1_subset_1(I,k1_ens_1(k2_orders_3(A)))
                                         => ( ( C = k4_tarski(k4_tarski(E,F),H)
                                              & D = k4_tarski(k4_tarski(F,G),I) )
                                           => k4_cat_1(B,C,D) = k4_tarski(k4_tarski(E,G),k5_relat_1(H,I)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d9_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ! [B] :
          ( ( v6_altcat_1(B)
            & l2_altcat_1(B) )
         => ( B = k4_orders_3(A)
          <=> ( u1_struct_0(B) = A
              & ! [C] :
                  ( m1_orders_3(C,A)
                 => ! [D] :
                      ( m1_orders_3(D,A)
                     => ( k1_binop_1(u1_altcat_1(B),C,D) = k1_orders_3(C,D)
                        & ! [E] :
                            ( m1_orders_3(E,A)
                           => ! [F] :
                                ( m1_orders_3(F,A)
                               => ! [G] :
                                    ( m1_orders_3(G,A)
                                   => k1_multop_1(u2_altcat_1(B),E,F,G) = k6_altcat_1(k1_orders_3(E,F),k1_orders_3(F,G)) ) ) ) ) ) ) ) ) ) ) )).

fof(t12_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k4_orders_3(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k4_orders_3(A)))
             => ! [D] :
                  ( m1_orders_3(D,A)
                 => ! [E] :
                      ( m1_orders_3(E,A)
                     => ( ( B = D
                          & C = E )
                       => r1_tarski(k1_altcat_1(k4_orders_3(A),B,C),k1_fraenkel(u1_struct_0(D),u1_struct_0(E))) ) ) ) ) ) ) )).

fof(s1_orders_3,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s1_orders_3)
       => ! [B] :
            ( m1_subset_1(B,f1_s1_orders_3)
           => ! [C] :
                ( m1_subset_1(C,f1_s1_orders_3)
               => ! [D] :
                    ( ( v1_relat_1(D)
                      & v1_funct_1(D) )
                   => ! [E] :
                        ( ( v1_relat_1(E)
                          & v1_funct_1(E) )
                       => ( ( r2_hidden(D,f2_s1_orders_3(A,B))
                            & r2_hidden(E,f2_s1_orders_3(B,C)) )
                         => r2_hidden(k5_relat_1(D,E),f2_s1_orders_3(A,C)) ) ) ) ) ) )
   => ? [A] :
        ( v6_altcat_1(A)
        & l2_altcat_1(A)
        & u1_struct_0(A) = f1_s1_orders_3
        & ! [B] :
            ( m1_subset_1(B,f1_s1_orders_3)
           => ! [C] :
                ( m1_subset_1(C,f1_s1_orders_3)
               => ( k1_binop_1(u1_altcat_1(A),B,C) = f2_s1_orders_3(B,C)
                  & ! [D] :
                      ( m1_subset_1(D,f1_s1_orders_3)
                     => ! [E] :
                          ( m1_subset_1(E,f1_s1_orders_3)
                         => ! [F] :
                              ( m1_subset_1(F,f1_s1_orders_3)
                             => k1_multop_1(u2_altcat_1(A),D,E,F) = k6_altcat_1(f2_s1_orders_3(D,E),f2_s1_orders_3(E,F)) ) ) ) ) ) ) ) )).

fof(s2_orders_3,axiom,(
    ! [A] :
      ( ( v6_altcat_1(A)
        & l2_altcat_1(A) )
     => ! [B] :
          ( ( v6_altcat_1(B)
            & l2_altcat_1(B) )
         => ( ( u1_struct_0(A) = f1_s2_orders_3
              & ! [C] :
                  ( m1_subset_1(C,f1_s2_orders_3)
                 => ! [D] :
                      ( m1_subset_1(D,f1_s2_orders_3)
                     => ( k1_binop_1(u1_altcat_1(A),C,D) = f2_s2_orders_3(C,D)
                        & ! [E] :
                            ( m1_subset_1(E,f1_s2_orders_3)
                           => ! [F] :
                                ( m1_subset_1(F,f1_s2_orders_3)
                               => ! [G] :
                                    ( m1_subset_1(G,f1_s2_orders_3)
                                   => k1_multop_1(u2_altcat_1(A),E,F,G) = k6_altcat_1(f2_s2_orders_3(E,F),f2_s2_orders_3(F,G)) ) ) ) ) ) )
              & u1_struct_0(B) = f1_s2_orders_3
              & ! [C] :
                  ( m1_subset_1(C,f1_s2_orders_3)
                 => ! [D] :
                      ( m1_subset_1(D,f1_s2_orders_3)
                     => ( k1_binop_1(u1_altcat_1(B),C,D) = f2_s2_orders_3(C,D)
                        & ! [E] :
                            ( m1_subset_1(E,f1_s2_orders_3)
                           => ! [F] :
                                ( m1_subset_1(F,f1_s2_orders_3)
                               => ! [G] :
                                    ( m1_subset_1(G,f1_s2_orders_3)
                                   => k1_multop_1(u2_altcat_1(B),E,F,G) = k6_altcat_1(f2_s2_orders_3(E,F),f2_s2_orders_3(F,G)) ) ) ) ) ) ) )
           => A = B ) ) ) )).

fof(dt_m1_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ! [B] :
          ( m1_orders_3(B,A)
         => ( ~ v3_struct_0(B)
            & v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & l1_orders_2(B) ) ) ) )).

fof(existence_m1_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ? [B] : m1_orders_3(B,A) ) )).

fof(redefinition_m1_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ! [B] :
          ( m1_orders_3(B,A)
        <=> m1_subset_1(B,A) ) ) )).

fof(dt_k1_orders_3,axiom,(
    $true )).

fof(dt_k2_orders_3,axiom,(
    $true )).

fof(dt_k3_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ( v1_cat_1(k3_orders_3(A))
        & v2_cat_1(k3_orders_3(A))
        & v1_cat_5(k3_orders_3(A))
        & l1_cat_1(k3_orders_3(A)) ) ) )).

fof(dt_k4_orders_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_orders_3(A) )
     => ( v6_altcat_1(k4_orders_3(A))
        & l2_altcat_1(k4_orders_3(A)) ) ) )).
%------------------------------------------------------------------------------