SET007 Axioms: SET007+285.ax


%------------------------------------------------------------------------------
% File     : SET007+285 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Metric-Affine Configurations in Metric Affine Planes - Part II
% 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    : conmetr [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   23 (   0 unit)
%            Number of atoms       :  361 (  29 equality)
%            Maximal formula depth :   37 (  18 average)
%            Number of connectives :  374 (  36 ~  ;  46  |; 149  &)
%                                         (   8 <=>; 135 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   25 (   0 propositional; 1-5 arity)
%            Number of functors    :    3 (   0 constant; 1-1 arity)
%            Number of variables   :  119 (   0 singleton; 116 !;   3 ?)
%            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(d1_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v1_conmetr(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(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(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ! [H] :
                                    ( m1_subset_1(H,u1_struct_0(A))
                                   => ! [I] :
                                        ( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(A)))
                                       => ! [J] :
                                            ( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
                                           => ( ( r2_hidden(B,I)
                                                & r2_hidden(C,I)
                                                & r2_hidden(D,I)
                                                & r2_hidden(E,I)
                                                & r2_hidden(B,J)
                                                & r2_hidden(F,J)
                                                & r2_hidden(G,J)
                                                & r2_hidden(H,J)
                                                & r11_analmetr(A,I,J)
                                                & r4_analmetr(A,E,G,D,F)
                                                & r4_analmetr(A,E,H,C,F) )
                                             => ( r2_hidden(G,I)
                                                | r2_hidden(E,J)
                                                | B = C
                                                | B = D
                                                | B = E
                                                | B = F
                                                | B = G
                                                | B = H
                                                | r4_analmetr(A,C,G,D,H) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d2_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v2_conmetr(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(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(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ! [H] :
                                    ( m1_subset_1(H,u1_struct_0(A))
                                   => ! [I] :
                                        ( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(A)))
                                       => ! [J] :
                                            ( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
                                           => ( ( v4_analmetr(I,A)
                                                & v4_analmetr(J,A)
                                                & r2_hidden(B,I)
                                                & r2_hidden(C,I)
                                                & r2_hidden(D,I)
                                                & r2_hidden(E,I)
                                                & r2_hidden(B,J)
                                                & r2_hidden(F,J)
                                                & r2_hidden(G,J)
                                                & r2_hidden(H,J)
                                                & r4_analmetr(A,E,G,D,F)
                                                & r4_analmetr(A,E,H,C,F) )
                                             => ( r2_hidden(G,I)
                                                | r2_hidden(E,J)
                                                | B = C
                                                | B = D
                                                | B = E
                                                | B = F
                                                | B = G
                                                | B = H
                                                | r4_analmetr(A,C,G,D,H) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d3_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v3_conmetr(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(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(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ! [H] :
                                    ( m1_subset_1(H,u1_struct_0(A))
                                   => ! [I] :
                                        ( m1_subset_1(I,u1_struct_0(A))
                                       => ! [J] :
                                            ( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
                                           => ! [K] :
                                                ( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
                                               => ( ( r11_analmetr(A,J,K)
                                                    & r2_hidden(B,J)
                                                    & r2_hidden(D,J)
                                                    & r2_hidden(F,J)
                                                    & r2_hidden(H,J)
                                                    & r2_hidden(C,K)
                                                    & r2_hidden(E,K)
                                                    & r2_hidden(G,K)
                                                    & r2_hidden(I,K)
                                                    & r5_analmetr(A,B,C,F,G)
                                                    & r5_analmetr(A,C,D,G,H)
                                                    & r5_analmetr(A,D,E,H,I) )
                                                 => ( r2_hidden(C,J)
                                                    | r2_hidden(E,J)
                                                    | r5_analmetr(A,B,E,F,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d4_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v4_conmetr(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(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(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ! [H] :
                                    ( m1_subset_1(H,u1_struct_0(A))
                                   => ! [I] :
                                        ( m1_subset_1(I,u1_struct_0(A))
                                       => ! [J] :
                                            ( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
                                           => ! [K] :
                                                ( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
                                               => ( ( r11_analmetr(A,J,K)
                                                    & r2_hidden(B,J)
                                                    & r2_hidden(D,J)
                                                    & r2_hidden(G,J)
                                                    & r2_hidden(I,J)
                                                    & r2_hidden(C,K)
                                                    & r2_hidden(E,K)
                                                    & r2_hidden(F,K)
                                                    & r2_hidden(H,K)
                                                    & r5_analmetr(A,B,C,F,G)
                                                    & r5_analmetr(A,C,D,G,H)
                                                    & r5_analmetr(A,D,E,H,I) )
                                                 => ( r2_hidden(C,J)
                                                    | r2_hidden(E,J)
                                                    | r5_analmetr(A,B,E,F,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d5_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v5_conmetr(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(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(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ! [H] :
                                    ( m1_subset_1(H,u1_struct_0(A))
                                   => ( ( r6_analmetr(A,B,C,D)
                                        & r6_analmetr(A,B,E,F)
                                        & r6_analmetr(A,B,G,H)
                                        & r4_analmetr(A,C,E,D,F)
                                        & r4_analmetr(A,C,E,B,G)
                                        & r4_analmetr(A,E,G,F,H) )
                                     => ( B = C
                                        | B = D
                                        | B = E
                                        | B = F
                                        | B = G
                                        | B = H
                                        | r6_analmetr(A,E,F,C)
                                        | r6_analmetr(A,E,F,G)
                                        | r4_analmetr(A,C,G,D,H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d6_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v6_conmetr(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(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(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ! [H] :
                                    ( m1_subset_1(H,u1_struct_0(A))
                                   => ! [I] :
                                        ( m1_subset_1(I,u1_struct_0(A))
                                       => ! [J] :
                                            ( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
                                           => ! [K] :
                                                ( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
                                               => ( ( v4_analmetr(J,A)
                                                    & v4_analmetr(K,A)
                                                    & r2_hidden(B,J)
                                                    & r2_hidden(D,J)
                                                    & r2_hidden(F,J)
                                                    & r2_hidden(H,J)
                                                    & r2_hidden(C,K)
                                                    & r2_hidden(E,K)
                                                    & r2_hidden(G,K)
                                                    & r2_hidden(I,K)
                                                    & r4_analmetr(A,D,C,H,G)
                                                    & r4_analmetr(A,C,B,G,F)
                                                    & r4_analmetr(A,B,E,F,I) )
                                                 => ( r2_hidden(E,J)
                                                    | r2_hidden(C,J)
                                                    | r2_hidden(G,J)
                                                    | r2_hidden(I,J)
                                                    | r2_hidden(B,K)
                                                    | r2_hidden(D,K)
                                                    | r2_hidden(F,K)
                                                    | r2_hidden(H,K)
                                                    | r4_analmetr(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d7_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v7_conmetr(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(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(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ! [H] :
                                    ( m1_subset_1(H,u1_struct_0(A))
                                   => ! [I] :
                                        ( m1_subset_1(I,u1_struct_0(A))
                                       => ! [J] :
                                            ( m1_subset_1(J,k1_zfmisc_1(u1_struct_0(A)))
                                           => ! [K] :
                                                ( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
                                               => ( ( r11_analmetr(A,J,K)
                                                    & r2_hidden(B,J)
                                                    & r2_hidden(D,J)
                                                    & r2_hidden(F,J)
                                                    & r2_hidden(H,J)
                                                    & r2_hidden(C,K)
                                                    & r2_hidden(E,K)
                                                    & r2_hidden(G,K)
                                                    & r2_hidden(I,K)
                                                    & r4_analmetr(A,D,C,H,G)
                                                    & r4_analmetr(A,C,B,G,F)
                                                    & r4_analmetr(A,B,E,F,I) )
                                                 => ( r2_hidden(E,J)
                                                    | r2_hidden(C,J)
                                                    | r2_hidden(G,J)
                                                    | r2_hidden(I,J)
                                                    | r2_hidden(B,K)
                                                    | r2_hidden(D,K)
                                                    | r2_hidden(F,K)
                                                    | r2_hidden(H,K)
                                                    | r4_analmetr(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d8_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v8_conmetr(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(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(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(A))
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ( ( r4_analmetr(A,B,C,D,E)
                                    & r4_analmetr(A,B,C,F,G)
                                    & r4_analmetr(A,B,D,C,E)
                                    & r4_analmetr(A,B,F,C,G) )
                                 => ( r6_analmetr(A,B,C,D)
                                    | r6_analmetr(A,B,C,F)
                                    | r4_analmetr(A,D,F,E,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t1_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ? [B] :
          ( m1_subset_1(B,u1_struct_0(A))
          & ? [C] :
              ( m1_subset_1(C,u1_struct_0(A))
              & ? [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                  & r6_analmetr(A,B,C,D)
                  & B != C
                  & C != D
                  & D != B ) ) ) ) )).

fof(t2_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => ~ ( r6_analmetr(A,B,C,D)
                          & B != D
                          & C != D ) ) ) ) ) ) )).

fof(t3_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( v4_analmetr(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ~ ( r2_hidden(C,D)
                          & r11_analmetr(A,B,D) ) ) ) ) ) ) )).

fof(t4_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                     => ( ( v4_analmetr(E,A)
                          & r2_hidden(B,E)
                          & r2_hidden(C,E)
                          & r2_hidden(D,E) )
                       => r6_analmetr(A,B,C,D) ) ) ) ) ) ) )).

fof(t5_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                     => ~ ( v4_analmetr(D,A)
                          & v4_analmetr(E,A)
                          & r2_hidden(B,D)
                          & r2_hidden(C,D)
                          & r2_hidden(B,E)
                          & r2_hidden(C,E)
                          & B != C
                          & D != E ) ) ) ) ) ) )).

fof(t6_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(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(A))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(A)))
                         => ! [G] :
                              ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k3_analmetr(A))))
                             => ! [H] :
                                  ( m1_subset_1(H,u1_struct_0(k3_analmetr(A)))
                                 => ! [I] :
                                      ( m1_subset_1(I,u1_struct_0(k3_analmetr(A)))
                                     => ( ( D = H
                                          & E = I
                                          & F = G
                                          & r2_hidden(B,F)
                                          & r2_hidden(C,F)
                                          & r2_aff_1(k3_analmetr(A),H,I,G) )
                                       => r4_analmetr(A,D,E,B,C) ) ) ) ) ) ) ) ) ) ) )).

fof(t7_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v5_conmetr(A)
       => v7_aff_2(k3_analmetr(A)) ) ) )).

fof(t8_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v11_aff_2(k3_analmetr(A))
       => v8_conmetr(A) ) ) )).

fof(t9_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v3_conmetr(A)
       => v7_conmetr(A) ) ) )).

fof(t10_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v4_conmetr(A)
       => v7_conmetr(A) ) ) )).

fof(t11_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v2_conaffm(A)
       => v5_conmetr(A) ) ) )).

fof(t12_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( ( v7_conmetr(A)
          & v5_conmetr(A) )
       => v6_conmetr(A) ) ) )).

fof(t13_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( ( v1_conmetr(A)
          & v1_conaffm(A) )
       => v2_conmetr(A) ) ) )).

fof(t14_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( ( v3_conmetr(A)
          & v4_conmetr(A) )
       => v1_conmetr(A) ) ) )).

fof(t15_conmetr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v3_conaffm(A)
       => v1_conmetr(A) ) ) )).
%------------------------------------------------------------------------------