SET007 Axioms: SET007+284.ax


%------------------------------------------------------------------------------
% File     : SET007+284 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Metric-Affine Configurations in Metric Affine Planes - Part I
% 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    : conaffm [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   13 (   0 unit)
%            Number of atoms       :  187 (  27 equality)
%            Maximal formula depth :   27 (  16 average)
%            Number of connectives :  190 (  16 ~  ;  36  |;  60  &)
%                                         (   7 <=>;  71 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   15 (   0 propositional; 1-5 arity)
%            Number of functors    :    1 (   0 constant; 1-1 arity)
%            Number of variables   :   59 (   0 singleton;  59 !;   0 ?)
%            Maximal term depth    :    2 (   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_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v1_conaffm(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,G,D,H) )
                                     => ( B = C
                                        | B = D
                                        | B = E
                                        | B = F
                                        | B = G
                                        | B = H
                                        | r6_analmetr(A,E,F,C)
                                        | r6_analmetr(A,C,D,G)
                                        | r4_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d2_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v2_conaffm(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))
                                   => ( ( r5_analmetr(A,B,C,B,D)
                                        & r5_analmetr(A,B,E,B,F)
                                        & r5_analmetr(A,B,G,B,H)
                                        & r5_analmetr(A,C,E,D,F)
                                        & r4_analmetr(A,B,C,E,G)
                                        & r5_analmetr(A,C,G,D,H) )
                                     => ( r4_analmetr(A,B,G,B,C)
                                        | r4_analmetr(A,B,C,B,E)
                                        | r5_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d3_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v3_conaffm(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)
                        & ! [E] :
                            ( m1_subset_1(E,u1_struct_0(A))
                           => ~ ( r5_analmetr(A,E,B,C,D)
                                & r5_analmetr(A,E,C,B,D)
                                & r5_analmetr(A,E,D,B,C) ) ) ) ) ) ) ) ) )).

fof(d4_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v4_conaffm(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))
                                   => ( ( r5_analmetr(A,B,C,B,D)
                                        & r5_analmetr(A,B,E,B,F)
                                        & r5_analmetr(A,B,G,B,H)
                                        & r5_analmetr(A,C,E,D,F)
                                        & r5_analmetr(A,C,G,D,H) )
                                     => ( r4_analmetr(A,B,G,B,C)
                                        | r4_analmetr(A,B,C,B,E)
                                        | r5_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d5_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v5_conaffm(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))
                                   => ( ( r5_analmetr(A,B,G,B,H)
                                        & r5_analmetr(A,B,C,B,D)
                                        & r5_analmetr(A,B,E,B,F)
                                        & r6_analmetr(A,B,C,E)
                                        & r6_analmetr(A,B,D,F)
                                        & r5_analmetr(A,C,G,D,H)
                                        & r5_analmetr(A,E,G,F,H) )
                                     => ( B = C
                                        | B = D
                                        | B = E
                                        | B = F
                                        | B = G
                                        | B = H
                                        | C = E
                                        | r6_analmetr(A,B,G,C)
                                        | r4_analmetr(A,C,D,E,F) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d6_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v6_conaffm(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))
                                   => ( ( r5_analmetr(A,B,G,B,H)
                                        & r5_analmetr(A,B,C,B,D)
                                        & r5_analmetr(A,B,E,B,F)
                                        & r6_analmetr(A,B,C,E)
                                        & r6_analmetr(A,B,D,F)
                                        & r5_analmetr(A,C,G,D,H)
                                        & r4_analmetr(A,C,D,E,F) )
                                     => ( B = C
                                        | B = D
                                        | B = E
                                        | B = F
                                        | B = G
                                        | B = H
                                        | C = E
                                        | r6_analmetr(A,B,G,C)
                                        | r5_analmetr(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d7_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v7_conaffm(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))
                                   => ( ( r4_analmetr(A,C,D,E,F)
                                        & r5_analmetr(A,B,C,B,D)
                                        & r5_analmetr(A,B,E,B,F)
                                        & r6_analmetr(A,B,C,E)
                                        & r6_analmetr(A,B,D,F)
                                        & r5_analmetr(A,C,G,D,H)
                                        & r5_analmetr(A,E,G,F,H) )
                                     => ( B = C
                                        | B = D
                                        | B = E
                                        | B = F
                                        | B = G
                                        | B = H
                                        | C = E
                                        | r6_analmetr(A,B,G,C)
                                        | r5_analmetr(A,B,G,B,H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t1_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v4_conaffm(A)
       => v1_conaffm(A) ) ) )).

fof(t2_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v4_conaffm(A)
       => v2_conaffm(A) ) ) )).

fof(t3_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v5_conaffm(A)
       => v6_conaffm(A) ) ) )).

fof(t4_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v6_conaffm(A)
       => v7_conaffm(A) ) ) )).

fof(t5_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v5_conaffm(A)
       => v4_conaffm(A) ) ) )).

fof(t6_conaffm,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v5_conaffm(A)
       => v3_conaffm(A) ) ) )).
%------------------------------------------------------------------------------