SET007 Axioms: SET007+287.ax


%------------------------------------------------------------------------------
% File     : SET007+287 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Elementary Variants of Affine Configurational Theorems
% 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    : pardepap [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    6 (   1 unit)
%            Number of atoms       :  125 (   0 equality)
%            Maximal formula depth :   27 (  18 average)
%            Number of connectives :  133 (  14 ~  ;   8  |;  51  &)
%                                         (   0 <=>;  60 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   1 propositional; 0-5 arity)
%            Number of functors    :    1 (   0 constant; 1-1 arity)
%            Number of variables   :   53 (   0 singleton;  51 !;   2 ?)
%            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(t1_pardepap,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v2_aff_2(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))
                               => ( ( r2_analoaf(A,B,C,B,D)
                                    & r2_analoaf(A,E,F,E,G)
                                    & r2_analoaf(A,B,F,C,E)
                                    & r2_analoaf(A,C,G,D,F) )
                                 => r2_analoaf(A,D,E,B,G) ) ) ) ) ) ) ) ) ) )).

fof(t2_pardepap,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v4_aff_2(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))
                                   => ( ( r2_analoaf(A,B,C,B,D)
                                        & r2_analoaf(A,B,E,B,F)
                                        & r2_analoaf(A,B,G,B,H)
                                        & r2_analoaf(A,C,E,D,F)
                                        & r2_analoaf(A,C,G,D,H) )
                                     => ( r2_analoaf(A,B,C,B,E)
                                        | r2_analoaf(A,B,C,B,G)
                                        | r2_analoaf(A,E,G,F,H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t3_pardepap,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v11_aff_2(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))
                               => ( ( r2_analoaf(A,B,C,D,E)
                                    & r2_analoaf(A,B,C,F,G)
                                    & r2_analoaf(A,B,D,C,E)
                                    & r2_analoaf(A,B,F,C,G) )
                                 => ( r2_analoaf(A,B,C,B,D)
                                    | r2_analoaf(A,B,C,B,F)
                                    | r2_analoaf(A,D,F,E,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t4_pardepap,axiom,(
    $true )).

fof(t5_pardepap,axiom,(
    ? [A] :
      ( ~ v3_struct_0(A)
      & ~ v3_realset2(A)
      & v1_diraf(A)
      & v2_diraf(A)
      & l1_analoaf(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))
                                 => ( ( r2_analoaf(A,B,C,B,D)
                                      & r2_analoaf(A,B,E,B,F)
                                      & r2_analoaf(A,B,G,B,H)
                                      & r2_analoaf(A,C,E,D,F)
                                      & r2_analoaf(A,C,G,D,H) )
                                   => ( r2_analoaf(A,B,C,B,E)
                                      | r2_analoaf(A,B,C,B,G)
                                      | r2_analoaf(A,E,G,F,H) ) ) ) ) ) ) ) ) )
      & ! [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))
                             => ( ( r2_analoaf(A,B,C,D,E)
                                  & r2_analoaf(A,B,C,F,G)
                                  & r2_analoaf(A,B,D,C,E)
                                  & r2_analoaf(A,B,F,C,G) )
                               => ( r2_analoaf(A,B,C,B,D)
                                  | r2_analoaf(A,B,C,B,F)
                                  | r2_analoaf(A,D,F,E,G) ) ) ) ) ) ) ) )
      & ! [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))
                             => ( ( r2_analoaf(A,B,C,B,D)
                                  & r2_analoaf(A,E,F,E,G)
                                  & r2_analoaf(A,B,F,C,E)
                                  & r2_analoaf(A,C,G,D,F) )
                               => r2_analoaf(A,D,E,B,G) ) ) ) ) ) ) )
      & ! [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))
                     => ~ ( ~ r2_analoaf(A,B,C,B,D)
                          & r2_analoaf(A,B,C,D,E)
                          & r2_analoaf(A,B,D,C,E)
                          & r2_analoaf(A,B,E,C,D) ) ) ) ) ) ) )).

fof(t6_pardepap,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(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))
                         => ( r2_analoaf(A,B,C,B,D)
                            & ~ ! [G] :
                                  ( m1_subset_1(G,u1_struct_0(A))
                                 => ( r2_analoaf(A,B,D,B,E)
                                    & ~ ( r2_analoaf(A,B,F,B,G)
                                        & r2_analoaf(A,D,F,E,G) ) ) ) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------