SET007 Axioms: SET007+296.ax


%------------------------------------------------------------------------------
% File     : SET007+296 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Shear Theorems and Their Role in Affine Geometry
% 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    : conmetr1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   23 (   0 unit)
%            Number of atoms       :  343 (   0 equality)
%            Maximal formula depth :   41 (  15 average)
%            Number of connectives :  361 (  41 ~  ;  48  |; 158  &)
%                                         (  17 <=>;  97 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   29 (   0 propositional; 1-5 arity)
%            Number of functors    :    3 (   0 constant; 1-1 arity)
%            Number of variables   :   85 (   0 singleton;  85 !;   0 ?)
%            Maximal term depth    :    3 (   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_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v1_conmetr1(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)))
                                               => ( ( r4_aff_1(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)
                                                    & r2_analoaf(A,D,C,H,G)
                                                    & r2_analoaf(A,C,B,G,F)
                                                    & r2_analoaf(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)
                                                    | r2_analoaf(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d2_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v2_conmetr1(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,u1_struct_0(A))
                                           => ! [K] :
                                                ( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
                                               => ! [L] :
                                                    ( m1_subset_1(L,k1_zfmisc_1(u1_struct_0(A)))
                                                   => ( ( v1_aff_1(K,A)
                                                        & v1_aff_1(L,A)
                                                        & r2_hidden(B,K)
                                                        & r2_hidden(B,L)
                                                        & r2_hidden(C,K)
                                                        & r2_hidden(E,K)
                                                        & r2_hidden(G,K)
                                                        & r2_hidden(I,K)
                                                        & r2_hidden(D,L)
                                                        & r2_hidden(F,L)
                                                        & r2_hidden(H,L)
                                                        & r2_hidden(J,L)
                                                        & r2_analoaf(A,E,D,I,H)
                                                        & r2_analoaf(A,D,C,H,G)
                                                        & r2_analoaf(A,C,F,G,J) )
                                                     => ( r2_hidden(F,K)
                                                        | r2_hidden(D,K)
                                                        | r2_hidden(H,K)
                                                        | r2_hidden(J,K)
                                                        | r2_hidden(C,L)
                                                        | r2_hidden(E,L)
                                                        | r2_hidden(G,L)
                                                        | r2_hidden(I,L)
                                                        | r2_analoaf(A,E,F,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d3_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v3_conmetr1(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)))
                                               => ( ( v1_aff_1(J,A)
                                                    & v1_aff_1(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)
                                                    & r2_analoaf(A,D,C,H,G)
                                                    & r2_analoaf(A,C,B,G,F)
                                                    & r2_analoaf(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)
                                                    | r2_analoaf(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d4_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v4_conmetr1(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)))
                                               => ( ( v1_aff_1(J,A)
                                                    & v1_aff_1(K,A)
                                                    & 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)
                                                    & r2_analoaf(A,D,C,H,G)
                                                    & r2_analoaf(A,C,B,G,F)
                                                    & r2_analoaf(A,B,E,F,I) )
                                                 => ( r2_hidden(E,J)
                                                    | r2_hidden(C,J)
                                                    | r2_hidden(F,J)
                                                    | r2_hidden(H,J)
                                                    | r2_hidden(B,K)
                                                    | r2_hidden(D,K)
                                                    | r2_hidden(G,K)
                                                    | r2_hidden(I,K)
                                                    | r2_analoaf(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d5_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v5_conmetr1(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)))
                                               => ( ( r4_aff_1(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)
                                                    & r2_analoaf(A,D,C,H,G)
                                                    & r2_analoaf(A,C,B,G,F)
                                                    & r2_analoaf(A,B,E,F,I) )
                                                 => ( r2_hidden(E,J)
                                                    | r2_hidden(C,J)
                                                    | r2_hidden(F,J)
                                                    | r2_hidden(H,J)
                                                    | r2_hidden(B,K)
                                                    | r2_hidden(D,K)
                                                    | r2_hidden(G,K)
                                                    | r2_hidden(I,K)
                                                    | r2_analoaf(A,D,E,H,I) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(d6_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v6_conmetr1(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,u1_struct_0(A))
                                           => ! [K] :
                                                ( m1_subset_1(K,k1_zfmisc_1(u1_struct_0(A)))
                                               => ! [L] :
                                                    ( m1_subset_1(L,k1_zfmisc_1(u1_struct_0(A)))
                                                   => ( ( v1_aff_1(K,A)
                                                        & v1_aff_1(L,A)
                                                        & r2_hidden(B,K)
                                                        & r2_hidden(B,L)
                                                        & r2_hidden(C,K)
                                                        & r2_hidden(E,K)
                                                        & r2_hidden(H,K)
                                                        & r2_hidden(J,K)
                                                        & r2_hidden(D,L)
                                                        & r2_hidden(F,L)
                                                        & r2_hidden(G,L)
                                                        & r2_hidden(I,L)
                                                        & r2_analoaf(A,E,D,I,H)
                                                        & r2_analoaf(A,D,C,H,G)
                                                        & r2_analoaf(A,C,F,G,J) )
                                                     => ( r2_hidden(F,K)
                                                        | r2_hidden(D,K)
                                                        | r2_hidden(G,K)
                                                        | r2_hidden(I,K)
                                                        | r2_hidden(C,L)
                                                        | r2_hidden(E,L)
                                                        | r2_hidden(H,L)
                                                        | r2_hidden(J,L)
                                                        | r2_analoaf(A,E,F,I,J) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t1_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v4_conmetr1(A)
      <=> ( v5_conmetr1(A)
          & v6_conmetr1(A) ) ) ) )).

fof(t2_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v3_conmetr1(A)
      <=> ( v1_conmetr1(A)
          & v2_conmetr1(A) ) ) ) )).

fof(t3_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v5_conmetr1(A)
       => v1_conmetr1(A) ) ) )).

fof(t4_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v6_conmetr1(A)
       => v2_conmetr1(A) ) ) )).

fof(t5_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v4_conmetr1(A)
       => v3_conmetr1(A) ) ) )).

fof(t6_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v11_aff_2(A)
       => v1_conmetr1(A) ) ) )).

fof(t7_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v4_aff_2(A)
       => v2_conmetr1(A) ) ) )).

fof(t8_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v4_aff_2(A)
      <=> v3_conmetr1(A) ) ) )).

fof(t9_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v13_aff_2(A)
      <=> v5_conmetr1(A) ) ) )).

fof(t10_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v2_aff_2(A)
      <=> v6_conmetr1(A) ) ) )).

fof(t11_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v1_aff_2(A)
      <=> v4_conmetr1(A) ) ) )).

fof(t12_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v3_realset2(A)
        & v1_diraf(A)
        & v2_diraf(A)
        & l1_analoaf(A) )
     => ( v6_conmetr1(A)
       => v5_conmetr1(A) ) ) )).

fof(t13_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v3_conmetr1(k3_analmetr(A))
      <=> v6_conmetr(A) ) ) )).

fof(t14_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v5_conmetr(A)
      <=> v7_aff_2(k3_analmetr(A)) ) ) )).

fof(t15_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v11_aff_2(k3_analmetr(A))
      <=> v8_conmetr(A) ) ) )).

fof(t16_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v2_conmetr(A)
      <=> v2_aff_2(k3_analmetr(A)) ) ) )).

fof(t17_conmetr1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v3_analmetr(A)
        & l1_analmetr(A) )
     => ( v1_conaffm(A)
      <=> v4_aff_2(k3_analmetr(A)) ) ) )).
%------------------------------------------------------------------------------