SET007 Axioms: SET007+232.ax


%------------------------------------------------------------------------------
% File     : SET007+232 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Collinearity Structure
% 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    : collsp [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   54 (  14 unit)
%            Number of atoms       :  346 (  36 equality)
%            Maximal formula depth :   17 (   8 average)
%            Number of connectives :  351 (  59 ~  ;   7  |; 147  &)
%                                         (   8 <=>; 130 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   1 propositional; 0-4 arity)
%            Number of functors    :   11 (   1 constant; 0-6 arity)
%            Number of variables   :  140 (   0 singleton; 128 !;  12 ?)
%            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(rc1_collsp,axiom,(
    ? [A] :
      ( l1_collsp(A)
      & v1_collsp(A) ) )).

fof(rc2_collsp,axiom,(
    ? [A] :
      ( l1_collsp(A)
      & ~ v3_struct_0(A)
      & v1_collsp(A) ) )).

fof(rc3_collsp,axiom,(
    ? [A] :
      ( l1_collsp(A)
      & ~ v3_struct_0(A)
      & v1_collsp(A)
      & v2_collsp(A)
      & v3_collsp(A) ) )).

fof(rc4_collsp,axiom,(
    ? [A] :
      ( l1_collsp(A)
      & ~ v3_struct_0(A)
      & v1_collsp(A)
      & v2_collsp(A)
      & v3_collsp(A)
      & v4_collsp(A) ) )).

fof(d1_collsp,axiom,(
    ! [A,B] :
      ( m1_collsp(B,A)
    <=> r1_tarski(B,k3_zfmisc_1(A,A,A)) ) )).

fof(t1_collsp,axiom,(
    $true )).

fof(t2_collsp,axiom,(
    ! [A] :
      ~ ( A != k1_xboole_0
        & ! [B] :
            ( k1_tarski(B) != A
            & ! [C,D] :
                ~ ( C != D
                  & r2_hidden(C,A)
                  & r2_hidden(D,A) ) ) ) )).

fof(d2_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_collsp(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))
                 => ( r1_collsp(A,B,C,D)
                  <=> r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,D),u1_collsp(A)) ) ) ) ) ) )).

fof(d3_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_collsp(A) )
     => ( v2_collsp(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))
                   => ( ~ ( B != C
                          & B != D
                          & C != D )
                     => r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,D),u1_collsp(A)) ) ) ) ) ) ) )).

fof(d4_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_collsp(A) )
     => ( v3_collsp(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_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,D),u1_collsp(A))
                                & r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,E),u1_collsp(A))
                                & r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),B,C,F),u1_collsp(A)) )
                             => ( B = C
                                | r2_hidden(k4_domain_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),D,E,F),u1_collsp(A)) ) ) ) ) ) ) ) ) ) )).

fof(t3_collsp,axiom,(
    $true )).

fof(t4_collsp,axiom,(
    $true )).

fof(t5_collsp,axiom,(
    $true )).

fof(t6_collsp,axiom,(
    $true )).

fof(t7_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(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))
                 => ( ~ ( B != C
                        & B != D
                        & C != D )
                   => r1_collsp(A,B,C,D) ) ) ) ) ) )).

fof(t8_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(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))
                         => ( ( r1_collsp(A,B,C,D)
                              & r1_collsp(A,B,C,E)
                              & r1_collsp(A,B,C,F) )
                           => ( B = C
                              | r1_collsp(A,D,E,F) ) ) ) ) ) ) ) ) )).

fof(t9_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(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))
                 => ( r1_collsp(A,B,C,D)
                   => ( r1_collsp(A,C,B,D)
                      & r1_collsp(A,B,D,C) ) ) ) ) ) ) )).

fof(t10_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_collsp(A,B,C,B) ) ) ) )).

fof(t11_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(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))
                     => ( ( r1_collsp(A,B,C,D)
                          & r1_collsp(A,B,C,E) )
                       => ( B = C
                          | r1_collsp(A,B,D,E) ) ) ) ) ) ) ) )).

fof(t12_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(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))
                 => ( r1_collsp(A,B,C,D)
                   => r1_collsp(A,C,B,D) ) ) ) ) ) )).

fof(t13_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(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))
                 => ( r1_collsp(A,B,C,D)
                   => r1_collsp(A,C,D,B) ) ) ) ) ) )).

fof(t14_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(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))
                         => ( ( r1_collsp(A,D,E,B)
                              & r1_collsp(A,D,E,C)
                              & r1_collsp(A,B,C,F) )
                           => ( B = C
                              | r1_collsp(A,D,E,F) ) ) ) ) ) ) ) ) )).

fof(t15_collsp,axiom,(
    $true )).

fof(t16_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r2_hidden(B,k1_collsp(A,B,C))
                & r2_hidden(C,k1_collsp(A,B,C)) ) ) ) ) )).

fof(t17_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(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))
                 => ( r1_collsp(A,B,C,D)
                  <=> r2_hidden(D,k1_collsp(A,B,C)) ) ) ) ) ) )).

fof(d6_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_collsp(A) )
     => ( v4_collsp(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))
                     => r1_collsp(A,B,C,D) ) ) ) ) ) )).

fof(t18_collsp,axiom,(
    $true )).

fof(t19_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(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))
                     => r1_collsp(A,B,C,D) ) ) ) ) ) )).

fof(d7_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m2_collsp(B,A)
        <=> ? [C] :
              ( m1_subset_1(C,u1_struct_0(A))
              & ? [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                  & C != D
                  & B = k1_collsp(A,C,D) ) ) ) ) )).

fof(t20_collsp,axiom,(
    $true )).

fof(t21_collsp,axiom,(
    $true )).

fof(t22_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( B = C
               => k1_collsp(A,B,C) = u1_struct_0(A) ) ) ) ) )).

fof(t23_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m2_collsp(B,A)
         => ? [C] :
              ( m1_subset_1(C,u1_struct_0(A))
              & ? [D] :
                  ( m1_subset_1(D,u1_struct_0(A))
                  & C != D
                  & r2_hidden(C,B)
                  & r2_hidden(D,B) ) ) ) ) )).

fof(t24_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & ! [D] :
                      ( m2_collsp(D,A)
                     => ~ ( r2_hidden(B,D)
                          & r2_hidden(C,D) ) ) ) ) ) ) )).

fof(t25_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(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] :
                      ( m2_collsp(E,A)
                     => ( ( r2_hidden(B,E)
                          & r2_hidden(C,E)
                          & r2_hidden(D,E) )
                       => r1_collsp(A,B,C,D) ) ) ) ) ) ) )).

fof(t26_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m2_collsp(B,A)
         => ! [C] :
              ( m2_collsp(C,A)
             => ( r1_tarski(B,C)
               => B = C ) ) ) ) )).

fof(t27_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_collsp(D,A)
                 => ( ( r2_hidden(B,D)
                      & r2_hidden(C,D) )
                   => ( B = C
                      | r1_tarski(k1_collsp(A,B,C),D) ) ) ) ) ) ) )).

fof(t28_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_collsp(D,A)
                 => ( ( r2_hidden(B,D)
                      & r2_hidden(C,D) )
                   => ( B = C
                      | k1_collsp(A,B,C) = D ) ) ) ) ) ) )).

fof(t29_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m2_collsp(D,A)
                 => ! [E] :
                      ( m2_collsp(E,A)
                     => ( ( r2_hidden(B,D)
                          & r2_hidden(C,D)
                          & r2_hidden(B,E)
                          & r2_hidden(C,E) )
                       => ( B = C
                          | D = E ) ) ) ) ) ) ) )).

fof(t30_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m2_collsp(B,A)
         => ! [C] :
              ( m2_collsp(C,A)
             => ~ ( B != C
                  & ~ r1_xboole_0(B,C)
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k3_xboole_0(B,C) != k1_struct_0(A,D) ) ) ) ) ) )).

fof(t31_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & k1_collsp(A,B,C) = u1_struct_0(A) ) ) ) ) )).

fof(dt_m1_collsp,axiom,(
    $true )).

fof(existence_m1_collsp,axiom,(
    ! [A] :
    ? [B] : m1_collsp(B,A) )).

fof(dt_m2_collsp,axiom,(
    $true )).

fof(existence_m2_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & v4_collsp(A)
        & l1_collsp(A) )
     => ? [B] : m2_collsp(B,A) ) )).

fof(dt_l1_collsp,axiom,(
    ! [A] :
      ( l1_collsp(A)
     => l1_struct_0(A) ) )).

fof(existence_l1_collsp,axiom,(
    ? [A] : l1_collsp(A) )).

fof(abstractness_v1_collsp,axiom,(
    ! [A] :
      ( l1_collsp(A)
     => ( v1_collsp(A)
       => A = g1_collsp(u1_struct_0(A),u1_collsp(A)) ) ) )).

fof(dt_k1_collsp,axiom,(
    $true )).

fof(dt_u1_collsp,axiom,(
    ! [A] :
      ( l1_collsp(A)
     => m1_collsp(u1_collsp(A),u1_struct_0(A)) ) )).

fof(dt_g1_collsp,axiom,(
    ! [A,B] :
      ( m1_collsp(B,A)
     => ( v1_collsp(g1_collsp(A,B))
        & l1_collsp(g1_collsp(A,B)) ) ) )).

fof(free_g1_collsp,axiom,(
    ! [A,B] :
      ( m1_collsp(B,A)
     => ! [C,D] :
          ( g1_collsp(A,B) = g1_collsp(C,D)
         => ( A = C
            & B = D ) ) ) )).

fof(d5_collsp,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_collsp(A)
        & v3_collsp(A)
        & l1_collsp(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k1_collsp(A,B,C) = a_3_0_collsp(A,B,C) ) ) ) )).

fof(fraenkel_a_3_0_collsp,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_collsp(B)
        & v3_collsp(B)
        & l1_collsp(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_0_collsp(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r1_collsp(B,C,D,E) ) ) ) )).
%------------------------------------------------------------------------------