SET007 Axioms: SET007+265.ax


%------------------------------------------------------------------------------
% File     : SET007+265 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Submetric Spaces - 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    : sub_metr [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   75 (  39 unit)
%            Number of atoms       :  268 (  19 equality)
%            Maximal formula depth :   14 (   4 average)
%            Number of connectives :  233 (  40 ~  ;   0  |; 102  &)
%                                         (   8 <=>;  83 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   1 propositional; 0-4 arity)
%            Number of functors    :   26 (   6 constant; 0-5 arity)
%            Number of variables   :   87 (   0 singleton;  84 !;   3 ?)
%            Maximal term depth    :    5 (   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(fc1_sub_metr,axiom,
    ( ~ v3_struct_0(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
    & v1_metric_1(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
    & v6_metric_1(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
    & v8_metric_1(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
    & v9_metric_1(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1))
    & v2_sub_metr(g1_metric_1(k1_tarski(k1_xboole_0),k3_metric_1)) )).

fof(rc1_sub_metr,axiom,(
    ? [A] :
      ( l1_metric_1(A)
      & ~ v3_struct_0(A)
      & v6_metric_1(A)
      & v8_metric_1(A)
      & v9_metric_1(A)
      & v2_sub_metr(A) ) )).

fof(rc2_sub_metr,axiom,(
    ? [A] :
      ( l1_metric_1(A)
      & ~ v3_struct_0(A)
      & v1_metric_1(A)
      & v6_metric_1(A)
      & v8_metric_1(A)
      & v2_sub_metr(A)
      & v3_sub_metr(A) ) )).

fof(cc1_sub_metr,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( ( ~ v3_struct_0(A)
          & v6_metric_1(A)
          & v7_metric_1(A)
          & v8_metric_1(A)
          & v9_metric_1(A) )
       => ( ~ v3_struct_0(A)
          & v6_metric_1(A)
          & v7_metric_1(A)
          & v8_metric_1(A)
          & v9_metric_1(A)
          & v2_sub_metr(A) ) ) ) )).

fof(cc2_sub_metr,axiom,(
    ! [A] :
      ( l1_metric_1(A)
     => ( ( ~ v3_struct_0(A)
          & v6_metric_1(A)
          & v8_metric_1(A)
          & v2_sub_metr(A)
          & v3_sub_metr(A) )
       => ( ~ v3_struct_0(A)
          & v6_metric_1(A)
          & v7_metric_1(A)
          & v8_metric_1(A)
          & v9_metric_1(A)
          & v2_sub_metr(A)
          & v3_sub_metr(A) ) ) ) )).

fof(fc2_sub_metr,axiom,
    ( ~ v3_struct_0(k2_sub_metr)
    & v1_metric_1(k2_sub_metr) )).

fof(fc3_sub_metr,axiom,
    ( ~ v3_struct_0(k2_sub_metr)
    & v1_metric_1(k2_sub_metr)
    & v6_metric_1(k2_sub_metr)
    & v8_metric_1(k2_sub_metr)
    & v9_metric_1(k2_sub_metr) )).

fof(t1_sub_metr,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ( r1_xreal_0(np__0,A)
              & r1_xreal_0(np__0,B) )
           => r1_xreal_0(k2_square_1(A,B),k2_xcmplx_0(A,B)) ) ) ) )).

fof(t2_sub_metr,axiom,(
    ! [A] :
      ( ( v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ~ ( B != C
                  & r1_xreal_0(k4_metric_1(A,B,C),np__0) ) ) ) ) )).

fof(t3_sub_metr,axiom,(
    $true )).

fof(t4_sub_metr,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_tarski(k1_xboole_0))
     => k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,A) = np__0 ) )).

fof(t5_sub_metr,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_tarski(k1_xboole_0))
     => ! [B] :
          ( m1_subset_1(B,k1_tarski(k1_xboole_0))
         => ~ ( A != B
              & r1_xreal_0(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,B),np__0) ) ) ) )).

fof(t6_sub_metr,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_tarski(k1_xboole_0))
     => ! [B] :
          ( m1_subset_1(B,k1_tarski(k1_xboole_0))
         => k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,B) = k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,B,A) ) ) )).

fof(t7_sub_metr,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_tarski(k1_xboole_0))
     => ! [B] :
          ( m1_subset_1(B,k1_tarski(k1_xboole_0))
         => ! [C] :
              ( m1_subset_1(C,k1_tarski(k1_xboole_0))
             => r1_xreal_0(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,C),k2_xcmplx_0(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,B),k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,B,C))) ) ) ) )).

fof(t8_sub_metr,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_tarski(k1_xboole_0))
     => ! [B] :
          ( m1_subset_1(B,k1_tarski(k1_xboole_0))
         => ! [C] :
              ( m1_subset_1(C,k1_tarski(k1_xboole_0))
             => r1_xreal_0(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,C),k4_square_1(k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,A,B),k1_metric_1(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0),k3_metric_1,B,C))) ) ) ) )).

fof(d1_sub_metr,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(A,A),k1_numbers)
            & m2_relset_1(B,k2_zfmisc_1(A,A),k1_numbers) )
         => ( v1_sub_metr(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,A)
               => ! [D] :
                    ( m1_subset_1(D,A)
                   => ~ ( C != D
                        & r1_xreal_0(k1_metric_1(A,A,B,C,D),np__0) ) ) ) ) ) ) )).

fof(d2_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ( v2_sub_metr(A)
      <=> v1_sub_metr(u1_metric_1(A),u1_struct_0(A)) ) ) )).

fof(t9_sub_metr,axiom,(
    $true )).

fof(t10_sub_metr,axiom,(
    $true )).

fof(t11_sub_metr,axiom,(
    $true )).

fof(t12_sub_metr,axiom,(
    $true )).

fof(t13_sub_metr,axiom,(
    $true )).

fof(t14_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ( ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ~ ( B != C
                    & r1_xreal_0(k2_metric_1(A,B,C),np__0) ) ) )
      <=> v2_sub_metr(A) ) ) )).

fof(t15_sub_metr,axiom,(
    $true )).

fof(t16_sub_metr,axiom,(
    $true )).

fof(t17_sub_metr,axiom,(
    $true )).

fof(t18_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v2_sub_metr(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_xreal_0(np__0,k2_metric_1(A,B,C)) ) ) ) )).

fof(t19_sub_metr,axiom,(
    $true )).

fof(t20_sub_metr,axiom,(
    $true )).

fof(t21_sub_metr,axiom,(
    $true )).

fof(t22_sub_metr,axiom,(
    $true )).

fof(t23_sub_metr,axiom,(
    $true )).

fof(t24_sub_metr,axiom,(
    $true )).

fof(t25_sub_metr,axiom,(
    $true )).

fof(d3_sub_metr,axiom,(
    $true )).

fof(d4_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ( v3_sub_metr(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_xreal_0(k2_metric_1(A,B,D),k4_square_1(k2_metric_1(A,B,C),k2_metric_1(A,C,D))) ) ) ) ) ) )).

fof(t26_sub_metr,axiom,(
    $true )).

fof(t27_sub_metr,axiom,(
    $true )).

fof(t28_sub_metr,axiom,(
    $true )).

fof(t29_sub_metr,axiom,(
    $true )).

fof(t30_sub_metr,axiom,(
    $true )).

fof(t31_sub_metr,axiom,(
    $true )).

fof(t32_sub_metr,axiom,(
    $true )).

fof(t33_sub_metr,axiom,(
    $true )).

fof(t34_sub_metr,axiom,(
    $true )).

fof(t35_sub_metr,axiom,(
    $true )).

fof(t36_sub_metr,axiom,(
    $true )).

fof(t37_sub_metr,axiom,(
    $true )).

fof(t38_sub_metr,axiom,(
    $true )).

fof(d5_sub_metr,axiom,(
    k1_sub_metr = k2_funcop_1(k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))),np__0) )).

fof(t39_sub_metr,axiom,
    ( r2_hidden(k4_tarski(k1_xboole_0,k1_xboole_0),k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))))
    & r2_hidden(k4_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))))
    & r2_hidden(k4_tarski(k1_tarski(k1_xboole_0),k1_xboole_0),k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))))
    & r2_hidden(k4_tarski(k1_tarski(k1_xboole_0),k1_tarski(k1_xboole_0)),k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))) )).

fof(t40_sub_metr,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
     => ! [B] :
          ( m1_subset_1(B,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
         => k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,A,B) = np__0 ) ) )).

fof(t41_sub_metr,axiom,(
    $true )).

fof(t42_sub_metr,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
     => ! [B] :
          ( m1_subset_1(B,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
         => k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,A,B) = k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,B,A) ) ) )).

fof(t43_sub_metr,axiom,(
    ! [A] :
      ( m1_subset_1(A,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
     => ! [B] :
          ( m1_subset_1(B,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
         => ! [C] :
              ( m1_subset_1(C,k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)))
             => r1_xreal_0(k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,A,C),k2_xcmplx_0(k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,A,B),k1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr,B,C))) ) ) ) )).

fof(d6_sub_metr,axiom,(
    k2_sub_metr = g1_metric_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k1_sub_metr) )).

fof(d7_sub_metr,axiom,(
    ! [A] :
      ( l1_metric_1(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_sub_metr(A,B,C,D)
                  <=> ( B != C
                      & B != D
                      & C != D
                      & k2_metric_1(A,B,D) = k2_xcmplx_0(k2_metric_1(A,B,C),k2_metric_1(A,C,D)) ) ) ) ) ) ) )).

fof(t44_sub_metr,axiom,(
    $true )).

fof(t45_sub_metr,axiom,(
    $true )).

fof(t46_sub_metr,axiom,(
    $true )).

fof(t47_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(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_sub_metr(A,B,C,D)
                   => r1_sub_metr(A,D,C,B) ) ) ) ) ) )).

fof(t48_sub_metr,axiom,(
    ! [A] :
      ( ( v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(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_sub_metr(A,B,C,D)
                   => ( ~ r1_sub_metr(A,C,B,D)
                      & ~ r1_sub_metr(A,B,D,C) ) ) ) ) ) ) )).

fof(t49_sub_metr,axiom,(
    ! [A] :
      ( ( v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(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_sub_metr(A,B,C,D)
                          & r1_sub_metr(A,B,D,E) )
                       => ( r1_sub_metr(A,B,C,E)
                          & r1_sub_metr(A,C,D,E) ) ) ) ) ) ) ) )).

fof(t50_sub_metr,axiom,(
    $true )).

fof(t51_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(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))
                 => ( r2_hidden(D,k3_sub_metr(A,B,C))
                  <=> r1_sub_metr(A,B,D,C) ) ) ) ) ) )).

fof(t52_sub_metr,axiom,(
    $true )).

fof(t53_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(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))
                 => ( r2_hidden(D,k4_sub_metr(A,B,C))
                  <=> ~ ( ~ r1_sub_metr(A,B,D,C)
                        & D != B
                        & D != C ) ) ) ) ) ) )).

fof(t54_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => r1_tarski(k3_sub_metr(A,B,C),k4_sub_metr(A,B,C)) ) ) ) )).

fof(dt_k1_sub_metr,axiom,
    ( v1_funct_1(k1_sub_metr)
    & v1_funct_2(k1_sub_metr,k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))),k1_numbers)
    & m2_relset_1(k1_sub_metr,k2_zfmisc_1(k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0)),k2_tarski(k1_xboole_0,k1_tarski(k1_xboole_0))),k1_numbers) )).

fof(dt_k2_sub_metr,axiom,(
    l1_metric_1(k2_sub_metr) )).

fof(dt_k3_sub_metr,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k3_sub_metr(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(dt_k4_sub_metr,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A)
        & m1_subset_1(B,u1_struct_0(A))
        & m1_subset_1(C,u1_struct_0(A)) )
     => m1_subset_1(k4_sub_metr(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(d8_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k3_sub_metr(A,B,C) = a_3_0_sub_metr(A,B,C) ) ) ) )).

fof(d9_sub_metr,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => k4_sub_metr(A,B,C) = k2_xboole_0(a_3_0_sub_metr(A,B,C),k2_struct_0(A,B,C)) ) ) ) )).

fof(fraenkel_a_3_0_sub_metr,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & l1_metric_1(B)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => ( r2_hidden(A,a_3_0_sub_metr(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,u1_struct_0(B))
            & A = E
            & r1_sub_metr(B,C,E,D) ) ) ) )).
%------------------------------------------------------------------------------