SET007 Axioms: SET007+345.ax


%------------------------------------------------------------------------------
% File     : SET007+345 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Paracompactness of Metrizable Spaces
% 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    : pcomps_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   29 (   7 unit)
%            Number of atoms       :  191 (  19 equality)
%            Maximal formula depth :   22 (   8 average)
%            Number of connectives :  183 (  21 ~  ;   0  |; 102  &)
%                                         (   8 <=>;  52 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   38 (   1 propositional; 0-4 arity)
%            Number of functors    :   46 (  11 constant; 0-4 arity)
%            Number of variables   :   75 (   0 singleton;  65 !;  10 ?)
%            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(t1_pcomps_2,axiom,(
    $true )).

fof(t2_pcomps_2,axiom,(
    $true )).

fof(t3_pcomps_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(B,np__0)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ r1_xreal_0(k7_xcmplx_0(B,k3_newton(np__2,C)),A) ) ) ) ) )).

fof(t4_pcomps_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(C,B)
                  & r1_xreal_0(np__1,A) )
               => r1_xreal_0(k2_newton(A,C),k2_newton(A,B)) ) ) ) ) )).

fof(t5_pcomps_2,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ! [B] :
          ( r2_wellord1(A,B)
         => ( r2_wellord1(k2_wellord1(A,B),B)
            & B = k3_relat_1(k2_wellord1(A,B)) ) ) ) )).

fof(d1_pcomps_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ! [C] :
          ( m1_subset_1(C,A)
         => k1_pcomps_2(A,B,C) = k3_tarski(k1_wellord1(B,C)) ) ) )).

fof(d2_pcomps_2,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ! [C] :
          ( C = k2_pcomps_2(A,B)
        <=> ! [D] :
              ( r2_hidden(D,C)
            <=> ? [E] :
                  ( m1_subset_1(E,A)
                  & r2_hidden(E,A)
                  & D = k4_xboole_0(E,k1_pcomps_2(A,B,E)) ) ) ) ) )).

fof(d3_pcomps_2,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,k1_pcomps_1(A))
            & m2_relset_1(C,k5_numbers,k1_pcomps_1(A)) )
         => k3_pcomps_2(A,B,C) = k3_tarski(k9_relat_1(C,k4_xboole_0(k2_finseq_1(B),k1_tarski(B)))) ) ) )).

fof(t6_pcomps_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v4_compts_1(A)
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( r1_pre_topc(A,B)
                & v1_tops_2(B,A)
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                   => ~ ( v1_tops_2(C,A)
                        & r1_pre_topc(A,C)
                        & ! [D] :
                            ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                           => ~ ( r2_hidden(D,C)
                                & ! [E] :
                                    ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                                   => ~ ( r2_hidden(E,B)
                                        & r1_tarski(k6_pre_topc(A,D),E) ) ) ) ) ) ) ) ) ) ) )).

fof(t7_pcomps_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ~ ( v3_compts_1(A)
              & v2_pcomps_1(A)
              & r1_pre_topc(A,B)
              & v1_tops_2(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                 => ~ ( v1_tops_2(C,A)
                      & r1_pre_topc(A,C)
                      & r1_setfam_1(k3_pcomps_1(A,C),B)
                      & v1_pcomps_1(C,A) ) ) ) ) ) )).

fof(t8_pcomps_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
             => ( ( r1_pcomps_1(u1_struct_0(A),C)
                  & B = k6_pcomps_1(u1_struct_0(A),C) )
               => u1_struct_0(B) = u1_struct_0(A) ) ) ) ) )).

fof(t9_pcomps_2,axiom,(
    $true )).

fof(t10_pcomps_2,axiom,(
    $true )).

fof(t11_pcomps_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
                    & m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
                 => ( ( r1_pcomps_1(u1_struct_0(A),D)
                      & B = k6_pcomps_1(u1_struct_0(A),D) )
                   => ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                    <=> m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) ) ) ) ) ) ) )).

fof(t12_pcomps_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_pcomps_1(A)
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( r1_pre_topc(A,B)
                & v1_tops_2(B,A)
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                   => ~ ( v1_tops_2(C,A)
                        & r1_pre_topc(A,C)
                        & r1_setfam_1(C,B)
                        & v1_pcomps_1(C,A) ) ) ) ) ) ) )).

fof(t13_pcomps_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_pcomps_1(A)
       => v2_pcomps_1(A) ) ) )).

fof(s1_pcomps_2,axiom,
    ( ( r2_wellord1(f2_s1_pcomps_2,f1_s1_pcomps_2)
      & ? [A] :
          ( r2_hidden(A,f1_s1_pcomps_2)
          & p1_s1_pcomps_2(A) ) )
   => ? [A] :
        ( r2_hidden(A,f1_s1_pcomps_2)
        & p1_s1_pcomps_2(A)
        & ! [B] :
            ( ( r2_hidden(B,f1_s1_pcomps_2)
              & p1_s1_pcomps_2(B) )
           => r2_hidden(k4_tarski(A,B),f2_s1_pcomps_2) ) ) )).

fof(dt_k1_pcomps_2,axiom,(
    $true )).

fof(dt_k2_pcomps_2,axiom,(
    $true )).

fof(dt_k3_pcomps_2,axiom,(
    $true )).

fof(dt_k4_pcomps_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k3_finseq_2(k1_pcomps_1(k1_pcomps_1(A))))
        & m1_relset_1(B,k5_numbers,k3_finseq_2(k1_pcomps_1(k1_pcomps_1(A))))
        & m1_subset_1(C,k5_numbers) )
     => m2_finseq_1(k4_pcomps_2(A,B,C),k1_pcomps_1(k1_pcomps_1(A))) ) )).

fof(redefinition_k4_pcomps_2,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k3_finseq_2(k1_pcomps_1(k1_pcomps_1(A))))
        & m1_relset_1(B,k5_numbers,k3_finseq_2(k1_pcomps_1(k1_pcomps_1(A))))
        & m1_subset_1(C,k5_numbers) )
     => k4_pcomps_2(A,B,C) = k1_funct_1(B,C) ) )).

fof(s2_pcomps_2,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
      & m2_relset_1(A,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
      & k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)),A,np__0) = f2_s2_pcomps_2
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)),A,k1_nat_1(B,np__1)) = a_2_0_pcomps_2(A,B) ) ) )).

fof(s3_pcomps_2,axiom,(
    ? [A] :
      ( v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
      & m2_relset_1(A,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
      & k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)),A,np__0) = f2_s3_pcomps_2
      & ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)),A,k1_nat_1(B,np__1)) = a_2_1_pcomps_2(A,B) ) ) )).

fof(fraenkel_a_2_0_pcomps_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
        & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
        & m2_subset_1(C,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_2_0_pcomps_2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(f1_s2_pcomps_2))
            & A = k3_tarski(a_3_0_pcomps_2(B,C,D))
            & p1_s2_pcomps_2(D) ) ) ) )).

fof(fraenkel_a_3_0_pcomps_2,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
        & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
        & m2_subset_1(C,k1_numbers,k5_numbers)
        & m1_subset_1(D,k1_zfmisc_1(f1_s2_pcomps_2)) )
     => ( r2_hidden(A,a_3_0_pcomps_2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,f1_s2_pcomps_2)
            & A = f3_s2_pcomps_2(E,C)
            & ! [F] :
                ( m1_subset_1(F,k1_zfmisc_1(k1_zfmisc_1(f1_s2_pcomps_2)))
               => ! [G] :
                    ( m2_subset_1(G,k1_numbers,k5_numbers)
                   => ( ( r1_xreal_0(G,C)
                        & F = k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)),B,G) )
                     => p2_s2_pcomps_2(E,D,C,F) ) ) ) ) ) ) )).

fof(fraenkel_a_2_1_pcomps_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
        & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
        & m2_subset_1(C,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_2_1_pcomps_2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(f1_s3_pcomps_2))
            & A = k3_tarski(a_3_1_pcomps_2(B,C,D))
            & p1_s3_pcomps_2(D) ) ) ) )).

fof(fraenkel_a_3_1_pcomps_2,axiom,(
    ! [A,B,C,D] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
        & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
        & m2_subset_1(C,k1_numbers,k5_numbers)
        & m1_subset_1(D,k1_zfmisc_1(f1_s3_pcomps_2)) )
     => ( r2_hidden(A,a_3_1_pcomps_2(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,f1_s3_pcomps_2)
            & A = f3_s3_pcomps_2(E,C)
            & p2_s3_pcomps_2(E,D,C)
            & ~ r2_hidden(E,k3_tarski(a_2_2_pcomps_2(B,C))) ) ) ) )).

fof(fraenkel_a_2_2_pcomps_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
        & m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
        & m2_subset_1(C,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_2_2_pcomps_2(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = k5_setfam_1(f1_s3_pcomps_2,k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)),B,D))
            & r1_xreal_0(D,C) ) ) ) )).
%------------------------------------------------------------------------------