SET007 Axioms: SET007+426.ax


%------------------------------------------------------------------------------
% File     : SET007+426 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Cantor Set
% 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    : cantor_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   39 (   3 unit)
%            Number of atoms       :  153 (  17 equality)
%            Maximal formula depth :   17 (   6 average)
%            Number of connectives :  129 (  15 ~  ;   0  |;  45  &)
%                                         (  10 <=>;  59 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   1 propositional; 0-3 arity)
%            Number of functors    :   21 (   5 constant; 0-5 arity)
%            Number of variables   :   91 (   0 singleton;  82 !;   9 ?)
%            Maximal term depth    :    5 (   2 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_cantor_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( v1_relat_1(k2_funcop_1(A,B))
        & v2_relat_1(k2_funcop_1(A,B))
        & v1_funct_1(k2_funcop_1(A,B)) ) ) )).

fof(fc2_cantor_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ~ v1_xboole_0(u1_pre_topc(A)) ) )).

fof(fc3_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ~ v1_xboole_0(k2_cantor_1(A,B)) ) )).

fof(d1_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( C = k1_cantor_1(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,C)
                <=> ? [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A)))
                      & r1_tarski(E,B)
                      & D = k5_setfam_1(A,E) ) ) ) ) ) ) )).

fof(d2_cantor_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( m1_cantor_1(B,A)
          <=> ( r1_tarski(B,u1_pre_topc(A))
              & r1_tarski(u1_pre_topc(A),k1_cantor_1(u1_struct_0(A),B)) ) ) ) ) )).

fof(t1_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => r1_tarski(B,k1_cantor_1(A,B)) ) )).

fof(t2_cantor_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => m1_cantor_1(u1_pre_topc(A),A) ) )).

fof(t3_cantor_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => v1_tops_2(u1_pre_topc(A),A) ) )).

fof(d3_cantor_1,axiom,(
    $true )).

fof(d4_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( C = k2_cantor_1(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,C)
                <=> ? [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A)))
                      & r1_tarski(E,B)
                      & v1_finset_1(E)
                      & D = k8_setfam_1(A,E) ) ) ) ) ) ) )).

fof(t4_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => r1_tarski(B,k2_cantor_1(A,B)) ) )).

fof(t5_cantor_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => u1_pre_topc(A) = k2_cantor_1(u1_struct_0(A),u1_pre_topc(A)) ) )).

fof(t6_cantor_1,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => u1_pre_topc(A) = k1_cantor_1(u1_struct_0(A),u1_pre_topc(A)) ) )).

fof(t7_cantor_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => u1_pre_topc(A) = k1_cantor_1(u1_struct_0(A),k2_cantor_1(u1_struct_0(A),u1_pre_topc(A))) ) )).

fof(t8_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => r2_hidden(A,k2_cantor_1(A,B)) ) )).

fof(t9_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( r1_tarski(B,C)
           => r1_tarski(k1_cantor_1(A,B),k1_cantor_1(A,C)) ) ) ) )).

fof(t10_cantor_1,axiom,(
    $true )).

fof(t11_cantor_1,axiom,(
    $true )).

fof(t13_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => k2_cantor_1(A,B) = k2_cantor_1(A,k2_cantor_1(A,B)) ) )).

fof(t14_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C,D] :
          ( ( r2_hidden(C,k2_cantor_1(A,B))
            & r2_hidden(D,k2_cantor_1(A,B)) )
         => r2_hidden(k3_xboole_0(C,D),k2_cantor_1(A,B)) ) ) )).

fof(t15_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C,D] :
          ( ( r1_tarski(C,k2_cantor_1(A,B))
            & r1_tarski(D,k2_cantor_1(A,B)) )
         => r1_tarski(k3_setfam_1(C,D),k2_cantor_1(A,B)) ) ) )).

fof(t16_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( r1_tarski(B,C)
           => r1_tarski(k2_cantor_1(A,B),k2_cantor_1(A,C)) ) ) ) )).

fof(t17_cantor_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => v2_pre_topc(g1_pre_topc(A,k1_cantor_1(A,k2_cantor_1(A,B)))) ) ) )).

fof(d5_cantor_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( m2_cantor_1(B,A)
          <=> ( r1_tarski(B,u1_pre_topc(A))
              & ? [C] :
                  ( m1_cantor_1(C,A)
                  & r1_tarski(C,k2_cantor_1(u1_struct_0(A),B)) ) ) ) ) ) )).

fof(t18_cantor_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => m1_cantor_1(B,g1_pre_topc(A,k1_cantor_1(A,B))) ) ) )).

fof(t19_cantor_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_pre_topc(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v1_pre_topc(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( m2_cantor_1(C,A)
             => ( ( u1_struct_0(A) = u1_struct_0(B)
                  & m2_cantor_1(C,B) )
               => A = B ) ) ) ) )).

fof(t20_cantor_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => m2_cantor_1(B,g1_pre_topc(A,k1_cantor_1(A,k2_cantor_1(A,B)))) ) ) )).

fof(d6_cantor_1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_pre_topc(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( A = k4_cantor_1
      <=> ( u1_struct_0(A) = k4_card_3(k2_funcop_1(k5_numbers,k2_tarski(np__0,np__1)))
          & ? [B] :
              ( m2_cantor_1(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(k4_card_3(k2_funcop_1(k5_numbers,k2_tarski(np__0,np__1)))))
                 => ( r2_hidden(C,B)
                  <=> ? [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                        & ? [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                            & ! [F] :
                                ( m1_subset_1(F,k4_card_3(k2_funcop_1(k5_numbers,k2_tarski(np__0,np__1))))
                               => ( r2_hidden(F,C)
                                <=> k1_funct_1(F,D) = E ) ) ) ) ) ) ) ) ) ) )).

fof(dt_m1_cantor_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_cantor_1(B,A)
         => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) )).

fof(existence_m1_cantor_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ? [B] : m1_cantor_1(B,A) ) )).

fof(dt_m2_cantor_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m2_cantor_1(B,A)
         => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) )).

fof(existence_m2_cantor_1,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ? [B] : m2_cantor_1(B,A) ) )).

fof(dt_k1_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k1_cantor_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(dt_k2_cantor_1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => m1_subset_1(k2_cantor_1(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(dt_k3_cantor_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
        & v1_funct_1(D)
        & v1_funct_2(D,B,k1_zfmisc_1(C))
        & m1_relset_1(D,B,k1_zfmisc_1(C)) )
     => m1_subset_1(k3_cantor_1(A,B,C,D,E),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(redefinition_k3_cantor_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
        & v1_funct_1(D)
        & v1_funct_2(D,B,k1_zfmisc_1(C))
        & m1_relset_1(D,B,k1_zfmisc_1(C)) )
     => k3_cantor_1(A,B,C,D,E) = k1_funct_1(D,E) ) )).

fof(dt_k4_cantor_1,axiom,
    ( ~ v3_struct_0(k4_cantor_1)
    & v1_pre_topc(k4_cantor_1)
    & v2_pre_topc(k4_cantor_1)
    & l1_pre_topc(k4_cantor_1) )).

fof(t12_cantor_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_zfmisc_1(A)))) )
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( C = a_2_0_cantor_1(A,B)
           => k8_setfam_1(A,C) = k8_setfam_1(A,k5_setfam_1(k1_zfmisc_1(A),B)) ) ) ) )).

fof(fraenkel_a_2_0_cantor_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k1_zfmisc_1(B)))) )
     => ( r2_hidden(A,a_2_0_cantor_1(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(B)),C)
            & A = k8_setfam_1(B,D) ) ) ) )).
%------------------------------------------------------------------------------