SET007 Axioms: SET007+712.ax


%------------------------------------------------------------------------------
% File     : SET007+712 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Hierarchies and Classifications of Sets
% 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    : taxonom2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   6 unit)
%            Number of atoms       :  241 (  14 equality)
%            Maximal formula depth :   22 (   9 average)
%            Number of connectives :  258 (  52 ~  ;   5  |; 111  &)
%                                         (  10 <=>;  80 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   0 propositional; 1-3 arity)
%            Number of functors    :   13 (   1 constant; 0-3 arity)
%            Number of variables   :  109 (   0 singleton; 102 !;   7 ?)
%            Maximal term depth    :    4 (   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_taxonom2,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & ~ v3_struct_0(A)
      & v1_orders_2(A)
      & v2_orders_2(A)
      & v3_orders_2(A)
      & v4_orders_2(A)
      & v1_taxonom2(A)
      & v2_taxonom2(A) ) )).

fof(cc1_taxonom2,axiom,(
    ! [A] :
      ( v1_realset1(A)
     => v3_taxonom2(A) ) )).

fof(rc2_taxonom2,axiom,(
    ? [A] :
      ( ~ v1_realset1(A)
      & v3_taxonom2(A) ) )).

fof(rc3_taxonom2,axiom,(
    ! [A] :
    ? [B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
      & v4_taxonom2(B) ) )).

fof(rc4_taxonom2,axiom,(
    ! [A] :
    ? [B] :
      ( m1_taxonom2(B,A)
      & v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
      & v5_taxonom2(B,A) ) )).

fof(d1_taxonom2,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_taxonom2(A)
      <=> ? [B] :
            ( m1_subset_1(B,u1_struct_0(A))
            & r8_orders_1(u1_orders_2(A),B) ) ) ) )).

fof(d2_taxonom2,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v2_taxonom2(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_orders_2(A,D,B)
                        & r1_orders_2(A,D,C) )
                    & ~ r1_orders_2(A,B,C)
                    & ~ r1_orders_2(A,C,B) ) ) ) ) ) )).

fof(t1_taxonom2,axiom,(
    ! [A] :
      ( ~ v3_struct_0(k2_yellow_1(k1_tarski(k1_tarski(A))))
      & v2_orders_2(k2_yellow_1(k1_tarski(k1_tarski(A))))
      & v3_orders_2(k2_yellow_1(k1_tarski(k1_tarski(A))))
      & v4_orders_2(k2_yellow_1(k1_tarski(k1_tarski(A))))
      & v1_taxonom2(k2_yellow_1(k1_tarski(k1_tarski(A))))
      & v2_taxonom2(k2_yellow_1(k1_tarski(k1_tarski(A)))) ) )).

fof(t2_taxonom2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C,D,E] :
              ( ( r2_hidden(E,k6_eqrel_1(A,B,C))
                & r2_hidden(E,k6_eqrel_1(A,B,D)) )
             => k6_eqrel_1(A,B,C) = k6_eqrel_1(A,B,D) ) ) ) )).

fof(t3_taxonom2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C,D,E] :
              ( ( r2_hidden(C,B)
                & r2_hidden(D,B)
                & r2_hidden(E,C)
                & r2_hidden(E,D) )
             => C = D ) ) ) )).

fof(t4_taxonom2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( ( m1_taxonom1(B,A)
            & r2_hidden(C,k3_tarski(B)) )
         => r1_tarski(C,A) ) ) )).

fof(t5_taxonom2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_taxonom1(B,A)
         => ( v2_orders_2(k2_yellow_1(k3_tarski(B)))
            & v3_orders_2(k2_yellow_1(k3_tarski(B)))
            & v4_orders_2(k2_yellow_1(k3_tarski(B)))
            & v1_taxonom2(k2_yellow_1(k3_tarski(B)))
            & v2_taxonom2(k2_yellow_1(k3_tarski(B)))
            & l1_orders_2(k2_yellow_1(k3_tarski(B))) ) ) ) )).

fof(d3_taxonom2,axiom,(
    ! [A] :
      ( v3_taxonom2(A)
    <=> ! [B,C] :
          ~ ( r2_hidden(B,A)
            & r2_hidden(C,A)
            & ~ r1_tarski(B,C)
            & ~ r1_tarski(C,B)
            & ~ r1_xboole_0(B,C) ) ) )).

fof(t6_taxonom2,axiom,(
    v3_taxonom2(k1_xboole_0) )).

fof(t7_taxonom2,axiom,(
    v3_taxonom2(k1_tarski(k1_xboole_0)) )).

fof(d4_taxonom2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( m1_taxonom2(B,A)
      <=> v3_taxonom2(B) ) ) )).

fof(d5_taxonom2,axiom,(
    ! [A] :
      ( v4_taxonom2(A)
    <=> ! [B,C] :
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => ( B = C
            | r1_xboole_0(B,C) ) ) ) )).

fof(t8_taxonom2,axiom,(
    v4_taxonom2(k1_xboole_0) )).

fof(t9_taxonom2,axiom,(
    v4_taxonom2(k1_tarski(k1_xboole_0)) )).

fof(t10_taxonom2,axiom,(
    ! [A] : v4_taxonom2(k1_tarski(A)) )).

fof(d6_taxonom2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v5_taxonom2(B,A)
      <=> ! [C] :
            ( m1_subset_1(C,k1_zfmisc_1(A))
           => ( r2_hidden(C,B)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ~ ( ~ r2_hidden(D,C)
                      & ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(A))
                         => ~ ( r2_hidden(D,E)
                              & r2_hidden(E,B)
                              & r1_xboole_0(C,E) ) ) ) ) ) ) ) ) )).

fof(t11_taxonom2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( B = k1_xboole_0
       => v5_taxonom2(B,A) ) ) )).

fof(d7_taxonom2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v6_taxonom2(B,A)
      <=> ! [C] :
            ~ ( C != k1_xboole_0
              & r1_tarski(C,B)
              & v6_ordinal1(C)
              & ! [D] :
                  ~ ( r2_hidden(D,B)
                    & r1_tarski(D,k1_setfam_1(C)) ) ) ) ) )).

fof(t12_taxonom2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( ( v4_taxonom2(C)
            & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( ! [D] :
                ( r2_hidden(D,C)
               => ( r1_xboole_0(B,D)
                  & A != k1_xboole_0 ) )
           => ( v4_taxonom2(k2_xboole_0(C,k1_tarski(B)))
              & m1_subset_1(k2_xboole_0(C,k1_tarski(B)),k1_zfmisc_1(k1_zfmisc_1(A)))
              & ~ ( B != k1_xboole_0
                  & k3_tarski(k2_xboole_0(C,k1_tarski(B))) = k5_setfam_1(A,C) ) ) ) ) ) )).

fof(d8_taxonom2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v7_taxonom2(B,A)
      <=> ! [C] :
            ( m1_subset_1(C,k1_zfmisc_1(A))
           => ~ ( r2_hidden(C,B)
                & ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(A))
                   => ~ ( r1_tarski(C,D)
                        & r2_hidden(D,B)
                        & ! [E] :
                            ( m1_subset_1(E,k1_zfmisc_1(A))
                           => ( ( r1_tarski(D,E)
                                & r2_hidden(E,B) )
                             => E = A ) ) ) ) ) ) ) ) )).

fof(t13_taxonom2,axiom,(
    ! [A,B] :
      ( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_taxonom2(B,A) )
     => ~ ( v7_taxonom2(B,A)
          & ! [C] :
              ( m1_eqrel_1(C,A)
             => ~ r1_tarski(C,B) ) ) ) )).

fof(t14_taxonom2,axiom,(
    ! [A,B] :
      ( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_taxonom2(B,A) )
     => ! [C] :
          ( ( v4_taxonom2(C)
            & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( ( r1_tarski(C,B)
              & ! [D] :
                  ( ( v4_taxonom2(D)
                    & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
                 => ( ( r1_tarski(D,B)
                      & r1_tarski(k5_setfam_1(A,C),k5_setfam_1(A,D)) )
                   => C = D ) ) )
           => m1_eqrel_1(C,A) ) ) ) )).

fof(t15_taxonom2,axiom,(
    ! [A,B] :
      ( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & v5_taxonom2(B,A)
        & m1_taxonom2(B,A) )
     => ( v6_taxonom2(B,A)
       => ( r2_hidden(k1_xboole_0,B)
          | ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( ( v4_taxonom2(D)
                    & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
                 => ( ( r2_hidden(C,D)
                      & r1_tarski(D,B)
                      & ! [E] :
                          ( ( v4_taxonom2(E)
                            & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A))) )
                         => ( ( r2_hidden(C,E)
                              & r1_tarski(E,B)
                              & r1_tarski(k5_setfam_1(A,D),k5_setfam_1(A,E)) )
                           => k5_setfam_1(A,D) = k5_setfam_1(A,E) ) ) )
                   => m1_eqrel_1(D,A) ) ) ) ) ) ) )).

fof(t16_taxonom2,axiom,(
    ! [A,B] :
      ( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & v5_taxonom2(B,A)
        & m1_taxonom2(B,A) )
     => ( v6_taxonom2(B,A)
       => ( r2_hidden(k1_xboole_0,B)
          | ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( ( v4_taxonom2(D)
                    & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
                 => ( ( r2_hidden(C,D)
                      & r1_tarski(D,B)
                      & ! [E] :
                          ( ( v4_taxonom2(E)
                            & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(A))) )
                         => ( ( r2_hidden(C,E)
                              & r1_tarski(E,B)
                              & r1_tarski(D,E) )
                           => D = E ) ) )
                   => m1_eqrel_1(D,A) ) ) ) ) ) ) )).

fof(t17_taxonom2,axiom,(
    ! [A,B] :
      ( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & v5_taxonom2(B,A)
        & m1_taxonom2(B,A) )
     => ( v6_taxonom2(B,A)
       => ( r2_hidden(k1_xboole_0,B)
          | ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ~ ( r2_hidden(C,B)
                  & ! [D] :
                      ( m1_eqrel_1(D,A)
                     => ~ ( r2_hidden(C,D)
                          & r1_tarski(D,B) ) ) ) ) ) ) ) )).

fof(t18_taxonom2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( ( r2_hidden(D,C)
                    & r1_tarski(B,D) )
                 => ! [E] :
                      ( m1_eqrel_1(E,A)
                     => ( ( r2_hidden(B,E)
                          & ! [F] :
                              ~ ( r2_hidden(F,E)
                                & ~ r1_tarski(F,D)
                                & ~ r1_tarski(D,F)
                                & ~ r1_xboole_0(D,F) ) )
                       => ! [F] :
                            ( ! [G] :
                                ( r2_hidden(G,F)
                              <=> ( r2_hidden(G,E)
                                  & r1_tarski(G,D) ) )
                           => ( m1_eqrel_1(k2_xboole_0(F,k4_xboole_0(C,k1_tarski(D))),A)
                              & r1_setfam_1(k2_xboole_0(F,k4_xboole_0(C,k1_tarski(D))),C) ) ) ) ) ) ) ) ) )).

fof(t19_taxonom2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ( r1_tarski(B,A)
           => ! [C] :
                ( m1_eqrel_1(C,A)
               => ( ! [D] :
                      ~ ( r2_hidden(D,C)
                        & ~ r1_tarski(D,B)
                        & ~ r1_tarski(B,D)
                        & ~ r1_xboole_0(B,D) )
                 => ( ! [D] :
                        ~ ( r2_hidden(D,C)
                          & r1_tarski(D,B) )
                    | ! [D] :
                        ( ! [E] :
                            ( r2_hidden(E,D)
                          <=> ( r2_hidden(E,C)
                              & r1_xboole_0(E,B) ) )
                       => ( m1_eqrel_1(k2_xboole_0(D,k1_tarski(B)),A)
                          & r1_setfam_1(C,k2_xboole_0(D,k1_tarski(B)))
                          & ! [E] :
                              ( m1_eqrel_1(E,A)
                             => ( r1_setfam_1(C,E)
                               => ! [F] :
                                    ( ( r2_hidden(F,E)
                                      & r1_tarski(B,F) )
                                   => r1_setfam_1(k2_xboole_0(D,k1_tarski(B)),E) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t20_taxonom2,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
            & v5_taxonom2(B,A)
            & m1_taxonom2(B,A) )
         => ~ ( v6_taxonom2(B,A)
              & ~ r2_hidden(k1_xboole_0,B)
              & ! [C] :
                  ~ ( C != k1_xboole_0
                    & r1_tarski(C,k1_partit1(A))
                    & ! [D,E] :
                        ~ ( r2_hidden(D,C)
                          & r2_hidden(E,C)
                          & ~ r1_setfam_1(D,E)
                          & ~ r1_setfam_1(E,D) )
                    & ! [D,E] :
                        ~ ( r2_hidden(D,C)
                          & r2_hidden(E,C)
                          & ! [F] :
                              ( r2_hidden(F,C)
                             => ( r1_setfam_1(F,E)
                                & r1_setfam_1(D,F) ) ) ) )
              & ! [C] :
                  ( m1_taxonom1(C,A)
                 => k3_tarski(C) != B ) ) ) ) )).

fof(dt_m1_taxonom2,axiom,(
    ! [A,B] :
      ( m1_taxonom2(B,A)
     => m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(existence_m1_taxonom2,axiom,(
    ! [A] :
    ? [B] : m1_taxonom2(B,A) )).
%------------------------------------------------------------------------------