SET007 Axioms: SET007+776.ax


%------------------------------------------------------------------------------
% File     : SET007+776 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Class of Series-Parallel Graphs. Part II
% 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    : neckla_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   22 (   5 unit)
%            Number of atoms       :  126 (  12 equality)
%            Maximal formula depth :   17 (   6 average)
%            Number of connectives :  117 (  13 ~  ;   3  |;  58  &)
%                                         (   6 <=>;  37 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-3 arity)
%            Number of functors    :   19 (   8 constant; 0-6 arity)
%            Number of variables   :   38 (   0 singleton;  36 !;   2 ?)
%            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(cc1_neckla_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( m1_subset_1(B,k4_classes1(A))
         => v1_finset_1(B) ) ) )).

fof(cc2_neckla_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k13_classes2)
     => v1_finset_1(A) ) )).

fof(cc3_neckla_2,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v1_finset_1(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_card_1(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) )).

fof(rc1_neckla_2,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & ~ v3_struct_0(A)
      & v1_orders_2(A)
      & v6_group_1(A)
      & v1_neckla_2(A) ) )).

fof(fc1_neckla_2,axiom,(
    ~ v1_xboole_0(k3_neckla_2) )).

fof(fc2_neckla_2,axiom,(
    ~ v1_xboole_0(k4_neckla_2) )).

fof(t1_neckla_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_classes2(A) )
     => ! [B,C] :
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => ! [D] :
              ( m2_relset_1(D,B,C)
             => r2_hidden(D,A) ) ) ) )).

fof(t2_neckla_2,axiom,(
    u1_orders_2(k4_necklace(np__4)) = k4_enumset1(k4_tarski(np__0,np__1),k4_tarski(np__1,np__0),k4_tarski(np__1,np__2),k4_tarski(np__2,np__1),k4_tarski(np__2,np__3),k4_tarski(np__3,np__2)) )).

fof(t3_neckla_2,axiom,(
    ! [A] :
      ( r2_hidden(A,k13_classes2)
     => v1_finset_1(A) ) )).

fof(d1_neckla_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ( v1_neckla_2(A)
      <=> ~ r3_necklace(k4_necklace(np__4),A) ) ) )).

fof(d2_neckla_2,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C] :
              ( ( v1_orders_2(C)
                & l1_orders_2(C) )
             => ( C = k1_neckla_2(A,B)
              <=> ( u1_struct_0(C) = k2_xboole_0(u1_struct_0(A),u1_struct_0(B))
                  & u1_orders_2(C) = k2_xboole_0(u1_orders_2(A),u1_orders_2(B)) ) ) ) ) ) )).

fof(d3_neckla_2,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ! [C] :
              ( ( v1_orders_2(C)
                & l1_orders_2(C) )
             => ( C = k2_neckla_2(A,B)
              <=> ( u1_struct_0(C) = k2_xboole_0(u1_struct_0(A),u1_struct_0(B))
                  & u1_orders_2(C) = k2_xboole_0(k2_xboole_0(k2_xboole_0(u1_orders_2(A),u1_orders_2(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A))) ) ) ) ) ) )).

fof(d4_neckla_2,axiom,(
    ! [A] :
      ( A = k3_neckla_2
    <=> ! [B] :
          ( r2_hidden(B,A)
        <=> ? [C] :
              ( v1_orders_2(C)
              & l1_orders_2(C)
              & B = C
              & r2_hidden(u1_struct_0(C),k13_classes2) ) ) ) )).

fof(d5_neckla_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k3_neckla_2))
     => ( A = k4_neckla_2
      <=> ( ! [B] :
              ( ( v1_orders_2(B)
                & l1_orders_2(B) )
             => ( ( v1_realset1(u1_struct_0(B))
                  & r2_hidden(u1_struct_0(B),k13_classes2) )
               => ( v1_xboole_0(u1_struct_0(B))
                  | r2_hidden(B,A) ) ) )
          & ! [B] :
              ( ( v1_orders_2(B)
                & l1_orders_2(B) )
             => ! [C] :
                  ( ( v1_orders_2(C)
                    & l1_orders_2(C) )
                 => ( ( r1_xboole_0(u1_struct_0(B),u1_struct_0(C))
                      & r2_hidden(B,A)
                      & r2_hidden(C,A) )
                   => ( r2_hidden(k1_neckla_2(B,C),A)
                      & r2_hidden(k2_neckla_2(B,C),A) ) ) ) )
          & ! [B] :
              ( m1_subset_1(B,k1_zfmisc_1(k3_neckla_2))
             => ( ( ! [C] :
                      ( ( v1_orders_2(C)
                        & l1_orders_2(C) )
                     => ( ( v1_realset1(u1_struct_0(C))
                          & r2_hidden(u1_struct_0(C),k13_classes2) )
                       => ( v1_xboole_0(u1_struct_0(C))
                          | r2_hidden(C,B) ) ) )
                  & ! [C] :
                      ( ( v1_orders_2(C)
                        & l1_orders_2(C) )
                     => ! [D] :
                          ( ( v1_orders_2(D)
                            & l1_orders_2(D) )
                         => ( ( r1_xboole_0(u1_struct_0(C),u1_struct_0(D))
                              & r2_hidden(C,B)
                              & r2_hidden(D,B) )
                           => ( r2_hidden(k1_neckla_2(C,D),B)
                              & r2_hidden(k2_neckla_2(C,D),B) ) ) ) ) )
               => r1_tarski(A,B) ) ) ) ) ) )).

fof(t4_neckla_2,axiom,(
    ! [A] :
      ( r2_hidden(A,k4_neckla_2)
     => ( ~ v3_struct_0(A)
        & v1_orders_2(A)
        & v6_group_1(A)
        & l1_orders_2(A) ) ) )).

fof(t5_neckla_2,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( r2_hidden(A,k4_neckla_2)
       => r2_hidden(u1_struct_0(A),k13_classes2) ) ) )).

fof(t6_neckla_2,axiom,(
    ! [A] :
      ~ ( r2_hidden(A,k4_neckla_2)
        & ~ ( ~ v3_struct_0(A)
            & v1_orders_2(A)
            & v3_realset2(A)
            & l1_orders_2(A) )
        & ! [B] :
            ( ( v1_orders_2(B)
              & l1_orders_2(B) )
           => ! [C] :
                ( ( v1_orders_2(C)
                  & l1_orders_2(C) )
               => ~ ( r1_xboole_0(u1_struct_0(B),u1_struct_0(C))
                    & r2_hidden(B,k4_neckla_2)
                    & r2_hidden(C,k4_neckla_2)
                    & ( A = k1_neckla_2(B,C)
                      | A = k2_neckla_2(B,C) ) ) ) ) ) )).

fof(t7_neckla_2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_orders_2(A)
        & l1_orders_2(A) )
     => ( r2_hidden(A,k4_neckla_2)
       => v1_neckla_2(A) ) ) )).

fof(dt_k1_neckla_2,axiom,(
    ! [A,B] :
      ( ( l1_orders_2(A)
        & l1_orders_2(B) )
     => ( v1_orders_2(k1_neckla_2(A,B))
        & l1_orders_2(k1_neckla_2(A,B)) ) ) )).

fof(dt_k2_neckla_2,axiom,(
    ! [A,B] :
      ( ( l1_orders_2(A)
        & l1_orders_2(B) )
     => ( v1_orders_2(k2_neckla_2(A,B))
        & l1_orders_2(k2_neckla_2(A,B)) ) ) )).

fof(dt_k3_neckla_2,axiom,(
    $true )).

fof(dt_k4_neckla_2,axiom,(
    m1_subset_1(k4_neckla_2,k1_zfmisc_1(k3_neckla_2)) )).
%------------------------------------------------------------------------------