SET007 Axioms: SET007+750.ax


%------------------------------------------------------------------------------
% File     : SET007+750 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Class of Series - Parallel Graphs. 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    : necklace [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   66 (   7 unit)
%            Number of atoms       :  276 (  54 equality)
%            Maximal formula depth :   17 (   6 average)
%            Number of connectives :  264 (  54 ~  ;   3  |; 100  &)
%                                         (  16 <=>;  91 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   0 propositional; 1-5 arity)
%            Number of functors    :   50 (  12 constant; 0-9 arity)
%            Number of variables   :  118 (   0 singleton; 111 !;   7 ?)
%            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_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => v1_card_1(A) ) )).

fof(fc1_necklace,axiom,(
    ! [A] :
      ( v1_relat_1(k11_funct_3(A))
      & v1_funct_1(k11_funct_3(A))
      & v2_funct_1(k11_funct_3(A))
      & v1_setfam_1(k11_funct_3(A)) ) )).

fof(fc2_necklace,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_realset1(A) )
     => v1_realset1(k1_relat_1(A)) ) )).

fof(cc2_necklace,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_realset1(A) )
     => ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A)
        & v1_setfam_1(A) ) ) )).

fof(fc3_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_orders_2(k2_necklace(A))
        & v1_necklace(k2_necklace(A)) ) ) )).

fof(rc1_necklace,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & ~ v3_struct_0(A)
      & v1_necklace(A) ) )).

fof(fc4_necklace,axiom,(
    ! [A] :
      ( ( v1_necklace(A)
        & l1_orders_2(A) )
     => ( v1_relat_1(u1_orders_2(A))
        & v3_relat_2(u1_orders_2(A))
        & v1_setfam_1(u1_orders_2(A)) ) ) )).

fof(fc5_necklace,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ( ~ v3_struct_0(k3_necklace(A))
        & v1_orders_2(k3_necklace(A)) ) ) )).

fof(fc6_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_orders_2(k4_necklace(A))
        & v1_necklace(k4_necklace(A)) ) ) )).

fof(fc7_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_orders_2(k4_necklace(A))
        & v1_necklace(k4_necklace(A))
        & v3_necklace(k4_necklace(A)) ) ) )).

fof(fc8_necklace,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ( ~ v3_struct_0(k4_necklace(A))
        & v1_orders_2(k4_necklace(A))
        & v1_necklace(k4_necklace(A))
        & v3_necklace(k4_necklace(A)) ) ) )).

fof(fc9_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => v1_finset_1(u1_struct_0(k4_necklace(A))) ) )).

fof(d1_necklace,axiom,(
    ! [A,B,C,D,E] :
      ( r1_necklace(A,B,C,D,E)
    <=> ( A != B
        & A != C
        & A != D
        & A != E
        & B != C
        & B != D
        & B != E
        & C != D
        & C != E
        & D != E ) ) )).

fof(t1_necklace,axiom,(
    ! [A,B,C,D,E] :
      ( r1_necklace(A,B,C,D,E)
     => k4_card_1(k3_enumset1(A,B,C,D,E)) = np__5 ) )).

fof(t2_necklace,axiom,(
    np__4 = k2_enumset1(np__0,np__1,np__2,np__3) )).

fof(t3_necklace,axiom,(
    ! [A,B,C,D,E,F] : k2_zfmisc_1(k1_enumset1(A,B,C),k1_enumset1(D,E,F)) = k1_bvfunc24(k4_tarski(A,D),k4_tarski(A,E),k4_tarski(A,F),k4_tarski(B,D),k4_tarski(B,E),k4_tarski(B,F),k4_tarski(C,D),k4_tarski(C,E),k4_tarski(C,F)) )).

fof(t4_necklace,axiom,(
    ! [A,B] :
      ( v4_ordinal2(B)
     => ( r2_hidden(A,B)
       => v4_ordinal2(A) ) ) )).

fof(t5_necklace,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => r2_hidden(np__0,A) ) )).

fof(t6_necklace,axiom,(
    ! [A] : k1_card_1(k6_partfun1(A)) = k1_card_1(A) )).

fof(t7_necklace,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
           => k2_relat_1(k1_funct_4(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ) )).

fof(t8_necklace,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v2_funct_1(B) )
         => ( ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
              & r1_xboole_0(k2_relat_1(A),k2_relat_1(B)) )
           => k2_funct_1(k1_funct_4(A,B)) = k1_funct_4(k2_funct_1(A),k2_funct_1(B)) ) ) ) )).

fof(t9_necklace,axiom,(
    ! [A,B,C] : k1_funct_4(k2_funcop_1(A,B),k2_funcop_1(A,C)) = k2_funcop_1(A,C) )).

fof(t10_necklace,axiom,(
    ! [A,B] : k2_funct_1(k3_cqc_lang(A,B)) = k3_cqc_lang(B,A) )).

fof(t11_necklace,axiom,(
    ! [A,B,C,D] :
      ~ ( ( A = B
         => C = D )
        & ( C = D
         => A = B )
        & k2_funct_1(k4_funct_4(A,B,C,D)) != k4_funct_4(C,D,A,B) ) )).

fof(t12_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( r2_hidden(B,C)
               => ( r1_xreal_0(B,A)
                  | r2_hidden(A,C) ) ) ) ) ) )).

fof(d2_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ( r2_necklace(A,B)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
                & v2_funct_1(C)
                & ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ( r2_hidden(k4_tarski(D,E),u1_orders_2(A))
                        <=> r2_hidden(k4_tarski(k1_funct_1(C,D),k1_funct_1(C,E)),u1_orders_2(B)) ) ) ) ) ) ) ) )).

fof(t13_necklace,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & l1_orders_2(C) )
             => ( ( r3_necklace(B,A)
                  & r3_necklace(C,B) )
               => r3_necklace(C,A) ) ) ) ) )).

fof(d3_necklace,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ( r4_necklace(A,B)
          <=> ( r3_necklace(B,A)
              & r3_necklace(A,B) ) ) ) ) )).

fof(t14_necklace,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & l1_orders_2(C) )
             => ( ( r4_necklace(A,B)
                  & r4_necklace(B,C) )
               => r4_necklace(A,C) ) ) ) ) )).

fof(d4_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_necklace(A)
      <=> r3_relat_2(u1_orders_2(A),u1_struct_0(A)) ) ) )).

fof(d5_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v2_necklace(A)
      <=> v5_relat_2(u1_orders_2(A)) ) ) )).

fof(t15_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v2_necklace(A)
       => r1_xboole_0(u1_orders_2(A),k6_relset_1(u1_struct_0(A),u1_struct_0(A),u1_orders_2(A))) ) ) )).

fof(d6_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v3_necklace(A)
      <=> ! [B] :
            ~ ( r2_hidden(B,u1_struct_0(A))
              & r2_hidden(k4_tarski(B,B),u1_orders_2(A)) ) ) ) )).

fof(t16_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => v2_necklace(k1_necklace(A)) ) )).

fof(t17_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k1_card_1(u1_orders_2(k1_necklace(A))) = k6_xcmplx_0(A,np__1) ) ) )).

fof(d8_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( ( v1_orders_2(B)
            & l1_orders_2(B) )
         => ( B = k2_necklace(A)
          <=> ( u1_struct_0(B) = u1_struct_0(A)
              & u1_orders_2(B) = k1_relset_1(u1_struct_0(A),u1_struct_0(A),u1_orders_2(A),k6_relset_1(u1_struct_0(A),u1_struct_0(A),u1_orders_2(A))) ) ) ) ) )).

fof(d9_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( ( v1_orders_2(B)
            & l1_orders_2(B) )
         => ( B = k3_necklace(A)
          <=> ( u1_struct_0(B) = u1_struct_0(A)
              & u1_orders_2(B) = k6_subset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k3_subset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_orders_2(A)),k6_partfun1(u1_struct_0(A))) ) ) ) ) )).

fof(t18_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( l1_orders_2(B)
         => ( r5_waybel_1(A,B)
           => k1_card_1(u1_orders_2(A)) = k1_card_1(u1_orders_2(B)) ) ) ) )).

fof(d10_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k4_necklace(A) = k2_necklace(k1_necklace(A)) ) )).

fof(t20_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( r2_hidden(B,u1_orders_2(k4_necklace(A)))
        <=> ? [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
              & ~ r1_xreal_0(A,k1_nat_1(C,np__1))
              & ( B = k4_tarski(C,k1_nat_1(C,np__1))
                | B = k4_tarski(k1_nat_1(C,np__1),C) ) ) ) ) )).

fof(t21_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => u1_struct_0(k4_necklace(A)) = A ) )).

fof(t22_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(A,k2_xcmplx_0(B,np__1))
           => r2_hidden(k4_tarski(B,k2_xcmplx_0(B,np__1)),u1_orders_2(k4_necklace(A))) ) ) ) )).

fof(t23_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( r2_hidden(B,u1_struct_0(k4_necklace(A)))
              & r1_xreal_0(A,B) ) ) ) )).

fof(t24_necklace,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ~ v3_orders_3(k4_necklace(A)) ) )).

fof(t25_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ~ ( r2_hidden(k4_tarski(B,C),u1_orders_2(k4_necklace(A)))
                  & B != k2_xcmplx_0(C,np__1)
                  & C != k2_xcmplx_0(B,np__1) ) ) ) ) )).

fof(t26_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( ( r2_hidden(B,u1_struct_0(k4_necklace(A)))
                  & r2_hidden(C,u1_struct_0(k4_necklace(A))) )
               => ( ( B != k2_xcmplx_0(C,np__1)
                    & C != k2_xcmplx_0(B,np__1) )
                  | r2_hidden(k4_tarski(B,C),u1_orders_2(k4_necklace(A))) ) ) ) ) ) )).

fof(t28_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k1_card_1(u1_orders_2(k4_necklace(A))) = k3_xcmplx_0(np__2,k6_xcmplx_0(A,np__1)) ) ) )).

fof(t29_necklace,axiom,(
    r5_waybel_1(k4_necklace(np__1),k3_necklace(k4_necklace(np__1))) )).

fof(t30_necklace,axiom,(
    r5_waybel_1(k4_necklace(np__4),k3_necklace(k4_necklace(np__4))) )).

fof(t31_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ~ ( r5_waybel_1(k4_necklace(A),k3_necklace(k4_necklace(A)))
          & A != np__0
          & A != np__1
          & A != np__4 ) ) )).

fof(reflexivity_r3_necklace,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A)
        & ~ v3_struct_0(B)
        & l1_orders_2(B) )
     => r3_necklace(B,B) ) )).

fof(redefinition_r3_necklace,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A)
        & ~ v3_struct_0(B)
        & l1_orders_2(B) )
     => ( r3_necklace(A,B)
      <=> r2_necklace(A,B) ) ) )).

fof(symmetry_r4_necklace,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A)
        & ~ v3_struct_0(B)
        & l1_orders_2(B) )
     => ( r4_necklace(A,B)
       => r4_necklace(B,A) ) ) )).

fof(reflexivity_r4_necklace,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A)
        & ~ v3_struct_0(B)
        & l1_orders_2(B) )
     => r4_necklace(A,A) ) )).

fof(dt_k1_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_orders_2(k1_necklace(A))
        & l1_orders_2(k1_necklace(A)) ) ) )).

fof(dt_k2_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_orders_2(k2_necklace(A))
        & l1_orders_2(k2_necklace(A)) ) ) )).

fof(dt_k3_necklace,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_orders_2(k3_necklace(A))
        & l1_orders_2(k3_necklace(A)) ) ) )).

fof(dt_k4_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_orders_2(k4_necklace(A))
        & l1_orders_2(k4_necklace(A)) ) ) )).

fof(d7_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( ( v1_orders_2(B)
            & l1_orders_2(B) )
         => ( B = k1_necklace(A)
          <=> ( u1_struct_0(B) = A
              & u1_orders_2(B) = a_1_0_necklace(A) ) ) ) ) )).

fof(t19_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => u1_orders_2(k4_necklace(A)) = k2_xboole_0(a_1_0_necklace(A),a_1_1_necklace(A)) ) )).

fof(t27_necklace,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k1_card_1(a_1_1_necklace(A)) = k6_xcmplx_0(A,np__1) ) ) )).

fof(s1_necklace,axiom,
    ( f2_s1_necklace = a_0_0_necklace
   => k4_relat_1(f2_s1_necklace) = a_0_1_necklace )).

fof(fraenkel_a_1_0_necklace,axiom,(
    ! [A,B] :
      ( v4_ordinal2(B)
     => ( r2_hidden(A,a_1_0_necklace(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k4_tarski(C,k1_nat_1(C,np__1))
            & ~ r1_xreal_0(B,k1_nat_1(C,np__1)) ) ) ) )).

fof(fraenkel_a_1_1_necklace,axiom,(
    ! [A,B] :
      ( v4_ordinal2(B)
     => ( r2_hidden(A,a_1_1_necklace(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k4_tarski(k1_nat_1(C,np__1),C)
            & ~ r1_xreal_0(B,k1_nat_1(C,np__1)) ) ) ) )).

fof(fraenkel_a_0_0_necklace,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_necklace)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_necklace)
          & A = k4_tarski(f4_s1_necklace(B),f3_s1_necklace(B))
          & p1_s1_necklace(B) ) ) )).

fof(fraenkel_a_0_1_necklace,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_necklace)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_necklace)
          & A = k4_tarski(f3_s1_necklace(B),f4_s1_necklace(B))
          & p1_s1_necklace(B) ) ) )).
%------------------------------------------------------------------------------