SET007 Axioms: SET007+561.ax


%------------------------------------------------------------------------------
% File     : SET007+561 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Representation Theorem for Free Continuous Lattices
% 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    : waybel22 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   32 (   5 unit)
%            Number of atoms       :  327 (  25 equality)
%            Maximal formula depth :   17 (   9 average)
%            Number of connectives :  324 (  29 ~  ;   0  |; 236  &)
%                                         (   7 <=>;  52 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   40 (   0 propositional; 1-3 arity)
%            Number of functors    :   25 (   0 constant; 1-5 arity)
%            Number of variables   :   92 (   0 singleton;  83 !;   9 ?)
%            Maximal term depth    :    6 (   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_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v2_yellow_0(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ( ~ v3_struct_0(k2_yellow_1(k9_waybel_0(A)))
        & v1_orders_2(k2_yellow_1(k9_waybel_0(A)))
        & v2_orders_2(k2_yellow_1(k9_waybel_0(A)))
        & v3_orders_2(k2_yellow_1(k9_waybel_0(A)))
        & v4_orders_2(k2_yellow_1(k9_waybel_0(A)))
        & v1_yellow_0(k2_yellow_1(k9_waybel_0(A)))
        & v2_yellow_0(k2_yellow_1(k9_waybel_0(A)))
        & v3_yellow_0(k2_yellow_1(k9_waybel_0(A)))
        & v24_waybel_0(k2_yellow_1(k9_waybel_0(A)))
        & v25_waybel_0(k2_yellow_1(k9_waybel_0(A)))
        & v1_lattice3(k2_yellow_1(k9_waybel_0(A)))
        & v2_lattice3(k2_yellow_1(k9_waybel_0(A)))
        & v3_lattice3(k2_yellow_1(k9_waybel_0(A)))
        & v3_waybel_3(k2_yellow_1(k9_waybel_0(A))) ) ) )).

fof(cc1_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v2_yellow_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k2_yellow_1(k9_waybel_0(A))))
         => ~ v1_xboole_0(B) ) ) )).

fof(fc2_waybel22,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v3_lattice3(B)
        & v3_waybel_3(B)
        & l1_orders_2(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
        & m1_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
     => ( v1_relat_1(k2_waybel22(A,B,C))
        & v1_funct_1(k2_waybel22(A,B,C))
        & v1_funct_2(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
        & v22_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B) ) ) )).

fof(fc3_waybel22,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v3_lattice3(B)
        & v3_waybel_3(B)
        & l1_orders_2(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
        & m1_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
     => ( v1_relat_1(k2_waybel22(A,B,C))
        & v1_funct_1(k2_waybel22(A,B,C))
        & v1_funct_2(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
        & v17_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
        & v19_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
        & v21_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
        & v22_waybel_0(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B) ) ) )).

fof(t1_waybel22,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v2_yellow_0(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_waybel_0(B,k2_yellow_1(k9_waybel_0(A)))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k2_yellow_1(k9_waybel_0(A))))) )
         => k1_yellow_0(k2_yellow_1(k9_waybel_0(A)),B) = k3_tarski(B) ) ) )).

fof(t2_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v3_lattice3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & v3_lattice3(B)
            & l1_orders_2(B) )
         => ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_orders_2(C)
                & v3_orders_2(C)
                & v4_orders_2(C)
                & v3_lattice3(C)
                & l1_orders_2(C) )
             => ! [D] :
                  ( m1_waybel16(D,A,B)
                 => ! [E] :
                      ( m1_waybel16(E,B,C)
                     => m1_waybel16(k7_funct_2(u1_struct_0(A),u1_struct_0(B),u1_struct_0(C),D,E),A,C) ) ) ) ) ) )).

fof(t3_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => v17_waybel_0(k7_grcat_1(A),A,A) ) )).

fof(t4_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => v22_waybel_0(k7_grcat_1(A),A,A) ) )).

fof(t5_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v3_lattice3(A)
        & l1_orders_2(A) )
     => m1_waybel16(k7_grcat_1(A),A,A) ) )).

fof(t6_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v2_yellow_0(A)
        & v2_lattice3(A)
        & l1_orders_2(A) )
     => ( ~ v3_struct_0(k2_yellow_1(k9_waybel_0(A)))
        & v4_yellow_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A)))
        & v7_yellow_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A)))
        & v4_waybel_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A)))
        & m1_yellow_0(k2_yellow_1(k9_waybel_0(A)),k3_yellow_1(u1_struct_0(A))) ) ) )).

fof(d1_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v3_lattice3(A)
        & v3_waybel_3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( r1_waybel22(A,B)
        <=> ! [C] :
              ( ( ~ v3_struct_0(C)
                & v2_orders_2(C)
                & v3_orders_2(C)
                & v4_orders_2(C)
                & v3_lattice3(C)
                & v3_waybel_3(C)
                & l1_orders_2(C) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,B,u1_struct_0(C))
                    & m2_relset_1(D,B,u1_struct_0(C)) )
                 => ? [E] :
                      ( m1_waybel16(E,A,C)
                      & k7_relat_1(E,B) = D
                      & ! [F] :
                          ( m1_waybel16(F,A,C)
                         => ( k7_relat_1(F,B) = D
                           => F = E ) ) ) ) ) ) ) )).

fof(t7_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v3_lattice3(A)
        & v3_waybel_3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( r1_waybel22(A,B)
         => m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) ) ) )).

fof(t8_waybel22,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v3_lattice3(A)
        & v3_waybel_3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( r1_waybel22(A,B)
         => ! [C] :
              ( m1_waybel16(C,A,A)
             => ( k7_relat_1(C,B) = k1_pralg_3(B)
               => C = k7_grcat_1(A) ) ) ) ) )).

fof(t9_waybel22,axiom,(
    ! [A] : r1_tarski(k1_waybel22(A),k9_waybel_0(k3_yellow_1(A))) )).

fof(t10_waybel22,axiom,(
    ! [A] : k1_card_1(k1_waybel22(A)) = k1_card_1(A) )).

fof(t12_waybel22,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v3_lattice3(B)
        & v3_waybel_3(B)
        & l1_orders_2(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
            & m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
         => v5_orders_3(k2_waybel22(A,B,C),k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B) ) ) )).

fof(t13_waybel22,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v3_lattice3(B)
        & v3_waybel_3(B)
        & l1_orders_2(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
            & m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
         => k1_waybel_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B,k2_waybel22(A,B,C),k4_yellow_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))))) = k4_yellow_0(B) ) ) )).

fof(t14_waybel22,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v3_lattice3(B)
        & v3_waybel_3(B)
        & l1_orders_2(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
            & m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
         => k7_relat_1(k2_waybel22(A,B,C),k1_waybel22(A)) = C ) ) )).

fof(t15_waybel22,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v3_lattice3(B)
        & v3_waybel_3(B)
        & l1_orders_2(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
            & m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
         => ! [D] :
              ( m1_waybel16(D,k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B)
             => ( k7_relat_1(D,k1_waybel22(A)) = C
               => D = k2_waybel22(A,B,C) ) ) ) ) )).

fof(t16_waybel22,axiom,(
    ! [A] : r1_waybel22(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),k1_waybel22(A)) )).

fof(t17_waybel22,axiom,(
    ! [A] :
      ( ( v2_orders_2(A)
        & v3_orders_2(A)
        & v4_orders_2(A)
        & v1_lattice3(A)
        & v2_lattice3(A)
        & v3_lattice3(A)
        & v3_waybel_3(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ( v2_orders_2(B)
            & v3_orders_2(B)
            & v4_orders_2(B)
            & v1_lattice3(B)
            & v2_lattice3(B)
            & v3_lattice3(B)
            & v3_waybel_3(B)
            & l1_orders_2(B) )
         => ! [C,D] :
              ( ( r1_waybel22(A,C)
                & r1_waybel22(B,D)
                & k1_card_1(C) = k1_card_1(D) )
             => r5_waybel_1(A,B) ) ) ) )).

fof(t18_waybel22,axiom,(
    ! [A,B] :
      ( ( v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v1_lattice3(B)
        & v2_lattice3(B)
        & v3_lattice3(B)
        & v3_waybel_3(B)
        & l1_orders_2(B) )
     => ! [C] :
          ( ( r1_waybel22(B,C)
            & k1_card_1(C) = k1_card_1(A) )
         => r5_waybel_1(B,k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))) ) ) )).

fof(dt_k1_waybel22,axiom,(
    ! [A] : m1_subset_1(k1_waybel22(A),k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k3_yellow_1(A))))) )).

fof(dt_k2_waybel22,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v3_lattice3(B)
        & v3_waybel_3(B)
        & l1_orders_2(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
        & m1_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
     => ( v1_funct_1(k2_waybel22(A,B,C))
        & v1_funct_2(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
        & m2_relset_1(k2_waybel22(A,B,C),u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B)) ) ) )).

fof(d2_waybel22,axiom,(
    ! [A] : k1_waybel22(A) = a_1_0_waybel22(A) )).

fof(t11_waybel22,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v2_waybel_0(B,k3_yellow_1(A))
        & v13_waybel_0(B,k3_yellow_1(A))
        & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k3_yellow_1(A)))) )
     => B = k1_yellow_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),a_2_0_waybel22(A,B)) ) )).

fof(d3_waybel22,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(B)
        & v2_orders_2(B)
        & v3_orders_2(B)
        & v4_orders_2(B)
        & v3_lattice3(B)
        & v3_waybel_3(B)
        & l1_orders_2(B) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k1_waybel22(A),u1_struct_0(B))
            & m2_relset_1(C,k1_waybel22(A),u1_struct_0(B)) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B))
                & m2_relset_1(D,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))),u1_struct_0(B)) )
             => ( D = k2_waybel22(A,B,C)
              <=> ! [E] :
                    ( m1_subset_1(E,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A)))))
                   => k1_waybel_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(A))),B,D,E) = k1_yellow_0(B,a_4_0_waybel22(A,B,C,E)) ) ) ) ) ) )).

fof(fraenkel_a_1_0_waybel22,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_0_waybel22(B))
    <=> ? [C] :
          ( m1_subset_1(C,u1_struct_0(k3_yellow_1(B)))
          & A = k7_waybel_0(k3_yellow_1(B),C)
          & ? [D] :
              ( m1_subset_1(D,B)
              & C = k1_tarski(D) ) ) ) )).

fof(fraenkel_a_2_0_waybel22,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & v2_waybel_0(C,k3_yellow_1(B))
        & v13_waybel_0(C,k3_yellow_1(B))
        & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k3_yellow_1(B)))) )
     => ( r2_hidden(A,a_2_0_waybel22(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(B))
            & A = k2_yellow_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(B))),a_2_1_waybel22(B,D))
            & r2_hidden(D,C) ) ) ) )).

fof(fraenkel_a_2_1_waybel22,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(B))
     => ( r2_hidden(A,a_2_1_waybel22(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_struct_0(k3_yellow_1(B)))
            & A = k7_waybel_0(k3_yellow_1(B),D)
            & ? [E] :
                ( m1_subset_1(E,B)
                & D = k1_tarski(E)
                & r2_hidden(E,C) ) ) ) ) )).

fof(fraenkel_a_4_0_waybel22,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(C)
        & v2_orders_2(C)
        & v3_orders_2(C)
        & v4_orders_2(C)
        & v3_lattice3(C)
        & v3_waybel_3(C)
        & l1_orders_2(C)
        & v1_funct_1(D)
        & v1_funct_2(D,k1_waybel22(B),u1_struct_0(C))
        & m2_relset_1(D,k1_waybel22(B),u1_struct_0(C))
        & m1_subset_1(E,u1_struct_0(k2_yellow_1(k9_waybel_0(k3_yellow_1(B))))) )
     => ( r2_hidden(A,a_4_0_waybel22(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,k1_zfmisc_1(B))
            & A = k2_yellow_0(C,a_4_1_waybel22(B,C,D,F))
            & r2_hidden(F,E) ) ) ) )).

fof(fraenkel_a_4_1_waybel22,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v3_struct_0(C)
        & v2_orders_2(C)
        & v3_orders_2(C)
        & v4_orders_2(C)
        & v3_lattice3(C)
        & v3_waybel_3(C)
        & l1_orders_2(C)
        & v1_funct_1(D)
        & v1_funct_2(D,k1_waybel22(B),u1_struct_0(C))
        & m2_relset_1(D,k1_waybel22(B),u1_struct_0(C))
        & m1_subset_1(E,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_4_1_waybel22(B,C,D,E))
      <=> ? [F] :
            ( m1_subset_1(F,u1_struct_0(k3_yellow_1(B)))
            & A = k1_funct_1(D,k7_waybel_0(k3_yellow_1(B),F))
            & ? [G] :
                ( m1_subset_1(G,B)
                & F = k1_tarski(G)
                & r2_hidden(G,E) ) ) ) ) )).
%------------------------------------------------------------------------------