SET007 Axioms: SET007+508.ax


%------------------------------------------------------------------------------
% File     : SET007+508 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Same Equivalents of Well-foundedness
% 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    : wellfnd1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   1 unit)
%            Number of atoms       :  213 (  13 equality)
%            Maximal formula depth :   14 (   7 average)
%            Number of connectives :  200 (  22 ~  ;   1  |;  84  &)
%                                         (  12 <=>;  81 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   1 propositional; 0-4 arity)
%            Number of functors    :   32 (   8 constant; 0-4 arity)
%            Number of variables   :   86 (   0 singleton;  79 !;   7 ?)
%            Maximal term depth    :    6 (   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(fc1_wellfnd1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(k4_partfun1(A,B))
      & v1_fraenkel(k4_partfun1(A,B)) ) )).

fof(fc2_wellfnd1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k2_card_1(A))
      & v1_ordinal1(k2_card_1(A))
      & v2_ordinal1(k2_card_1(A))
      & v3_ordinal1(k2_card_1(A))
      & v1_card_1(k2_card_1(A)) ) )).

fof(rc1_wellfnd1,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & ~ v1_finset_1(A)
      & v1_card_1(A)
      & v1_card_5(A) ) )).

fof(rc2_wellfnd1,axiom,(
    ? [A] :
      ( l1_orders_2(A)
      & ~ v3_struct_0(A)
      & v1_wellfnd1(A) ) )).

fof(rc3_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & v2_wellfnd1(B,A) ) ) )).

fof(fc3_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v12_waybel_0(k2_wellfnd1(A),A)
        & v2_wellfnd1(k2_wellfnd1(A),A) ) ) )).

fof(t1_wellfnd1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( ( r1_tarski(B,C)
              & r1_tarski(A,k1_relat_1(B)) )
           => k7_relat_1(B,A) = k7_relat_1(C,A) ) ) ) )).

fof(t2_wellfnd1,axiom,(
    ! [A] :
      ( v1_fraenkel(A)
     => ( ! [B] :
            ( ( v1_relat_1(B)
              & v1_funct_1(B) )
           => ! [C] :
                ( ( v1_relat_1(C)
                  & v1_funct_1(C) )
               => ( ( r2_hidden(B,A)
                    & r2_hidden(C,A) )
                 => r1_partfun1(B,C) ) ) )
       => ( v1_relat_1(k3_tarski(A))
          & v1_funct_1(k3_tarski(A)) ) ) ) )).

fof(t3_wellfnd1,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & v1_card_5(A) )
     => ! [B] :
          ( ( r1_tarski(B,A)
            & r2_hidden(k1_card_1(B),A) )
         => r2_hidden(k7_ordinal2(B),A) ) ) )).

fof(t4_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] : r1_tarski(k1_wellord1(u1_orders_2(A),B),u1_struct_0(A)) ) )).

fof(d1_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v12_waybel_0(B,A)
          <=> ! [C,D] :
                ( ( r2_hidden(C,B)
                  & r2_hidden(k4_tarski(D,C),u1_orders_2(A)) )
               => r2_hidden(D,B) ) ) ) ) )).

fof(t5_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( ( v12_waybel_0(B,A)
                & r2_hidden(C,B) )
             => r1_tarski(k1_wellord1(u1_orders_2(A),C),B) ) ) ) )).

fof(t6_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( ( v12_waybel_0(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ! [D] :
                  ( ( C = k2_xboole_0(B,k1_tarski(D))
                    & r1_tarski(k1_wellord1(u1_orders_2(A),D),B) )
                 => v12_waybel_0(C,A) ) ) ) ) )).

fof(d2_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_wellfnd1(A)
      <=> r1_wellord1(u1_orders_2(A),u1_struct_0(A)) ) ) )).

fof(d3_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_wellfnd1(B,A)
          <=> r1_wellord1(u1_orders_2(A),B) ) ) ) )).

fof(t7_wellfnd1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( v2_wellfnd1(k1_struct_0(A,B),A)
            & m1_subset_1(k1_struct_0(A,B),k1_zfmisc_1(u1_struct_0(A))) ) ) ) )).

fof(t8_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( ( v2_wellfnd1(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => ! [C] :
              ( ( v2_wellfnd1(C,A)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
             => ( v12_waybel_0(B,A)
               => ( v2_wellfnd1(k4_subset_1(u1_struct_0(A),B,C),A)
                  & m1_subset_1(k4_subset_1(u1_struct_0(A),B,C),k1_zfmisc_1(u1_struct_0(A))) ) ) ) ) ) )).

fof(t9_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ( v1_wellfnd1(A)
      <=> k2_wellfnd1(A) = u1_struct_0(A) ) ) )).

fof(t10_wellfnd1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ( r1_tarski(k1_wellord1(u1_orders_2(A),B),k2_wellfnd1(A))
           => r2_hidden(B,k2_wellfnd1(A)) ) ) ) )).

fof(t11_wellfnd1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ( v1_wellfnd1(A)
      <=> ! [B] :
            ( ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( r1_tarski(k1_wellord1(u1_orders_2(A),C),B)
                 => r2_hidden(C,B) ) )
           => r1_tarski(u1_struct_0(A),B) ) ) ) )).

fof(d5_wellfnd1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B)
                & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B) )
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D) )
                 => ( r1_wellfnd1(A,B,C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => k1_funct_1(D,E) = k1_funct_1(C,k4_tarski(E,k7_relat_1(D,k1_wellord1(u1_orders_2(A),E)))) ) ) ) ) ) ) )).

fof(t12_wellfnd1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ( v1_wellfnd1(A)
      <=> ! [B] :
            ( ~ v1_xboole_0(B)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B)
                  & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B) )
               => ? [D] :
                    ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),B)
                    & m2_relset_1(D,u1_struct_0(A),B)
                    & r1_wellfnd1(A,B,C,D) ) ) ) ) ) )).

fof(t13_wellfnd1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ~ v1_realset1(B)
         => ( ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B)
                  & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B) )
               => ! [D] :
                    ( ( v1_funct_1(D)
                      & v1_funct_2(D,u1_struct_0(A),B)
                      & m2_relset_1(D,u1_struct_0(A),B) )
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(A),B)
                          & m2_relset_1(E,u1_struct_0(A),B) )
                       => ( ( r1_wellfnd1(A,B,C,D)
                            & r1_wellfnd1(A,B,C,E) )
                         => D = E ) ) ) )
           => v1_wellfnd1(A) ) ) ) )).

fof(t14_wellfnd1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_wellfnd1(A)
        & l1_orders_2(A) )
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B)
                & m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),k4_rfunct_3(u1_struct_0(A),B)),B) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),B)
                    & m2_relset_1(D,u1_struct_0(A),B) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(A),B)
                        & m2_relset_1(E,u1_struct_0(A),B) )
                     => ( ( r1_wellfnd1(A,B,C,D)
                          & r1_wellfnd1(A,B,C,E) )
                       => D = E ) ) ) ) ) ) )).

fof(d6_wellfnd1,axiom,(
    $true )).

fof(d7_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,u1_struct_0(A))
            & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
         => ( v3_wellfnd1(B,A)
          <=> ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)) != k8_funct_2(k5_numbers,u1_struct_0(A),B,C)
                  & r2_hidden(k4_tarski(k8_funct_2(k5_numbers,u1_struct_0(A),B,k1_nat_1(C,np__1)),k8_funct_2(k5_numbers,u1_struct_0(A),B,C)),u1_orders_2(A)) ) ) ) ) ) )).

fof(t15_wellfnd1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_orders_2(A) )
     => ( v1_wellfnd1(A)
      <=> ! [B] :
            ( ( v1_funct_1(B)
              & v1_funct_2(B,k5_numbers,u1_struct_0(A))
              & m2_relset_1(B,k5_numbers,u1_struct_0(A)) )
           => ~ v3_wellfnd1(B,A) ) ) ) )).

fof(s1_wellfnd1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k4_rfunct_3(f1_s1_wellfnd1,f2_s1_wellfnd1)))
      & ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,f1_s1_wellfnd1,f2_s1_wellfnd1) )
         => ( r2_hidden(B,A)
          <=> p1_s1_wellfnd1(B) ) ) ) )).

fof(s2_wellfnd1,axiom,
    ( ( p1_s2_wellfnd1(f2_s2_wellfnd1)
      & v1_wellfnd1(f1_s2_wellfnd1) )
   => ? [A] :
        ( m1_subset_1(A,u1_struct_0(f1_s2_wellfnd1))
        & p1_s2_wellfnd1(A)
        & ! [B] :
            ( m1_subset_1(B,u1_struct_0(f1_s2_wellfnd1))
           => ~ ( A != B
                & p1_s2_wellfnd1(B)
                & r2_hidden(k4_tarski(B,A),u1_orders_2(f1_s2_wellfnd1)) ) ) ) )).

fof(s3_wellfnd1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s3_wellfnd1))
         => ( ! [B] :
                ( m1_subset_1(B,u1_struct_0(f1_s3_wellfnd1))
               => ( r2_hidden(k4_tarski(B,A),u1_orders_2(f1_s3_wellfnd1))
                 => ( B = A
                    | p1_s3_wellfnd1(B) ) ) )
           => p1_s3_wellfnd1(A) ) )
      & v1_wellfnd1(f1_s3_wellfnd1) )
   => ! [A] :
        ( m1_subset_1(A,u1_struct_0(f1_s3_wellfnd1))
       => p1_s3_wellfnd1(A) ) )).

fof(dt_k1_wellfnd1,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & v1_funct_1(C)
        & m1_relset_1(C,u1_struct_0(A),B) )
     => m1_subset_1(k1_wellfnd1(A,B,C),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(redefinition_k1_wellfnd1,axiom,(
    ! [A,B,C] :
      ( ( l1_struct_0(A)
        & v1_funct_1(C)
        & m1_relset_1(C,u1_struct_0(A),B) )
     => k1_wellfnd1(A,B,C) = k1_relat_1(C) ) )).

fof(dt_k2_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => m1_subset_1(k2_wellfnd1(A),k1_zfmisc_1(u1_struct_0(A))) ) )).

fof(d4_wellfnd1,axiom,(
    ! [A] :
      ( l1_orders_2(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( B = k2_wellfnd1(A)
          <=> B = k3_tarski(a_1_0_wellfnd1(A)) ) ) ) )).

fof(fraenkel_a_1_0_wellfnd1,axiom,(
    ! [A,B] :
      ( l1_orders_2(B)
     => ( r2_hidden(A,a_1_0_wellfnd1(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
            & A = C
            & v2_wellfnd1(C,B)
            & v12_waybel_0(C,B) ) ) ) )).
%------------------------------------------------------------------------------