SET007 Axioms: SET007+421.ax


%------------------------------------------------------------------------------
% File     : SET007+421 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : A Scheme for Extensions of Homomorphisms of Many Sorted Algebras
% 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    : msafree1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   29 (   1 unit)
%            Number of atoms       :  223 (  24 equality)
%            Maximal formula depth :   18 (   8 average)
%            Number of connectives :  229 (  35 ~  ;   0  |;  99  &)
%                                         (   6 <=>;  89 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   0 propositional; 1-4 arity)
%            Number of functors    :   56 (  14 constant; 0-6 arity)
%            Number of variables   :   96 (   0 singleton;  93 !;   3 ?)
%            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_msafree1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v2_relat_1(B)
        & m1_pboole(B,A) )
     => ( ~ v1_xboole_0(k2_relat_1(B))
        & v1_setfam_1(k2_relat_1(B)) ) ) )).

fof(fc2_msafree1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_setfam_1(A) )
     => ~ v1_xboole_0(k3_tarski(A)) ) )).

fof(cc1_msafree1,axiom,(
    ! [A,B] :
      ( m1_pboole(B,A)
     => ( v3_relat_1(B)
       => v1_prob_2(B) ) ) )).

fof(rc1_msafree1,axiom,(
    ! [A] :
    ? [B] :
      ( m1_pboole(B,A)
      & v1_relat_1(B)
      & v1_funct_1(B)
      & v1_prob_2(B) ) )).

fof(rc2_msafree1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ? [B] :
          ( l3_msualg_1(B,A)
          & v5_msualg_1(B,A)
          & v1_msafree1(B,A) ) ) )).

fof(fc3_msafree1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ( v4_msualg_1(k2_msafree1(A),A)
        & v5_msualg_1(k2_msafree1(A),A)
        & v1_msafree1(k2_msafree1(A),A) ) ) )).

fof(fc4_msafree1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A)
        & v1_msafree1(B,A)
        & l3_msualg_1(B,A) )
     => ( v1_relat_1(u4_msualg_1(A,B))
        & v1_funct_1(u4_msualg_1(A,B))
        & v1_prob_2(u4_msualg_1(A,B)) ) ) )).

fof(fc5_msafree1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => ( v1_relat_1(k8_msafree(A,B))
        & v2_relat_1(k8_msafree(A,B))
        & ~ v3_relat_1(k8_msafree(A,B))
        & v1_funct_1(k8_msafree(A,B))
        & v1_prob_2(k8_msafree(A,B)) ) ) )).

fof(fc6_msafree1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => ( v4_msualg_1(k11_msafree(A,B),A)
        & v5_msualg_1(k11_msafree(A,B),A) ) ) )).

fof(fc7_msafree1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & m1_subset_1(B,u1_msualg_1(A))
        & v5_msualg_1(C,A)
        & l3_msualg_1(C,A) )
     => ( ~ v1_xboole_0(k3_msualg_1(A,B,C))
        & v1_fraenkel(k3_msualg_1(A,B,C)) ) ) )).

fof(fc8_msafree1,axiom,(
    ! [A,B,C] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & m1_subset_1(B,u1_msualg_1(A))
        & v5_msualg_1(C,A)
        & l3_msualg_1(C,A) )
     => ~ v1_xboole_0(k4_msualg_1(A,B,C)) ) )).

fof(fc9_msafree1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => ( v1_relat_1(u4_msualg_1(A,k11_msafree(A,B)))
        & v2_relat_1(u4_msualg_1(A,k11_msafree(A,B)))
        & ~ v3_relat_1(u4_msualg_1(A,k11_msafree(A,B)))
        & v1_funct_1(u4_msualg_1(A,k11_msafree(A,B)))
        & v1_prob_2(u4_msualg_1(A,k11_msafree(A,B))) ) ) )).

fof(fc10_msafree1,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A)
        & v2_relat_1(B)
        & m1_pboole(B,u1_struct_0(A)) )
     => ( v4_msualg_1(k11_msafree(A,B),A)
        & v5_msualg_1(k11_msafree(A,B),A)
        & v1_msafree1(k11_msafree(A,B),A) ) ) )).

fof(t1_msafree1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( r2_hidden(B,k4_card_3(A))
           => r1_tarski(k2_relat_1(B),k3_card_3(A)) ) ) ) )).

fof(t2_msafree1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( m1_pboole(B,u1_struct_0(A))
         => ! [C,D] :
              ( r2_hidden(k4_tarski(C,D),k4_msafree(A,B))
             => ( r2_hidden(C,k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))))
                & r2_hidden(D,k13_finseq_1(k2_xboole_0(k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))),k3_card_3(k3_msafree(u1_struct_0(A),B))))) ) ) ) ) )).

fof(t3_msafree1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( m1_pboole(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_msualg_1(A))
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v1_finseq_1(D) )
                 => ( r2_hidden(k4_tarski(k4_tarski(C,u1_struct_0(A)),D),k4_msafree(A,B))
                   => ( k3_finseq_1(D) = k3_finseq_1(k1_msualg_1(A,C))
                      & ! [E] :
                          ( r2_hidden(E,k4_finseq_1(D))
                         => ( ( r2_hidden(k1_funct_1(D,E),k2_zfmisc_1(u1_msualg_1(A),k1_tarski(u1_struct_0(A))))
                             => ! [F] :
                                  ( m1_subset_1(F,u1_msualg_1(A))
                                 => ( k4_tarski(F,u1_struct_0(A)) = k1_funct_1(D,E)
                                   => k2_msualg_1(A,F) = k1_funct_1(k1_msualg_1(A,C),E) ) ) )
                            & ( r2_hidden(k1_funct_1(D,E),k3_card_3(k3_msafree(u1_struct_0(A),B)))
                             => r2_hidden(k1_funct_1(D,E),k2_msafree(u1_struct_0(A),B,k1_funct_1(k1_msualg_1(A,C),E))) ) ) ) ) ) ) ) ) ) )).

fof(d1_msafree1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_prob_2(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( ( v2_relat_1(C)
                & m1_pboole(C,A) )
             => ! [D] :
                  ( m3_pboole(D,A,B,C)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k3_card_3(B),k3_card_3(C))
                        & m2_relset_1(E,k3_card_3(B),k3_card_3(C)) )
                     => ( E = k1_msafree1(A,B,C,D)
                      <=> ! [F] :
                            ( m1_subset_1(F,A)
                           => ! [G] :
                                ( r2_hidden(G,k1_funct_1(B,F))
                               => k1_funct_1(E,G) = k1_funct_1(k1_msualg_3(A,B,C,D,F),G) ) ) ) ) ) ) ) ) )).

fof(t4_msafree1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_prob_2(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( ( v2_relat_1(C)
                & m1_pboole(C,A) )
             => ! [D] :
                  ( m3_pboole(D,A,B,C)
                 => ! [E] :
                      ( m3_pboole(E,A,B,C)
                     => ( k1_msafree1(A,B,C,D) = k1_msafree1(A,B,C,E)
                       => r6_pboole(A,D,E) ) ) ) ) ) ) )).

fof(d2_msafree1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( l3_msualg_1(B,A)
         => ( v1_msafree1(B,A)
          <=> v1_prob_2(u4_msualg_1(A,B)) ) ) ) )).

fof(d3_msafree1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( ( v4_msualg_1(B,A)
            & l3_msualg_1(B,A) )
         => ( B = k2_msafree1(A)
          <=> ! [C] :
                ( r2_hidden(C,u1_struct_0(A))
               => k1_funct_1(u4_msualg_1(A,B),C) = k1_tarski(C) ) ) ) ) )).

fof(t5_msafree1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & ~ v2_msualg_1(A)
        & l1_msualg_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_msualg_1(A))
         => ! [C] :
              ( ( v5_msualg_1(C,A)
                & v1_msafree1(C,A)
                & l3_msualg_1(C,A) )
             => ! [D] :
                  ( ( v5_msualg_1(D,A)
                    & l3_msualg_1(D,A) )
                 => ! [E] :
                      ( m3_pboole(E,u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D))
                     => ! [F] :
                          ( m1_subset_1(F,k3_msualg_1(A,B,C))
                         => k5_relat_1(F,k1_msafree1(u1_struct_0(A),u4_msualg_1(A,C),u4_msualg_1(A,D),E)) = k6_msualg_3(A,C,D,B,E,F) ) ) ) ) ) ) )).

fof(s1_msafree1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s1_msafree1))
         => ( r2_hidden(A,k5_lang1(f1_s1_msafree1))
           => k1_funct_1(f5_s1_msafree1,k2_trees_4(u1_struct_0(f1_s1_msafree1),A)) = f3_s1_msafree1(A) ) )
      & ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s1_msafree1))
         => ! [B] :
              ( m1_trees_4(B,k5_trees_3(u1_struct_0(f1_s1_msafree1)),k4_dtconstr(f1_s1_msafree1))
             => ( r1_lang1(f1_s1_msafree1,A,k1_dtconstr(u1_struct_0(f1_s1_msafree1),k5_trees_3(u1_struct_0(f1_s1_msafree1)),B))
               => ! [C] :
                    ( m2_finseq_1(C,f2_s1_msafree1)
                   => ( C = k5_relat_1(B,f5_s1_msafree1)
                     => k1_funct_1(f5_s1_msafree1,k12_trees_4(u1_struct_0(f1_s1_msafree1),A,B)) = f4_s1_msafree1(A,B,C) ) ) ) ) )
      & ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s1_msafree1))
         => ( r2_hidden(A,k5_lang1(f1_s1_msafree1))
           => k1_funct_1(f6_s1_msafree1,k2_trees_4(u1_struct_0(f1_s1_msafree1),A)) = f3_s1_msafree1(A) ) )
      & ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s1_msafree1))
         => ! [B] :
              ( m1_trees_4(B,k5_trees_3(u1_struct_0(f1_s1_msafree1)),k4_dtconstr(f1_s1_msafree1))
             => ( r1_lang1(f1_s1_msafree1,A,k1_dtconstr(u1_struct_0(f1_s1_msafree1),k5_trees_3(u1_struct_0(f1_s1_msafree1)),B))
               => ! [C] :
                    ( m2_finseq_1(C,f2_s1_msafree1)
                   => ( C = k5_relat_1(B,f6_s1_msafree1)
                     => k1_funct_1(f6_s1_msafree1,k12_trees_4(u1_struct_0(f1_s1_msafree1),A,B)) = f4_s1_msafree1(A,B,C) ) ) ) ) ) )
   => f5_s1_msafree1 = f6_s1_msafree1 )).

fof(s2_msafree1,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,u1_msualg_1(f1_s2_msafree1))
         => ! [B] :
              ( m1_subset_1(B,k3_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)))
             => ! [C] :
                  ( m1_msafree1(C,k3_card_3(f3_s2_msafree1))
                 => ( C = k5_relat_1(B,k1_msafree1(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f6_s2_msafree1))
                   => k1_funct_1(k1_msualg_3(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f6_s2_msafree1,k2_msualg_1(f1_s2_msafree1,A)),k8_funct_2(k3_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),k4_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),k5_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),B)) = f5_s2_msafree1(A,B,C) ) ) ) )
      & ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s2_msafree1))
         => ! [B] :
              ( r2_hidden(B,k12_msafree(f1_s2_msafree1,f2_s2_msafree1,A))
             => k1_funct_1(k1_msualg_3(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f6_s2_msafree1,A),B) = f4_s2_msafree1(B) ) )
      & ! [A] :
          ( m1_subset_1(A,u1_msualg_1(f1_s2_msafree1))
         => ! [B] :
              ( m1_subset_1(B,k3_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)))
             => ! [C] :
                  ( m1_msafree1(C,k3_card_3(f3_s2_msafree1))
                 => ( C = k5_relat_1(B,k1_msafree1(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f7_s2_msafree1))
                   => k1_funct_1(k1_msualg_3(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f7_s2_msafree1,k2_msualg_1(f1_s2_msafree1,A)),k8_funct_2(k3_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),k4_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),k5_msualg_1(f1_s2_msafree1,A,k11_msafree(f1_s2_msafree1,f2_s2_msafree1)),B)) = f5_s2_msafree1(A,B,C) ) ) ) )
      & ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s2_msafree1))
         => ! [B] :
              ( r2_hidden(B,k12_msafree(f1_s2_msafree1,f2_s2_msafree1,A))
             => k1_funct_1(k1_msualg_3(u1_struct_0(f1_s2_msafree1),k8_msafree(f1_s2_msafree1,f2_s2_msafree1),f3_s2_msafree1,f7_s2_msafree1,A),B) = f4_s2_msafree1(B) ) ) )
   => r6_pboole(u1_struct_0(f1_s2_msafree1),f6_s2_msafree1,f7_s2_msafree1) )).

fof(s3_msafree1,axiom,
    ( ( r1_msualg_3(f1_s3_msafree1,k11_msafree(f1_s3_msafree1,f2_s3_msafree1),f3_s3_msafree1,f4_s3_msafree1)
      & ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s3_msafree1))
         => ! [B,C] :
              ( r2_hidden(C,k12_msafree(f1_s3_msafree1,f2_s3_msafree1,A))
             => ( k1_funct_1(k1_msualg_3(u1_struct_0(f1_s3_msafree1),u4_msualg_1(f1_s3_msafree1,k11_msafree(f1_s3_msafree1,f2_s3_msafree1)),u4_msualg_1(f1_s3_msafree1,f3_s3_msafree1),f4_s3_msafree1,A),C) = B
              <=> p1_s3_msafree1(A,B,C) ) ) )
      & r1_msualg_3(f1_s3_msafree1,k11_msafree(f1_s3_msafree1,f2_s3_msafree1),f3_s3_msafree1,f5_s3_msafree1)
      & ! [A] :
          ( m1_subset_1(A,u1_struct_0(f1_s3_msafree1))
         => ! [B,C] :
              ( r2_hidden(C,k12_msafree(f1_s3_msafree1,f2_s3_msafree1,A))
             => ( k1_funct_1(k1_msualg_3(u1_struct_0(f1_s3_msafree1),u4_msualg_1(f1_s3_msafree1,k11_msafree(f1_s3_msafree1,f2_s3_msafree1)),u4_msualg_1(f1_s3_msafree1,f3_s3_msafree1),f5_s3_msafree1,A),C) = B
              <=> p1_s3_msafree1(A,B,C) ) ) ) )
   => r6_pboole(u1_struct_0(f1_s3_msafree1),f4_s3_msafree1,f5_s3_msafree1) )).

fof(dt_m1_msafree1,axiom,(
    ! [A,B] :
      ( m1_msafree1(B,A)
     => m1_subset_1(B,k13_finseq_1(A)) ) )).

fof(existence_m1_msafree1,axiom,(
    ! [A] :
    ? [B] : m1_msafree1(B,A) )).

fof(redefinition_m1_msafree1,axiom,(
    ! [A,B] :
      ( m1_msafree1(B,A)
    <=> m1_finseq_1(B,A) ) )).

fof(dt_k1_msafree1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & v1_prob_2(B)
        & m1_pboole(B,A)
        & v2_relat_1(C)
        & m1_pboole(C,A)
        & m3_pboole(D,A,B,C) )
     => ( v1_funct_1(k1_msafree1(A,B,C,D))
        & v1_funct_2(k1_msafree1(A,B,C,D),k3_card_3(B),k3_card_3(C))
        & m2_relset_1(k1_msafree1(A,B,C,D),k3_card_3(B),k3_card_3(C)) ) ) )).

fof(dt_k2_msafree1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_msualg_1(A) )
     => ( v4_msualg_1(k2_msafree1(A),A)
        & l3_msualg_1(k2_msafree1(A),A) ) ) )).
%------------------------------------------------------------------------------