SET007 Axioms: SET007+568.ax


%------------------------------------------------------------------------------
% File     : SET007+568 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : A Theory of Partitions. 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    : partit1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   60 (   7 unit)
%            Number of atoms       :  314 (  42 equality)
%            Maximal formula depth :   33 (   8 average)
%            Number of connectives :  316 (  62 ~  ;   3  |;  89  &)
%                                         (  12 <=>; 150 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   1 propositional; 0-4 arity)
%            Number of functors    :   30 (   4 constant; 0-3 arity)
%            Number of variables   :  183 (   0 singleton; 175 !;   8 ?)
%            Maximal term depth    :    4 (   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_partit1,axiom,(
    ! [A] : ~ v1_xboole_0(k1_partit1(A)) )).

fof(t1_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C,D] :
              ( ( r2_hidden(C,B)
                & r2_hidden(D,B)
                & r1_tarski(C,D) )
             => C = D ) ) ) )).

fof(t2_partit1,axiom,(
    $true )).

fof(t3_partit1,axiom,(
    ! [A] : k3_tarski(k4_xboole_0(A,k1_tarski(k1_xboole_0))) = k3_tarski(A) )).

fof(t4_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ( ( r1_setfam_1(C,B)
                  & r1_setfam_1(B,C) )
               => r1_tarski(C,B) ) ) ) ) )).

fof(t5_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ( ( r1_setfam_1(C,B)
                  & r1_setfam_1(B,C) )
               => B = C ) ) ) ) )).

fof(t6_partit1,axiom,(
    $true )).

fof(t7_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ( r1_setfam_1(C,B)
               => r2_setfam_1(C,B) ) ) ) ) )).

fof(d1_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( r1_partit1(A,B,C)
            <=> ? [D] :
                  ( r1_tarski(D,B)
                  & D != k1_xboole_0
                  & C = k3_tarski(D) ) ) ) ) )).

fof(d2_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( r2_partit1(A,B,C,D)
                <=> ( r1_partit1(A,B,D)
                    & r1_partit1(A,C,D)
                    & ! [E] :
                        ( ( r1_tarski(E,D)
                          & r1_partit1(A,B,E)
                          & r1_partit1(A,C,E) )
                       => E = D ) ) ) ) ) ) )).

fof(t8_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ( r1_setfam_1(C,B)
               => ! [D] :
                    ( r2_hidden(D,B)
                   => r1_partit1(A,C,D) ) ) ) ) ) )).

fof(t9_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => r1_partit1(A,B,A) ) ) )).

fof(t10_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ( ! [D] :
                    ( r2_hidden(D,C)
                   => r1_partit1(A,B,D) )
               => ( k8_setfam_1(A,C) = k1_xboole_0
                  | r1_partit1(A,B,k8_setfam_1(A,C)) ) ) ) ) ) )).

fof(t11_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( ( r1_partit1(A,B,C)
                      & r1_partit1(A,B,D) )
                   => ( r1_xboole_0(C,D)
                      | r1_partit1(A,B,k5_subset_1(A,C,D)) ) ) ) ) ) ) )).

fof(t12_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( r1_partit1(A,B,C)
               => ( C = A
                  | r1_partit1(A,B,k3_subset_1(A,C)) ) ) ) ) ) )).

fof(t13_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ? [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(A))
                      & r2_hidden(D,E)
                      & r2_partit1(A,B,C,E) ) ) ) ) ) )).

fof(t14_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ? [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                  & r2_hidden(C,D)
                  & r2_hidden(D,B) ) ) ) ) )).

fof(d3_partit1,axiom,(
    ! [A,B] :
      ( B = k1_partit1(A)
    <=> ! [C] :
          ( r2_hidden(C,B)
        <=> m1_eqrel_1(C,A) ) ) )).

fof(d4_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => k2_partit1(A,B,C) = k4_xboole_0(k3_setfam_1(B,C),k1_tarski(k1_xboole_0)) ) ) ) )).

fof(t15_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => k2_partit1(A,B,B) = B ) ) )).

fof(t16_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => k2_partit1(A,k2_partit1(A,B,C),D) = k2_partit1(A,B,k2_partit1(A,C,D)) ) ) ) ) )).

fof(t17_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => r1_setfam_1(k2_partit1(A,B,C),B) ) ) ) )).

fof(d5_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ( D = k3_partit1(A,B,C)
                  <=> ! [E] :
                        ( r2_hidden(E,D)
                      <=> r2_partit1(A,B,C,E) ) ) ) ) ) ) )).

fof(t18_partit1,axiom,(
    $true )).

fof(t19_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => r1_setfam_1(B,k3_partit1(A,B,C)) ) ) ) )).

fof(t20_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => k3_partit1(A,B,B) = B ) ) )).

fof(t21_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C,D,E] :
          ( m1_eqrel_1(E,A)
         => ! [F] :
              ( m1_eqrel_1(F,A)
             => ( ( r1_setfam_1(E,F)
                  & r2_hidden(B,F)
                  & r2_hidden(C,E)
                  & r2_hidden(D,B)
                  & r2_hidden(D,C) )
               => r1_tarski(C,B) ) ) ) ) )).

fof(t22_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C,D,E] :
          ( m1_eqrel_1(E,A)
         => ! [F] :
              ( m1_eqrel_1(F,A)
             => ( ( r2_hidden(B,k3_partit1(A,E,F))
                  & r2_hidden(C,E)
                  & r2_hidden(D,B)
                  & r2_hidden(D,C) )
               => r1_tarski(C,B) ) ) ) ) )).

fof(t23_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ? [C] :
              ( v3_relat_2(C)
              & v8_relat_2(C)
              & v1_partfun1(C,A,A)
              & m2_relset_1(C,A,A)
              & ! [D,E] :
                  ( r2_hidden(k4_tarski(D,E),C)
                <=> ? [F] :
                      ( m1_subset_1(F,k1_zfmisc_1(A))
                      & r2_hidden(F,B)
                      & r2_hidden(D,F)
                      & r2_hidden(E,F) ) ) ) ) ) )).

fof(d6_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( ( v3_relat_2(C)
                & v8_relat_2(C)
                & v1_partfun1(C,A,A)
                & m2_relset_1(C,A,A) )
             => ( C = k4_partit1(A,B)
              <=> ! [D,E] :
                    ( r2_hidden(k4_tarski(D,E),C)
                  <=> ? [F] :
                        ( m1_subset_1(F,k1_zfmisc_1(A))
                        & r2_hidden(F,B)
                        & r2_hidden(D,F)
                        & r2_hidden(E,F) ) ) ) ) ) ) )).

fof(d7_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( B = k5_partit1(A)
          <=> ( k1_relat_1(B) = k1_partit1(A)
              & ! [C] :
                  ~ ( r2_hidden(C,k1_partit1(A))
                    & ! [D] :
                        ( m1_eqrel_1(D,A)
                       => ~ ( D = C
                            & k1_funct_1(B,C) = k4_partit1(A,D) ) ) ) ) ) ) ) )).

fof(t24_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ( r1_setfam_1(B,C)
              <=> r1_tarski(k4_partit1(A,B),k4_partit1(A,C)) ) ) ) ) )).

fof(t25_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D,E,F,G] :
                  ( m2_finseq_1(G,A)
                 => ( ( r1_tarski(D,A)
                      & r2_hidden(E,D)
                      & k1_funct_1(G,np__1) = E
                      & k1_funct_1(G,k3_finseq_1(G)) = F
                      & r1_xreal_0(np__1,k3_finseq_1(G))
                      & ! [H] :
                          ( m2_subset_1(H,k1_numbers,k5_numbers)
                         => ~ ( r1_xreal_0(np__1,H)
                              & ~ r1_xreal_0(k3_finseq_1(G),H)
                              & ! [I,J,K] :
                                  ~ ( r2_hidden(I,B)
                                    & r2_hidden(J,C)
                                    & r2_hidden(k1_funct_1(G,H),I)
                                    & r2_hidden(K,I)
                                    & r2_hidden(K,J)
                                    & r2_hidden(k1_funct_1(G,k1_nat_1(H,np__1)),J) ) ) )
                      & r1_partit1(A,B,D)
                      & r1_partit1(A,C,D) )
                   => r2_hidden(F,D) ) ) ) ) ) )).

fof(t26_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v3_relat_2(B)
            & v8_relat_2(B)
            & v1_partfun1(B,A,A)
            & m2_relset_1(B,A,A) )
         => ! [C] :
              ( ( v3_relat_2(C)
                & v8_relat_2(C)
                & v1_partfun1(C,A,A)
                & m2_relset_1(C,A,A) )
             => ! [D] :
                  ( m2_finseq_1(D,A)
                 => ! [E,F] :
                      ( ( r2_hidden(E,A)
                        & r2_hidden(F,A)
                        & k1_funct_1(D,np__1) = E
                        & k1_funct_1(D,k3_finseq_1(D)) = F
                        & r1_xreal_0(np__1,k3_finseq_1(D))
                        & ! [G] :
                            ( m2_subset_1(G,k1_numbers,k5_numbers)
                           => ~ ( r1_xreal_0(np__1,G)
                                & ~ r1_xreal_0(k3_finseq_1(D),G)
                                & ! [H] :
                                    ~ ( r2_hidden(H,A)
                                      & r2_hidden(k4_tarski(k1_funct_1(D,G),H),k3_eqrel_1(A,B,C))
                                      & r2_hidden(k4_tarski(H,k1_funct_1(D,k1_nat_1(G,np__1))),k3_eqrel_1(A,B,C)) ) ) ) )
                     => r2_hidden(k4_tarski(E,F),k5_eqrel_1(A,B,C)) ) ) ) ) ) )).

fof(t27_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => k4_partit1(A,k3_partit1(A,B,C)) = k5_eqrel_1(A,k4_partit1(A,B),k4_partit1(A,C)) ) ) ) )).

fof(t28_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => k4_partit1(A,k2_partit1(A,B,C)) = k4_eqrel_1(A,k4_partit1(A,B),k4_partit1(A,C)) ) ) ) )).

fof(t29_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ( k4_partit1(A,B) = k4_partit1(A,C)
               => B = C ) ) ) ) )).

fof(t30_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => k3_partit1(A,k3_partit1(A,B,C),D) = k3_partit1(A,B,k3_partit1(A,C,D)) ) ) ) ) )).

fof(t31_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => k2_partit1(A,B,k3_partit1(A,B,C)) = B ) ) ) )).

fof(t32_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => k3_partit1(A,B,k2_partit1(A,B,C)) = B ) ) ) )).

fof(t33_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ( ( r1_setfam_1(B,D)
                      & r1_setfam_1(C,D) )
                   => r1_setfam_1(k3_partit1(A,B,C),D) ) ) ) ) ) )).

fof(t34_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ! [C] :
              ( m1_eqrel_1(C,A)
             => ! [D] :
                  ( m1_eqrel_1(D,A)
                 => ( ( r1_setfam_1(D,B)
                      & r1_setfam_1(D,C) )
                   => r1_setfam_1(D,k2_partit1(A,B,C)) ) ) ) ) ) )).

fof(d8_partit1,axiom,(
    $true )).

fof(d9_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k6_partit1(A) = k1_tarski(A) ) )).

fof(t36_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ( r1_setfam_1(B,k6_partit1(A))
            & r1_setfam_1(k3_pua2mss1(A),B) ) ) ) )).

fof(t37_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_partit1(A,k6_partit1(A)) = k1_eqrel_1(A) ) )).

fof(t38_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_partit1(A,k3_pua2mss1(A)) = k6_relat_1(A) ) )).

fof(t39_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => r1_setfam_1(k3_pua2mss1(A),k6_partit1(A)) ) )).

fof(t40_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ( k3_partit1(A,k6_partit1(A),B) = k6_partit1(A)
            & k2_partit1(A,k6_partit1(A),B) = B ) ) ) )).

fof(t41_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_eqrel_1(B,A)
         => ( k3_partit1(A,k3_pua2mss1(A),B) = B
            & k2_partit1(A,k3_pua2mss1(A),B) = k3_pua2mss1(A) ) ) ) )).

fof(dt_k1_partit1,axiom,(
    $true )).

fof(dt_k2_partit1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_eqrel_1(B,A)
        & m1_eqrel_1(C,A) )
     => m1_eqrel_1(k2_partit1(A,B,C),A) ) )).

fof(commutativity_k2_partit1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_eqrel_1(B,A)
        & m1_eqrel_1(C,A) )
     => k2_partit1(A,B,C) = k2_partit1(A,C,B) ) )).

fof(dt_k3_partit1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_eqrel_1(B,A)
        & m1_eqrel_1(C,A) )
     => m1_eqrel_1(k3_partit1(A,B,C),A) ) )).

fof(commutativity_k3_partit1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_eqrel_1(B,A)
        & m1_eqrel_1(C,A) )
     => k3_partit1(A,B,C) = k3_partit1(A,C,B) ) )).

fof(dt_k4_partit1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_eqrel_1(B,A) )
     => ( v3_relat_2(k4_partit1(A,B))
        & v8_relat_2(k4_partit1(A,B))
        & v1_partfun1(k4_partit1(A,B),A,A)
        & m2_relset_1(k4_partit1(A,B),A,A) ) ) )).

fof(dt_k5_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_relat_1(k5_partit1(A))
        & v1_funct_1(k5_partit1(A)) ) ) )).

fof(dt_k6_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_eqrel_1(k6_partit1(A),A) ) )).

fof(t35_partit1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k3_pua2mss1(A) = a_1_0_partit1(A) ) )).

fof(fraenkel_a_1_0_partit1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_0_partit1(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(B))
            & A = C
            & ? [D] :
                ( C = k1_tarski(D)
                & r2_hidden(D,B) ) ) ) ) )).
%------------------------------------------------------------------------------