SET007 Axioms: SET007+409.ax


%------------------------------------------------------------------------------
% File     : SET007+409 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Formalization of Simple Graphs
% 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    : sgraph1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   97 (  31 unit)
%            Number of atoms       :  382 (  64 equality)
%            Maximal formula depth :   24 (   6 average)
%            Number of connectives :  323 (  38 ~  ;   7  |; 134  &)
%                                         (  24 <=>; 120 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   1 propositional; 0-5 arity)
%            Number of functors    :   42 (   9 constant; 0-4 arity)
%            Number of variables   :  229 (   0 singleton; 201 !;  28 ?)
%            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_sgraph1,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => v1_finset_1(k1_sgraph1(A,B)) ) )).

fof(cc1_sgraph1,axiom,(
    ! [A] :
      ( v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_xboole_0(B)
            & v1_finset_1(B) ) ) ) )).

fof(rc1_sgraph1,axiom,(
    ? [A] :
      ( l1_sgraph1(A)
      & v1_sgraph1(A) ) )).

fof(fc2_sgraph1,axiom,(
    ! [A] : ~ v1_xboole_0(k3_sgraph1(A)) )).

fof(t1_sgraph1,axiom,(
    $true )).

fof(t2_sgraph1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( r2_hidden(C,k1_sgraph1(A,B))
            <=> ? [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                  & C = D
                  & r1_xreal_0(A,D)
                  & r1_xreal_0(D,B) ) ) ) ) )).

fof(t3_sgraph1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k1_sgraph1(A,B))
              <=> ( r1_xreal_0(A,C)
                  & r1_xreal_0(C,B) ) ) ) ) ) )).

fof(t4_sgraph1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_sgraph1(np__1,A) = k2_finseq_1(A) ) )).

fof(t5_sgraph1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => r1_tarski(k1_sgraph1(A,B),k2_finseq_1(B)) ) ) ) )).

fof(t6_sgraph1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ~ r1_xreal_0(B,A)
               => r1_xboole_0(k2_finseq_1(A),k1_sgraph1(B,C)) ) ) ) ) )).

fof(t7_sgraph1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(A,B)
           => k1_sgraph1(A,B) = k1_xboole_0 ) ) ) )).

fof(d2_sgraph1,axiom,(
    $true )).

fof(d3_sgraph1,axiom,(
    $true )).

fof(t8_sgraph1,axiom,(
    $true )).

fof(t9_sgraph1,axiom,(
    ! [A,B] :
      ( r2_hidden(B,k2_sgraph1(A))
    <=> ? [C] :
          ( v1_finset_1(C)
          & m1_subset_1(C,k1_zfmisc_1(A))
          & B = C
          & k4_card_1(C) = np__2 ) ) )).

fof(t10_sgraph1,axiom,(
    ! [A,B] :
      ( r2_hidden(B,k2_sgraph1(A))
    <=> ( v1_finset_1(B)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & ? [C,D] :
            ( r2_hidden(C,A)
            & r2_hidden(D,A)
            & C != D
            & B = k2_tarski(C,D) ) ) ) )).

fof(t11_sgraph1,axiom,(
    ! [A] : r1_tarski(k2_sgraph1(A),k1_zfmisc_1(A)) )).

fof(t12_sgraph1,axiom,(
    ! [A,B,C] :
      ( r2_hidden(k2_tarski(B,C),k2_sgraph1(A))
     => ( r2_hidden(B,A)
        & r2_hidden(C,A)
        & B != C ) ) )).

fof(t13_sgraph1,axiom,(
    k2_sgraph1(k1_xboole_0) = k1_xboole_0 )).

fof(t14_sgraph1,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r1_tarski(k2_sgraph1(A),k2_sgraph1(B)) ) )).

fof(t15_sgraph1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => v1_finset_1(k2_sgraph1(A)) ) )).

fof(t16_sgraph1,axiom,(
    ! [A] :
      ( ~ v1_realset1(A)
     => ~ v1_xboole_0(k2_sgraph1(A)) ) )).

fof(t17_sgraph1,axiom,(
    ! [A] : k2_sgraph1(k1_tarski(A)) = k1_xboole_0 )).

fof(d5_sgraph1,axiom,(
    $true )).

fof(t18_sgraph1,axiom,(
    $true )).

fof(t19_sgraph1,axiom,(
    ! [A] : r2_hidden(g1_sgraph1(k1_xboole_0,k1_subset_1(k2_sgraph1(k1_xboole_0))),k3_sgraph1(A)) )).

fof(d7_sgraph1,axiom,(
    ! [A,B] :
      ( ( v1_sgraph1(B)
        & l1_sgraph1(B) )
     => ( m1_sgraph1(B,A)
      <=> m1_subset_1(B,k3_sgraph1(A)) ) ) )).

fof(t20_sgraph1,axiom,(
    $true )).

fof(t21_sgraph1,axiom,(
    ! [A,B] :
      ( r2_hidden(B,k3_sgraph1(A))
    <=> ? [C] :
          ( v1_finset_1(C)
          & m1_subset_1(C,k1_zfmisc_1(A))
          & ? [D] :
              ( v1_finset_1(D)
              & m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
              & B = g1_sgraph1(C,D) ) ) ) )).

fof(t22_sgraph1,axiom,(
    $true )).

fof(t23_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ( r1_tarski(u1_struct_0(B),A)
        & r1_tarski(u1_sgraph1(B),k2_sgraph1(u1_struct_0(B))) ) ) )).

fof(t24_sgraph1,axiom,(
    $true )).

fof(t25_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C] :
          ~ ( r2_hidden(C,u1_sgraph1(B))
            & ! [D,E] :
                ~ ( r2_hidden(D,u1_struct_0(B))
                  & r2_hidden(E,u1_struct_0(B))
                  & D != E
                  & C = k2_tarski(D,E) ) ) ) )).

fof(t26_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C,D] :
          ( r2_hidden(k2_tarski(C,D),u1_sgraph1(B))
         => ( r2_hidden(C,u1_struct_0(B))
            & r2_hidden(D,u1_struct_0(B))
            & C != D ) ) ) )).

fof(t27_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ( v1_finset_1(u1_struct_0(B))
        & m1_subset_1(u1_struct_0(B),k1_zfmisc_1(A))
        & v1_finset_1(u1_sgraph1(B))
        & m1_subset_1(u1_sgraph1(B),k1_zfmisc_1(k2_sgraph1(u1_struct_0(B)))) ) ) )).

fof(d8_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C] :
          ( m1_sgraph1(C,A)
         => ( r1_sgraph1(A,B,C)
          <=> ? [D] :
                ( v1_funct_1(D)
                & v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
                & m2_relset_1(D,u1_struct_0(B),u1_struct_0(C))
                & v3_funct_2(D,u1_struct_0(B),u1_struct_0(C))
                & ! [E] :
                    ( m1_subset_1(E,u1_struct_0(B))
                   => ! [F] :
                        ( m1_subset_1(F,u1_struct_0(B))
                       => ( r2_hidden(k2_tarski(E,F),u1_sgraph1(B))
                        <=> r2_hidden(k2_tarski(k1_funct_1(D,E),k1_funct_1(D,F)),u1_sgraph1(B)) ) ) ) ) ) ) ) )).

fof(t28_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ~ ( B != g1_sgraph1(k1_xboole_0,k1_subset_1(k2_sgraph1(k1_xboole_0)))
          & ! [C,D] :
              ( m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
             => ~ ( ~ v1_xboole_0(C)
                  & B = g1_sgraph1(C,D) ) ) ) ) )).

fof(t29_sgraph1,axiom,(
    $true )).

fof(t30_sgraph1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k2_sgraph1(B)))
         => ! [D,E] :
              ( ( v1_finset_1(E)
                & m1_subset_1(E,k1_zfmisc_1(k2_sgraph1(k2_xboole_0(B,k1_tarski(D))))) )
             => ( ( r2_hidden(g1_sgraph1(B,C),k3_sgraph1(A))
                  & r2_hidden(D,A) )
               => ( r2_hidden(D,B)
                  | r2_hidden(g1_sgraph1(k2_xboole_0(B,k1_tarski(D)),E),k3_sgraph1(A)) ) ) ) ) ) )).

fof(t31_sgraph1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(k2_sgraph1(B)))
         => ! [D,E] :
              ~ ( r2_hidden(D,B)
                & r2_hidden(E,B)
                & D != E
                & r2_hidden(g1_sgraph1(B,C),k3_sgraph1(A))
                & ! [F] :
                    ( ( v1_finset_1(F)
                      & m1_subset_1(F,k1_zfmisc_1(k2_sgraph1(B))) )
                   => ~ ( F = k2_xboole_0(C,k1_tarski(k2_tarski(D,E)))
                        & r2_hidden(g1_sgraph1(B,F),k3_sgraph1(A)) ) ) ) ) ) )).

fof(d9_sgraph1,axiom,(
    ! [A,B] :
      ( r2_sgraph1(A,B)
    <=> ( r2_hidden(g1_sgraph1(k1_xboole_0,k1_subset_1(k2_sgraph1(k1_xboole_0))),B)
        & ! [C] :
            ( m1_subset_1(C,k1_zfmisc_1(A))
           => ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
               => ! [E,F] :
                    ( ( v1_finset_1(F)
                      & m1_subset_1(F,k1_zfmisc_1(k2_sgraph1(k2_xboole_0(C,k1_tarski(E))))) )
                   => ( ( r2_hidden(g1_sgraph1(C,D),B)
                        & r2_hidden(E,A) )
                     => ( r2_hidden(E,C)
                        | r2_hidden(g1_sgraph1(k2_xboole_0(C,k1_tarski(E)),F),B) ) ) ) ) )
        & ! [C] :
            ( m1_subset_1(C,k1_zfmisc_1(A))
           => ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
               => ! [E,F] :
                    ~ ( r2_hidden(g1_sgraph1(C,D),B)
                      & r2_hidden(E,C)
                      & r2_hidden(F,C)
                      & E != F
                      & ~ r2_hidden(k2_tarski(E,F),D)
                      & ! [G] :
                          ( ( v1_finset_1(G)
                            & m1_subset_1(G,k1_zfmisc_1(k2_sgraph1(C))) )
                         => ~ ( G = k2_xboole_0(D,k1_tarski(k2_tarski(E,F)))
                              & r2_hidden(g1_sgraph1(C,G),B) ) ) ) ) ) ) ) )).

fof(t32_sgraph1,axiom,(
    $true )).

fof(t33_sgraph1,axiom,(
    $true )).

fof(t34_sgraph1,axiom,(
    $true )).

fof(t35_sgraph1,axiom,(
    ! [A] : r2_sgraph1(A,k3_sgraph1(A)) )).

fof(t36_sgraph1,axiom,(
    ! [A,B] :
      ( r2_sgraph1(A,B)
     => r1_tarski(k3_sgraph1(A),B) ) )).

fof(t37_sgraph1,axiom,(
    ! [A] :
      ( r2_sgraph1(A,k3_sgraph1(A))
      & ! [B] :
          ( r2_sgraph1(A,B)
         => r1_tarski(k3_sgraph1(A),B) ) ) )).

fof(d10_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C] :
          ( m1_sgraph1(C,A)
         => ( m2_sgraph1(C,A,B)
          <=> ( r1_tarski(u1_struct_0(C),u1_struct_0(B))
              & r1_tarski(u1_sgraph1(C),u1_sgraph1(B)) ) ) ) ) )).

fof(d11_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C,D] :
          ( m2_subset_1(D,k1_numbers,k5_numbers)
         => ( D = k4_sgraph1(A,B,C)
          <=> ? [E] :
                ( v1_finset_1(E)
                & ! [F] :
                    ( r2_hidden(F,E)
                  <=> ( r2_hidden(F,u1_sgraph1(B))
                      & r2_hidden(C,F) ) )
                & D = k4_card_1(E) ) ) ) ) )).

fof(t38_sgraph1,axiom,(
    $true )).

fof(t40_sgraph1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_sgraph1(B,A)
         => ! [C] :
              ~ ( r2_hidden(C,u1_struct_0(B))
                & ! [D] :
                    ( v1_finset_1(D)
                   => ~ ( D = u1_struct_0(B)
                        & ~ r1_xreal_0(k4_card_1(D),k4_sgraph1(A,B,C)) ) ) ) ) ) )).

fof(t41_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C,D] :
          ~ ( r2_hidden(C,u1_struct_0(B))
            & r2_hidden(D,u1_sgraph1(B))
            & k4_sgraph1(A,B,C) = np__0
            & r2_hidden(C,D) ) ) )).

fof(t42_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C,D] :
          ( v1_finset_1(D)
         => ( ( D = u1_struct_0(B)
              & r2_hidden(C,D)
              & k1_nat_1(np__1,k4_sgraph1(A,B,C)) = k4_card_1(D) )
           => ! [E] :
                ( m1_subset_1(E,D)
               => ~ ( C != E
                    & ! [F] :
                        ~ ( r2_hidden(F,u1_sgraph1(B))
                          & F = k2_tarski(C,E) ) ) ) ) ) ) )).

fof(d12_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(B))
             => ! [E] :
                  ( m2_finseq_1(E,u1_struct_0(B))
                 => ( r3_sgraph1(A,B,C,D,E)
                  <=> ( k1_funct_1(E,np__1) = C
                      & k1_funct_1(E,k3_finseq_1(E)) = D
                      & ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( r1_xreal_0(np__1,F)
                           => ( r1_xreal_0(k3_finseq_1(E),F)
                              | r2_hidden(k2_tarski(k1_funct_1(E,F),k1_funct_1(E,k1_nat_1(F,np__1))),u1_sgraph1(B)) ) ) )
                      & ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ! [G] :
                              ( m2_subset_1(G,k1_numbers,k5_numbers)
                             => ( r1_xreal_0(np__1,F)
                               => ( r1_xreal_0(k3_finseq_1(E),F)
                                  | r1_xreal_0(G,F)
                                  | r1_xreal_0(k3_finseq_1(E),G)
                                  | ( k1_funct_1(E,F) != k1_funct_1(E,G)
                                    & k2_tarski(k1_funct_1(E,F),k1_funct_1(E,k1_nat_1(F,np__1))) != k2_tarski(k1_funct_1(E,G),k1_funct_1(E,k1_nat_1(G,np__1))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t43_sgraph1,axiom,(
    $true )).

fof(t44_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(B))
             => ! [E] :
                  ( r2_hidden(E,k5_sgraph1(A,B,C,D))
                <=> ? [F] :
                      ( m2_finseq_2(F,u1_struct_0(B),k3_finseq_2(u1_struct_0(B)))
                      & E = F
                      & r3_sgraph1(A,B,C,D,F) ) ) ) ) ) )).

fof(t45_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(B))
             => ! [E] :
                  ( m2_finseq_2(E,u1_struct_0(B),k3_finseq_2(u1_struct_0(B)))
                 => ( r3_sgraph1(A,B,C,D,E)
                   => r2_hidden(E,k5_sgraph1(A,B,C,D)) ) ) ) ) ) )).

fof(d14_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C] :
          ( r4_sgraph1(A,B,C)
        <=> ? [D] :
              ( m1_subset_1(D,u1_struct_0(B))
              & r2_hidden(C,k5_sgraph1(A,B,D,D)) ) ) ) )).

fof(d15_sgraph1,axiom,(
    $true )).

fof(d18_sgraph1,axiom,(
    k8_sgraph1 = k7_sgraph1(np__3) )).

fof(t46_sgraph1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k2_sgraph1(k2_finseq_1(np__3))))
      & A = k4_setwiseo(k5_finsub_1(k5_numbers),k3_setwiseo(k5_numbers,np__1,np__2),k3_setwiseo(k5_numbers,np__2,np__3),k3_setwiseo(k5_numbers,np__3,np__1))
      & k8_sgraph1 = g1_sgraph1(k2_finseq_1(np__3),A) ) )).

fof(t47_sgraph1,axiom,
    ( u1_struct_0(k8_sgraph1) = k2_finseq_1(np__3)
    & u1_sgraph1(k8_sgraph1) = k4_setwiseo(k5_finsub_1(k5_numbers),k3_setwiseo(k5_numbers,np__1,np__2),k3_setwiseo(k5_numbers,np__2,np__3),k3_setwiseo(k5_numbers,np__3,np__1)) )).

fof(t48_sgraph1,axiom,
    ( r2_hidden(k3_setwiseo(k5_numbers,np__1,np__2),u1_sgraph1(k8_sgraph1))
    & r2_hidden(k3_setwiseo(k5_numbers,np__2,np__3),u1_sgraph1(k8_sgraph1))
    & r2_hidden(k3_setwiseo(k5_numbers,np__3,np__1),u1_sgraph1(k8_sgraph1)) )).

fof(t49_sgraph1,axiom,(
    r4_sgraph1(k5_numbers,k8_sgraph1,k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,np__1),k12_finseq_1(k5_numbers,np__2)),k12_finseq_1(k5_numbers,np__3)),k12_finseq_1(k5_numbers,np__1))) )).

fof(s1_sgraph1,axiom,
    ( ( p1_s1_sgraph1(g1_sgraph1(k1_xboole_0,k1_subset_1(k2_sgraph1(k1_xboole_0))))
      & ! [A] :
          ( m1_sgraph1(A,f1_s1_sgraph1)
         => ! [B] :
              ( ( r2_hidden(A,k3_sgraph1(f1_s1_sgraph1))
                & p1_s1_sgraph1(A)
                & r2_hidden(B,f1_s1_sgraph1) )
             => ( r2_hidden(B,u1_struct_0(A))
                | p1_s1_sgraph1(g1_sgraph1(k2_xboole_0(u1_struct_0(A),k1_tarski(B)),k1_subset_1(k2_sgraph1(k2_xboole_0(u1_struct_0(A),k1_tarski(B)))))) ) ) )
      & ! [A] :
          ( m1_sgraph1(A,f1_s1_sgraph1)
         => ! [B] :
              ~ ( p1_s1_sgraph1(A)
                & r2_hidden(B,k2_sgraph1(u1_struct_0(A)))
                & ~ r2_hidden(B,u1_sgraph1(A))
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(k2_sgraph1(u1_struct_0(A))))
                   => ~ ( C = k2_xboole_0(u1_sgraph1(A),k1_tarski(B))
                        & p1_s1_sgraph1(g1_sgraph1(u1_struct_0(A),C)) ) ) ) ) )
   => ! [A] :
        ( r2_hidden(A,k3_sgraph1(f1_s1_sgraph1))
       => p1_s1_sgraph1(A) ) )).

fof(dt_m1_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ( v1_sgraph1(B)
        & l1_sgraph1(B) ) ) )).

fof(existence_m1_sgraph1,axiom,(
    ! [A] :
    ? [B] : m1_sgraph1(B,A) )).

fof(dt_m2_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C] :
          ( m2_sgraph1(C,A,B)
         => m1_sgraph1(C,A) ) ) )).

fof(existence_m2_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ? [C] : m2_sgraph1(C,A,B) ) )).

fof(dt_l1_sgraph1,axiom,(
    ! [A] :
      ( l1_sgraph1(A)
     => l1_struct_0(A) ) )).

fof(existence_l1_sgraph1,axiom,(
    ? [A] : l1_sgraph1(A) )).

fof(abstractness_v1_sgraph1,axiom,(
    ! [A] :
      ( l1_sgraph1(A)
     => ( v1_sgraph1(A)
       => A = g1_sgraph1(u1_struct_0(A),u1_sgraph1(A)) ) ) )).

fof(dt_k1_sgraph1,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => m1_subset_1(k1_sgraph1(A,B),k1_zfmisc_1(k5_numbers)) ) )).

fof(dt_k2_sgraph1,axiom,(
    $true )).

fof(dt_k3_sgraph1,axiom,(
    $true )).

fof(dt_k4_sgraph1,axiom,(
    ! [A,B,C] :
      ( m1_sgraph1(B,A)
     => m2_subset_1(k4_sgraph1(A,B,C),k1_numbers,k5_numbers) ) )).

fof(dt_k5_sgraph1,axiom,(
    ! [A,B,C,D] :
      ( ( m1_sgraph1(B,A)
        & m1_subset_1(C,u1_struct_0(B))
        & m1_subset_1(D,u1_struct_0(B)) )
     => m1_subset_1(k5_sgraph1(A,B,C,D),k1_zfmisc_1(k3_finseq_2(u1_struct_0(B)))) ) )).

fof(dt_k6_sgraph1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_sgraph1(k6_sgraph1(A,B),k5_numbers) ) )).

fof(dt_k7_sgraph1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m1_sgraph1(k7_sgraph1(A),k5_numbers) ) )).

fof(dt_k8_sgraph1,axiom,(
    m1_sgraph1(k8_sgraph1,k5_numbers) )).

fof(dt_u1_sgraph1,axiom,(
    ! [A] :
      ( l1_sgraph1(A)
     => m1_subset_1(u1_sgraph1(A),k1_zfmisc_1(k2_sgraph1(u1_struct_0(A)))) ) )).

fof(dt_g1_sgraph1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k2_sgraph1(A)))
     => ( v1_sgraph1(g1_sgraph1(A,B))
        & l1_sgraph1(g1_sgraph1(A,B)) ) ) )).

fof(free_g1_sgraph1,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k2_sgraph1(A)))
     => ! [C,D] :
          ( g1_sgraph1(A,B) = g1_sgraph1(C,D)
         => ( A = C
            & B = D ) ) ) )).

fof(d1_sgraph1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => k1_sgraph1(A,B) = a_2_0_sgraph1(A,B) ) ) )).

fof(d4_sgraph1,axiom,(
    ! [A] : k2_sgraph1(A) = a_1_0_sgraph1(A) )).

fof(d6_sgraph1,axiom,(
    ! [A] : k3_sgraph1(A) = a_1_1_sgraph1(A) )).

fof(t39_sgraph1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_sgraph1(B,A)
         => ! [C] :
            ? [D] :
              ( v1_finset_1(D)
              & D = a_3_0_sgraph1(A,B,C)
              & k4_sgraph1(A,B,C) = k4_card_1(D) ) ) ) )).

fof(d13_sgraph1,axiom,(
    ! [A,B] :
      ( m1_sgraph1(B,A)
     => ! [C] :
          ( m1_subset_1(C,u1_struct_0(B))
         => ! [D] :
              ( m1_subset_1(D,u1_struct_0(B))
             => k5_sgraph1(A,B,C,D) = a_4_0_sgraph1(A,B,C,D) ) ) ) )).

fof(d16_sgraph1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_sgraph1(C,k5_numbers)
             => ( C = k6_sgraph1(A,B)
              <=> ? [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(k2_finseq_1(k1_nat_1(B,A)))))
                    & D = a_2_1_sgraph1(A,B)
                    & C = g1_sgraph1(k2_finseq_1(k1_nat_1(B,A)),D) ) ) ) ) ) )).

fof(d17_sgraph1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_sgraph1(B,k5_numbers)
         => ( B = k7_sgraph1(A)
          <=> ? [C] :
                ( v1_finset_1(C)
                & m1_subset_1(C,k1_zfmisc_1(k2_sgraph1(k2_finseq_1(A))))
                & C = a_1_2_sgraph1(A)
                & B = g1_sgraph1(k2_finseq_1(A),C) ) ) ) ) )).

fof(fraenkel_a_2_0_sgraph1,axiom,(
    ! [A,B,C] :
      ( ( v4_ordinal2(B)
        & v4_ordinal2(C) )
     => ( r2_hidden(A,a_2_0_sgraph1(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = D
            & r1_xreal_0(B,D)
            & r1_xreal_0(D,C) ) ) ) )).

fof(fraenkel_a_1_0_sgraph1,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_0_sgraph1(B))
    <=> ? [C] :
          ( v1_finset_1(C)
          & m1_subset_1(C,k1_zfmisc_1(B))
          & A = C
          & k4_card_1(C) = np__2 ) ) )).

fof(fraenkel_a_1_1_sgraph1,axiom,(
    ! [A,B] :
      ( r2_hidden(A,a_1_1_sgraph1(B))
    <=> ? [C,D] :
          ( v1_finset_1(C)
          & m1_subset_1(C,k1_zfmisc_1(B))
          & v1_finset_1(D)
          & m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
          & A = g1_sgraph1(C,D) ) ) )).

fof(fraenkel_a_3_0_sgraph1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_sgraph1(C,B) )
     => ( r2_hidden(A,a_3_0_sgraph1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & r2_hidden(E,u1_struct_0(C))
            & r2_hidden(k2_tarski(D,E),u1_sgraph1(C)) ) ) ) )).

fof(fraenkel_a_4_0_sgraph1,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_sgraph1(C,B)
        & m1_subset_1(D,u1_struct_0(C))
        & m1_subset_1(E,u1_struct_0(C)) )
     => ( r2_hidden(A,a_4_0_sgraph1(B,C,D,E))
      <=> ? [F] :
            ( m2_finseq_2(F,u1_struct_0(C),k3_finseq_2(u1_struct_0(C)))
            & A = F
            & r3_sgraph1(B,C,D,E,F) ) ) ) )).

fof(fraenkel_a_2_1_sgraph1,axiom,(
    ! [A,B,C] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & m2_subset_1(C,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_2_1_sgraph1(B,C))
      <=> ? [D,E] :
            ( m1_subset_1(D,k5_numbers)
            & m1_subset_1(E,k5_numbers)
            & A = k3_setwiseo(k5_numbers,D,E)
            & r2_hidden(D,k2_finseq_1(C))
            & r2_hidden(E,k1_sgraph1(k1_nat_1(C,np__1),k1_nat_1(C,B))) ) ) ) )).

fof(fraenkel_a_1_2_sgraph1,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_2_sgraph1(B))
      <=> ? [C,D] :
            ( m1_subset_1(C,k5_numbers)
            & m1_subset_1(D,k5_numbers)
            & A = k3_setwiseo(k5_numbers,C,D)
            & r2_hidden(C,k2_finseq_1(B))
            & r2_hidden(D,k2_finseq_1(B))
            & C != D ) ) ) )).
%------------------------------------------------------------------------------