SET007 Axioms: SET007+191.ax


%------------------------------------------------------------------------------
% File     : SET007+191 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Euler's Function
% 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    : euler_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   2 unit)
%            Number of atoms       :  140 (  35 equality)
%            Maximal formula depth :   20 (   8 average)
%            Number of connectives :  133 (  18 ~  ;  14  |;  33  &)
%                                         (   3 <=>;  65 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   0 propositional; 1-3 arity)
%            Number of functors    :   19 (   5 constant; 0-2 arity)
%            Number of variables   :   60 (   0 singleton;  52 !;   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(t1_euler_1,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r2_hidden(A,B)
          <=> ~ r1_xreal_0(B,A) ) ) ) )).

fof(t2_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r2_int_2(A,A)
      <=> A = np__1 ) ) )).

fof(t3_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( v1_int_2(B)
           => ( A = np__0
              | r1_xreal_0(B,A)
              | r2_int_2(A,B) ) ) ) ) )).

fof(t5_euler_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( r2_hidden(B,A)
         => k4_card_1(k4_xboole_0(A,k1_tarski(B))) = k6_xcmplx_0(k4_card_1(A),k4_card_1(k1_tarski(B))) ) ) )).

fof(t6_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k6_nat_1(A,B) = np__1
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k6_nat_1(k2_nat_1(A,C),k2_nat_1(B,C)) = C ) ) ) ) )).

fof(t7_euler_1,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)
             => ( k6_nat_1(k2_nat_1(A,C),k2_nat_1(B,C)) = C
               => ( A = np__0
                  | B = np__0
                  | C = np__0
                  | r2_int_2(A,B) ) ) ) ) ) )).

fof(t8_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k6_nat_1(A,B) = np__1
           => k6_nat_1(k1_nat_1(A,B),B) = np__1 ) ) ) )).

fof(t9_euler_1,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)
             => k6_nat_1(k1_nat_1(A,k2_nat_1(B,C)),B) = k6_nat_1(A,B) ) ) ) )).

fof(t10_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( r2_int_2(A,B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( ? [D] :
                          ( v1_int_1(D)
                          & ? [E] :
                              ( v1_int_1(E)
                              & C = k2_xcmplx_0(k3_xcmplx_0(D,A),k3_xcmplx_0(E,B))
                              & ~ r1_xreal_0(C,np__0) ) )
                      & ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ( ? [E] :
                                ( v1_int_1(E)
                                & ? [F] :
                                    ( v1_int_1(F)
                                    & D = k2_xcmplx_0(k3_xcmplx_0(E,A),k3_xcmplx_0(F,B))
                                    & ~ r1_xreal_0(D,np__0) ) )
                           => r1_xreal_0(C,D) ) ) ) ) ) ) ) )).

fof(t11_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_int_2(A,B)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ? [D] :
                    ( v1_int_1(D)
                    & ? [E] :
                        ( v1_int_1(E)
                        & k2_xcmplx_0(k3_xcmplx_0(D,A),k3_xcmplx_0(E,B)) = C ) ) ) ) ) ) )).

fof(t12_euler_1,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_finset_1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v1_finset_1(B) )
         => ( ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B)
                & v2_funct_1(C)
                & v2_funct_2(C,A,B) )
           => k1_card_1(A) = k1_card_1(B) ) ) ) )).

fof(t13_euler_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => k6_int_1(k2_xcmplx_0(A,k3_xcmplx_0(B,C)),C) = k6_int_1(A,C) ) ) ) )).

fof(t14_euler_1,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_nat_1(C,k2_nat_1(A,B))
                  & r2_int_2(A,C) )
               => ( A = np__0
                  | B = np__0
                  | C = np__0
                  | r1_nat_1(C,B) ) ) ) ) ) )).

fof(t15_euler_1,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_int_2(A,C)
                  & r2_int_2(B,C) )
               => ( A = np__0
                  | B = np__0
                  | C = np__0
                  | r2_int_2(k2_nat_1(A,B),C) ) ) ) ) ) )).

fof(t16_euler_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ~ ( A != np__0
                  & B != np__0
                  & ~ r1_xreal_0(C,np__0)
                  & k3_int_2(k3_xcmplx_0(C,A),k3_xcmplx_0(C,B)) != k3_xcmplx_0(C,k3_int_2(A,B)) ) ) ) ) )).

fof(t17_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( v1_int_1(C)
             => ~ ( A != np__0
                  & B != np__0
                  & k3_int_2(k2_xcmplx_0(A,k3_xcmplx_0(C,B)),B) != k3_int_2(A,B) ) ) ) ) )).

fof(t18_euler_1,axiom,(
    k1_euler_1(np__1) = np__1 )).

fof(t19_euler_1,axiom,(
    k1_euler_1(np__2) = np__1 )).

fof(t20_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__1)
       => r1_xreal_0(k1_euler_1(A),k6_xcmplx_0(A,np__1)) ) ) )).

fof(t21_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( v1_int_2(A)
       => k1_euler_1(A) = k6_xcmplx_0(A,np__1) ) ) )).

fof(t22_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_int_2(A,B)
           => ( r1_xreal_0(A,np__1)
              | r1_xreal_0(B,np__1)
              | k1_euler_1(k2_nat_1(A,B)) = k2_nat_1(k1_euler_1(A),k1_euler_1(B)) ) ) ) ) )).

fof(dt_k1_euler_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m2_subset_1(k1_euler_1(A),k1_numbers,k5_numbers) ) )).

fof(t4_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( ( v1_int_2(A)
                & r2_hidden(B,a_1_0_euler_1(A)) )
             => ( v1_int_2(A)
                & r2_hidden(B,A)
                & ~ r2_hidden(B,k1_tarski(np__0)) ) )
            & ( ( v1_int_2(A)
                & r2_hidden(B,A) )
             => ( r2_hidden(B,k1_tarski(np__0))
                | ( v1_int_2(A)
                  & r2_hidden(B,a_1_0_euler_1(A)) ) ) ) ) ) ) )).

fof(d1_euler_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_euler_1(A) = k1_card_1(a_1_0_euler_1(A)) ) )).

fof(fraenkel_a_1_0_euler_1,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_0_euler_1(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & r2_int_2(B,C)
            & r1_xreal_0(np__1,C)
            & r1_xreal_0(C,B) ) ) ) )).
%------------------------------------------------------------------------------