SET007 Axioms: SET007+35.ax


%------------------------------------------------------------------------------
% File     : SET007+35 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Subsets of Complex Numbers
% 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    : numbers [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   45 (  44 unit)
%            Number of atoms       :   48 (  17 equality)
%            Maximal formula depth :    6 (   1 average)
%            Number of connectives :   17 (  14 ~  ;   0  |;   2  &)
%                                         (   1 <=>;   0 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    7 (   1 propositional; 0-2 arity)
%            Number of functors    :   20 (  11 constant; 0-2 arity)
%            Number of variables   :    2 (   0 singleton;   1 !;   1 ?)
%            Maximal term depth    :    5 (   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_numbers,axiom,(
    ~ v1_xboole_0(k1_numbers) )).

fof(fc2_numbers,axiom,(
    ~ v1_xboole_0(k2_numbers) )).

fof(fc3_numbers,axiom,(
    ~ v1_xboole_0(k3_numbers) )).

fof(fc4_numbers,axiom,(
    ~ v1_xboole_0(k4_numbers) )).

fof(d1_numbers,axiom,(
    k1_numbers = k4_xboole_0(k2_xboole_0(k2_arytm_2,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)),k1_tarski(k4_tarski(np__0,np__0))) )).

fof(d3_numbers,axiom,(
    k3_numbers = k4_xboole_0(k2_xboole_0(k6_arytm_3,k2_zfmisc_1(k1_tarski(np__0),k6_arytm_3)),k1_tarski(k4_tarski(np__0,np__0))) )).

fof(d4_numbers,axiom,(
    k4_numbers = k4_xboole_0(k2_xboole_0(k5_ordinal2,k2_zfmisc_1(k1_tarski(np__0),k5_ordinal2)),k1_tarski(k4_tarski(np__0,np__0))) )).

fof(t1_numbers,axiom,(
    r2_xboole_0(k1_numbers,k2_numbers) )).

fof(t2_numbers,axiom,(
    r2_xboole_0(k3_numbers,k1_numbers) )).

fof(t3_numbers,axiom,(
    r2_xboole_0(k3_numbers,k2_numbers) )).

fof(t4_numbers,axiom,(
    r2_xboole_0(k4_numbers,k3_numbers) )).

fof(t5_numbers,axiom,(
    r2_xboole_0(k4_numbers,k1_numbers) )).

fof(t6_numbers,axiom,(
    r2_xboole_0(k4_numbers,k2_numbers) )).

fof(t7_numbers,axiom,(
    r2_xboole_0(k5_numbers,k4_numbers) )).

fof(t8_numbers,axiom,(
    r2_xboole_0(k5_numbers,k3_numbers) )).

fof(t9_numbers,axiom,(
    r2_xboole_0(k5_numbers,k1_numbers) )).

fof(t10_numbers,axiom,(
    r2_xboole_0(k5_numbers,k2_numbers) )).

fof(t11_numbers,axiom,(
    r1_tarski(k1_numbers,k2_numbers) )).

fof(t12_numbers,axiom,(
    r1_tarski(k3_numbers,k1_numbers) )).

fof(t13_numbers,axiom,(
    r1_tarski(k3_numbers,k2_numbers) )).

fof(t14_numbers,axiom,(
    r1_tarski(k4_numbers,k3_numbers) )).

fof(t15_numbers,axiom,(
    r1_tarski(k4_numbers,k1_numbers) )).

fof(t16_numbers,axiom,(
    r1_tarski(k4_numbers,k2_numbers) )).

fof(t17_numbers,axiom,(
    r1_tarski(k5_numbers,k4_numbers) )).

fof(t18_numbers,axiom,(
    r1_tarski(k5_numbers,k3_numbers) )).

fof(t19_numbers,axiom,(
    r1_tarski(k5_numbers,k1_numbers) )).

fof(t20_numbers,axiom,(
    r1_tarski(k5_numbers,k2_numbers) )).

fof(t21_numbers,axiom,(
    k1_numbers != k2_numbers )).

fof(t22_numbers,axiom,(
    k3_numbers != k1_numbers )).

fof(t23_numbers,axiom,(
    k3_numbers != k2_numbers )).

fof(t24_numbers,axiom,(
    k4_numbers != k3_numbers )).

fof(t25_numbers,axiom,(
    k4_numbers != k1_numbers )).

fof(t26_numbers,axiom,(
    k4_numbers != k2_numbers )).

fof(t27_numbers,axiom,(
    k5_numbers != k4_numbers )).

fof(t28_numbers,axiom,(
    k5_numbers != k3_numbers )).

fof(t29_numbers,axiom,(
    k5_numbers != k1_numbers )).

fof(t30_numbers,axiom,(
    k5_numbers != k2_numbers )).

fof(dt_k1_numbers,axiom,(
    $true )).

fof(dt_k2_numbers,axiom,(
    $true )).

fof(dt_k3_numbers,axiom,(
    $true )).

fof(dt_k4_numbers,axiom,(
    $true )).

fof(dt_k5_numbers,axiom,(
    m1_subset_1(k5_numbers,k1_zfmisc_1(k1_numbers)) )).

fof(redefinition_k5_numbers,axiom,(
    k5_numbers = k5_ordinal2 )).

fof(d2_numbers,axiom,(
    k2_numbers = k2_xboole_0(k4_xboole_0(k1_funct_2(k2_tarski(np__0,k13_arytm_3),k1_numbers),a_0_0_numbers),k1_numbers) )).

fof(fraenkel_a_0_0_numbers,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_numbers)
    <=> ? [B] :
          ( m1_subset_1(B,k1_funct_2(k2_tarski(np__0,k13_arytm_3),k1_numbers))
          & A = B
          & k1_funct_1(B,k13_arytm_3) = np__0 ) ) )).
%------------------------------------------------------------------------------