SET007 Axioms: SET007+191.ax
%------------------------------------------------------------------------------
% File : SET007+191 : TPTP v9.0.0. 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 unt; 0 def)
% Number of atoms : 140 ( 35 equ)
% Maximal formula atoms : 14 ( 5 avg)
% Number of connectives : 133 ( 18 ~; 14 |; 33 &)
% ( 3 <=>; 65 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 17 ( 16 usr; 0 prp; 1-3 aty)
% Number of functors : 19 ( 19 usr; 5 con; 0-2 aty)
% Number of variables : 60 ( 52 !; 8 ?)
% 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) ) ) ) ).
%------------------------------------------------------------------------------