SET007 Axioms: SET007+683.ax
%------------------------------------------------------------------------------
% File : SET007+683 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Properties of Dyadic Numbers and Intervals
% 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 : urysohn2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 36 ( 4 unt; 0 def)
% Number of atoms : 234 ( 58 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 249 ( 51 ~; 6 |; 81 &)
% ( 0 <=>; 111 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 8 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 16 ( 14 usr; 1 prp; 0-3 aty)
% Number of functors : 34 ( 34 usr; 14 con; 0-3 aty)
% Number of variables : 77 ( 77 !; 0 ?)
% 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_urysohn2,axiom,
! [A,B] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
& m1_subset_1(B,k1_numbers) )
=> ( v5_measure5(k1_integra2(A,B))
& v1_membered(k1_integra2(A,B))
& v2_membered(k1_integra2(A,B)) ) ) ).
fof(t1_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( A != k6_measure5
=> ( ( ~ r1_supinf_1(k7_measure6(A),k6_measure6(A))
=> k5_measure5(A) = k4_supinf_2(k7_measure6(A),k6_measure6(A)) )
& ( k7_measure6(A) = k6_measure6(A)
=> k5_measure5(A) = k1_supinf_2 ) ) ) ) ).
fof(t2_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( B != np__0
=> k1_integra2(k1_integra2(A,B),k2_real_1(B)) = A ) ) ) ).
fof(t3_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( A != np__0
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k6_supinf_1))
=> ( B = k6_supinf_1
=> k1_integra2(B,A) = B ) ) ) ) ).
fof(t4_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
=> ( A != k6_measure5
=> k1_integra2(A,np__0) = k1_tarski(np__0) ) ) ).
fof(t5_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> k1_integra2(k6_measure5,A) = k6_measure5 ) ).
fof(t6_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ~ ( r1_supinf_1(A,B)
& ~ ( A = k4_measure6
& B = k4_measure6 )
& ~ ( A = k4_measure6
& r2_hidden(B,k6_supinf_1) )
& ~ ( A = k4_measure6
& B = k5_measure6 )
& ~ ( r2_hidden(A,k6_supinf_1)
& r2_hidden(B,k6_supinf_1) )
& ~ ( r2_hidden(A,k6_supinf_1)
& B = k5_measure6 )
& ~ ( A = k5_measure6
& B = k5_measure6 ) ) ) ) ).
fof(t7_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ( v5_measure5(k1_measure5(A,A))
& m1_subset_1(k1_measure5(A,A),k1_zfmisc_1(k6_supinf_1)) ) ) ).
fof(t8_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( v5_measure5(k1_integra2(A,np__0))
& m1_subset_1(k1_integra2(A,np__0),k1_zfmisc_1(k6_supinf_1)) ) ) ).
fof(t9_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v1_measure5(A)
=> ( B = np__0
| v1_measure5(k1_integra2(A,B)) ) ) ) ) ).
fof(t10_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v2_measure5(A)
=> ( B = np__0
| v2_measure5(k1_integra2(A,B)) ) ) ) ) ).
fof(t11_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v3_measure5(A)
=> ( r1_xreal_0(B,np__0)
| v3_measure5(k1_integra2(A,B)) ) ) ) ) ).
fof(t12_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v3_measure5(A)
=> ( r1_xreal_0(np__0,B)
| v4_measure5(k1_integra2(A,B)) ) ) ) ) ).
fof(t13_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v4_measure5(A)
=> ( r1_xreal_0(B,np__0)
| v4_measure5(k1_integra2(A,B)) ) ) ) ) ).
fof(t14_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v4_measure5(A)
=> ( r1_xreal_0(np__0,B)
| v3_measure5(k1_integra2(A,B)) ) ) ) ) ).
fof(t15_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( A != k6_measure5
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> ! [C] :
( ( v5_measure5(C)
& m1_subset_1(C,k1_zfmisc_1(k6_supinf_1)) )
=> ( ( C = k1_integra2(A,B)
& A = k1_measure5(k6_measure6(A),k7_measure6(A)) )
=> ( C = k1_measure5(k6_measure6(C),k7_measure6(C))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( D = k6_measure6(A)
& E = k7_measure6(A) )
=> ( k6_measure6(C) = k4_real_1(B,D)
& k7_measure6(C) = k4_real_1(B,E) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t16_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( A != k6_measure5
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> ! [C] :
( ( v5_measure5(C)
& m1_subset_1(C,k1_zfmisc_1(k6_supinf_1)) )
=> ( ( C = k1_integra2(A,B)
& A = k3_measure5(k6_measure6(A),k7_measure6(A)) )
=> ( C = k3_measure5(k6_measure6(C),k7_measure6(C))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( D = k6_measure6(A)
& E = k7_measure6(A) )
=> ( k6_measure6(C) = k4_real_1(B,D)
& k7_measure6(C) = k4_real_1(B,E) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t17_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( A != k6_measure5
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> ! [C] :
( ( v5_measure5(C)
& m1_subset_1(C,k1_zfmisc_1(k6_supinf_1)) )
=> ( ( C = k1_integra2(A,B)
& A = k2_measure5(k6_measure6(A),k7_measure6(A)) )
=> ( C = k2_measure5(k6_measure6(C),k7_measure6(C))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( D = k6_measure6(A)
& E = k7_measure6(A) )
=> ( k6_measure6(C) = k4_real_1(B,D)
& k7_measure6(C) = k4_real_1(B,E) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ( A != k6_measure5
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( ~ r1_xreal_0(B,np__0)
=> ! [C] :
( ( v5_measure5(C)
& m1_subset_1(C,k1_zfmisc_1(k6_supinf_1)) )
=> ( ( C = k1_integra2(A,B)
& A = k4_measure5(k6_measure6(A),k7_measure6(A)) )
=> ( C = k4_measure5(k6_measure6(C),k7_measure6(C))
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( D = k6_measure6(A)
& E = k7_measure6(A) )
=> ( k6_measure6(C) = k4_real_1(B,D)
& k7_measure6(C) = k4_real_1(B,E) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t19_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( v5_measure5(k1_integra2(A,B))
& m1_subset_1(k1_integra2(A,B),k1_zfmisc_1(k6_supinf_1)) ) ) ) ).
fof(t20_urysohn2,axiom,
! [A] :
( ( v5_measure5(A)
& m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ( r1_xreal_0(np__0,B)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( C = k5_measure5(A)
=> k4_real_1(B,C) = k5_measure5(k1_integra2(A,B)) ) ) ) ) ) ).
fof(t21_urysohn2,axiom,
$true ).
fof(t22_urysohn2,axiom,
$true ).
fof(t23_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(k4_real_1(k3_newton(np__2,B),A),np__1) ) ) ) ).
fof(t24_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(k5_real_1(B,A),np__1)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(B,C) ) ) ) ) ) ).
fof(t25_urysohn2,axiom,
$true ).
fof(t26_urysohn2,axiom,
$true ).
fof(t27_urysohn2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> r1_tarski(k3_urysohn1(A),k4_urysohn1) ) ).
fof(t28_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,A)
& r1_xreal_0(np__0,A)
& r1_xreal_0(B,np__1)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r2_hidden(C,k4_urysohn1)
& ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(B,C) ) ) ) ) ) ).
fof(t29_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,A)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r2_hidden(C,k5_urysohn1)
& ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(B,C) ) ) ) ) ) ).
fof(t30_urysohn2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( r1_tarski(A,k1_measure5(B,C))
=> ( r1_supinf_1(B,k10_supinf_1(A))
& r1_supinf_1(k9_supinf_1(A),C) ) ) ) ) ) ).
fof(t31_urysohn2,axiom,
( r2_hidden(np__0,k4_urysohn1)
& r2_hidden(np__1,k4_urysohn1) ) ).
fof(t32_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ( ( A = np__0
& B = np__1 )
=> r1_tarski(k4_urysohn1,k1_measure5(A,B)) ) ) ) ).
fof(t33_urysohn2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> r1_tarski(k3_urysohn1(A),k3_urysohn1(B)) ) ) ) ).
fof(t34_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(B,C)
& ~ r1_xreal_0(D,A)
& ~ r1_xreal_0(B,D)
& r1_xreal_0(k5_real_1(B,A),k18_complex1(k5_real_1(D,C))) ) ) ) ) ) ).
fof(t35_urysohn2,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ~ ( ~ r1_xreal_0(B,np__0)
& r1_xreal_0(B,np__1)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( r2_hidden(C,k4_subset_1(k1_numbers,k4_urysohn1,k2_urysohn1))
& r2_hidden(D,k4_subset_1(k1_numbers,k4_urysohn1,k2_urysohn1))
& ~ r1_xreal_0(C,np__0)
& ~ r1_xreal_0(B,C)
& ~ r1_xreal_0(D,B)
& ~ r1_xreal_0(A,k5_real_1(D,C)) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------