SET007 Axioms: SET007+90.ax
%------------------------------------------------------------------------------
% File : SET007+90 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Topological Properties of Subsets in Real 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 : rcomp_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 66 ( 10 unt; 0 def)
% Number of atoms : 310 ( 16 equ)
% Maximal formula atoms : 18 ( 4 avg)
% Number of connectives : 277 ( 33 ~; 3 |; 83 &)
% ( 12 <=>; 146 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 25 ( 23 usr; 1 prp; 0-3 aty)
% Number of functors : 24 ( 24 usr; 5 con; 0-4 aty)
% Number of variables : 137 ( 129 !; 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(fc1_rcomp_1,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k2_rcomp_1(A,B))
& v2_membered(k2_rcomp_1(A,B))
& v3_rcomp_1(k2_rcomp_1(A,B)) ) ) ).
fof(fc2_rcomp_1,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> ( v1_membered(k1_rcomp_1(A,B))
& v2_membered(k1_rcomp_1(A,B))
& v2_rcomp_1(k1_rcomp_1(A,B)) ) ) ).
fof(rc1_rcomp_1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& v1_membered(A)
& v2_membered(A)
& v3_rcomp_1(A) ) ).
fof(cc1_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_rcomp_1(B,A)
=> ( v1_membered(B)
& v2_membered(B)
& v3_rcomp_1(B) ) ) ) ).
fof(t1_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,A)
=> r2_hidden(C,B) ) )
=> r1_tarski(A,B) ) ) ) ).
fof(t2_rcomp_1,axiom,
$true ).
fof(t3_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( ( r1_tarski(A,B)
& v2_seq_4(B) )
=> v2_seq_4(A) ) ) ) ).
fof(t4_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( ( r1_tarski(A,B)
& v1_seq_4(B) )
=> v1_seq_4(A) ) ) ) ).
fof(t5_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( ( r1_tarski(A,B)
& v3_seq_4(B) )
=> v3_seq_4(A) ) ) ) ).
fof(t6_rcomp_1,axiom,
$true ).
fof(t7_rcomp_1,axiom,
$true ).
fof(t8_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(A,k2_rcomp_1(k6_xcmplx_0(B,C),k2_xcmplx_0(B,C)))
<=> ~ r1_xreal_0(C,k18_complex1(k6_xcmplx_0(A,B))) ) ) ) ) ).
fof(t9_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(A,k1_rcomp_1(B,C))
<=> r1_xreal_0(k18_complex1(k6_xcmplx_0(k2_xcmplx_0(B,C),k3_xcmplx_0(np__2,A))),k6_xcmplx_0(C,B)) ) ) ) ) ).
fof(t10_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(A,k2_rcomp_1(B,C))
<=> ~ r1_xreal_0(k6_xcmplx_0(C,B),k18_complex1(k6_xcmplx_0(k2_xcmplx_0(B,C),k3_xcmplx_0(np__2,A)))) ) ) ) ) ).
fof(t11_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k1_rcomp_1(A,B) = k2_xboole_0(k2_rcomp_1(A,B),k2_tarski(A,B)) ) ) ) ).
fof(t12_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k2_rcomp_1(B,A) = k1_xboole_0 ) ) ) ).
fof(t13_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ r1_xreal_0(B,A)
=> k1_rcomp_1(B,A) = k1_xboole_0 ) ) ) ).
fof(t14_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> k1_rcomp_1(A,A) = k1_seq_4(A) ) ).
fof(t15_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ~ ( ~ r1_xreal_0(B,A)
& k2_rcomp_1(A,B) = k1_xboole_0 )
& ( r1_xreal_0(A,B)
=> ( r2_hidden(A,k1_rcomp_1(A,B))
& r2_hidden(B,k1_rcomp_1(A,B)) ) )
& r1_tarski(k2_rcomp_1(A,B),k1_rcomp_1(A,B)) ) ) ) ).
fof(t16_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r2_hidden(A,k1_rcomp_1(B,C))
& r2_hidden(D,k1_rcomp_1(B,C)) )
=> r1_tarski(k1_rcomp_1(A,D),k1_rcomp_1(B,C)) ) ) ) ) ) ).
fof(t17_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r2_hidden(A,k2_rcomp_1(B,C))
& r2_hidden(D,k2_rcomp_1(B,C)) )
=> r1_tarski(k1_rcomp_1(A,D),k2_rcomp_1(B,C)) ) ) ) ) ) ).
fof(t18_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> k1_rcomp_1(A,B) = k4_subset_1(k1_numbers,k1_rcomp_1(A,B),k1_rcomp_1(B,A)) ) ) ) ).
fof(d3_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_rcomp_1(A)
<=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ~ ( r1_tarski(k2_relat_1(B),A)
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ~ ( m1_seqm_3(C,B)
& v4_seq_2(C)
& r2_hidden(k2_seq_2(C),A) ) ) ) ) ) ) ).
fof(d4_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v2_rcomp_1(A)
<=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( ( r1_tarski(k2_relat_1(B),A)
& v4_seq_2(B) )
=> r2_hidden(k2_seq_2(B),A) ) ) ) ) ).
fof(d5_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v3_rcomp_1(A)
<=> v2_rcomp_1(k3_subset_1(k1_numbers,A)) ) ) ).
fof(t19_rcomp_1,axiom,
$true ).
fof(t20_rcomp_1,axiom,
$true ).
fof(t21_rcomp_1,axiom,
$true ).
fof(t22_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ( r1_tarski(k2_relat_1(C),k1_rcomp_1(A,B))
=> v3_seq_2(C) ) ) ) ) ).
fof(t23_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> v2_rcomp_1(k1_rcomp_1(A,B)) ) ) ).
fof(t24_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> v1_rcomp_1(k1_rcomp_1(A,B)) ) ) ).
fof(t25_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> v3_rcomp_1(k2_rcomp_1(A,B)) ) ) ).
fof(t26_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_rcomp_1(A)
=> v2_rcomp_1(A) ) ) ).
fof(t27_rcomp_1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( ! [C] :
( v1_xreal_0(C)
=> ~ ( r2_hidden(C,B)
& ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(F,E)
& r1_xreal_0(k18_complex1(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,F),C)),D) ) ) ) ) ) ) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_numbers)
& m2_relset_1(C,k5_numbers,k1_numbers) )
=> ~ ( m1_seqm_3(C,A)
& v4_seq_2(C)
& r2_hidden(k2_seq_2(C),B) ) ) ) ) ) ).
fof(t28_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_rcomp_1(A)
=> v3_seq_4(A) ) ) ).
fof(t29_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( ( v3_seq_4(A)
& v2_rcomp_1(A) )
=> v1_rcomp_1(A) ) ) ).
fof(t30_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( ( v2_rcomp_1(A)
& v1_seq_4(A) )
=> ( A = k1_xboole_0
| r2_hidden(k4_seq_4(A),A) ) ) ) ).
fof(t31_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( ( v2_rcomp_1(A)
& v2_seq_4(A) )
=> ( A = k1_xboole_0
| r2_hidden(k5_seq_4(A),A) ) ) ) ).
fof(t32_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_rcomp_1(A)
=> ( A = k1_xboole_0
| ( r2_hidden(k4_seq_4(A),A)
& r2_hidden(k5_seq_4(A),A) ) ) ) ) ).
fof(t33_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ~ ( v1_rcomp_1(A)
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r1_tarski(k1_rcomp_1(B,C),A) ) ) )
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> A != k1_rcomp_1(B,C) ) ) ) ) ).
fof(d6_rcomp_1,axiom,
$true ).
fof(d7_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ( m1_rcomp_1(B,A)
<=> ? [C] :
( v1_xreal_0(C)
& ~ r1_xreal_0(C,np__0)
& B = k2_rcomp_1(k6_xcmplx_0(A,C),k2_xcmplx_0(A,C)) ) ) ) ) ).
fof(t34_rcomp_1,axiom,
$true ).
fof(t35_rcomp_1,axiom,
$true ).
fof(t36_rcomp_1,axiom,
$true ).
fof(t37_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_rcomp_1(B,A)
=> r2_hidden(A,B) ) ) ).
fof(t38_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_rcomp_1(B,A)
=> ! [C] :
( m1_rcomp_1(C,A)
=> ? [D] :
( m1_rcomp_1(D,A)
& r1_tarski(D,B)
& r1_tarski(D,C) ) ) ) ) ).
fof(t39_rcomp_1,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r2_hidden(B,A)
& ! [C] :
( m1_rcomp_1(C,B)
=> ~ r1_tarski(C,A) ) ) ) ) ).
fof(t40_rcomp_1,axiom,
! [A] :
( ( v3_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r2_hidden(B,A)
& ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& r1_tarski(k2_rcomp_1(k6_xcmplx_0(B,C),k2_xcmplx_0(B,C)),A) ) ) ) ) ) ).
fof(t41_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( ! [B] :
( v1_xreal_0(B)
=> ~ ( r2_hidden(B,A)
& ! [C] :
( m1_rcomp_1(C,B)
=> ~ r1_tarski(C,A) ) ) )
=> v3_rcomp_1(A) ) ) ).
fof(t42_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( ! [B] :
( v1_xreal_0(B)
=> ~ ( r2_hidden(B,A)
& ! [C] :
( m1_rcomp_1(C,B)
=> ~ r1_tarski(C,A) ) ) )
<=> v3_rcomp_1(A) ) ) ).
fof(t43_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ~ ( v3_rcomp_1(A)
& v1_seq_4(A)
& r2_hidden(k4_seq_4(A),A) ) ) ).
fof(t44_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ~ ( v3_rcomp_1(A)
& v2_seq_4(A)
& r2_hidden(k5_seq_4(A),A) ) ) ).
fof(t45_rcomp_1,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ~ ( v3_rcomp_1(A)
& v3_seq_4(A)
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> r1_tarski(k1_rcomp_1(B,C),A) ) ) )
& ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> A != k2_rcomp_1(B,C) ) ) ) ) ).
fof(t46_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> r1_xboole_0(k2_rcomp_1(A,B),k2_tarski(A,B)) ) ) ).
fof(t47_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,k2_rcomp_1(A,B))
<=> ( ~ r1_xreal_0(C,A)
& ~ r1_xreal_0(B,C) ) ) ) ) ) ).
fof(t48_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ( r2_hidden(C,k1_rcomp_1(A,B))
<=> ( r1_xreal_0(A,C)
& r1_xreal_0(C,B) ) ) ) ) ) ).
fof(s1_rcomp_1,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ? [B] :
( v1_xreal_0(B)
& p1_s1_rcomp_1(A,B) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> p1_s1_rcomp_1(B,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) ) ) ).
fof(dt_m1_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m1_rcomp_1(B,A)
=> m1_subset_1(B,k1_zfmisc_1(k1_numbers)) ) ) ).
fof(existence_m1_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ? [B] : m1_rcomp_1(B,A) ) ).
fof(dt_k1_rcomp_1,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> m1_subset_1(k1_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ).
fof(dt_k2_rcomp_1,axiom,
! [A,B] :
( ( v1_xreal_0(A)
& v1_xreal_0(B) )
=> m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(k1_numbers)) ) ).
fof(d1_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k1_rcomp_1(A,B) = a_2_0_rcomp_1(A,B) ) ) ).
fof(d2_rcomp_1,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> k2_rcomp_1(A,B) = a_2_1_rcomp_1(A,B) ) ) ).
fof(fraenkel_a_2_0_rcomp_1,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v1_xreal_0(C) )
=> ( r2_hidden(A,a_2_0_rcomp_1(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = D
& r1_xreal_0(B,D)
& r1_xreal_0(D,C) ) ) ) ).
fof(fraenkel_a_2_1_rcomp_1,axiom,
! [A,B,C] :
( ( v1_xreal_0(B)
& v1_xreal_0(C) )
=> ( r2_hidden(A,a_2_1_rcomp_1(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_numbers)
& A = D
& ~ r1_xreal_0(D,B)
& ~ r1_xreal_0(C,D) ) ) ) ).
%------------------------------------------------------------------------------