SET007 Axioms: SET007+669.ax
%------------------------------------------------------------------------------
% File : SET007+669 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Lemmas for the Jordan Curve Theorem
% 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 : jct_misc [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 35 ( 5 unt; 0 def)
% Number of atoms : 244 ( 18 equ)
% Maximal formula atoms : 23 ( 6 avg)
% Number of connectives : 249 ( 40 ~; 0 |; 109 &)
% ( 6 <=>; 94 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 44 ( 44 usr; 10 con; 0-4 aty)
% Number of variables : 115 ( 104 !; 11 ?)
% 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_jct_misc,axiom,
$true ).
fof(t2_jct_misc,axiom,
$true ).
fof(t3_jct_misc,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( ( r1_tarski(A,k1_relat_1(C))
& r1_tarski(k9_relat_1(C,A),B) )
=> r1_tarski(A,k10_relat_1(C,B)) ) ) ).
fof(t4_jct_misc,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B,C] :
( r1_xboole_0(B,C)
=> r1_xboole_0(k10_relat_1(A,B),k10_relat_1(A,C)) ) ) ).
fof(t5_jct_misc,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> ( ( B = k1_xboole_0
=> A = k1_xboole_0 )
=> k3_subset_1(A,k3_funct_2(A,B,C,D)) = k3_funct_2(A,B,C,k3_subset_1(B,D)) ) ) ) ).
fof(t6_jct_misc,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),B)
& m2_relset_1(C,u1_struct_0(A),B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> k3_subset_1(u1_struct_0(A),k3_funct_2(u1_struct_0(A),B,C,D)) = k3_funct_2(u1_struct_0(A),B,C,k3_subset_1(B,D)) ) ) ) ) ).
fof(t7_jct_misc,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(A,B)
=> k5_binarith(B,k5_binarith(B,A)) = A ) ) ) ).
fof(t8_jct_misc,axiom,
$true ).
fof(t9_jct_misc,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(C,D))
& r2_hidden(B,k1_rcomp_1(C,D)) )
=> r2_hidden(k7_xcmplx_0(k2_xcmplx_0(A,B),np__2),k1_rcomp_1(C,D)) ) ) ) ) ) ).
fof(t10_jct_misc,axiom,
$true ).
fof(t11_jct_misc,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> r1_xreal_0(k18_complex1(k5_real_1(k18_complex1(k6_xcmplx_0(A,B)),k18_complex1(k6_xcmplx_0(C,D)))),k3_real_1(k18_complex1(k6_xcmplx_0(A,C)),k18_complex1(k6_xcmplx_0(B,D)))) ) ) ) ) ).
fof(t12_jct_misc,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(k4_square_1(k18_complex1(B),k18_complex1(C)),k18_complex1(A)) ) ) ) ) ).
fof(d1_jct_misc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k2_zfmisc_1(B,C))
& m2_relset_1(D,A,k2_zfmisc_1(B,C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( E = k1_jct_misc(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,A)
=> k8_funct_2(A,B,E,F) = k1_mcart_1(k8_funct_2(A,k2_zfmisc_1(B,C),D,F)) ) ) ) ) ) ) ) ).
fof(d2_jct_misc,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,k2_zfmisc_1(B,C))
& m2_relset_1(D,A,k2_zfmisc_1(B,C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,C)
& m2_relset_1(E,A,C) )
=> ( E = k2_jct_misc(A,B,C,D)
<=> ! [F] :
( m1_subset_1(F,A)
=> k8_funct_2(A,C,E,F) = k2_mcart_1(k8_funct_2(A,k2_zfmisc_1(B,C),D,F)) ) ) ) ) ) ) ) ).
fof(t13_jct_misc,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_pre_topc(B)
& l1_pre_topc(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_borsuk_1(A,B))))
=> ( ! [D] :
( m1_subset_1(D,u1_struct_0(k6_borsuk_1(A,B)))
=> ~ ( r2_hidden(D,C)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(B)))
=> ~ ( v3_pre_topc(E,A)
& v3_pre_topc(F,B)
& r2_hidden(D,k7_borsuk_1(A,B,E,F))
& r1_tarski(k7_borsuk_1(A,B,E,F),C) ) ) ) ) )
=> v3_pre_topc(C,k6_borsuk_1(A,B)) ) ) ) ) ).
fof(t14_jct_misc,axiom,
! [A] :
( ( v1_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> v1_rcomp_1(k5_subset_1(k1_numbers,A,B)) ) ) ).
fof(d3_jct_misc,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ( v1_jct_misc(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) ) ) ) ) ) ).
fof(t15_jct_misc,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(A),k1_numbers)
& v9_pscomp_1(B,A)
& m2_relset_1(B,u1_struct_0(A),k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_connsp_1(C,A)
=> v1_jct_misc(k2_funct_2(u1_struct_0(A),k1_numbers,B,C)) ) ) ) ) ).
fof(t16_jct_misc,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)
=> ! [D] :
( v1_xreal_0(D)
=> ( ( r2_hidden(C,A)
& r2_hidden(D,B) )
=> r1_xreal_0(k3_jct_misc(A,B),k18_complex1(k6_xcmplx_0(C,D))) ) ) ) ) ) ).
fof(t17_jct_misc,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(k1_numbers)) )
=> ( ( r1_tarski(C,A)
& r1_tarski(D,B) )
=> r1_xreal_0(k3_jct_misc(A,B),k3_jct_misc(C,D)) ) ) ) ) ) ).
fof(t18_jct_misc,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ? [C] :
( v1_xreal_0(C)
& ? [D] :
( v1_xreal_0(D)
& r2_hidden(C,A)
& r2_hidden(D,B)
& k3_jct_misc(A,B) = k18_complex1(k6_xcmplx_0(C,D)) ) ) ) ) ).
fof(t19_jct_misc,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> r1_xreal_0(np__0,k3_jct_misc(A,B)) ) ) ).
fof(t20_jct_misc,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_rcomp_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_rcomp_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ~ ( r1_subset_1(A,B)
& r1_xreal_0(k3_jct_misc(A,B),np__0) ) ) ) ).
fof(t21_jct_misc,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_rcomp_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ! [D] :
( ( v1_rcomp_1(D)
& m1_subset_1(D,k1_zfmisc_1(k1_numbers)) )
=> ( ( r1_xboole_0(C,D)
& r1_tarski(C,k1_rcomp_1(A,B))
& r1_tarski(D,k1_rcomp_1(A,B)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_zfmisc_1(k1_numbers))
& m2_relset_1(E,k5_numbers,k1_zfmisc_1(k1_numbers)) )
=> ~ ( ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( v1_jct_misc(k8_funct_2(k5_numbers,k1_zfmisc_1(k1_numbers),E,F))
& ~ r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(k1_numbers),E,F),C)
& ~ r1_xboole_0(k8_funct_2(k5_numbers,k1_zfmisc_1(k1_numbers),E,F),D) ) )
& ! [F] :
( v1_xreal_0(F)
=> ~ ( r2_hidden(F,k1_rcomp_1(A,B))
& ~ r2_hidden(F,k4_subset_1(k1_numbers,C,D))
& ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ? [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
& r1_xreal_0(G,H)
& r2_hidden(F,k8_funct_2(k5_numbers,k1_zfmisc_1(k1_numbers),E,H)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(s2_jct_misc,axiom,
( ! [A] :
( m1_subset_1(A,f1_s2_jct_misc)
=> ? [B] :
( m1_subset_1(B,f2_s2_jct_misc)
& ? [C] :
( m1_subset_1(C,f3_s2_jct_misc)
& p1_s2_jct_misc(A,B,C) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f1_s2_jct_misc,f2_s2_jct_misc)
& m2_relset_1(A,f1_s2_jct_misc,f2_s2_jct_misc)
& ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,f1_s2_jct_misc,f3_s2_jct_misc)
& m2_relset_1(B,f1_s2_jct_misc,f3_s2_jct_misc)
& ! [C] :
( m1_subset_1(C,f1_s2_jct_misc)
=> p1_s2_jct_misc(C,k8_funct_2(f1_s2_jct_misc,f2_s2_jct_misc,A,C),k8_funct_2(f1_s2_jct_misc,f3_s2_jct_misc,B,C)) ) ) ) ) ).
fof(dt_k1_jct_misc,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,k2_zfmisc_1(B,C))
& m1_relset_1(D,A,k2_zfmisc_1(B,C)) )
=> ( v1_funct_1(k1_jct_misc(A,B,C,D))
& v1_funct_2(k1_jct_misc(A,B,C,D),A,B)
& m2_relset_1(k1_jct_misc(A,B,C,D),A,B) ) ) ).
fof(redefinition_k1_jct_misc,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,k2_zfmisc_1(B,C))
& m1_relset_1(D,A,k2_zfmisc_1(B,C)) )
=> k1_jct_misc(A,B,C,D) = k15_mcart_1(D) ) ).
fof(dt_k2_jct_misc,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,k2_zfmisc_1(B,C))
& m1_relset_1(D,A,k2_zfmisc_1(B,C)) )
=> ( v1_funct_1(k2_jct_misc(A,B,C,D))
& v1_funct_2(k2_jct_misc(A,B,C,D),A,C)
& m2_relset_1(k2_jct_misc(A,B,C,D),A,C) ) ) ).
fof(redefinition_k2_jct_misc,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,k2_zfmisc_1(B,C))
& m1_relset_1(D,A,k2_zfmisc_1(B,C)) )
=> k2_jct_misc(A,B,C,D) = k16_mcart_1(D) ) ).
fof(dt_k3_jct_misc,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> v1_xreal_0(k3_jct_misc(A,B)) ) ).
fof(commutativity_k3_jct_misc,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> k3_jct_misc(A,B) = k3_jct_misc(B,A) ) ).
fof(d4_jct_misc,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)
=> ( C = k3_jct_misc(A,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_numbers))
& D = a_2_0_jct_misc(A,B)
& C = k5_seq_4(D) ) ) ) ) ) ).
fof(s1_jct_misc,axiom,
~ v1_xboole_0(a_0_0_jct_misc) ).
fof(fraenkel_a_2_0_jct_misc,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ( r2_hidden(A,a_2_0_jct_misc(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_numbers)
& m1_subset_1(E,k1_numbers)
& A = k18_complex1(k5_real_1(D,E))
& r2_hidden(D,B)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_0_0_jct_misc,axiom,
! [A] :
( r2_hidden(A,a_0_0_jct_misc)
<=> ? [B] :
( m1_subset_1(B,f1_s1_jct_misc)
& A = f2_s1_jct_misc(B) ) ) ).
%------------------------------------------------------------------------------