SET007 Axioms: SET007+345.ax
%------------------------------------------------------------------------------
% File : SET007+345 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Paracompactness of Metrizable Spaces
% 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 : pcomps_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 29 ( 7 unt; 0 def)
% Number of atoms : 191 ( 19 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 183 ( 21 ~; 0 |; 102 &)
% ( 8 <=>; 52 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 38 ( 36 usr; 1 prp; 0-4 aty)
% Number of functors : 46 ( 46 usr; 11 con; 0-4 aty)
% Number of variables : 75 ( 65 !; 10 ?)
% 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_pcomps_2,axiom,
$true ).
fof(t2_pcomps_2,axiom,
$true ).
fof(t3_pcomps_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r1_xreal_0(A,np__0)
& ~ r1_xreal_0(B,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k7_xcmplx_0(B,k3_newton(np__2,C)),A) ) ) ) ) ).
fof(t4_pcomps_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(C,B)
& r1_xreal_0(np__1,A) )
=> r1_xreal_0(k2_newton(A,C),k2_newton(A,B)) ) ) ) ) ).
fof(t5_pcomps_2,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( r2_wellord1(A,B)
=> ( r2_wellord1(k2_wellord1(A,B),B)
& B = k3_relat_1(k2_wellord1(A,B)) ) ) ) ).
fof(d1_pcomps_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( m1_subset_1(C,A)
=> k1_pcomps_2(A,B,C) = k3_tarski(k1_wellord1(B,C)) ) ) ).
fof(d2_pcomps_2,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( C = k2_pcomps_2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( m1_subset_1(E,A)
& r2_hidden(E,A)
& D = k4_xboole_0(E,k1_pcomps_2(A,B,E)) ) ) ) ) ).
fof(d3_pcomps_2,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k1_pcomps_1(A))
& m2_relset_1(C,k5_numbers,k1_pcomps_1(A)) )
=> k3_pcomps_2(A,B,C) = k3_tarski(k9_relat_1(C,k4_xboole_0(k2_finseq_1(B),k1_tarski(B)))) ) ) ).
fof(t6_pcomps_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v4_compts_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_pre_topc(A,B)
& v1_tops_2(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_tops_2(C,A)
& r1_pre_topc(A,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(D,C)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( r2_hidden(E,B)
& r1_tarski(k6_pre_topc(A,D),E) ) ) ) ) ) ) ) ) ) ) ).
fof(t7_pcomps_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v3_compts_1(A)
& v2_pcomps_1(A)
& r1_pre_topc(A,B)
& v1_tops_2(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_tops_2(C,A)
& r1_pre_topc(A,C)
& r1_setfam_1(k3_pcomps_1(A,C),B)
& v1_pcomps_1(C,A) ) ) ) ) ) ).
fof(t8_pcomps_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(C,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
=> ( ( r1_pcomps_1(u1_struct_0(A),C)
& B = k6_pcomps_1(u1_struct_0(A),C) )
=> u1_struct_0(B) = u1_struct_0(A) ) ) ) ) ).
fof(t9_pcomps_2,axiom,
$true ).
fof(t10_pcomps_2,axiom,
$true ).
fof(t11_pcomps_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( ( v6_metric_1(B)
& v7_metric_1(B)
& v8_metric_1(B)
& v9_metric_1(B)
& l1_metric_1(B) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers)
& m2_relset_1(D,k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k1_numbers) )
=> ( ( r1_pcomps_1(u1_struct_0(A),D)
& B = k6_pcomps_1(u1_struct_0(A),D) )
=> ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
<=> m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(B)))) ) ) ) ) ) ) ).
fof(t12_pcomps_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_pcomps_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( r1_pre_topc(A,B)
& v1_tops_2(B,A)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
=> ~ ( v1_tops_2(C,A)
& r1_pre_topc(A,C)
& r1_setfam_1(C,B)
& v1_pcomps_1(C,A) ) ) ) ) ) ) ).
fof(t13_pcomps_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ( v3_pcomps_1(A)
=> v2_pcomps_1(A) ) ) ).
fof(s1_pcomps_2,axiom,
( ( r2_wellord1(f2_s1_pcomps_2,f1_s1_pcomps_2)
& ? [A] :
( r2_hidden(A,f1_s1_pcomps_2)
& p1_s1_pcomps_2(A) ) )
=> ? [A] :
( r2_hidden(A,f1_s1_pcomps_2)
& p1_s1_pcomps_2(A)
& ! [B] :
( ( r2_hidden(B,f1_s1_pcomps_2)
& p1_s1_pcomps_2(B) )
=> r2_hidden(k4_tarski(A,B),f2_s1_pcomps_2) ) ) ) ).
fof(dt_k1_pcomps_2,axiom,
$true ).
fof(dt_k2_pcomps_2,axiom,
$true ).
fof(dt_k3_pcomps_2,axiom,
$true ).
fof(dt_k4_pcomps_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k3_finseq_2(k1_pcomps_1(k1_pcomps_1(A))))
& m1_relset_1(B,k5_numbers,k3_finseq_2(k1_pcomps_1(k1_pcomps_1(A))))
& m1_subset_1(C,k5_numbers) )
=> m2_finseq_1(k4_pcomps_2(A,B,C),k1_pcomps_1(k1_pcomps_1(A))) ) ).
fof(redefinition_k4_pcomps_2,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k3_finseq_2(k1_pcomps_1(k1_pcomps_1(A))))
& m1_relset_1(B,k5_numbers,k3_finseq_2(k1_pcomps_1(k1_pcomps_1(A))))
& m1_subset_1(C,k5_numbers) )
=> k4_pcomps_2(A,B,C) = k1_funct_1(B,C) ) ).
fof(s2_pcomps_2,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
& m2_relset_1(A,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
& k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)),A,np__0) = f2_s2_pcomps_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)),A,k1_nat_1(B,np__1)) = a_2_0_pcomps_2(A,B) ) ) ).
fof(s3_pcomps_2,axiom,
? [A] :
( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
& m2_relset_1(A,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
& k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)),A,np__0) = f2_s3_pcomps_2
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)),A,k1_nat_1(B,np__1)) = a_2_1_pcomps_2(A,B) ) ) ).
fof(fraenkel_a_2_0_pcomps_2,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
& m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_0_pcomps_2(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(f1_s2_pcomps_2))
& A = k3_tarski(a_3_0_pcomps_2(B,C,D))
& p1_s2_pcomps_2(D) ) ) ) ).
fof(fraenkel_a_3_0_pcomps_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
& m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)))
& m2_subset_1(C,k1_numbers,k5_numbers)
& m1_subset_1(D,k1_zfmisc_1(f1_s2_pcomps_2)) )
=> ( r2_hidden(A,a_3_0_pcomps_2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,f1_s2_pcomps_2)
& A = f3_s2_pcomps_2(E,C)
& ! [F] :
( m1_subset_1(F,k1_zfmisc_1(k1_zfmisc_1(f1_s2_pcomps_2)))
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(G,C)
& F = k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s2_pcomps_2)),B,G) )
=> p2_s2_pcomps_2(E,D,C,F) ) ) ) ) ) ) ).
fof(fraenkel_a_2_1_pcomps_2,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
& m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_1_pcomps_2(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(f1_s3_pcomps_2))
& A = k3_tarski(a_3_1_pcomps_2(B,C,D))
& p1_s3_pcomps_2(D) ) ) ) ).
fof(fraenkel_a_3_1_pcomps_2,axiom,
! [A,B,C,D] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
& m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
& m2_subset_1(C,k1_numbers,k5_numbers)
& m1_subset_1(D,k1_zfmisc_1(f1_s3_pcomps_2)) )
=> ( r2_hidden(A,a_3_1_pcomps_2(B,C,D))
<=> ? [E] :
( m1_subset_1(E,f1_s3_pcomps_2)
& A = f3_s3_pcomps_2(E,C)
& p2_s3_pcomps_2(E,D,C)
& ~ r2_hidden(E,k3_tarski(a_2_2_pcomps_2(B,C))) ) ) ) ).
fof(fraenkel_a_2_2_pcomps_2,axiom,
! [A,B,C] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
& m2_relset_1(B,k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)))
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_2_pcomps_2(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k5_setfam_1(f1_s3_pcomps_2,k8_funct_2(k5_numbers,k1_pcomps_1(k1_pcomps_1(f1_s3_pcomps_2)),B,D))
& r1_xreal_0(D,C) ) ) ) ).
%------------------------------------------------------------------------------