SET007 Axioms: SET007+529.ax
%------------------------------------------------------------------------------
% File : SET007+529 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Intermediate Value Theorem and Thickness of Simple Closed Curves
% 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 : topreal5 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 25 ( 5 unt; 0 def)
% Number of atoms : 226 ( 29 equ)
% Maximal formula atoms : 23 ( 9 avg)
% Number of connectives : 260 ( 59 ~; 0 |; 118 &)
% ( 0 <=>; 83 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 11 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-3 aty)
% Number of functors : 30 ( 30 usr; 8 con; 0-4 aty)
% Number of variables : 77 ( 73 !; 4 ?)
% 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_topreal5,axiom,
$true ).
fof(t2_topreal5,axiom,
$true ).
fof(t3_topreal5,axiom,
$true ).
fof(t4_topreal5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v3_pre_topc(C,A)
& v3_pre_topc(D,A)
& ~ r1_xboole_0(C,B)
& ~ r1_xboole_0(D,B)
& r1_tarski(B,k4_subset_1(u1_struct_0(A),C,D))
& r1_xboole_0(C,D)
& v2_connsp_1(B,A) ) ) ) ) ) ).
fof(t5_topreal5,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] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& v5_pre_topc(C,A,B)
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_connsp_1(D,A)
=> v2_connsp_1(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) ).
fof(t6_topreal5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( r1_xreal_0(A,B)
=> v2_connsp_1(k2_pre_topc(k4_topmetr(A,B)),k4_topmetr(A,B)) ) ) ) ).
fof(t7_topreal5,axiom,
$true ).
fof(t8_topreal5,axiom,
$true ).
fof(t9_topreal5,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( ~ r2_hidden(B,A)
& ? [C] :
( v1_xreal_0(C)
& ? [D] :
( v1_xreal_0(D)
& r2_hidden(C,A)
& r2_hidden(D,A)
& ~ r1_xreal_0(B,C)
& ~ r1_xreal_0(D,B) ) )
& v2_connsp_1(A,k3_topmetr) ) ) ) ).
fof(t10_topreal5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,u1_struct_0(A),u1_struct_0(k3_topmetr))
& v5_pre_topc(G,A,k3_topmetr)
& m2_relset_1(G,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ~ ( v1_connsp_1(A)
& k1_funct_1(G,B) = D
& k1_funct_1(G,C) = E
& r1_xreal_0(D,F)
& r1_xreal_0(F,E)
& ! [H] :
( m1_subset_1(H,u1_struct_0(A))
=> k1_funct_1(G,H) != F ) ) ) ) ) ) ) ) ) ).
fof(t11_topreal5,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_pre_topc(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ! [H] :
( ( v1_funct_1(H)
& v1_funct_2(H,u1_struct_0(A),u1_struct_0(k3_topmetr))
& v5_pre_topc(H,A,k3_topmetr)
& m2_relset_1(H,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
=> ~ ( v2_connsp_1(D,A)
& k1_funct_1(H,B) = E
& k1_funct_1(H,C) = F
& r1_xreal_0(E,G)
& r1_xreal_0(G,F)
& r2_hidden(B,D)
& r2_hidden(C,D)
& ! [I] :
( m1_subset_1(I,u1_struct_0(A))
=> ~ ( r2_hidden(I,D)
& k1_funct_1(H,I) = G ) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_topreal5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ~ r1_xreal_0(B,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr))
& v5_pre_topc(E,k4_topmetr(A,B),k3_topmetr)
& m2_relset_1(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr)) )
=> ! [F] :
( v1_xreal_0(F)
=> ~ ( k1_funct_1(E,A) = C
& k1_funct_1(E,B) = D
& ~ r1_xreal_0(F,C)
& ~ r1_xreal_0(D,F)
& ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( k1_funct_1(E,G) = F
& ~ r1_xreal_0(G,A)
& ~ r1_xreal_0(B,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t13_topreal5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ! [D] :
( v1_xreal_0(D)
=> ( ~ r1_xreal_0(B,A)
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr))
& v5_pre_topc(E,k4_topmetr(A,B),k3_topmetr)
& m2_relset_1(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr)) )
=> ! [F] :
( v1_xreal_0(F)
=> ~ ( k1_funct_1(E,A) = C
& k1_funct_1(E,B) = D
& ~ r1_xreal_0(C,F)
& ~ r1_xreal_0(F,D)
& ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( k1_funct_1(E,G) = F
& ~ r1_xreal_0(G,A)
& ~ r1_xreal_0(B,G) ) ) ) ) ) ) ) ) ) ) ).
fof(t14_topreal5,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr))
& v5_pre_topc(C,k4_topmetr(A,B),k3_topmetr)
& m2_relset_1(C,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr)) )
=> ! [D] :
( v1_xreal_0(D)
=> ! [E] :
( v1_xreal_0(E)
=> ~ ( ~ r1_xreal_0(B,A)
& ~ r1_xreal_0(np__0,k3_xcmplx_0(D,E))
& D = k1_funct_1(C,A)
& E = k1_funct_1(C,B)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ~ ( k1_funct_1(C,F) = np__0
& ~ r1_xreal_0(F,A)
& ~ r1_xreal_0(B,F) ) ) ) ) ) ) ) ) ).
fof(t15_topreal5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr))
& m2_relset_1(A,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr)) )
=> ! [B] :
( v1_xreal_0(B)
=> ! [C] :
( v1_xreal_0(C)
=> ~ ( v5_pre_topc(A,k22_borsuk_1,k3_topmetr)
& k1_funct_1(A,np__0) != k1_funct_1(A,np__1)
& B = k1_funct_1(A,np__0)
& C = k1_funct_1(A,np__1)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ~ ( ~ r1_xreal_0(D,np__0)
& ~ r1_xreal_0(np__1,D)
& k1_funct_1(A,D) = k7_xcmplx_0(k2_xcmplx_0(B,C),np__2) ) ) ) ) ) ) ).
fof(t16_topreal5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
& m2_relset_1(A,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) )
=> ( A = k16_pscomp_1
=> v5_pre_topc(A,k15_euclid(np__2),k3_topmetr) ) ) ).
fof(t17_topreal5,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
& m2_relset_1(A,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) )
=> ( A = k17_pscomp_1
=> v5_pre_topc(A,k15_euclid(np__2),k3_topmetr) ) ) ).
fof(t18_topreal5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(B,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
=> ~ ( v5_pre_topc(B,k22_borsuk_1,k3_pre_topc(k15_euclid(np__2),A))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr))
& m2_relset_1(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr)) )
=> ~ ( v5_pre_topc(C,k22_borsuk_1,k3_topmetr)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(D,u1_struct_0(k22_borsuk_1))
& E = k1_funct_1(B,D) )
=> k21_euclid(E) = k1_funct_1(C,D) ) ) ) ) ) ) ) ) ).
fof(t19_topreal5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
& m2_relset_1(B,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
=> ~ ( v5_pre_topc(B,k22_borsuk_1,k3_pre_topc(k15_euclid(np__2),A))
& ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr))
& m2_relset_1(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr)) )
=> ~ ( v5_pre_topc(C,k22_borsuk_1,k3_topmetr)
& ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r2_hidden(D,u1_struct_0(k22_borsuk_1))
& E = k1_funct_1(B,D) )
=> k22_euclid(E) = k1_funct_1(C,D) ) ) ) ) ) ) ) ) ).
fof(t20_topreal5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v1_topreal2(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& r2_hidden(C,A)
& k22_euclid(C) != B ) ) ) ) ).
fof(t21_topreal5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ( v1_topreal2(A)
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
& r2_hidden(C,A)
& k21_euclid(C) != B ) ) ) ) ).
fof(t22_topreal5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( v1_topreal2(A)
& r1_xreal_0(k19_pscomp_1(A),k21_pscomp_1(A)) ) ) ).
fof(t23_topreal5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( v1_topreal2(A)
& r1_xreal_0(k20_pscomp_1(A),k18_pscomp_1(A)) ) ) ).
fof(t24_topreal5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( v1_topreal2(A)
& k37_pscomp_1(A) = k33_pscomp_1(A) ) ) ).
fof(t25_topreal5,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v6_compts_1(A,k15_euclid(np__2))
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ~ ( v1_topreal2(A)
& k30_pscomp_1(A) = k34_pscomp_1(A) ) ) ).
%------------------------------------------------------------------------------