SET007 Axioms: SET007+733.ax
%------------------------------------------------------------------------------
% File : SET007+733 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Ordering of Points on a Curve. Part III
% 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 : jordan17 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 28 ( 0 unt; 0 def)
% Number of atoms : 297 ( 58 equ)
% Maximal formula atoms : 18 ( 10 avg)
% Number of connectives : 336 ( 67 ~; 18 |; 105 &)
% ( 1 <=>; 145 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 15 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-5 aty)
% Number of functors : 15 ( 15 usr; 6 con; 0-2 aty)
% Number of variables : 132 ( 132 !; 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(t1_jordan17,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
=> ~ ( r2_hidden(B,E)
& r1_topreal1(k15_euclid(A),C,D,E)
& ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(A),E)))
& m2_relset_1(F,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(A),E))) )
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ~ ( v3_tops_2(F,k22_borsuk_1,k3_pre_topc(k15_euclid(A),E))
& k1_funct_1(F,np__0) = C
& k1_funct_1(F,np__1) = D
& r1_xreal_0(np__0,G)
& r1_xreal_0(G,np__1)
& k1_funct_1(F,G) = B ) ) ) ) ) ) ) ) ) ).
fof(t2_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> r1_jordan6(A,k30_pscomp_1(A),k34_pscomp_1(A)) ) ).
fof(t3_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jordan6(A,B,k34_pscomp_1(A))
=> r2_hidden(B,k8_jordan6(A)) ) ) ) ).
fof(t4_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jordan6(A,k34_pscomp_1(A),B)
=> r2_hidden(B,k9_jordan6(A)) ) ) ) ).
fof(t5_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jordan6(A,B,k30_pscomp_1(A))
=> r2_hidden(B,k9_jordan6(A)) ) ) ) ).
fof(t6_jordan17,axiom,
! [A] :
( m1_subset_1(A,u1_struct_0(k15_euclid(np__2)))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ~ ( A != B
& r1_topreal1(k15_euclid(np__2),C,D,E)
& r1_jordan5c(E,C,D,A,B)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( A != F
& B != F
& r1_jordan5c(E,C,D,A,F)
& r1_jordan5c(E,C,D,F,B) ) ) ) ) ) ) ) ) ).
fof(t7_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(B,A)
& ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan6(A,B,C) ) ) ) ) ) ).
fof(t8_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan6(A,B,C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ~ ( D != B
& D != C
& r1_jordan6(A,B,D)
& r1_jordan6(A,D,C) ) ) ) ) ) ) ).
fof(d1_jordan17,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jordan17(A,B,C,D,E)
<=> ~ ( ~ ( r1_jordan6(A,B,C)
& r1_jordan6(A,C,D)
& r1_jordan6(A,D,E) )
& ~ ( r1_jordan6(A,C,D)
& r1_jordan6(A,D,E)
& r1_jordan6(A,E,B) )
& ~ ( r1_jordan6(A,D,E)
& r1_jordan6(A,E,B)
& r1_jordan6(A,B,C) )
& ~ ( r1_jordan6(A,E,B)
& r1_jordan6(A,B,C)
& r1_jordan6(A,C,D) ) ) ) ) ) ) ) ) ).
fof(t9_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ( r2_hidden(B,A)
=> r1_jordan17(A,B,B,B,B) ) ) ) ).
fof(t10_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jordan17(A,B,C,D,E)
=> r1_jordan17(A,C,D,E,B) ) ) ) ) ) ) ).
fof(t11_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jordan17(A,B,C,D,E)
=> r1_jordan17(A,D,E,B,C) ) ) ) ) ) ) ).
fof(t12_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( r1_jordan17(A,B,C,D,E)
=> r1_jordan17(A,E,B,C,D) ) ) ) ) ) ) ).
fof(t13_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan17(A,B,C,D,E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( F != B
& F != C
& r1_jordan17(A,B,F,C,D) ) ) ) ) ) ) ) ) ).
fof(t14_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan17(A,B,C,D,E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( F != B
& F != C
& r1_jordan17(A,B,F,C,E) ) ) ) ) ) ) ) ) ).
fof(t15_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan17(A,D,B,C,E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( F != B
& F != C
& r1_jordan17(A,D,B,F,C) ) ) ) ) ) ) ) ) ).
fof(t16_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan17(A,D,B,C,E)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( F != B
& F != C
& r1_jordan17(A,B,F,C,E) ) ) ) ) ) ) ) ) ).
fof(t17_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan17(A,D,E,B,C)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( F != B
& F != C
& r1_jordan17(A,D,B,F,C) ) ) ) ) ) ) ) ) ).
fof(t18_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan17(A,D,E,B,C)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( F != B
& F != C
& r1_jordan17(A,E,B,F,C) ) ) ) ) ) ) ) ) ).
fof(t19_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan17(A,C,D,E,B)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( F != B
& F != C
& r1_jordan17(A,C,D,B,F) ) ) ) ) ) ) ) ) ).
fof(t20_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( B != C
& r1_jordan17(A,C,D,E,B)
& ! [F] :
( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
=> ~ ( F != B
& F != C
& r1_jordan17(A,C,E,B,F) ) ) ) ) ) ) ) ) ).
fof(t21_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan17(A,B,E,C,D)
& r1_jordan17(A,E,B,C,D) )
=> ( B = C
| B = D
| E = D
| B = E ) ) ) ) ) ) ) ).
fof(t22_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan17(A,B,C,D,E)
& r1_jordan17(A,D,C,B,E) )
=> ( B = C
| C = D
| D = E
| B = D ) ) ) ) ) ) ) ).
fof(t23_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan17(A,B,C,D,E)
& r1_jordan17(A,E,C,D,B) )
=> ( B = C
| B = D
| C = E
| B = E ) ) ) ) ) ) ) ).
fof(t24_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan17(A,B,E,C,D)
& r1_jordan17(A,B,C,E,D) )
=> ( B = C
| B = D
| E = D
| E = C ) ) ) ) ) ) ) ).
fof(t25_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan17(A,B,C,D,E)
& r1_jordan17(A,B,E,D,C) )
=> ( B = C
| C = D
| D = E
| C = E ) ) ) ) ) ) ) ).
fof(t26_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ( ( r1_jordan17(A,B,C,D,E)
& r1_jordan17(A,B,C,E,D) )
=> ( B = C
| B = D
| C = E
| D = E ) ) ) ) ) ) ) ).
fof(t27_jordan17,axiom,
! [A] :
( ( v1_topreal2(A)
& m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
=> ~ ( r2_hidden(B,A)
& r2_hidden(C,A)
& r2_hidden(D,A)
& r2_hidden(E,A)
& ~ r1_jordan17(A,B,C,D,E)
& ~ r1_jordan17(A,B,C,E,D)
& ~ r1_jordan17(A,B,D,C,E)
& ~ r1_jordan17(A,B,D,E,C)
& ~ r1_jordan17(A,B,E,C,D)
& ~ r1_jordan17(A,B,E,D,C) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------