SET007 Axioms: SET007+632.ax
%------------------------------------------------------------------------------
% File : SET007+632 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Darboux's 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 : integra3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 20 ( 0 unt; 0 def)
% Number of atoms : 228 ( 23 equ)
% Maximal formula atoms : 22 ( 11 avg)
% Number of connectives : 223 ( 15 ~; 6 |; 102 &)
% ( 0 <=>; 100 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 14 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 17 ( 16 usr; 0 prp; 1-4 aty)
% Number of functors : 38 ( 38 usr; 5 con; 0-5 aty)
% Number of variables : 87 ( 86 !; 1 ?)
% 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_integra3,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> ~ ( k3_integra1(A) != np__0
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(C,k4_finseq_1(B))
& ~ r1_xreal_0(k3_integra1(k2_integra1(A,k8_integra1(A),B,C)),np__0) ) ) ) ) ) ).
fof(t2_integra3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( m3_integra1(C,B,k8_integra1(B))
=> ~ ( r2_hidden(A,B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(C))
& r2_hidden(A,k2_integra1(B,k8_integra1(B),C,D)) ) ) ) ) ) ) ).
fof(t3_integra3,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> ! [C] :
( m3_integra1(C,A,k8_integra1(A))
=> ? [D] :
( m3_integra1(D,A,k8_integra1(A))
& r4_integra1(A,k8_integra1(A),B,D)
& r4_integra1(A,k8_integra1(A),C,D)
& k16_integra1(D) = k4_subset_1(k1_numbers,k16_integra1(B),k16_integra1(C)) ) ) ) ) ).
fof(t4_integra3,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> ! [C] :
( m3_integra1(C,A,k8_integra1(A))
=> ( ~ r1_xreal_0(k3_seq_4(k16_integra1(k14_integra1(A,k5_rfunct_1(A,A),k8_integra1(A),B))),k17_integra1(A,C))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r2_hidden(F,k4_finseq_1(C))
& r2_hidden(D,k5_subset_1(k1_numbers,k16_integra1(B),k2_integra1(A,k8_integra1(A),C,F)))
& r2_hidden(E,k5_subset_1(k1_numbers,k16_integra1(B),k2_integra1(A,k8_integra1(A),C,F))) )
=> D = E ) ) ) ) ) ) ) ) ).
fof(t5_integra3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_finseq_1(B,k1_numbers)
=> ( ( k5_relset_1(k5_numbers,k1_numbers,A) = k5_relset_1(k5_numbers,k1_numbers,B)
& v1_goboard1(A)
& v1_goboard1(B) )
=> A = B ) ) ) ).
fof(t6_integra3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_integra1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ! [D] :
( m3_integra1(D,C,k8_integra1(C))
=> ! [E] :
( m3_integra1(E,C,k8_integra1(C))
=> ( ( r4_integra1(C,k8_integra1(C),D,E)
& r2_hidden(A,k4_finseq_1(D))
& r2_hidden(B,k4_finseq_1(D))
& r1_xreal_0(A,B) )
=> ( r1_xreal_0(k18_integra1(C,k8_integra1(C),D,E,A),k18_integra1(C,k8_integra1(C),D,E,B))
& r2_hidden(k18_integra1(C,k8_integra1(C),D,E,A),k4_finseq_1(E)) ) ) ) ) ) ) ) ).
fof(t7_integra3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_integra1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
=> ! [D] :
( m3_integra1(D,C,k8_integra1(C))
=> ! [E] :
( m3_integra1(E,C,k8_integra1(C))
=> ~ ( r4_integra1(C,k8_integra1(C),D,E)
& r2_hidden(A,k4_finseq_1(D))
& r2_hidden(B,k4_finseq_1(D))
& ~ r1_xreal_0(B,A)
& r1_xreal_0(k18_integra1(C,k8_integra1(C),D,E,B),k18_integra1(C,k8_integra1(C),D,E,A)) ) ) ) ) ) ) ).
fof(t8_integra3,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> r1_xreal_0(np__0,k17_integra1(A,B)) ) ) ).
fof(t9_integra3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ! [E] :
( m3_integra1(E,B,k8_integra1(B))
=> ( ( r2_hidden(A,k2_integra1(B,k8_integra1(B),D,k3_finseq_1(D)))
& r1_xreal_0(np__2,k3_finseq_1(D))
& r4_integra1(B,k8_integra1(B),D,E)
& k16_integra1(E) = k4_subset_1(k1_numbers,k16_integra1(D),k1_seq_4(A))
& r3_rfunct_1(C,B) )
=> r1_xreal_0(k5_real_1(k15_rvsum_1(k15_integra1(B,C,k8_integra1(B),E)),k15_rvsum_1(k15_integra1(B,C,k8_integra1(B),D))),k4_real_1(k5_real_1(k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k17_integra1(B,D))) ) ) ) ) ) ) ).
fof(t10_integra3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ! [E] :
( m3_integra1(E,B,k8_integra1(B))
=> ( ( r2_hidden(A,k2_integra1(B,k8_integra1(B),D,k3_finseq_1(D)))
& r1_xreal_0(np__2,k3_finseq_1(D))
& r4_integra1(B,k8_integra1(B),D,E)
& k16_integra1(E) = k4_subset_1(k1_numbers,k16_integra1(D),k1_seq_4(A))
& r3_rfunct_1(C,B) )
=> r1_xreal_0(k5_real_1(k15_rvsum_1(k14_integra1(B,C,k8_integra1(B),D)),k15_rvsum_1(k14_integra1(B,C,k8_integra1(B),E))),k4_real_1(k5_real_1(k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k17_integra1(B,D))) ) ) ) ) ) ) ).
fof(t11_integra3,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(D,k4_finseq_1(B))
& r2_hidden(E,k4_finseq_1(B))
& r1_xreal_0(D,E)
& ~ r1_xreal_0(k1_goboard1(k1_jordan3(k1_numbers,B,D,E),np__1),C)
& ! [F] :
( ( v1_integra1(F)
& m1_subset_1(F,k1_zfmisc_1(k1_numbers)) )
=> ~ ( C = k4_pscomp_1(F)
& k3_pscomp_1(F) = k1_goboard1(k1_jordan3(k1_numbers,B,D,E),k3_finseq_1(k1_jordan3(k1_numbers,B,D,E)))
& k3_finseq_1(k1_jordan3(k1_numbers,B,D,E)) = k3_real_1(k5_real_1(E,D),np__1)
& m1_integra1(k1_jordan3(k1_numbers,B,D,E),F) ) ) ) ) ) ) ) ) ).
fof(t12_integra3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ! [E] :
( m3_integra1(E,B,k8_integra1(B))
=> ( ( r2_hidden(A,k2_integra1(B,k8_integra1(B),D,k3_finseq_1(D)))
& r4_integra1(B,k8_integra1(B),D,E)
& k16_integra1(E) = k4_subset_1(k1_numbers,k16_integra1(D),k1_seq_4(A))
& r3_rfunct_1(C,B) )
=> ( k3_integra1(B) = np__0
| r1_xreal_0(A,k4_pscomp_1(B))
| r1_xreal_0(k5_real_1(k15_rvsum_1(k15_integra1(B,C,k8_integra1(B),E)),k15_rvsum_1(k15_integra1(B,C,k8_integra1(B),D))),k4_real_1(k5_real_1(k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k17_integra1(B,D))) ) ) ) ) ) ) ) ).
fof(t13_integra3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ! [E] :
( m3_integra1(E,B,k8_integra1(B))
=> ( ( r2_hidden(A,k2_integra1(B,k8_integra1(B),D,k3_finseq_1(D)))
& r4_integra1(B,k8_integra1(B),D,E)
& k16_integra1(E) = k4_subset_1(k1_numbers,k16_integra1(D),k1_seq_4(A))
& r3_rfunct_1(C,B) )
=> ( k3_integra1(B) = np__0
| r1_xreal_0(A,k4_pscomp_1(B))
| r1_xreal_0(k5_real_1(k15_rvsum_1(k14_integra1(B,C,k8_integra1(B),D)),k15_rvsum_1(k14_integra1(B,C,k8_integra1(B),E))),k4_real_1(k5_real_1(k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k17_integra1(B,D))) ) ) ) ) ) ) ) ).
fof(t14_integra3,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> ! [C] :
( m3_integra1(C,A,k8_integra1(A))
=> ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(E,k4_finseq_1(B))
& r2_hidden(F,k4_finseq_1(B))
& r1_xreal_0(E,F)
& r4_integra1(A,k8_integra1(A),B,C)
& ~ r1_xreal_0(k1_goboard1(k1_jordan3(k1_numbers,C,k18_integra1(A,k8_integra1(A),B,C,E),k18_integra1(A,k8_integra1(A),B,C,F)),np__1),D)
& ! [G] :
( ( v1_integra1(G)
& m1_subset_1(G,k1_zfmisc_1(k1_numbers)) )
=> ! [H] :
( m3_integra1(H,G,k8_integra1(G))
=> ! [I] :
( m3_integra1(I,G,k8_integra1(G))
=> ~ ( D = k4_pscomp_1(G)
& k3_pscomp_1(G) = k1_goboard1(I,k3_finseq_1(I))
& k3_pscomp_1(G) = k1_goboard1(H,k3_finseq_1(H))
& r4_integra1(G,k8_integra1(G),H,I)
& H = k1_jordan3(k1_numbers,B,E,F)
& I = k1_jordan3(k1_numbers,C,k18_integra1(A,k8_integra1(A),B,C,E),k18_integra1(A,k8_integra1(A),B,C,F)) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_integra3,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( m3_integra1(C,B,k8_integra1(B))
=> ( r2_hidden(A,k16_integra1(C))
=> ( r1_xreal_0(k1_goboard1(C,np__1),A)
& r1_xreal_0(A,k1_goboard1(C,k3_finseq_1(C))) ) ) ) ) ) ).
fof(t16_integra3,axiom,
! [A] :
( m2_finseq_1(A,k1_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( v1_goboard1(A)
& r2_hidden(B,k4_finseq_1(A))
& r2_hidden(C,k4_finseq_1(A))
& r2_hidden(D,k4_finseq_1(A))
& r1_xreal_0(k1_goboard1(A,B),k1_goboard1(A,D))
& r1_xreal_0(k1_goboard1(A,D),k1_goboard1(A,C)) )
=> r2_hidden(k1_goboard1(A,D),k5_relset_1(k5_numbers,k1_numbers,k1_jordan3(k1_numbers,A,B,C))) ) ) ) ) ) ).
fof(t17_integra3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ( ( r3_rfunct_1(C,B)
& r2_hidden(A,k4_finseq_1(D)) )
=> r1_xreal_0(k4_pscomp_1(k5_relset_1(B,k1_numbers,k2_partfun1(B,k1_numbers,C,k2_integra1(B,k8_integra1(B),D,A)))),k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C))) ) ) ) ) ) ).
fof(t18_integra3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( m3_integra1(D,B,k8_integra1(B))
=> ( ( r3_rfunct_1(C,B)
& r2_hidden(A,k4_finseq_1(D)) )
=> r1_xreal_0(k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k3_pscomp_1(k5_relset_1(B,k1_numbers,k2_partfun1(B,k1_numbers,C,k2_integra1(B,k8_integra1(B),D,A))))) ) ) ) ) ) ).
fof(t19_integra3,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k8_integra1(A))
& m2_relset_1(C,k5_numbers,k8_integra1(A)) )
=> ( ( r3_rfunct_1(B,A)
& v1_fdiff_1(k2_integra2(A,C)) )
=> ( k3_integra1(A) = np__0
| ( v4_seq_2(k4_integra2(A,B,C))
& k2_seq_2(k4_integra2(A,B,C)) = k12_integra1(A,B) ) ) ) ) ) ) ).
fof(t20_integra3,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k8_integra1(A))
& m2_relset_1(C,k5_numbers,k8_integra1(A)) )
=> ( ( r3_rfunct_1(B,A)
& v1_fdiff_1(k2_integra2(A,C)) )
=> ( k3_integra1(A) = np__0
| ( v4_seq_2(k3_integra2(A,B,C))
& k2_seq_2(k3_integra2(A,B,C)) = k11_integra1(A,B) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------