SET007 Axioms: SET007+750.ax
%------------------------------------------------------------------------------
% File : SET007+750 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Class of Series - Parallel Graphs. Part I
% 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 : necklace [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 66 ( 7 unt; 0 def)
% Number of atoms : 276 ( 54 equ)
% Maximal formula atoms : 11 ( 4 avg)
% Number of connectives : 264 ( 54 ~; 3 |; 100 &)
% ( 16 <=>; 91 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 33 ( 32 usr; 0 prp; 1-5 aty)
% Number of functors : 50 ( 50 usr; 12 con; 0-9 aty)
% Number of variables : 118 ( 111 !; 7 ?)
% 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(cc1_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> v1_card_1(A) ) ).
fof(fc1_necklace,axiom,
! [A] :
( v1_relat_1(k11_funct_3(A))
& v1_funct_1(k11_funct_3(A))
& v2_funct_1(k11_funct_3(A))
& v1_setfam_1(k11_funct_3(A)) ) ).
fof(fc2_necklace,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_realset1(A) )
=> v1_realset1(k1_relat_1(A)) ) ).
fof(cc2_necklace,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_realset1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A)
& v1_setfam_1(A) ) ) ).
fof(fc3_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_orders_2(k2_necklace(A))
& v1_necklace(k2_necklace(A)) ) ) ).
fof(rc1_necklace,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v1_necklace(A) ) ).
fof(fc4_necklace,axiom,
! [A] :
( ( v1_necklace(A)
& l1_orders_2(A) )
=> ( v1_relat_1(u1_orders_2(A))
& v3_relat_2(u1_orders_2(A))
& v1_setfam_1(u1_orders_2(A)) ) ) ).
fof(fc5_necklace,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( ~ v3_struct_0(k3_necklace(A))
& v1_orders_2(k3_necklace(A)) ) ) ).
fof(fc6_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_orders_2(k4_necklace(A))
& v1_necklace(k4_necklace(A)) ) ) ).
fof(fc7_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_orders_2(k4_necklace(A))
& v1_necklace(k4_necklace(A))
& v3_necklace(k4_necklace(A)) ) ) ).
fof(fc8_necklace,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ( ~ v3_struct_0(k4_necklace(A))
& v1_orders_2(k4_necklace(A))
& v1_necklace(k4_necklace(A))
& v3_necklace(k4_necklace(A)) ) ) ).
fof(fc9_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> v1_finset_1(u1_struct_0(k4_necklace(A))) ) ).
fof(d1_necklace,axiom,
! [A,B,C,D,E] :
( r1_necklace(A,B,C,D,E)
<=> ( A != B
& A != C
& A != D
& A != E
& B != C
& B != D
& B != E
& C != D
& C != E
& D != E ) ) ).
fof(t1_necklace,axiom,
! [A,B,C,D,E] :
( r1_necklace(A,B,C,D,E)
=> k4_card_1(k3_enumset1(A,B,C,D,E)) = np__5 ) ).
fof(t2_necklace,axiom,
np__4 = k2_enumset1(np__0,np__1,np__2,np__3) ).
fof(t3_necklace,axiom,
! [A,B,C,D,E,F] : k2_zfmisc_1(k1_enumset1(A,B,C),k1_enumset1(D,E,F)) = k1_bvfunc24(k4_tarski(A,D),k4_tarski(A,E),k4_tarski(A,F),k4_tarski(B,D),k4_tarski(B,E),k4_tarski(B,F),k4_tarski(C,D),k4_tarski(C,E),k4_tarski(C,F)) ).
fof(t4_necklace,axiom,
! [A,B] :
( v4_ordinal2(B)
=> ( r2_hidden(A,B)
=> v4_ordinal2(A) ) ) ).
fof(t5_necklace,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> r2_hidden(np__0,A) ) ).
fof(t6_necklace,axiom,
! [A] : k1_card_1(k6_partfun1(A)) = k1_card_1(A) ).
fof(t7_necklace,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
=> k2_relat_1(k1_funct_4(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ) ).
fof(t8_necklace,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B) )
=> ( ( r1_xboole_0(k1_relat_1(A),k1_relat_1(B))
& r1_xboole_0(k2_relat_1(A),k2_relat_1(B)) )
=> k2_funct_1(k1_funct_4(A,B)) = k1_funct_4(k2_funct_1(A),k2_funct_1(B)) ) ) ) ).
fof(t9_necklace,axiom,
! [A,B,C] : k1_funct_4(k2_funcop_1(A,B),k2_funcop_1(A,C)) = k2_funcop_1(A,C) ).
fof(t10_necklace,axiom,
! [A,B] : k2_funct_1(k3_cqc_lang(A,B)) = k3_cqc_lang(B,A) ).
fof(t11_necklace,axiom,
! [A,B,C,D] :
~ ( ( A = B
=> C = D )
& ( C = D
=> A = B )
& k2_funct_1(k4_funct_4(A,B,C,D)) != k4_funct_4(C,D,A,B) ) ).
fof(t12_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( r2_hidden(B,C)
=> ( r1_xreal_0(B,A)
| r2_hidden(A,C) ) ) ) ) ) ).
fof(d2_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( r2_necklace(A,B)
<=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(C,u1_struct_0(A),u1_struct_0(B))
& v2_funct_1(C)
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r2_hidden(k4_tarski(D,E),u1_orders_2(A))
<=> r2_hidden(k4_tarski(k1_funct_1(C,D),k1_funct_1(C,E)),u1_orders_2(B)) ) ) ) ) ) ) ) ).
fof(t13_necklace,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_orders_2(C) )
=> ( ( r3_necklace(B,A)
& r3_necklace(C,B) )
=> r3_necklace(C,A) ) ) ) ) ).
fof(d3_necklace,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ( r4_necklace(A,B)
<=> ( r3_necklace(B,A)
& r3_necklace(A,B) ) ) ) ) ).
fof(t14_necklace,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& l1_orders_2(C) )
=> ( ( r4_necklace(A,B)
& r4_necklace(B,C) )
=> r4_necklace(A,C) ) ) ) ) ).
fof(d4_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_necklace(A)
<=> r3_relat_2(u1_orders_2(A),u1_struct_0(A)) ) ) ).
fof(d5_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ( v2_necklace(A)
<=> v5_relat_2(u1_orders_2(A)) ) ) ).
fof(t15_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ( v2_necklace(A)
=> r1_xboole_0(u1_orders_2(A),k6_relset_1(u1_struct_0(A),u1_struct_0(A),u1_orders_2(A))) ) ) ).
fof(d6_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_necklace(A)
<=> ! [B] :
~ ( r2_hidden(B,u1_struct_0(A))
& r2_hidden(k4_tarski(B,B),u1_orders_2(A)) ) ) ) ).
fof(t16_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> v2_necklace(k1_necklace(A)) ) ).
fof(t17_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k1_card_1(u1_orders_2(k1_necklace(A))) = k6_xcmplx_0(A,np__1) ) ) ).
fof(d8_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( ( v1_orders_2(B)
& l1_orders_2(B) )
=> ( B = k2_necklace(A)
<=> ( u1_struct_0(B) = u1_struct_0(A)
& u1_orders_2(B) = k1_relset_1(u1_struct_0(A),u1_struct_0(A),u1_orders_2(A),k6_relset_1(u1_struct_0(A),u1_struct_0(A),u1_orders_2(A))) ) ) ) ) ).
fof(d9_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( ( v1_orders_2(B)
& l1_orders_2(B) )
=> ( B = k3_necklace(A)
<=> ( u1_struct_0(B) = u1_struct_0(A)
& u1_orders_2(B) = k6_subset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),k3_subset_1(k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_orders_2(A)),k6_partfun1(u1_struct_0(A))) ) ) ) ) ).
fof(t18_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ( r5_waybel_1(A,B)
=> k1_card_1(u1_orders_2(A)) = k1_card_1(u1_orders_2(B)) ) ) ) ).
fof(d10_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> k4_necklace(A) = k2_necklace(k1_necklace(A)) ) ).
fof(t20_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( r2_hidden(B,u1_orders_2(k4_necklace(A)))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ~ r1_xreal_0(A,k1_nat_1(C,np__1))
& ( B = k4_tarski(C,k1_nat_1(C,np__1))
| B = k4_tarski(k1_nat_1(C,np__1),C) ) ) ) ) ).
fof(t21_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> u1_struct_0(k4_necklace(A)) = A ) ).
fof(t22_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,k2_xcmplx_0(B,np__1))
=> r2_hidden(k4_tarski(B,k2_xcmplx_0(B,np__1)),u1_orders_2(k4_necklace(A))) ) ) ) ).
fof(t23_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( r2_hidden(B,u1_struct_0(k4_necklace(A)))
& r1_xreal_0(A,B) ) ) ) ).
fof(t24_necklace,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ~ v3_orders_3(k4_necklace(A)) ) ).
fof(t25_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ~ ( r2_hidden(k4_tarski(B,C),u1_orders_2(k4_necklace(A)))
& B != k2_xcmplx_0(C,np__1)
& C != k2_xcmplx_0(B,np__1) ) ) ) ) ).
fof(t26_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( ( r2_hidden(B,u1_struct_0(k4_necklace(A)))
& r2_hidden(C,u1_struct_0(k4_necklace(A))) )
=> ( ( B != k2_xcmplx_0(C,np__1)
& C != k2_xcmplx_0(B,np__1) )
| r2_hidden(k4_tarski(B,C),u1_orders_2(k4_necklace(A))) ) ) ) ) ) ).
fof(t28_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k1_card_1(u1_orders_2(k4_necklace(A))) = k3_xcmplx_0(np__2,k6_xcmplx_0(A,np__1)) ) ) ).
fof(t29_necklace,axiom,
r5_waybel_1(k4_necklace(np__1),k3_necklace(k4_necklace(np__1))) ).
fof(t30_necklace,axiom,
r5_waybel_1(k4_necklace(np__4),k3_necklace(k4_necklace(np__4))) ).
fof(t31_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ~ ( r5_waybel_1(k4_necklace(A),k3_necklace(k4_necklace(A)))
& A != np__0
& A != np__1
& A != np__4 ) ) ).
fof(reflexivity_r3_necklace,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A)
& ~ v3_struct_0(B)
& l1_orders_2(B) )
=> r3_necklace(B,B) ) ).
fof(redefinition_r3_necklace,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A)
& ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ( r3_necklace(A,B)
<=> r2_necklace(A,B) ) ) ).
fof(symmetry_r4_necklace,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A)
& ~ v3_struct_0(B)
& l1_orders_2(B) )
=> ( r4_necklace(A,B)
=> r4_necklace(B,A) ) ) ).
fof(reflexivity_r4_necklace,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A)
& ~ v3_struct_0(B)
& l1_orders_2(B) )
=> r4_necklace(A,A) ) ).
fof(dt_k1_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_orders_2(k1_necklace(A))
& l1_orders_2(k1_necklace(A)) ) ) ).
fof(dt_k2_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_orders_2(k2_necklace(A))
& l1_orders_2(k2_necklace(A)) ) ) ).
fof(dt_k3_necklace,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_orders_2(k3_necklace(A))
& l1_orders_2(k3_necklace(A)) ) ) ).
fof(dt_k4_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_orders_2(k4_necklace(A))
& l1_orders_2(k4_necklace(A)) ) ) ).
fof(d7_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( ( v1_orders_2(B)
& l1_orders_2(B) )
=> ( B = k1_necklace(A)
<=> ( u1_struct_0(B) = A
& u1_orders_2(B) = a_1_0_necklace(A) ) ) ) ) ).
fof(t19_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> u1_orders_2(k4_necklace(A)) = k2_xboole_0(a_1_0_necklace(A),a_1_1_necklace(A)) ) ).
fof(t27_necklace,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ r1_xreal_0(A,np__0)
=> k1_card_1(a_1_1_necklace(A)) = k6_xcmplx_0(A,np__1) ) ) ).
fof(s1_necklace,axiom,
( f2_s1_necklace = a_0_0_necklace
=> k4_relat_1(f2_s1_necklace) = a_0_1_necklace ) ).
fof(fraenkel_a_1_0_necklace,axiom,
! [A,B] :
( v4_ordinal2(B)
=> ( r2_hidden(A,a_1_0_necklace(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k4_tarski(C,k1_nat_1(C,np__1))
& ~ r1_xreal_0(B,k1_nat_1(C,np__1)) ) ) ) ).
fof(fraenkel_a_1_1_necklace,axiom,
! [A,B] :
( v4_ordinal2(B)
=> ( r2_hidden(A,a_1_1_necklace(B))
<=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& A = k4_tarski(k1_nat_1(C,np__1),C)
& ~ r1_xreal_0(B,k1_nat_1(C,np__1)) ) ) ) ).
fof(fraenkel_a_0_0_necklace,axiom,
! [A] :
( r2_hidden(A,a_0_0_necklace)
<=> ? [B] :
( m1_subset_1(B,f1_s1_necklace)
& A = k4_tarski(f4_s1_necklace(B),f3_s1_necklace(B))
& p1_s1_necklace(B) ) ) ).
fof(fraenkel_a_0_1_necklace,axiom,
! [A] :
( r2_hidden(A,a_0_1_necklace)
<=> ? [B] :
( m1_subset_1(B,f1_s1_necklace)
& A = k4_tarski(f3_s1_necklace(B),f4_s1_necklace(B))
& p1_s1_necklace(B) ) ) ).
%------------------------------------------------------------------------------