SET007 Axioms: SET007+449.ax
%------------------------------------------------------------------------------
% File : SET007+449 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On the Category of Posets
% 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 : orders_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 43 ( 2 unt; 0 def)
% Number of atoms : 308 ( 40 equ)
% Maximal formula atoms : 29 ( 7 avg)
% Number of connectives : 303 ( 38 ~; 0 |; 141 &)
% ( 13 <=>; 111 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 40 ( 38 usr; 1 prp; 0-3 aty)
% Number of functors : 36 ( 36 usr; 3 con; 0-4 aty)
% Number of variables : 122 ( 108 !; 14 ?)
% 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(rc1_orders_3,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v1_orders_2(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_orders_3(A) ) ).
fof(rc2_orders_3,axiom,
? [A] :
( l1_orders_2(A)
& v3_struct_0(A)
& v1_orders_2(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_orders_3(A) ) ).
fof(fc1_orders_3,axiom,
( v3_struct_0(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0)))
& v1_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0)))
& v2_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0)))
& v3_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0)))
& v4_orders_2(g1_orders_2(k1_xboole_0,k6_partfun1(k1_xboole_0))) ) ).
fof(fc2_orders_3,axiom,
! [A] :
( ( v3_struct_0(A)
& l1_orders_2(A) )
=> ( v1_relat_1(u1_orders_2(A))
& v1_funct_1(u1_orders_2(A))
& v2_funct_1(u1_orders_2(A))
& v1_xboole_0(u1_orders_2(A)) ) ) ).
fof(cc1_orders_3,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_struct_0(A)
=> v1_orders_3(A) ) ) ).
fof(rc3_orders_3,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v1_orders_2(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& ~ v3_orders_3(A) ) ).
fof(rc4_orders_3,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v1_orders_2(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_orders_3(A)
& v3_orders_3(A) ) ).
fof(rc5_orders_3,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v4_orders_3(A) ) ).
fof(fc3_orders_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ~ v1_xboole_0(k1_orders_3(A,A)) ) ).
fof(fc4_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ~ v1_xboole_0(k2_orders_3(A)) ) ).
fof(fc5_orders_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& l1_orders_2(A)
& ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& l1_orders_2(B) )
=> v1_fraenkel(k1_orders_3(A,B)) ) ).
fof(fc6_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ( ~ v3_struct_0(k4_orders_3(A))
& v2_altcat_1(k4_orders_3(A))
& v6_altcat_1(k4_orders_3(A)) ) ) ).
fof(fc7_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ( ~ v3_struct_0(k4_orders_3(A))
& v2_altcat_1(k4_orders_3(A))
& v6_altcat_1(k4_orders_3(A))
& v11_altcat_1(k4_orders_3(A))
& v12_altcat_1(k4_orders_3(A)) ) ) ).
fof(d1_orders_3,axiom,
! [A] :
( l1_orders_2(A)
=> ( v1_orders_3(A)
<=> u1_orders_2(A) = k6_partfun1(u1_struct_0(A)) ) ) ).
fof(d2_orders_3,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_orders_3(B,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)))
& C != k1_xboole_0
& D != k1_xboole_0
& B = k4_subset_1(u1_struct_0(A),C,D)
& r1_xboole_0(C,D)
& u1_orders_2(A) = k2_xboole_0(k2_wellord1(u1_orders_2(A),C),k2_wellord1(u1_orders_2(A),D)) ) ) ) ) ) ).
fof(d3_orders_3,axiom,
! [A] :
( l1_orders_2(A)
=> ( v3_orders_3(A)
<=> v2_orders_3(k2_pre_topc(A),A) ) ) ).
fof(t1_orders_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_orders_3(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_orders_2(A,B,C)
<=> B = C ) ) ) ) ).
fof(t2_orders_3,axiom,
! [A] :
( v1_relat_1(A)
=> ! [B] :
( ( v1_relat_2(A)
& v4_relat_2(A)
& v8_relat_2(A)
& v1_partfun1(A,k1_tarski(B),k1_tarski(B))
& m2_relset_1(A,k1_tarski(B),k1_tarski(B)) )
=> A = k6_partfun1(k1_tarski(B)) ) ) ).
fof(t3_orders_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ( v2_orders_2(A)
& k2_pre_topc(A) = k1_struct_0(A,B) )
=> v1_orders_3(A) ) ) ) ).
fof(t4_orders_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ! [B] :
~ ( k2_pre_topc(A) = k1_tarski(B)
& v3_orders_3(A) ) ) ).
fof(t5_orders_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_orders_3(A)
& l1_orders_2(A) )
=> ( ~ ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> B = C ) )
=> v3_orders_3(A) ) ) ).
fof(d4_orders_3,axiom,
! [A] :
( v4_orders_3(A)
<=> ! [B] :
( r2_hidden(B,A)
=> ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& l1_orders_2(B) ) ) ) ).
fof(d5_orders_3,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(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)) )
=> ( v5_orders_3(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( r1_orders_2(A,D,E)
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(B))
=> ( ( F = k1_funct_1(C,D)
& G = k1_funct_1(C,E) )
=> r1_orders_2(B,F,G) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_orders_3,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C] :
( C = k1_orders_3(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
& m2_relset_1(E,u1_struct_0(A),u1_struct_0(B))
& D = E
& r2_hidden(E,k1_funct_2(u1_struct_0(A),u1_struct_0(B)))
& v5_orders_3(E,A,B) ) ) ) ) ) ).
fof(t6_orders_3,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) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(D,k1_orders_3(A,B))
& r2_hidden(E,k1_orders_3(B,C)) )
=> r2_hidden(k5_relat_1(D,E),k1_orders_3(A,C)) ) ) ) ) ) ) ).
fof(t7_orders_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> r2_hidden(k6_partfun1(u1_struct_0(A)),k1_orders_3(A,A)) ) ).
fof(d7_orders_3,axiom,
! [A,B] :
( B = k2_orders_3(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> ? [D] :
( l1_struct_0(D)
& r2_hidden(D,A)
& C = u1_struct_0(D) ) ) ) ).
fof(t8_orders_3,axiom,
! [A] :
( l1_struct_0(A)
=> k2_orders_3(k1_tarski(A)) = k1_tarski(u1_struct_0(A)) ) ).
fof(t9_orders_3,axiom,
! [A] :
( l1_struct_0(A)
=> ! [B] :
( l1_struct_0(B)
=> k2_orders_3(k2_tarski(A,B)) = k2_tarski(u1_struct_0(A),u1_struct_0(B)) ) ) ).
fof(t10_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ! [B] :
( m1_orders_3(B,A)
=> ! [C] :
( m1_orders_3(C,A)
=> r1_tarski(k1_orders_3(B,C),k1_ens_1(k2_orders_3(A))) ) ) ) ).
fof(t11_orders_3,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> r1_tarski(k1_orders_3(A,B),k1_funct_2(u1_struct_0(A),u1_struct_0(B))) ) ) ).
fof(d8_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ! [B] :
( ( v1_cat_1(B)
& v2_cat_1(B)
& v1_cat_5(B)
& l1_cat_1(B) )
=> ( B = k3_orders_3(A)
<=> ( u1_cat_1(B) = A
& ! [C] :
( m1_orders_3(C,A)
=> ! [D] :
( m1_orders_3(D,A)
=> ! [E] :
( m1_subset_1(E,k1_ens_1(k2_orders_3(A)))
=> ( r2_hidden(E,k1_orders_3(C,D))
=> m1_subset_1(k4_tarski(k4_tarski(C,D),E),u2_cat_1(B)) ) ) ) )
& ! [C] :
( m1_subset_1(C,u2_cat_1(B))
=> ? [D] :
( m1_orders_3(D,A)
& ? [E] :
( m1_orders_3(E,A)
& ? [F] :
( m1_subset_1(F,k1_ens_1(k2_orders_3(A)))
& C = k4_tarski(k4_tarski(D,E),F)
& r2_hidden(F,k1_orders_3(D,E)) ) ) ) )
& ! [C] :
( m1_subset_1(C,u2_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> ! [E] :
( m1_orders_3(E,A)
=> ! [F] :
( m1_orders_3(F,A)
=> ! [G] :
( m1_orders_3(G,A)
=> ! [H] :
( m1_subset_1(H,k1_ens_1(k2_orders_3(A)))
=> ! [I] :
( m1_subset_1(I,k1_ens_1(k2_orders_3(A)))
=> ( ( C = k4_tarski(k4_tarski(E,F),H)
& D = k4_tarski(k4_tarski(F,G),I) )
=> k4_cat_1(B,C,D) = k4_tarski(k4_tarski(E,G),k5_relat_1(H,I)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d9_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ! [B] :
( ( v6_altcat_1(B)
& l2_altcat_1(B) )
=> ( B = k4_orders_3(A)
<=> ( u1_struct_0(B) = A
& ! [C] :
( m1_orders_3(C,A)
=> ! [D] :
( m1_orders_3(D,A)
=> ( k1_binop_1(u1_altcat_1(B),C,D) = k1_orders_3(C,D)
& ! [E] :
( m1_orders_3(E,A)
=> ! [F] :
( m1_orders_3(F,A)
=> ! [G] :
( m1_orders_3(G,A)
=> k1_multop_1(u2_altcat_1(B),E,F,G) = k6_altcat_1(k1_orders_3(E,F),k1_orders_3(F,G)) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k4_orders_3(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k4_orders_3(A)))
=> ! [D] :
( m1_orders_3(D,A)
=> ! [E] :
( m1_orders_3(E,A)
=> ( ( B = D
& C = E )
=> r1_tarski(k1_altcat_1(k4_orders_3(A),B,C),k1_fraenkel(u1_struct_0(D),u1_struct_0(E))) ) ) ) ) ) ) ).
fof(s1_orders_3,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_orders_3)
=> ! [B] :
( m1_subset_1(B,f1_s1_orders_3)
=> ! [C] :
( m1_subset_1(C,f1_s1_orders_3)
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(D,f2_s1_orders_3(A,B))
& r2_hidden(E,f2_s1_orders_3(B,C)) )
=> r2_hidden(k5_relat_1(D,E),f2_s1_orders_3(A,C)) ) ) ) ) ) )
=> ? [A] :
( v6_altcat_1(A)
& l2_altcat_1(A)
& u1_struct_0(A) = f1_s1_orders_3
& ! [B] :
( m1_subset_1(B,f1_s1_orders_3)
=> ! [C] :
( m1_subset_1(C,f1_s1_orders_3)
=> ( k1_binop_1(u1_altcat_1(A),B,C) = f2_s1_orders_3(B,C)
& ! [D] :
( m1_subset_1(D,f1_s1_orders_3)
=> ! [E] :
( m1_subset_1(E,f1_s1_orders_3)
=> ! [F] :
( m1_subset_1(F,f1_s1_orders_3)
=> k1_multop_1(u2_altcat_1(A),D,E,F) = k6_altcat_1(f2_s1_orders_3(D,E),f2_s1_orders_3(E,F)) ) ) ) ) ) ) ) ) ).
fof(s2_orders_3,axiom,
! [A] :
( ( v6_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( v6_altcat_1(B)
& l2_altcat_1(B) )
=> ( ( u1_struct_0(A) = f1_s2_orders_3
& ! [C] :
( m1_subset_1(C,f1_s2_orders_3)
=> ! [D] :
( m1_subset_1(D,f1_s2_orders_3)
=> ( k1_binop_1(u1_altcat_1(A),C,D) = f2_s2_orders_3(C,D)
& ! [E] :
( m1_subset_1(E,f1_s2_orders_3)
=> ! [F] :
( m1_subset_1(F,f1_s2_orders_3)
=> ! [G] :
( m1_subset_1(G,f1_s2_orders_3)
=> k1_multop_1(u2_altcat_1(A),E,F,G) = k6_altcat_1(f2_s2_orders_3(E,F),f2_s2_orders_3(F,G)) ) ) ) ) ) )
& u1_struct_0(B) = f1_s2_orders_3
& ! [C] :
( m1_subset_1(C,f1_s2_orders_3)
=> ! [D] :
( m1_subset_1(D,f1_s2_orders_3)
=> ( k1_binop_1(u1_altcat_1(B),C,D) = f2_s2_orders_3(C,D)
& ! [E] :
( m1_subset_1(E,f1_s2_orders_3)
=> ! [F] :
( m1_subset_1(F,f1_s2_orders_3)
=> ! [G] :
( m1_subset_1(G,f1_s2_orders_3)
=> k1_multop_1(u2_altcat_1(B),E,F,G) = k6_altcat_1(f2_s2_orders_3(E,F),f2_s2_orders_3(F,G)) ) ) ) ) ) ) )
=> A = B ) ) ) ).
fof(dt_m1_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ! [B] :
( m1_orders_3(B,A)
=> ( ~ v3_struct_0(B)
& v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& l1_orders_2(B) ) ) ) ).
fof(existence_m1_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ? [B] : m1_orders_3(B,A) ) ).
fof(redefinition_m1_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ! [B] :
( m1_orders_3(B,A)
<=> m1_subset_1(B,A) ) ) ).
fof(dt_k1_orders_3,axiom,
$true ).
fof(dt_k2_orders_3,axiom,
$true ).
fof(dt_k3_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ( v1_cat_1(k3_orders_3(A))
& v2_cat_1(k3_orders_3(A))
& v1_cat_5(k3_orders_3(A))
& l1_cat_1(k3_orders_3(A)) ) ) ).
fof(dt_k4_orders_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_orders_3(A) )
=> ( v6_altcat_1(k4_orders_3(A))
& l2_altcat_1(k4_orders_3(A)) ) ) ).
%------------------------------------------------------------------------------