SET007 Axioms: SET007+507.ax
%------------------------------------------------------------------------------
% File : SET007+507 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Properties of Objects and Morphisms
% 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 : altcat_3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 52 ( 0 unt; 0 def)
% Number of atoms : 592 ( 67 equ)
% Maximal formula atoms : 17 ( 11 avg)
% Number of connectives : 651 ( 111 ~; 13 |; 302 &)
% ( 17 <=>; 208 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 14 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 28 ( 27 usr; 0 prp; 1-5 aty)
% Number of functors : 7 ( 7 usr; 1 con; 0-6 aty)
% Number of variables : 235 ( 208 !; 27 ?)
% 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_altcat_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ? [C] :
( m1_subset_1(C,k1_altcat_1(A,B,B))
& v1_altcat_3(C,A,B,B)
& v2_altcat_3(C,A,B,B)
& v4_altcat_3(C,A,B,B)
& v5_altcat_3(C,A,B,B) ) ) ).
fof(rc2_altcat_3,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ? [C] :
( m1_subset_1(C,k1_altcat_1(A,B,B))
& v1_altcat_3(C,A,B,B)
& v2_altcat_3(C,A,B,B)
& v3_altcat_3(C,A,B,B)
& v4_altcat_3(C,A,B,B)
& v5_altcat_3(C,A,B,B) ) ) ).
fof(fc1_altcat_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A))
& v4_altcat_3(C,A,B,B)
& m1_subset_1(C,k1_altcat_1(A,B,B))
& v4_altcat_3(D,A,B,B)
& m1_subset_1(D,k1_altcat_1(A,B,B)) )
=> v4_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) ).
fof(fc2_altcat_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A))
& v5_altcat_3(C,A,B,B)
& m1_subset_1(C,k1_altcat_1(A,B,B))
& v5_altcat_3(D,A,B,B)
& m1_subset_1(D,k1_altcat_1(A,B,B)) )
=> v5_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) ).
fof(fc3_altcat_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A))
& v3_altcat_3(C,A,B,B)
& m1_subset_1(C,k1_altcat_1(A,B,B))
& v3_altcat_3(D,A,B,B)
& m1_subset_1(D,k1_altcat_1(A,B,B)) )
=> v3_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) ).
fof(fc4_altcat_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A))
& v1_altcat_3(C,A,B,B)
& m1_subset_1(C,k1_altcat_1(A,B,B))
& v1_altcat_3(D,A,B,B)
& m1_subset_1(D,k1_altcat_1(A,B,B)) )
=> v1_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) ).
fof(fc5_altcat_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A))
& v2_altcat_3(C,A,B,B)
& m1_subset_1(C,k1_altcat_1(A,B,B))
& v2_altcat_3(D,A,B,B)
& m1_subset_1(D,k1_altcat_1(A,B,B)) )
=> v2_altcat_3(k5_altcat_1(A,B,B,B,D,C),A,B,B) ) ).
fof(d1_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ! [E] :
( m1_subset_1(E,k1_altcat_1(A,C,B))
=> ( r1_altcat_3(A,B,C,D,E)
<=> k5_altcat_1(A,C,B,C,E,D) = k8_altcat_1(A,C) ) ) ) ) ) ) ).
fof(d2_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ( v1_altcat_3(D,A,B,C)
<=> ? [E] :
( m1_subset_1(E,k1_altcat_1(A,C,B))
& r1_altcat_3(A,B,C,D,E) ) ) ) ) ) ) ).
fof(d3_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ( v2_altcat_3(D,A,B,C)
<=> ? [E] :
( m1_subset_1(E,k1_altcat_1(A,C,B))
& r1_altcat_3(A,C,B,E,D) ) ) ) ) ) ) ).
fof(t1_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v1_altcat_3(k8_altcat_1(A,B),A,B,B)
& v2_altcat_3(k8_altcat_1(A,B),A,B,B) ) ) ) ).
fof(d4_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,B) != k1_xboole_0
& ? [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
& v1_altcat_3(D,A,B,C)
& v2_altcat_3(D,A,B,C)
& ~ ! [E] :
( m1_subset_1(E,k1_altcat_1(A,C,B))
=> ( E = k1_altcat_3(A,B,C,D)
<=> ( r1_altcat_3(A,C,B,E,D)
& r1_altcat_3(A,B,C,D,E) ) ) ) ) ) ) ) ) ).
fof(t2_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,B) != k1_xboole_0
& ? [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
& v1_altcat_3(D,A,B,C)
& v2_altcat_3(D,A,B,C)
& ~ ( k5_altcat_1(A,B,C,B,D,k1_altcat_3(A,B,C,D)) = k8_altcat_1(A,B)
& k5_altcat_1(A,C,B,C,k1_altcat_3(A,B,C,D),D) = k8_altcat_1(A,C) ) ) ) ) ) ) ).
fof(t3_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,B) != k1_xboole_0
& ? [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
& v1_altcat_3(D,A,B,C)
& v2_altcat_3(D,A,B,C)
& k1_altcat_3(A,C,B,k1_altcat_3(A,B,C,D)) != D ) ) ) ) ) ).
fof(t4_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_altcat_3(A,B,B,k8_altcat_1(A,B)) = k8_altcat_1(A,B) ) ) ).
fof(d5_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ( v3_altcat_3(D,A,B,C)
<=> ( k5_altcat_1(A,C,B,C,k1_altcat_3(A,B,C,D),D) = k8_altcat_1(A,C)
& k5_altcat_1(A,B,C,B,D,k1_altcat_3(A,B,C,D)) = k8_altcat_1(A,B) ) ) ) ) ) ) ).
fof(t5_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ( v3_altcat_3(D,A,B,C)
=> ( v1_altcat_3(D,A,B,C)
& v2_altcat_3(D,A,B,C) ) ) ) ) ) ) ).
fof(t6_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,B) != k1_xboole_0
& ~ ! [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
=> ( v3_altcat_3(D,A,B,C)
<=> ( v1_altcat_3(D,A,B,C)
& v2_altcat_3(D,A,B,C) ) ) ) ) ) ) ) ).
fof(t7_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
=> ! [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
=> ( ( v3_altcat_3(E,A,B,C)
& v3_altcat_3(F,A,C,D) )
=> ( k1_altcat_1(A,B,C) = k1_xboole_0
| k1_altcat_1(A,C,D) = k1_xboole_0
| k1_altcat_1(A,D,B) = k1_xboole_0
| ( v3_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
& k1_altcat_3(A,B,D,k5_altcat_1(A,B,C,D,E,F)) = k5_altcat_1(A,D,C,B,k1_altcat_3(A,C,D,F),k1_altcat_3(A,B,C,E)) ) ) ) ) ) ) ) ) ) ).
fof(d6_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r2_altcat_3(A,B,C)
<=> ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,B) != k1_xboole_0
& ? [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
& v3_altcat_3(D,A,B,C) ) ) ) ) ) ) ).
fof(t8_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_altcat_3(A,B,C)
& r2_altcat_3(A,C,D) )
=> r2_altcat_3(A,B,D) ) ) ) ) ) ).
fof(d7_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ( v4_altcat_3(D,A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( k1_altcat_1(A,E,B) != k1_xboole_0
=> ! [F] :
( m1_subset_1(F,k1_altcat_1(A,E,B))
=> ! [G] :
( m1_subset_1(G,k1_altcat_1(A,E,B))
=> ( k5_altcat_1(A,E,B,C,F,D) = k5_altcat_1(A,E,B,C,G,D)
=> F = G ) ) ) ) ) ) ) ) ) ) ).
fof(d8_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ( v5_altcat_3(D,A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( k1_altcat_1(A,C,E) != k1_xboole_0
=> ! [F] :
( m1_subset_1(F,k1_altcat_1(A,C,E))
=> ! [G] :
( m1_subset_1(G,k1_altcat_1(A,C,E))
=> ( k5_altcat_1(A,B,C,E,D,F) = k5_altcat_1(A,B,C,E,D,G)
=> F = G ) ) ) ) ) ) ) ) ) ) ).
fof(t9_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,D) != k1_xboole_0
& ? [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
& ? [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
& v4_altcat_3(E,A,B,C)
& v4_altcat_3(F,A,C,D)
& ~ v4_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) ).
fof(t10_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,D) != k1_xboole_0
& ? [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
& ? [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
& v5_altcat_3(E,A,B,C)
& v5_altcat_3(F,A,C,D)
& ~ v5_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) ).
fof(t11_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,D) != k1_xboole_0
& ? [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
& ? [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
& v4_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
& ~ v4_altcat_3(E,A,B,C) ) ) ) ) ) ) ) ).
fof(t12_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,D) != k1_xboole_0
& ? [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
& ? [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
& v5_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
& ~ v5_altcat_3(F,A,C,D) ) ) ) ) ) ) ) ).
fof(t13_altcat_3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k7_altcat_1(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k7_altcat_1(A)))
=> ( k1_altcat_1(k7_altcat_1(A),B,C) != k1_xboole_0
=> ! [D] :
( m1_subset_1(D,k1_altcat_1(k7_altcat_1(A),B,C))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,C)
& m2_relset_1(E,B,C) )
=> ( E = D
=> ( v4_altcat_3(D,k7_altcat_1(A),B,C)
<=> v2_funct_1(E) ) ) ) ) ) ) ) ) ).
fof(t14_altcat_3,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_setfam_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k7_altcat_1(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k7_altcat_1(A)))
=> ( k1_altcat_1(k7_altcat_1(A),B,C) != k1_xboole_0
=> ! [D] :
( m1_subset_1(D,k1_altcat_1(k7_altcat_1(A),B,C))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,C)
& m2_relset_1(E,B,C) )
=> ( E = D
=> ( v5_altcat_3(D,k7_altcat_1(A),B,C)
<=> v2_funct_2(E,B,C) ) ) ) ) ) ) ) ) ).
fof(t15_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,B) != k1_xboole_0
& ? [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
& v1_altcat_3(D,A,B,C)
& ~ v5_altcat_3(D,A,B,C) ) ) ) ) ) ).
fof(t16_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,B) != k1_xboole_0
& ? [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
& v2_altcat_3(D,A,B,C)
& ~ v4_altcat_3(D,A,B,C) ) ) ) ) ) ).
fof(t17_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,B) != k1_xboole_0
& ? [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
& v3_altcat_3(D,A,B,C)
& ~ ( v4_altcat_3(D,A,B,C)
& v5_altcat_3(D,A,B,C) ) ) ) ) ) ) ).
fof(t18_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,D) != k1_xboole_0
& k1_altcat_1(A,D,B) != k1_xboole_0
& ? [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
& ? [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
& v1_altcat_3(E,A,B,C)
& v1_altcat_3(F,A,C,D)
& ~ v1_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) ).
fof(t19_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,D) != k1_xboole_0
& k1_altcat_1(A,D,B) != k1_xboole_0
& ? [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
& ? [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
& v2_altcat_3(E,A,B,C)
& v2_altcat_3(F,A,C,D)
& ~ v2_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) ).
fof(t20_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ( ( v1_altcat_3(D,A,B,C)
& v4_altcat_3(D,A,B,C) )
=> ( k1_altcat_1(A,B,C) = k1_xboole_0
| k1_altcat_1(A,C,B) = k1_xboole_0
| v3_altcat_3(D,A,B,C) ) ) ) ) ) ) ).
fof(t21_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ( ( v2_altcat_3(D,A,B,C)
& v5_altcat_3(D,A,B,C) )
=> ( k1_altcat_1(A,B,C) = k1_xboole_0
| k1_altcat_1(A,C,B) = k1_xboole_0
| v3_altcat_3(D,A,B,C) ) ) ) ) ) ) ).
fof(t22_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
=> ! [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
=> ( v1_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
=> ( k1_altcat_1(A,B,C) = k1_xboole_0
| k1_altcat_1(A,C,D) = k1_xboole_0
| k1_altcat_1(A,D,B) = k1_xboole_0
| v1_altcat_3(F,A,C,D) ) ) ) ) ) ) ) ) ).
fof(t23_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
=> ! [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
=> ( v2_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D)
=> ( k1_altcat_1(A,B,C) = k1_xboole_0
| k1_altcat_1(A,C,D) = k1_xboole_0
| k1_altcat_1(A,D,B) = k1_xboole_0
| v2_altcat_3(E,A,B,C) ) ) ) ) ) ) ) ) ).
fof(t24_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> v1_altcat_3(D,A,B,C) ) ) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,B) != k1_xboole_0
& ~ v3_altcat_3(D,A,B,C) ) ) ) ) ) ) ).
fof(d9_altcat_3,axiom,
! [A] :
( l1_altcat_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v6_altcat_3(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
& r2_hidden(D,k1_altcat_1(A,B,C))
& v1_realset1(k1_altcat_1(A,B,C)) ) ) ) ) ) ).
fof(t25_altcat_3,axiom,
! [A] :
( l1_altcat_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v6_altcat_3(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
& r2_hidden(D,k1_altcat_1(A,B,C))
& ! [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
=> ( r2_hidden(E,k1_altcat_1(A,B,C))
=> D = E ) ) ) ) ) ) ) ).
fof(t26_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( v6_altcat_3(B,A)
& v6_altcat_3(C,A) )
=> r2_altcat_3(A,B,C) ) ) ) ) ).
fof(d10_altcat_3,axiom,
! [A] :
( l1_altcat_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v7_altcat_3(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,k1_altcat_1(A,C,B))
& r2_hidden(D,k1_altcat_1(A,C,B))
& v1_realset1(k1_altcat_1(A,C,B)) ) ) ) ) ) ).
fof(t27_altcat_3,axiom,
! [A] :
( l1_altcat_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v7_altcat_3(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m1_subset_1(D,k1_altcat_1(A,C,B))
& r2_hidden(D,k1_altcat_1(A,C,B))
& ! [E] :
( m1_subset_1(E,k1_altcat_1(A,C,B))
=> ( r2_hidden(E,k1_altcat_1(A,C,B))
=> D = E ) ) ) ) ) ) ) ).
fof(t28_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( v7_altcat_3(B,A)
& v7_altcat_3(C,A) )
=> r2_altcat_3(A,B,C) ) ) ) ) ).
fof(d11_altcat_3,axiom,
! [A] :
( l1_altcat_1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( v8_altcat_3(B,A)
<=> ( v6_altcat_3(B,A)
& v7_altcat_3(B,A) ) ) ) ) ).
fof(t29_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( ( v8_altcat_3(B,A)
& v8_altcat_3(C,A) )
=> r2_altcat_3(A,B,C) ) ) ) ) ).
fof(d12_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(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_altcat_1(A,B,C))
=> ( v9_altcat_3(D,A,B,C)
<=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ( v8_altcat_3(E,A)
=> ! [F] :
( m1_subset_1(F,k1_altcat_1(A,B,E))
=> ! [G] :
( m1_subset_1(G,k1_altcat_1(A,E,C))
=> D = k5_altcat_1(A,B,E,C,F,G) ) ) ) ) ) ) ) ) ) ).
fof(t30_altcat_3,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_altcat_1(A,B,C))
=> ! [F] :
( m1_subset_1(F,k1_altcat_1(A,C,D))
=> ( ( v9_altcat_3(E,A,B,C)
& v9_altcat_3(F,A,C,D) )
=> v9_altcat_3(k5_altcat_1(A,B,C,D,E,F),A,B,D) ) ) ) ) ) ) ) ).
fof(symmetry_r2_altcat_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( r2_altcat_3(A,B,C)
=> r2_altcat_3(A,C,B) ) ) ).
fof(reflexivity_r2_altcat_3,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> r2_altcat_3(A,B,B) ) ).
fof(dt_k1_altcat_3,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A))
& m1_subset_1(D,k1_altcat_1(A,B,C)) )
=> m1_subset_1(k1_altcat_3(A,B,C,D),k1_altcat_1(A,C,B)) ) ).
%------------------------------------------------------------------------------