SET007 Axioms: SET007+427.ax
%------------------------------------------------------------------------------
% File : SET007+427 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Categories without Uniqueness of cod and dom
% 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_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 95 ( 15 unt; 0 def)
% Number of atoms : 515 ( 81 equ)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 503 ( 83 ~; 0 |; 208 &)
% ( 18 <=>; 194 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 36 ( 34 usr; 1 prp; 0-4 aty)
% Number of functors : 47 ( 47 usr; 12 con; 0-6 aty)
% Number of variables : 274 ( 250 !; 24 ?)
% 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_altcat_1,axiom,
! [A] :
( v1_fraenkel(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> v1_fraenkel(B) ) ) ).
fof(fc1_altcat_1,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> ( v1_relat_1(k7_relat_1(A,B))
& v1_funct_1(k7_relat_1(A,B))
& v1_funcop_1(k7_relat_1(A,B)) ) ) ).
fof(fc2_altcat_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( ~ v1_xboole_0(k1_tarski(A))
& v1_finset_1(k1_tarski(A))
& v1_fraenkel(k1_tarski(A))
& v1_realset1(k1_tarski(A)) ) ) ).
fof(rc1_altcat_1,axiom,
? [A] :
( l1_altcat_1(A)
& v1_altcat_1(A) ) ).
fof(rc2_altcat_1,axiom,
? [A] :
( l2_altcat_1(A)
& v6_altcat_1(A) ) ).
fof(rc3_altcat_1,axiom,
? [A] :
( l2_altcat_1(A)
& ~ v3_struct_0(A)
& v6_altcat_1(A) ) ).
fof(rc4_altcat_1,axiom,
! [A,B] :
( ( v1_fraenkel(A)
& v1_fraenkel(B) )
=> ? [C] :
( m1_pboole(C,k2_zfmisc_1(A,B))
& v1_relat_1(C)
& v1_funct_1(C)
& v1_funcop_1(C)
& v7_altcat_1(C) ) ) ).
fof(fc3_altcat_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,k2_zfmisc_1(A,A))
& m3_pboole(C,k3_zfmisc_1(A,A,A),k3_altcat_1(A,B,B),k2_altcat_1(A,B)) )
=> ( ~ v3_struct_0(g2_altcat_1(A,B,C))
& v6_altcat_1(g2_altcat_1(A,B,C)) ) ) ).
fof(rc5_altcat_1,axiom,
? [A] :
( l2_altcat_1(A)
& ~ v3_struct_0(A)
& v6_altcat_1(A)
& v10_altcat_1(A) ) ).
fof(rc6_altcat_1,axiom,
? [A] :
( l2_altcat_1(A)
& ~ v3_struct_0(A)
& v2_altcat_1(A)
& v6_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A) ) ).
fof(fc4_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k7_altcat_1(A))
& v2_altcat_1(k7_altcat_1(A))
& v6_altcat_1(k7_altcat_1(A))
& v10_altcat_1(k7_altcat_1(A))
& v11_altcat_1(k7_altcat_1(A))
& v12_altcat_1(k7_altcat_1(A)) ) ) ).
fof(cc2_altcat_1,axiom,
! [A] :
( l2_altcat_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v8_altcat_1(A)
& v9_altcat_1(A) )
=> ( ~ v3_struct_0(A)
& v10_altcat_1(A) ) ) ) ).
fof(cc3_altcat_1,axiom,
! [A] :
( l2_altcat_1(A)
=> ( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v10_altcat_1(A)
& v12_altcat_1(A) )
=> ( ~ v3_struct_0(A)
& v8_altcat_1(A)
& v9_altcat_1(A) ) ) ) ).
fof(cc4_altcat_1,axiom,
! [A] :
( l2_altcat_1(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> v13_altcat_1(A) ) ) ).
fof(rc7_altcat_1,axiom,
? [A] :
( l2_altcat_1(A)
& ~ v3_struct_0(A)
& v3_realset2(A)
& v2_altcat_1(A)
& v6_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& v13_altcat_1(A)
& v14_altcat_1(A) ) ).
fof(rc8_altcat_1,axiom,
? [A] :
( l2_altcat_1(A)
& ~ v3_struct_0(A)
& v3_realset2(A)
& v2_altcat_1(A)
& v6_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& v13_altcat_1(A)
& v14_altcat_1(A) ) ).
fof(cc5_altcat_1,axiom,
! [A] :
( l2_altcat_1(A)
=> ( v13_altcat_1(A)
=> v2_altcat_1(A) ) ) ).
fof(fc5_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k9_altcat_1(A))
& v2_altcat_1(k9_altcat_1(A))
& v6_altcat_1(k9_altcat_1(A))
& v8_altcat_1(k9_altcat_1(A))
& v9_altcat_1(k9_altcat_1(A))
& v10_altcat_1(k9_altcat_1(A))
& v11_altcat_1(k9_altcat_1(A))
& v12_altcat_1(k9_altcat_1(A))
& v13_altcat_1(k9_altcat_1(A))
& v14_altcat_1(k9_altcat_1(A)) ) ) ).
fof(t1_altcat_1,axiom,
$true ).
fof(t2_altcat_1,axiom,
! [A] : r2_hidden(k6_relat_1(A),k1_funct_2(A,A)) ).
fof(t3_altcat_1,axiom,
k1_funct_2(k1_xboole_0,k1_xboole_0) = k1_tarski(k6_relat_1(k1_xboole_0)) ).
fof(t4_altcat_1,axiom,
! [A,B,C,D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(D,k1_funct_2(A,B))
& r2_hidden(E,k1_funct_2(B,C)) )
=> r2_hidden(k5_relat_1(D,E),k1_funct_2(A,C)) ) ) ) ).
fof(t5_altcat_1,axiom,
! [A,B,C] :
~ ( k1_funct_2(A,B) != k1_xboole_0
& k1_funct_2(B,C) != k1_xboole_0
& k1_funct_2(A,C) = k1_xboole_0 ) ).
fof(t6_altcat_1,axiom,
$true ).
fof(t7_altcat_1,axiom,
! [A,B,C] :
( m1_pboole(C,k2_zfmisc_1(B,A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
=> ! [F,G] :
( ( r2_hidden(F,D)
& r2_hidden(G,E) )
=> k1_binop_1(C,G,F) = k1_binop_1(k7_relat_1(C,k12_mcart_1(B,A,E,D)),G,F) ) ) ) ) ).
fof(t8_altcat_1,axiom,
! [A,B,C] :
( m1_pboole(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_pboole(D,k2_zfmisc_1(A,B))
=> ( ! [E,F] :
( ( r2_hidden(E,A)
& r2_hidden(F,B) )
=> k1_binop_1(C,E,F) = k1_binop_1(D,E,F) )
=> r6_pboole(k2_zfmisc_1(A,B),D,C) ) ) ) ).
fof(t9_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_pboole(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_pboole(D,k2_zfmisc_1(A,B))
=> ( ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,B)
=> k1_binop_1(C,E,F) = k1_binop_1(D,E,F) ) )
=> r6_pboole(k2_zfmisc_1(A,B),D,C) ) ) ) ) ) ).
fof(t10_altcat_1,axiom,
! [A,B] :
( m1_pboole(B,k3_zfmisc_1(A,A,A))
=> ! [C] :
( m1_pboole(C,k3_zfmisc_1(A,A,A))
=> ( ! [D,E,F] :
( ( r2_hidden(D,A)
& r2_hidden(E,A)
& r2_hidden(F,A) )
=> k1_multop_1(B,D,E,F) = k1_multop_1(C,D,E,F) )
=> r6_pboole(k3_zfmisc_1(A,A,A),C,B) ) ) ) ).
fof(t11_altcat_1,axiom,
! [A,B,C] : k9_funct_2(A,B,C) = k3_cqc_lang(k4_tarski(A,B),C) ).
fof(t12_altcat_1,axiom,
! [A,B,C] : k1_binop_1(k9_funct_2(A,B,C),A,B) = C ).
fof(d1_altcat_1,axiom,
$true ).
fof(d2_altcat_1,axiom,
! [A] :
( l1_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_binop_1(u1_altcat_1(A),B,C) ) ) ) ).
fof(d3_altcat_1,axiom,
$true ).
fof(d4_altcat_1,axiom,
! [A] :
( l1_altcat_1(A)
=> ( v2_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,B,D) = k1_xboole_0 ) ) ) ) ) ) ).
fof(d5_altcat_1,axiom,
! [A,B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_pboole(C,k3_zfmisc_1(A,A,A))
=> ( C = k2_altcat_1(A,B)
<=> ! [D,E,F] :
( ( r2_hidden(D,A)
& r2_hidden(E,A)
& r2_hidden(F,A) )
=> k1_multop_1(C,D,E,F) = k1_binop_1(B,D,F) ) ) ) ) ).
fof(d6_altcat_1,axiom,
! [A,B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_pboole(C,k2_zfmisc_1(A,A))
=> ! [D] :
( m1_pboole(D,k3_zfmisc_1(A,A,A))
=> ( D = k3_altcat_1(A,B,C)
<=> ! [E,F,G] :
( ( r2_hidden(E,A)
& r2_hidden(F,A)
& r2_hidden(G,A) )
=> k1_multop_1(D,E,F,G) = k2_zfmisc_1(k1_binop_1(C,F,G),k1_binop_1(B,E,F)) ) ) ) ) ) ).
fof(d7_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m3_pboole(C,k3_zfmisc_1(A,A,A),k3_altcat_1(A,B,B),k2_altcat_1(A,B))
=> ( v3_altcat_1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( m1_subset_1(G,A)
=> ! [H,I,J] :
( ( r2_hidden(H,k1_binop_1(B,D,E))
& r2_hidden(I,k1_binop_1(B,E,F))
& r2_hidden(J,k1_binop_1(B,F,G)) )
=> k1_binop_1(k4_altcat_1(A,B,C,D,F,G),J,k1_binop_1(k4_altcat_1(A,B,C,D,E,F),I,H)) = k1_binop_1(k4_altcat_1(A,B,C,D,E,G),k1_binop_1(k4_altcat_1(A,B,C,E,F,G),J,I),H) ) ) ) ) ) ) ) ) ) ).
fof(d8_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m3_pboole(C,k3_zfmisc_1(A,A,A),k3_altcat_1(A,B,B),k2_altcat_1(A,B))
=> ( v4_altcat_1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ? [E] :
( r2_hidden(E,k1_binop_1(B,D,D))
& ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( r2_hidden(G,k1_binop_1(B,D,F))
=> k1_binop_1(k4_altcat_1(A,B,C,D,D,F),G,E) = G ) ) ) ) ) ) ) ) ).
fof(d9_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m3_pboole(C,k3_zfmisc_1(A,A,A),k3_altcat_1(A,B,B),k2_altcat_1(A,B))
=> ( v5_altcat_1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,A)
=> ? [E] :
( r2_hidden(E,k1_binop_1(B,D,D))
& ! [F] :
( m1_subset_1(F,A)
=> ! [G] :
( r2_hidden(G,k1_binop_1(B,F,D))
=> k1_binop_1(k4_altcat_1(A,B,C,F,D,D),E,G) = G ) ) ) ) ) ) ) ) ).
fof(d10_altcat_1,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,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))
=> k5_altcat_1(A,B,C,D,E,F) = k1_binop_1(k4_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A),B,C,D),F,E) ) ) ) ) ) ) ) ).
fof(d11_altcat_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ( v7_altcat_1(A)
<=> ! [B] :
~ ( r2_hidden(B,k1_relat_1(A))
& ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ~ ( B = k4_tarski(D,C)
& k1_funct_1(A,B) = k5_relat_1(C,D) ) ) ) ) ) ) ).
fof(t13_altcat_1,axiom,
! [A] :
( v1_fraenkel(A)
=> ! [B] :
( v1_fraenkel(B)
=> ! [C] :
( ( v7_altcat_1(C)
& m1_pboole(C,k2_zfmisc_1(A,B)) )
=> ! [D] :
( ( v1_relat_1(D)
& v1_funct_1(D) )
=> ! [E] :
( ( v1_relat_1(E)
& v1_funct_1(E) )
=> ( ( r2_hidden(D,A)
& r2_hidden(E,B) )
=> k1_binop_1(C,D,E) = k5_relat_1(E,D) ) ) ) ) ) ) ).
fof(d12_altcat_1,axiom,
! [A] :
( v1_fraenkel(A)
=> ! [B] :
( v1_fraenkel(B)
=> ! [C] :
( ( v1_funcop_1(C)
& v7_altcat_1(C)
& m1_pboole(C,k2_zfmisc_1(B,A)) )
=> C = k6_altcat_1(A,B) ) ) ) ).
fof(t14_altcat_1,axiom,
! [A,B,C] : r1_tarski(k2_relat_1(k6_altcat_1(k1_funct_2(A,B),k1_funct_2(B,C))),k1_funct_2(A,C)) ).
fof(t15_altcat_1,axiom,
! [A] : k6_altcat_1(k1_tarski(k6_relat_1(A)),k1_tarski(k6_relat_1(A))) = k9_funct_2(k6_relat_1(A),k6_relat_1(A),k6_relat_1(A)) ).
fof(t16_altcat_1,axiom,
! [A] :
( v1_fraenkel(A)
=> ! [B] :
( v1_fraenkel(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
=> k6_altcat_1(C,D) = k7_relat_1(k6_altcat_1(A,B),k12_mcart_1(B,A,D,C)) ) ) ) ) ).
fof(d13_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ( v8_altcat_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_tarski(k1_altcat_1(A,B,C),k1_funct_2(B,C)) ) ) ) ) ).
fof(d14_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ( v9_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,B,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))
& ? [G] :
( v1_relat_1(G)
& v1_funct_1(G)
& ? [H] :
( v1_relat_1(H)
& v1_funct_1(H)
& E = G
& F = H
& k5_altcat_1(A,B,C,D,E,F) != k5_relat_1(G,H) ) ) ) ) ) ) ) ) ) ) ).
fof(d15_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ( v10_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))
=> k4_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A),B,C,D) = k7_relat_1(k6_altcat_1(k1_funct_2(B,C),k1_funct_2(C,D)),k2_zfmisc_1(k1_altcat_1(A,C,D),k1_altcat_1(A,B,C))) ) ) ) ) ) ).
fof(t17_altcat_1,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,u1_struct_0(A))
=> ( v1_funct_1(k4_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A),B,C,D))
& v1_funct_2(k4_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A),B,C,D),k2_zfmisc_1(k1_altcat_1(A,C,D),k1_altcat_1(A,B,C)),k1_altcat_1(A,B,D))
& m2_relset_1(k4_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A),B,C,D),k2_zfmisc_1(k1_altcat_1(A,C,D),k1_altcat_1(A,B,C)),k1_altcat_1(A,B,D)) ) ) ) ) ) ).
fof(t18_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_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,B,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))
& ? [G] :
( v1_relat_1(G)
& v1_funct_1(G)
& ? [H] :
( v1_relat_1(H)
& v1_funct_1(H)
& E = G
& F = H
& k5_altcat_1(A,B,C,D,E,F) != k5_relat_1(G,H) ) ) ) ) ) ) ) ) ) ).
fof(d16_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_altcat_1(B)
& v10_altcat_1(B)
& l2_altcat_1(B) )
=> ( B = k7_altcat_1(A)
<=> ( u1_struct_0(B) = A
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k1_altcat_1(B,C,D) = k1_funct_2(C,D) ) ) ) ) ) ) ).
fof(d17_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ( v11_altcat_1(A)
<=> v3_altcat_1(u2_altcat_1(A),u1_struct_0(A),u1_altcat_1(A)) ) ) ).
fof(d18_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ( v12_altcat_1(A)
<=> ( v5_altcat_1(u2_altcat_1(A),u1_struct_0(A),u1_altcat_1(A))
& v4_altcat_1(u2_altcat_1(A),u1_struct_0(A),u1_altcat_1(A)) ) ) ) ).
fof(t19_altcat_1,axiom,
$true ).
fof(t20_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_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_relat_1(k4_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A),B,C,D)) = k2_zfmisc_1(k1_altcat_1(A,C,D),k1_altcat_1(A,B,C))
& r1_tarski(k2_relat_1(k4_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A),B,C,D)),k1_altcat_1(A,B,D)) ) ) ) ) ) ).
fof(t21_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_altcat_1(A,B,B) != k1_xboole_0 ) ) ).
fof(d19_altcat_1,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,k1_altcat_1(A,B,B))
=> ( C = k8_altcat_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( k1_altcat_1(A,B,D) != k1_xboole_0
=> ! [E] :
( m1_subset_1(E,k1_altcat_1(A,B,D))
=> k5_altcat_1(A,B,B,D,C,E) = E ) ) ) ) ) ) ) ).
fof(t22_altcat_1,axiom,
$true ).
fof(t23_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> r2_hidden(k8_altcat_1(A,B),k1_altcat_1(A,B,B)) ) ) ).
fof(t24_altcat_1,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))
=> ( k1_altcat_1(A,B,C) != k1_xboole_0
=> ! [D] :
( m1_subset_1(D,k1_altcat_1(A,B,C))
=> k5_altcat_1(A,B,C,C,D,k8_altcat_1(A,C)) = D ) ) ) ) ) ).
fof(t25_altcat_1,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))
=> ! [E] :
( m1_subset_1(E,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,E) != k1_xboole_0
& ~ ! [F] :
( m1_subset_1(F,k1_altcat_1(A,B,C))
=> ! [G] :
( m1_subset_1(G,k1_altcat_1(A,C,D))
=> ! [H] :
( m1_subset_1(H,k1_altcat_1(A,D,E))
=> k5_altcat_1(A,B,D,E,k5_altcat_1(A,B,C,D,F,G),H) = k5_altcat_1(A,B,C,E,F,k5_altcat_1(A,C,D,E,G,H)) ) ) ) ) ) ) ) ) ) ).
fof(d20_altcat_1,axiom,
! [A] :
( l2_altcat_1(A)
=> ( v13_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
=> B = C ) ) ) ) ) ).
fof(d21_altcat_1,axiom,
! [A] :
( l2_altcat_1(A)
=> ( v14_altcat_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v1_realset1(k1_altcat_1(A,B,B)) ) ) ) ).
fof(t26_altcat_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ( v14_altcat_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_altcat_1(A,B,B) = k1_tarski(k8_altcat_1(A,B)) ) ) ) ).
fof(t27_altcat_1,axiom,
( v14_altcat_1(k7_altcat_1(np__1))
& v3_realset2(k7_altcat_1(np__1)) ) ).
fof(d22_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v3_struct_0(B)
& v6_altcat_1(B)
& v13_altcat_1(B)
& l2_altcat_1(B) )
=> ( B = k9_altcat_1(A)
<=> ( u1_struct_0(B) = A
& ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> k1_altcat_1(B,C,C) = k1_tarski(k6_relat_1(C)) ) ) ) ) ) ).
fof(t28_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k9_altcat_1(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k9_altcat_1(A)))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(k9_altcat_1(A)))
=> ( ~ ( B = C
& C = D )
=> k4_altcat_1(u1_struct_0(k9_altcat_1(A)),u1_altcat_1(k9_altcat_1(A)),u2_altcat_1(k9_altcat_1(A)),B,C,D) = k1_xboole_0 ) ) ) ) ) ).
fof(t29_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(k9_altcat_1(A)))
=> k4_altcat_1(u1_struct_0(k9_altcat_1(A)),u1_altcat_1(k9_altcat_1(A)),u2_altcat_1(k9_altcat_1(A)),B,B,B) = k9_funct_2(k6_relat_1(B),k6_relat_1(B),k6_relat_1(B)) ) ) ).
fof(s1_altcat_1,axiom,
? [A] :
( m1_pboole(A,k2_zfmisc_1(f1_s1_altcat_1,f2_s1_altcat_1))
& ! [B,C] :
( ( r2_hidden(B,f1_s1_altcat_1)
& r2_hidden(C,f2_s1_altcat_1) )
=> k1_binop_1(A,B,C) = f3_s1_altcat_1(B,C) ) ) ).
fof(s2_altcat_1,axiom,
? [A] :
( m1_pboole(A,k2_zfmisc_1(f1_s2_altcat_1,f2_s2_altcat_1))
& ! [B] :
( m1_subset_1(B,f1_s2_altcat_1)
=> ! [C] :
( m1_subset_1(C,f2_s2_altcat_1)
=> k1_binop_1(A,B,C) = f3_s2_altcat_1(B,C) ) ) ) ).
fof(s3_altcat_1,axiom,
? [A] :
( m1_pboole(A,k3_zfmisc_1(f1_s3_altcat_1,f2_s3_altcat_1,f3_s3_altcat_1))
& ! [B,C,D] :
( ( r2_hidden(B,f1_s3_altcat_1)
& r2_hidden(C,f2_s3_altcat_1)
& r2_hidden(D,f3_s3_altcat_1) )
=> k1_multop_1(A,B,C,D) = f4_s3_altcat_1(B,C,D) ) ) ).
fof(s4_altcat_1,axiom,
? [A] :
( m1_pboole(A,k3_zfmisc_1(f1_s4_altcat_1,f2_s4_altcat_1,f3_s4_altcat_1))
& ! [B] :
( m1_subset_1(B,f1_s4_altcat_1)
=> ! [C] :
( m1_subset_1(C,f2_s4_altcat_1)
=> ! [D] :
( m1_subset_1(D,f3_s4_altcat_1)
=> k1_multop_1(A,B,C,D) = f4_s4_altcat_1(B,C,D) ) ) ) ) ).
fof(dt_l1_altcat_1,axiom,
! [A] :
( l1_altcat_1(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_altcat_1,axiom,
? [A] : l1_altcat_1(A) ).
fof(dt_l2_altcat_1,axiom,
! [A] :
( l2_altcat_1(A)
=> l1_altcat_1(A) ) ).
fof(existence_l2_altcat_1,axiom,
? [A] : l2_altcat_1(A) ).
fof(abstractness_v1_altcat_1,axiom,
! [A] :
( l1_altcat_1(A)
=> ( v1_altcat_1(A)
=> A = g1_altcat_1(u1_struct_0(A),u1_altcat_1(A)) ) ) ).
fof(abstractness_v6_altcat_1,axiom,
! [A] :
( l2_altcat_1(A)
=> ( v6_altcat_1(A)
=> A = g2_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A)) ) ) ).
fof(dt_k1_altcat_1,axiom,
$true ).
fof(dt_k2_altcat_1,axiom,
! [A,B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> m1_pboole(k2_altcat_1(A,B),k3_zfmisc_1(A,A,A)) ) ).
fof(dt_k3_altcat_1,axiom,
! [A,B,C] :
( ( m1_pboole(B,k2_zfmisc_1(A,A))
& m1_pboole(C,k2_zfmisc_1(A,A)) )
=> m1_pboole(k3_altcat_1(A,B,C),k3_zfmisc_1(A,A,A)) ) ).
fof(dt_k4_altcat_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,k2_zfmisc_1(A,A))
& m3_pboole(C,k3_zfmisc_1(A,A,A),k3_altcat_1(A,B,B),k2_altcat_1(A,B))
& m1_subset_1(D,A)
& m1_subset_1(E,A)
& m1_subset_1(F,A) )
=> ( v1_funct_1(k4_altcat_1(A,B,C,D,E,F))
& v1_funct_2(k4_altcat_1(A,B,C,D,E,F),k2_zfmisc_1(k1_binop_1(B,E,F),k1_binop_1(B,D,E)),k1_binop_1(B,D,F))
& m2_relset_1(k4_altcat_1(A,B,C,D,E,F),k2_zfmisc_1(k1_binop_1(B,E,F),k1_binop_1(B,D,E)),k1_binop_1(B,D,F)) ) ) ).
fof(redefinition_k4_altcat_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,k2_zfmisc_1(A,A))
& m3_pboole(C,k3_zfmisc_1(A,A,A),k3_altcat_1(A,B,B),k2_altcat_1(A,B))
& m1_subset_1(D,A)
& m1_subset_1(E,A)
& m1_subset_1(F,A) )
=> k4_altcat_1(A,B,C,D,E,F) = k1_multop_1(C,D,E,F) ) ).
fof(dt_k5_altcat_1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v3_struct_0(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,u1_struct_0(A))
& m1_subset_1(E,k1_altcat_1(A,B,C))
& m1_subset_1(F,k1_altcat_1(A,C,D)) )
=> m1_subset_1(k5_altcat_1(A,B,C,D,E,F),k1_altcat_1(A,B,D)) ) ).
fof(dt_k6_altcat_1,axiom,
! [A,B] :
( ( v1_fraenkel(A)
& v1_fraenkel(B) )
=> ( v1_funcop_1(k6_altcat_1(A,B))
& v7_altcat_1(k6_altcat_1(A,B))
& m1_pboole(k6_altcat_1(A,B),k2_zfmisc_1(B,A)) ) ) ).
fof(dt_k7_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k7_altcat_1(A))
& v6_altcat_1(k7_altcat_1(A))
& v10_altcat_1(k7_altcat_1(A))
& l2_altcat_1(k7_altcat_1(A)) ) ) ).
fof(dt_k8_altcat_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k8_altcat_1(A,B),k1_altcat_1(A,B,B)) ) ).
fof(dt_k9_altcat_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( ~ v3_struct_0(k9_altcat_1(A))
& v6_altcat_1(k9_altcat_1(A))
& v13_altcat_1(k9_altcat_1(A))
& l2_altcat_1(k9_altcat_1(A)) ) ) ).
fof(dt_u1_altcat_1,axiom,
! [A] :
( l1_altcat_1(A)
=> m1_pboole(u1_altcat_1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A))) ) ).
fof(dt_u2_altcat_1,axiom,
! [A] :
( l2_altcat_1(A)
=> m3_pboole(u2_altcat_1(A),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),k3_altcat_1(u1_struct_0(A),u1_altcat_1(A),u1_altcat_1(A)),k2_altcat_1(u1_struct_0(A),u1_altcat_1(A))) ) ).
fof(dt_g1_altcat_1,axiom,
! [A,B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> ( v1_altcat_1(g1_altcat_1(A,B))
& l1_altcat_1(g1_altcat_1(A,B)) ) ) ).
fof(free_g1_altcat_1,axiom,
! [A,B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> ! [C,D] :
( g1_altcat_1(A,B) = g1_altcat_1(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(dt_g2_altcat_1,axiom,
! [A,B,C] :
( ( m1_pboole(B,k2_zfmisc_1(A,A))
& m3_pboole(C,k3_zfmisc_1(A,A,A),k3_altcat_1(A,B,B),k2_altcat_1(A,B)) )
=> ( v6_altcat_1(g2_altcat_1(A,B,C))
& l2_altcat_1(g2_altcat_1(A,B,C)) ) ) ).
fof(free_g2_altcat_1,axiom,
! [A,B,C] :
( ( m1_pboole(B,k2_zfmisc_1(A,A))
& m3_pboole(C,k3_zfmisc_1(A,A,A),k3_altcat_1(A,B,B),k2_altcat_1(A,B)) )
=> ! [D,E,F] :
( g2_altcat_1(A,B,C) = g2_altcat_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
%------------------------------------------------------------------------------