SET007 Axioms: SET007+798.ax
%------------------------------------------------------------------------------
% File : SET007+798 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Transitive Closure of Fuzzy Relations
% 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 : lfuzzy_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 77 ( 0 unt; 0 def)
% Number of atoms : 416 ( 48 equ)
% Maximal formula atoms : 14 ( 5 avg)
% Number of connectives : 432 ( 93 ~; 1 |; 122 &)
% ( 17 <=>; 199 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 32 ( 31 usr; 0 prp; 1-4 aty)
% Number of functors : 40 ( 40 usr; 4 con; 0-5 aty)
% Number of variables : 237 ( 228 !; 9 ?)
% 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_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ( v1_relat_1(B)
& v1_seq_1(B) ) ) ) ).
fof(fc1_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v1_relat_1(k4_fuzzy_4(A,A))
& v1_funct_1(k4_fuzzy_4(A,A))
& v1_seq_1(k4_fuzzy_4(A,A))
& v1_lfuzzy_1(k4_fuzzy_4(A,A),A)
& v2_lfuzzy_1(k4_fuzzy_4(A,A),A)
& v3_lfuzzy_1(k4_fuzzy_4(A,A),A)
& v4_lfuzzy_1(k4_fuzzy_4(A,A),A) ) ) ).
fof(rc1_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
& v1_relat_1(B)
& v1_funct_1(B)
& v1_seq_1(B)
& v1_lfuzzy_1(B,A)
& v2_lfuzzy_1(B,A)
& v3_lfuzzy_1(B,A)
& v4_lfuzzy_1(B,A) ) ) ).
fof(fc2_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_lfuzzy_1(B,A)
& m1_fuzzy_1(B,k2_zfmisc_1(A,A))
& v2_lfuzzy_1(C,A)
& m1_fuzzy_1(C,k2_zfmisc_1(A,A)) )
=> ( v1_relat_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
& v1_funct_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
& v1_seq_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
& v2_lfuzzy_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C),A) ) ) ).
fof(fc3_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v2_lfuzzy_1(B,A)
& m1_fuzzy_1(B,k2_zfmisc_1(A,A))
& v2_lfuzzy_1(C,A)
& m1_fuzzy_1(C,k2_zfmisc_1(A,A)) )
=> ( v1_relat_1(k2_fuzzy_1(k2_zfmisc_1(A,A),B,C))
& v1_funct_1(k2_fuzzy_1(k2_zfmisc_1(A,A),B,C))
& v1_seq_1(k2_fuzzy_1(k2_zfmisc_1(A,A),B,C))
& v2_lfuzzy_1(k2_fuzzy_1(k2_zfmisc_1(A,A),B,C),A) ) ) ).
fof(fc4_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v3_lfuzzy_1(B,A)
& m1_fuzzy_1(B,k2_zfmisc_1(A,A))
& v3_lfuzzy_1(C,A)
& m1_fuzzy_1(C,k2_zfmisc_1(A,A)) )
=> ( v1_relat_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
& v1_funct_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
& v1_seq_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C))
& v3_lfuzzy_1(k1_fuzzy_1(k2_zfmisc_1(A,A),B,C),A) ) ) ).
fof(fc5_lfuzzy_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,k2_zfmisc_1(A,A)) )
=> ( v1_relat_1(k5_lfuzzy_1(A,B))
& v1_funct_1(k5_lfuzzy_1(A,B))
& v1_seq_1(k5_lfuzzy_1(A,B))
& v3_lfuzzy_1(k5_lfuzzy_1(A,B),A) ) ) ).
fof(d1_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ( r1_lfuzzy_1(A,B,C,D)
<=> ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,B)
=> r1_xreal_0(k7_lfuzzy_0(k2_zfmisc_1(A,B),C,k1_domain_1(A,B,E,F)),k7_lfuzzy_0(k2_zfmisc_1(A,B),D,k1_domain_1(A,B,E,F))) ) ) ) ) ) ) ) ).
fof(t1_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ( ! [D] :
( m1_subset_1(D,A)
=> k7_lfuzzy_0(A,B,D) = k7_lfuzzy_0(A,C,D) )
=> B = C ) ) ) ) ).
fof(t2_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ( ! [E] :
( m1_subset_1(E,A)
=> ! [F] :
( m1_subset_1(F,B)
=> k7_lfuzzy_0(k2_zfmisc_1(A,B),C,k1_domain_1(A,B,E,F)) = k7_lfuzzy_0(k2_zfmisc_1(A,B),D,k1_domain_1(A,B,E,F)) ) )
=> C = D ) ) ) ) ) ).
fof(t3_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ( B = C
<=> ( r1_fuzzy_1(B,C)
& r1_fuzzy_1(C,B) ) ) ) ) ) ).
fof(t4_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> r1_fuzzy_1(B,B) ) ) ).
fof(t5_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> ! [D] :
( m1_fuzzy_1(D,A)
=> ( ( r1_fuzzy_1(B,C)
& r1_fuzzy_1(C,D) )
=> r1_fuzzy_1(B,D) ) ) ) ) ) ).
fof(t6_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,B))
=> ! [E] :
( m1_fuzzy_1(E,k2_zfmisc_1(A,B))
=> ! [F] :
( m1_fuzzy_1(F,k2_zfmisc_1(B,C))
=> ! [G] :
( m1_fuzzy_1(G,k2_zfmisc_1(B,C))
=> ( ( r1_fuzzy_1(D,E)
& r1_fuzzy_1(F,G) )
=> r1_fuzzy_1(k3_fuzzy_4(A,B,C,D,F),k3_fuzzy_4(A,B,C,E,G)) ) ) ) ) ) ) ) ) ).
fof(t7_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(k1_lfuzzy_1(A,B,C),B) ) ) ) ).
fof(t8_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,A)
=> ! [C] :
( m1_fuzzy_1(C,A)
=> r1_fuzzy_1(B,k2_lfuzzy_1(A,B,C)) ) ) ) ).
fof(d2_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ( v1_lfuzzy_1(B,A)
<=> r1_fuzzy_1(k4_fuzzy_4(A,A),B) ) ) ) ).
fof(d3_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ( v1_lfuzzy_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,C)) = np__1 ) ) ) ) ).
fof(d4_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ( v2_lfuzzy_1(B,A)
<=> k1_fuzzy_4(A,A,B) = B ) ) ) ).
fof(d5_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ( v2_lfuzzy_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,D)) = k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,D,C)) ) ) ) ) ) ).
fof(d6_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ( v3_lfuzzy_1(B,A)
<=> r1_fuzzy_1(k3_fuzzy_4(A,A,A,B,B),B) ) ) ) ).
fof(d7_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ( v3_lfuzzy_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,A)
=> r1_orders_2(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k12_lattice3(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,D)),k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,D,E))),k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,E))) ) ) ) ) ) ) ).
fof(d8_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ( v4_lfuzzy_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ~ ( k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,D)) != np__0
& k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,D,C)) != np__0
& C != D ) ) ) ) ) ) ).
fof(d9_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ( v4_lfuzzy_1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ~ ( k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,C,D)) != np__0
& C != D
& k7_lfuzzy_0(k2_zfmisc_1(A,A),B,k1_domain_1(A,A,D,C)) != np__0 ) ) ) ) ) ) ).
fof(t9_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
=> ( ( v2_lfuzzy_1(B,A)
& v2_lfuzzy_1(C,A) )
=> k1_fuzzy_4(A,A,k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C)) = k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C) ) ) ) ) ).
fof(t10_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
=> ( ( v2_lfuzzy_1(B,A)
& v2_lfuzzy_1(C,A) )
=> k1_fuzzy_4(A,A,k2_lfuzzy_1(k2_zfmisc_1(A,A),B,C)) = k2_lfuzzy_1(k2_zfmisc_1(A,A),B,C) ) ) ) ) ).
fof(t11_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
=> ( ( v3_lfuzzy_1(B,A)
& v3_lfuzzy_1(C,A) )
=> r1_fuzzy_1(k3_fuzzy_4(A,A,A,k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C),k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C)),k1_lfuzzy_1(k2_zfmisc_1(A,A),B,C)) ) ) ) ) ).
fof(t12_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( r1_relat_2(B,A)
=> v1_lfuzzy_1(k3_lfuzzy_1(B,k2_zfmisc_1(A,A)),A) ) ) ) ).
fof(t13_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( v4_relat_2(B)
=> v4_lfuzzy_1(k3_lfuzzy_1(B,k2_zfmisc_1(A,A)),A) ) ) ) ).
fof(t14_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( v3_relat_2(B)
=> v2_lfuzzy_1(k3_lfuzzy_1(B,k2_zfmisc_1(A,A)),A) ) ) ) ).
fof(t15_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_relset_1(B,A,A)
=> ( v8_relat_2(B)
=> v3_lfuzzy_1(k3_lfuzzy_1(B,k2_zfmisc_1(A,A)),A) ) ) ) ).
fof(t16_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v2_lfuzzy_1(k1_fuzzy_3(A,A),A)
& v4_lfuzzy_1(k1_fuzzy_3(A,A),A)
& v3_lfuzzy_1(k1_fuzzy_3(A,A),A) ) ) ).
fof(t17_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( v2_lfuzzy_1(k2_fuzzy_3(A,A),A)
& v3_lfuzzy_1(k2_fuzzy_3(A,A),A)
& v1_lfuzzy_1(k2_fuzzy_3(A,A),A) ) ) ).
fof(t18_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> v2_lfuzzy_1(k2_lfuzzy_1(k2_zfmisc_1(A,A),B,k1_fuzzy_4(A,A,B)),A) ) ) ).
fof(t19_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> v2_lfuzzy_1(k1_lfuzzy_1(k2_zfmisc_1(A,A),B,k1_fuzzy_4(A,A,B)),A) ) ) ).
fof(t20_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
=> ( ( v2_lfuzzy_1(C,A)
& r1_fuzzy_1(B,C) )
=> r1_fuzzy_1(k2_lfuzzy_1(k2_zfmisc_1(A,A),B,k1_fuzzy_4(A,A,B)),C) ) ) ) ) ).
fof(t21_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
=> ( ( v2_lfuzzy_1(C,A)
& r1_fuzzy_1(C,B) )
=> r1_fuzzy_1(C,k1_lfuzzy_1(k2_zfmisc_1(A,A),B,k1_fuzzy_4(A,A,B))) ) ) ) ) ).
fof(d10_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( m1_fuzzy_1(D,k2_zfmisc_1(A,A))
=> ( D = k4_lfuzzy_1(A,B,C)
<=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_funct_2(k2_zfmisc_1(A,A),k1_rcomp_1(np__0,np__1)))
& m2_relset_1(E,k5_numbers,k1_funct_2(k2_zfmisc_1(A,A),k1_rcomp_1(np__0,np__1)))
& D = k1_funct_1(E,C)
& k8_funct_2(k5_numbers,k1_funct_2(k2_zfmisc_1(A,A),k1_rcomp_1(np__0,np__1)),E,np__0) = k4_fuzzy_4(A,A)
& ! [F] :
( v4_ordinal2(F)
=> ? [G] :
( m1_fuzzy_1(G,k2_zfmisc_1(A,A))
& k1_funct_1(E,F) = G
& k1_funct_1(E,k2_xcmplx_0(F,np__1)) = k3_fuzzy_4(A,A,A,G,B) ) ) ) ) ) ) ) ) ).
fof(t22_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> k3_fuzzy_4(A,A,A,k4_fuzzy_4(A,A),B) = B ) ) ).
fof(t23_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> k3_fuzzy_4(A,A,A,B,k4_fuzzy_4(A,A)) = B ) ) ).
fof(t24_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> k4_lfuzzy_1(A,B,np__0) = k4_fuzzy_4(A,A) ) ) ).
fof(t25_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> k4_lfuzzy_1(A,B,np__1) = B ) ) ).
fof(t26_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( v4_ordinal2(C)
=> k4_lfuzzy_1(A,B,k2_xcmplx_0(C,np__1)) = k3_fuzzy_4(A,A,A,k4_lfuzzy_1(A,B,C),B) ) ) ) ).
fof(t27_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> k4_lfuzzy_1(A,B,k2_xcmplx_0(C,D)) = k3_fuzzy_4(A,A,A,k4_lfuzzy_1(A,B,C),k4_lfuzzy_1(A,B,D)) ) ) ) ) ).
fof(t28_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> k4_lfuzzy_1(A,B,k3_xcmplx_0(C,D)) = k4_lfuzzy_1(A,k4_lfuzzy_1(A,B,D),C) ) ) ) ) ).
fof(t30_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> r1_fuzzy_1(B,k5_lfuzzy_1(A,B)) ) ) ).
fof(t31_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( v4_ordinal2(C)
=> ( ~ r1_xreal_0(C,np__0)
=> r1_fuzzy_1(k4_lfuzzy_1(A,B,C),k5_lfuzzy_1(A,B)) ) ) ) ) ).
fof(t32_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(A))))
=> ! [C] :
( m1_subset_1(C,A)
=> k8_lfuzzy_0(A,k1_yellow_0(k4_lfuzzy_0(A),B),C) = k1_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k5_card_3(C,B)) ) ) ) ).
fof(t37_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( v4_ordinal2(C)
=> ( v3_lfuzzy_1(B,A)
=> ( r1_xreal_0(C,np__0)
| r1_fuzzy_1(k4_lfuzzy_1(A,B,C),B) ) ) ) ) ) ).
fof(t38_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ( v3_lfuzzy_1(B,A)
=> B = k5_lfuzzy_1(A,B) ) ) ) ).
fof(t39_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
=> ! [D] :
( v4_ordinal2(D)
=> ( r1_fuzzy_1(B,C)
=> r1_fuzzy_1(k4_lfuzzy_1(A,B,D),k4_lfuzzy_1(A,C,D)) ) ) ) ) ) ).
fof(t40_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_fuzzy_1(C,k2_zfmisc_1(A,A))
=> ( ( v3_lfuzzy_1(C,A)
& r1_fuzzy_1(B,C) )
=> r1_fuzzy_1(k5_lfuzzy_1(A,B),C) ) ) ) ) ).
fof(reflexivity_r1_lfuzzy_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_fuzzy_1(C,k2_zfmisc_1(A,B))
& m1_fuzzy_1(D,k2_zfmisc_1(A,B)) )
=> r1_lfuzzy_1(A,B,C,C) ) ).
fof(redefinition_r1_lfuzzy_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_fuzzy_1(C,k2_zfmisc_1(A,B))
& m1_fuzzy_1(D,k2_zfmisc_1(A,B)) )
=> ( r1_lfuzzy_1(A,B,C,D)
<=> r1_fuzzy_1(C,D) ) ) ).
fof(dt_k1_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> m1_fuzzy_1(k1_lfuzzy_1(A,B,C),A) ) ).
fof(commutativity_k1_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> k1_lfuzzy_1(A,B,C) = k1_lfuzzy_1(A,C,B) ) ).
fof(idempotence_k1_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> k1_lfuzzy_1(A,B,B) = B ) ).
fof(redefinition_k1_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> k1_lfuzzy_1(A,B,C) = k1_fuzzy_1(A,B,C) ) ).
fof(dt_k2_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> m1_fuzzy_1(k2_lfuzzy_1(A,B,C),A) ) ).
fof(commutativity_k2_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> k2_lfuzzy_1(A,B,C) = k2_lfuzzy_1(A,C,B) ) ).
fof(idempotence_k2_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> k2_lfuzzy_1(A,B,B) = B ) ).
fof(redefinition_k2_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,A)
& m1_fuzzy_1(C,A) )
=> k2_lfuzzy_1(A,B,C) = k2_fuzzy_1(A,B,C) ) ).
fof(dt_k3_lfuzzy_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> m1_fuzzy_1(k3_lfuzzy_1(A,B),B) ) ).
fof(redefinition_k3_lfuzzy_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> k3_lfuzzy_1(A,B) = k4_funct_3(A,B) ) ).
fof(dt_k4_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,k2_zfmisc_1(A,A))
& v4_ordinal2(C) )
=> m1_fuzzy_1(k4_lfuzzy_1(A,B,C),k2_zfmisc_1(A,A)) ) ).
fof(dt_k5_lfuzzy_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_fuzzy_1(B,k2_zfmisc_1(A,A)) )
=> m1_fuzzy_1(k5_lfuzzy_1(A,B),k2_zfmisc_1(A,A)) ) ).
fof(d11_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> k5_lfuzzy_1(A,B) = k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),a_2_0_lfuzzy_1(A,B)) ) ) ).
fof(t29_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> k7_lfuzzy_0(k2_zfmisc_1(A,A),k5_lfuzzy_1(A,B),k1_domain_1(A,A,C,D)) = k1_yellow_0(k1_lfuzzy_0(k1_rcomp_1(np__0,np__1)),k5_card_3(k1_domain_1(A,A,C,D),a_2_0_lfuzzy_1(A,B))) ) ) ) ) ).
fof(t33_lfuzzy_1,axiom,
! [A] :
( ( v2_orders_2(A)
& v3_orders_2(A)
& v4_orders_2(A)
& v1_lattice3(A)
& v2_lattice3(A)
& v3_lattice3(A)
& v9_waybel_1(A)
& l1_orders_2(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k12_lattice3(A,C,k1_yellow_0(A,B)) = k1_yellow_0(A,a_3_0_lfuzzy_1(A,B,C)) ) ) ) ).
fof(t34_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)))))
=> k3_fuzzy_4(A,A,A,B,k5_lfuzzy_0(k2_zfmisc_1(A,A),k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),C))) = k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),a_3_1_lfuzzy_1(A,B,C)) ) ) ) ).
fof(t35_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)))))
=> k3_fuzzy_4(A,A,A,k5_lfuzzy_0(k2_zfmisc_1(A,A),k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),C)),B) = k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),a_3_2_lfuzzy_1(A,B,C)) ) ) ) ).
fof(t36_lfuzzy_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_fuzzy_1(B,k2_zfmisc_1(A,A))
=> k3_fuzzy_4(A,A,A,k5_lfuzzy_1(A,B),k5_lfuzzy_1(A,B)) = k1_yellow_0(k4_lfuzzy_0(k2_zfmisc_1(A,A)),a_2_1_lfuzzy_1(A,B)) ) ) ).
fof(fraenkel_a_2_0_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fuzzy_1(C,k2_zfmisc_1(B,B)) )
=> ( r2_hidden(A,a_2_0_lfuzzy_1(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = k4_lfuzzy_1(B,C,D)
& ~ r1_xreal_0(D,np__0) ) ) ) ).
fof(fraenkel_a_3_0_lfuzzy_1,axiom,
! [A,B,C,D] :
( ( v2_orders_2(B)
& v3_orders_2(B)
& v4_orders_2(B)
& v1_lattice3(B)
& v2_lattice3(B)
& v3_lattice3(B)
& v9_waybel_1(B)
& l1_orders_2(B)
& m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_lfuzzy_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(B))
& A = k12_lattice3(B,D,E)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_3_1_lfuzzy_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_fuzzy_1(C,k2_zfmisc_1(B,B))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(B,B))))) )
=> ( r2_hidden(A,a_3_1_lfuzzy_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(B,B))))
& A = k3_fuzzy_4(B,B,B,C,k5_lfuzzy_0(k2_zfmisc_1(B,B),E))
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_3_2_lfuzzy_1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_fuzzy_1(C,k2_zfmisc_1(B,B))
& m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(B,B))))) )
=> ( r2_hidden(A,a_3_2_lfuzzy_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,u1_struct_0(k4_lfuzzy_0(k2_zfmisc_1(B,B))))
& A = k3_fuzzy_4(B,B,B,k5_lfuzzy_0(k2_zfmisc_1(B,B),E),C)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_2_1_lfuzzy_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fuzzy_1(C,k2_zfmisc_1(B,B)) )
=> ( r2_hidden(A,a_2_1_lfuzzy_1(B,C))
<=> ? [D,E] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& m2_subset_1(E,k1_numbers,k5_numbers)
& A = k3_fuzzy_4(B,B,B,k4_lfuzzy_1(B,C,D),k4_lfuzzy_1(B,C,E))
& ~ r1_xreal_0(D,np__0)
& ~ r1_xreal_0(E,np__0) ) ) ) ).
%------------------------------------------------------------------------------