SET007 Axioms: SET007+745.ax
%------------------------------------------------------------------------------
% File : SET007+745 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Armstrong's Axioms
% 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 : armstrng [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 163 ( 15 unt; 0 def)
% Number of atoms : 866 ( 99 equ)
% Maximal formula atoms : 15 ( 5 avg)
% Number of connectives : 780 ( 77 ~; 8 |; 324 &)
% ( 54 <=>; 317 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 50 ( 48 usr; 1 prp; 0-4 aty)
% Number of functors : 71 ( 71 usr; 10 con; 0-4 aty)
% Number of variables : 476 ( 422 !; 54 ?)
% 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_armstrng,axiom,
? [A] :
( v1_relat_1(A)
& ~ v1_xboole_0(A)
& v1_relat_2(A)
& v4_relat_2(A)
& v8_relat_2(A) ) ).
fof(rc2_armstrng,axiom,
! [A] :
? [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& v2_finsub_1(B)
& v1_armstrng(B,A) ) ).
fof(cc1_armstrng,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& m1_subset_1(C,k5_numbers) )
=> ! [D] :
( m1_relset_1(D,A,k4_finseq_2(C,B))
=> ( ( v1_funct_1(D)
& v1_funct_2(D,A,k4_finseq_2(C,B)) )
=> ( v1_funct_1(D)
& v1_funct_2(D,A,k4_finseq_2(C,B))
& v1_matrlin(D) ) ) ) ) ).
fof(fc1_armstrng,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_matrlin(A) )
=> ( v1_relat_1(k1_funct_1(A,B))
& v1_funct_1(k1_funct_1(A,B))
& v1_finset_1(k1_funct_1(A,B))
& v1_finseq_1(k1_funct_1(A,B)) ) ) ).
fof(rc3_armstrng,axiom,
? [A] :
( l1_armstrng(A)
& v2_armstrng(A) ) ).
fof(rc4_armstrng,axiom,
! [A] :
? [B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v1_relat_1(B)
& ~ v1_xboole_0(B)
& v1_finset_1(B) ) ).
fof(fc2_armstrng,axiom,
! [A] :
( v1_relat_1(k4_armstrng(A))
& ~ v1_xboole_0(k4_armstrng(A)) ) ).
fof(fc3_armstrng,axiom,
! [A] :
( v1_relat_1(k7_armstrng(A))
& ~ v1_xboole_0(k7_armstrng(A)) ) ).
fof(fc4_armstrng,axiom,
! [A] :
( v1_relat_1(k7_armstrng(A))
& v1_relat_2(k7_armstrng(A))
& v4_relat_2(k7_armstrng(A))
& v8_relat_2(k7_armstrng(A))
& v1_partfun1(k7_armstrng(A),k4_armstrng(A),k4_armstrng(A)) ) ).
fof(rc5_armstrng,axiom,
! [A] :
? [B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v1_relat_1(B)
& ~ v1_xboole_0(B)
& v8_relat_2(B)
& v3_armstrng(B,A)
& v4_armstrng(B,A)
& v5_armstrng(B,A) ) ).
fof(rc6_armstrng,axiom,
! [A] :
? [B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v1_relat_1(B)
& v6_armstrng(B,A) ) ).
fof(rc7_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ? [B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v1_relat_1(B)
& v1_finset_1(B)
& v6_armstrng(B,A) ) ) ).
fof(cc2_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> v1_finset_1(B) ) ) ).
fof(cc3_armstrng,axiom,
! [A,B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v6_armstrng(B,A)
=> ( v8_relat_2(B)
& v3_armstrng(B,A)
& v4_armstrng(B,A)
& v5_armstrng(B,A) ) ) ) ).
fof(cc4_armstrng,axiom,
! [A,B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( v8_relat_2(B)
& v3_armstrng(B,A)
& v4_armstrng(B,A)
& v5_armstrng(B,A) )
=> v6_armstrng(B,A) ) ) ).
fof(cc5_armstrng,axiom,
! [A,B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( v3_armstrng(B,A)
& v4_armstrng(B,A) )
=> v7_armstrng(B,A) ) ) ).
fof(cc6_armstrng,axiom,
! [A,B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( v8_relat_2(B)
& v7_armstrng(B,A) )
=> ( v3_armstrng(B,A)
& v4_armstrng(B,A) ) ) ) ).
fof(rc8_armstrng,axiom,
! [A] :
? [B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
& v1_relat_1(B)
& ~ v1_xboole_0(B)
& v8_relat_2(B)
& v3_armstrng(B,A)
& v4_armstrng(B,A)
& v5_armstrng(B,A)
& v6_armstrng(B,A)
& v7_armstrng(B,A) ) ).
fof(cc7_armstrng,axiom,
! [A,B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v3_armstrng(B,A)
=> ~ v1_xboole_0(B) ) ) ).
fof(fc5_armstrng,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v6_armstrng(B,A)
& m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ( v1_relat_1(k8_armstrng(A,B))
& ~ v1_xboole_0(k8_armstrng(A,B))
& v1_finset_1(k8_armstrng(A,B)) ) ) ).
fof(fc6_armstrng,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> v1_finset_1(k9_armstrng(A,B)) ) ).
fof(t1_armstrng,axiom,
! [A] :
( v2_finsub_1(A)
=> ! [B,C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
=> ( ( r2_hidden(B,A)
& r1_tarski(C,A) )
=> r2_hidden(k8_setfam_1(B,C),A) ) ) ) ).
fof(t2_armstrng,axiom,
! [A] :
( ( v1_relat_1(A)
& ~ v1_xboole_0(A)
& v4_relat_2(A)
& v8_relat_2(A) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k3_relat_1(A))) )
=> ~ ( B != k1_xboole_0
& ! [C] :
( m1_subset_1(C,B)
=> ~ r2_waybel_4(B,C,A) ) ) ) ) ).
fof(d1_armstrng,axiom,
! [A,B] :
( v1_relat_1(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( C = k1_armstrng(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> r2_waybel_4(A,D,B) ) ) ) ) ).
fof(d2_armstrng,axiom,
! [A,B] :
( r1_armstrng(A,B)
<=> ( r2_hidden(A,B)
& ! [C,D] :
~ ( r2_hidden(C,B)
& r2_hidden(D,B)
& A = k3_xboole_0(C,D)
& A != C
& A != D ) ) ) ).
fof(d3_armstrng,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( B = k2_armstrng(A)
<=> ! [C] :
( r2_hidden(C,B)
<=> r1_armstrng(C,A) ) ) ) ) ).
fof(t3_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ? [C] :
( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A))
& B = k1_setfam_1(C)
& ! [D] :
( r2_hidden(D,C)
=> r1_armstrng(D,A) ) ) ) ) ).
fof(d4_armstrng,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( v1_armstrng(B,A)
<=> r2_hidden(A,B) ) ) ).
fof(t4_armstrng,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( v2_finsub_1(B)
=> ! [C] :
( m2_subset_1(C,k1_zfmisc_1(A),B)
=> ( r1_armstrng(C,B)
=> ( C = A
| ! [D] :
( ( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( r1_tarski(D,B)
& C = k8_setfam_1(A,D) )
=> r2_hidden(C,D) ) ) ) ) ) ) ) ).
fof(d5_armstrng,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [C] :
( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ! [D] :
( m2_finseq_2(D,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> ( D = k3_armstrng(A,B,C)
<=> ! [E] :
( r2_hidden(E,k2_finseq_1(A))
=> k1_funct_1(D,E) = k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,E),k4_finseq_4(k5_numbers,k6_margrel1,C,E)) ) ) ) ) ) ) ).
fof(t5_armstrng,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> k3_armstrng(A,k1_binari_3(A,np__0),B) = k1_binari_3(A,np__0) ) ) ).
fof(t6_armstrng,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
=> k3_armstrng(A,k6_binarith(A,k1_binari_3(A,np__0)),B) = B ) ) ).
fof(t7_armstrng,axiom,
$true ).
fof(t8_armstrng,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,B)
=> ( k1_funct_1(k1_binari_3(A,k3_series_1(np__2,B)),k1_nat_1(B,np__1)) = np__1
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k2_finseq_1(A))
=> ( C = k1_nat_1(B,np__1)
| k1_funct_1(k1_binari_3(A,k3_series_1(np__2,B)),C) = np__0 ) ) ) ) ) ) ) ).
fof(d6_armstrng,axiom,
$true ).
fof(d7_armstrng,axiom,
! [A] : k4_armstrng(A) = k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(A)) ).
fof(t9_armstrng,axiom,
! [A,B] :
( r2_hidden(B,k4_armstrng(A))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
& ? [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
& B = k4_tarski(C,D) ) ) ) ).
fof(t10_armstrng,axiom,
! [A,B,C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ~ ( r2_hidden(B,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> B != k4_tarski(D,E) ) ) ) ) ).
fof(d8_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_armstrng(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_armstrng(A)))
=> ( r2_armstrng(A,B,C)
<=> ! [D] :
( m1_subset_1(D,u3_armstrng(A))
=> ! [E] :
( m1_subset_1(E,u3_armstrng(A))
=> ( k7_relat_1(D,B) = k7_relat_1(E,B)
=> k7_relat_1(D,C) = k7_relat_1(E,C) ) ) ) ) ) ) ) ).
fof(t11_armstrng,axiom,
$true ).
fof(t12_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_armstrng(A)))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_armstrng(A)))
=> ( r2_hidden(k4_tarski(B,C),k5_armstrng(A))
<=> r2_armstrng(A,B,C) ) ) ) ) ).
fof(d10_armstrng,axiom,
! [A,B] :
( m1_subset_1(B,k4_armstrng(A))
=> ! [C] :
( m1_subset_1(C,k4_armstrng(A))
=> ( r3_armstrng(A,B,C)
<=> ( r1_tarski(k1_mcart_1(B),k1_mcart_1(C))
& r1_tarski(k2_mcart_1(C),k2_mcart_1(B)) ) ) ) ) ).
fof(t13_armstrng,axiom,
! [A,B] :
( m1_subset_1(B,k4_armstrng(A))
=> ! [C] :
( m1_subset_1(C,k4_armstrng(A))
=> ( ( r3_armstrng(A,C,B)
& r3_armstrng(A,B,C) )
=> B = C ) ) ) ).
fof(t14_armstrng,axiom,
! [A,B] :
( m1_subset_1(B,k4_armstrng(A))
=> ! [C] :
( m1_subset_1(C,k4_armstrng(A))
=> ! [D] :
( m1_subset_1(D,k4_armstrng(A))
=> ( ( r3_armstrng(A,C,B)
& r3_armstrng(A,D,C) )
=> r3_armstrng(A,D,B) ) ) ) ) ).
fof(t15_armstrng,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( r3_armstrng(A,k6_armstrng(A,B,C),k6_armstrng(A,D,E))
<=> ( r1_tarski(B,D)
& r1_tarski(E,C) ) ) ) ) ) ) ).
fof(t16_armstrng,axiom,
! [A,B] :
( r2_hidden(B,k7_armstrng(A))
<=> ? [C] :
( m1_subset_1(C,k4_armstrng(A))
& ? [D] :
( m1_subset_1(D,k4_armstrng(A))
& B = k4_tarski(C,D)
& r3_armstrng(A,D,C) ) ) ) ).
fof(t17_armstrng,axiom,
! [A] : k4_relset_1(k4_armstrng(A),k4_armstrng(A),k7_armstrng(A)) = k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(A)) ).
fof(t18_armstrng,axiom,
! [A] : k5_relset_1(k4_armstrng(A),k4_armstrng(A),k7_armstrng(A)) = k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(A)) ).
fof(t19_armstrng,axiom,
! [A] : k3_relat_1(k7_armstrng(A)) = k2_zfmisc_1(k1_zfmisc_1(A),k1_zfmisc_1(A)) ).
fof(d12_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v3_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> r2_hidden(k6_armstrng(A,C,C),B) ) ) ) ).
fof(t20_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v8_relat_2(B)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( ( r2_hidden(k6_armstrng(A,C,D),B)
& r2_hidden(k6_armstrng(A,D,E),B) )
=> r2_hidden(k6_armstrng(A,C,E),B) ) ) ) ) ) ) ).
fof(d13_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v4_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
=> ( ( r2_hidden(k6_armstrng(A,C,D),B)
& r3_armstrng(A,k6_armstrng(A,C,D),k6_armstrng(A,E,F)) )
=> r2_hidden(k6_armstrng(A,E,F),B) ) ) ) ) ) ) ) ).
fof(d14_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v5_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
=> ( ( r2_hidden(k6_armstrng(A,C,D),B)
& r2_hidden(k6_armstrng(A,E,F),B) )
=> r2_hidden(k6_armstrng(A,k4_subset_1(A,C,E),k4_subset_1(A,D,F)),B) ) ) ) ) ) ) ) ).
fof(t21_armstrng,axiom,
! [A] :
( v3_armstrng(k4_armstrng(A),A)
& v8_relat_2(k4_armstrng(A))
& v4_armstrng(k4_armstrng(A),A)
& v5_armstrng(k4_armstrng(A),A) ) ).
fof(d15_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v6_armstrng(B,A)
<=> ( v3_armstrng(B,A)
& v8_relat_2(B)
& v4_armstrng(B,A)
& v5_armstrng(B,A) ) ) ) ).
fof(t22_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> v1_finset_1(B) ) ) ).
fof(d16_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v7_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r1_tarski(D,C)
=> r2_hidden(k6_armstrng(A,C,D),B) ) ) ) ) ) ).
fof(t23_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( v7_armstrng(B,A)
& v8_relat_2(B) )
=> ( v3_armstrng(B,A)
& v4_armstrng(B,A) ) ) ) ).
fof(t24_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( v3_armstrng(B,A)
& v4_armstrng(B,A) )
=> v7_armstrng(B,A) ) ) ).
fof(t25_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> v6_armstrng(k5_armstrng(A),u1_armstrng(A)) ) ).
fof(d17_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> k8_armstrng(A,B) = k1_armstrng(B,k7_armstrng(A)) ) ).
fof(t27_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> r1_tarski(k8_armstrng(A,B),B) ) ).
fof(d18_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C,D] :
( r4_armstrng(A,B,C,D)
<=> r2_hidden(k4_tarski(C,D),k8_armstrng(A,B)) ) ) ).
fof(t28_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k4_armstrng(A))
=> ! [C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ~ ( r2_hidden(B,C)
& ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ~ ( r2_hidden(k6_armstrng(A,D,E),k8_armstrng(A,C))
& r3_armstrng(A,k6_armstrng(A,D,E),B) ) ) ) ) ) ) ) ).
fof(t29_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r4_armstrng(A,B,C,D)
<=> ( r2_hidden(k6_armstrng(A,C,D),B)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
=> ~ ( r2_hidden(k6_armstrng(A,E,F),B)
& ~ ( C = E
& D = F )
& r3_armstrng(A,k6_armstrng(A,E,F),k6_armstrng(A,C,D)) ) ) ) ) ) ) ) ) ).
fof(d19_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v8_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
& r3_armstrng(A,k6_armstrng(A,D,E),k6_armstrng(A,C,C))
& r2_hidden(k6_armstrng(A,D,E),B) ) ) ) ) ) ).
fof(d20_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v9_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
=> ( ( r2_hidden(k6_armstrng(A,C,D),B)
& r2_hidden(k6_armstrng(A,E,F),B)
& r3_armstrng(A,k6_armstrng(A,C,D),k6_armstrng(A,E,F)) )
=> ( C = E
& D = F ) ) ) ) ) ) ) ) ).
fof(d21_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v10_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
=> ( ( r2_hidden(k6_armstrng(A,C,D),B)
& r2_hidden(k6_armstrng(A,E,F),B)
& r1_tarski(E,D) )
=> r1_tarski(F,D) ) ) ) ) ) ) ) ).
fof(t30_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v6_armstrng(B,A)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ( v8_armstrng(k8_armstrng(A,B),A)
& v9_armstrng(k8_armstrng(A,B),A)
& v10_armstrng(k8_armstrng(A,B),A) ) ) ) ).
fof(t33_armstrng,axiom,
! [A,B,C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( r2_hidden(B,k9_armstrng(A,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
& B = D
& r4_armstrng(A,C,E,D) ) ) ) ) ).
fof(t34_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v6_armstrng(B,A)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ( v1_armstrng(k9_armstrng(A,B),A)
& v2_finsub_1(k9_armstrng(A,B)) ) ) ) ).
fof(t35_armstrng,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> v6_armstrng(k10_armstrng(A,B),A) ) ) ).
fof(t36_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> r1_tarski(B,k9_armstrng(A,k10_armstrng(A,B))) ) ) ).
fof(t37_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( ( v1_armstrng(B,A)
& v2_finsub_1(B) )
=> ( B = k9_armstrng(A,k10_armstrng(A,B))
& ! [C] :
( ( v6_armstrng(C,A)
& m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ( B = k9_armstrng(A,C)
=> C = k10_armstrng(A,B) ) ) ) ) ) ) ).
fof(t38_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v1_armstrng(k11_armstrng(A,B),A)
& v2_finsub_1(k11_armstrng(A,B)) ) ) ) ).
fof(t39_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( r1_tarski(B,k10_armstrng(A,k11_armstrng(A,B)))
& ! [C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( r1_tarski(B,C)
& v6_armstrng(C,A) )
=> r1_tarski(k10_armstrng(A,k11_armstrng(A,B)),C) ) ) ) ) ) ).
fof(d25_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C] :
( ( v6_armstrng(C,A)
& m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ( C = k12_armstrng(A,B)
<=> ( r1_tarski(B,C)
& ! [D] :
( m2_relset_1(D,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( r1_tarski(B,D)
& v6_armstrng(D,A) )
=> r1_tarski(C,D) ) ) ) ) ) ) ) ).
fof(t40_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> k12_armstrng(A,B) = k10_armstrng(A,k11_armstrng(A,B)) ) ) ).
fof(t44_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> r5_armstrng(A,B,k9_armstrng(A,k10_armstrng(A,B))) ) ) ).
fof(t45_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v6_armstrng(B,A)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
& r5_armstrng(A,C,k9_armstrng(A,B))
& B = k10_armstrng(A,C) ) ) ) ).
fof(t46_armstrng,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( v1_armstrng(B,A)
& v2_finsub_1(B) )
=> r5_armstrng(A,k2_armstrng(B),B) ) ) ).
fof(t47_armstrng,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( ( v1_armstrng(C,A)
& v2_finsub_1(C)
& r5_armstrng(A,B,C) )
=> r1_tarski(k2_armstrng(C),k2_xboole_0(B,k1_tarski(A))) ) ) ).
fof(t48_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( ( v6_armstrng(B,A)
& m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ? [C] :
( l1_armstrng(C)
& u1_armstrng(C) = A
& ! [D] :
( m1_subset_1(D,A)
=> k1_funct_1(u2_armstrng(C),D) = k4_numbers )
& B = k5_armstrng(C) ) ) ) ).
fof(d28_armstrng,axiom,
! [A] :
( v11_armstrng(A)
<=> ! [B,C] :
( ( r2_hidden(B,A)
& r2_hidden(C,A)
& r1_tarski(B,C) )
=> B = C ) ) ).
fof(t50_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> ( ~ v1_xboole_0(k13_armstrng(u1_armstrng(A),k5_armstrng(A)))
& v11_armstrng(k13_armstrng(u1_armstrng(A),k5_armstrng(A))) ) ) ).
fof(t51_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ~ ( ~ v1_xboole_0(B)
& v11_armstrng(B)
& ! [C] :
( ( v6_armstrng(C,A)
& m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> B != k13_armstrng(A,C) ) ) ) ) ).
fof(t53_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ~ ( ~ v1_xboole_0(B)
& v11_armstrng(B)
& ! [C] :
( l1_armstrng(C)
=> ~ ( u1_armstrng(C) = A
& B = k13_armstrng(u1_armstrng(C),k5_armstrng(C)) ) ) ) ) ) ).
fof(d29_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v12_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( ( r2_hidden(k6_armstrng(A,C,D),B)
& r2_hidden(k6_armstrng(A,C,E),B) )
=> r2_hidden(k6_armstrng(A,C,k4_subset_1(A,D,E)),B) ) ) ) ) ) ) ).
fof(d30_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v13_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
=> ( ( r2_hidden(k6_armstrng(A,C,D),B)
& r2_hidden(k6_armstrng(A,k4_subset_1(A,D,E),F),B) )
=> r2_hidden(k6_armstrng(A,k4_subset_1(A,C,E),F),B) ) ) ) ) ) ) ) ).
fof(d31_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( v14_armstrng(B,A)
<=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( r2_hidden(k6_armstrng(A,C,D),B)
=> r2_hidden(k6_armstrng(A,k4_subset_1(A,C,E),D),B) ) ) ) ) ) ) ).
fof(t54_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( ( v3_armstrng(B,A)
& v8_relat_2(B)
& v4_armstrng(B,A)
& v5_armstrng(B,A) )
=> ( v8_relat_2(B)
& v7_armstrng(B,A)
& v5_armstrng(B,A) ) )
& ( ( v8_relat_2(B)
& v7_armstrng(B,A)
& v5_armstrng(B,A) )
=> ( v3_armstrng(B,A)
& v8_relat_2(B)
& v4_armstrng(B,A)
& v5_armstrng(B,A) ) ) ) ) ).
fof(t55_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( ( v3_armstrng(B,A)
& v8_relat_2(B)
& v4_armstrng(B,A)
& v5_armstrng(B,A) )
=> ( v8_relat_2(B)
& v7_armstrng(B,A)
& v12_armstrng(B,A) ) )
& ( ( v8_relat_2(B)
& v7_armstrng(B,A)
& v12_armstrng(B,A) )
=> ( v3_armstrng(B,A)
& v8_relat_2(B)
& v4_armstrng(B,A)
& v5_armstrng(B,A) ) ) ) ) ).
fof(t56_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( ( v3_armstrng(B,A)
& v8_relat_2(B)
& v4_armstrng(B,A)
& v5_armstrng(B,A) )
=> ( v3_armstrng(B,A)
& v13_armstrng(B,A)
& v14_armstrng(B,A) ) )
& ( ( v3_armstrng(B,A)
& v13_armstrng(B,A)
& v14_armstrng(B,A) )
=> ( v3_armstrng(B,A)
& v8_relat_2(B)
& v4_armstrng(B,A)
& v5_armstrng(B,A) ) ) ) ) ).
fof(t57_armstrng,axiom,
! [A,B,C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( r2_hidden(B,k14_armstrng(A,C))
=> ( m1_subset_1(B,k1_zfmisc_1(A))
& ? [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
& r2_hidden(k6_armstrng(A,D,E),C)
& r1_tarski(D,B)
& ~ r1_tarski(E,B) ) ) ) ) ) ).
fof(t58_armstrng,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ? [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
& r2_hidden(k6_armstrng(A,D,E),C)
& r1_tarski(D,B)
& ~ r1_tarski(E,B) ) )
=> r2_hidden(B,k14_armstrng(A,C)) ) ) ) ).
fof(t59_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(k6_armstrng(A,C,D),k12_armstrng(A,B))
<=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( r1_tarski(C,E)
=> ( r1_tarski(D,E)
| r2_hidden(E,k14_armstrng(A,B)) ) ) ) ) ) )
& k9_armstrng(A,k12_armstrng(A,B)) = k4_xboole_0(k1_zfmisc_1(A),k14_armstrng(A,B)) ) ) ) ).
fof(t60_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( k14_armstrng(A,B) = k14_armstrng(A,C)
=> k12_armstrng(A,B) = k12_armstrng(A,C) ) ) ) ) ).
fof(t61_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> k14_armstrng(A,B) = k14_armstrng(A,k12_armstrng(A,B)) ) ) ).
fof(d33_armstrng,axiom,
! [A,B,C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( r6_armstrng(A,B,C)
<=> ( ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r1_tarski(B,D)
=> ( D = A
| r2_hidden(D,k14_armstrng(A,C)) ) ) )
& ! [D] :
~ ( r2_xboole_0(D,B)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ~ ( r1_tarski(D,E)
& E != A
& ~ r2_hidden(E,k14_armstrng(A,C)) ) ) ) ) ) ) ).
fof(t62_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,k13_armstrng(A,k12_armstrng(A,B)))
<=> r6_armstrng(A,C,B) ) ) ) ) ).
fof(s1_armstrng,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(f1_s1_armstrng))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(f1_s1_armstrng))
=> ( ( ! [C] :
( r2_hidden(C,A)
<=> p1_s1_armstrng(C) )
& ! [C] :
( r2_hidden(C,B)
<=> p1_s1_armstrng(C) ) )
=> A = B ) ) ) ).
fof(s2_armstrng,axiom,
( ( ! [A] :
( r1_armstrng(A,f1_s2_armstrng)
=> p1_s2_armstrng(A) )
& ! [A,B] :
( ( r2_hidden(A,f1_s2_armstrng)
& r2_hidden(B,f1_s2_armstrng)
& p1_s2_armstrng(A)
& p1_s2_armstrng(B) )
=> p1_s2_armstrng(k3_xboole_0(A,B)) ) )
=> ! [A] :
( r2_hidden(A,f1_s2_armstrng)
=> p1_s2_armstrng(A) ) ) ).
fof(dt_m1_armstrng,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ! [C] :
( m1_armstrng(C,A,B)
=> m1_subset_1(C,k4_armstrng(A)) ) ) ).
fof(existence_m1_armstrng,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ? [C] : m1_armstrng(C,A,B) ) ).
fof(redefinition_m1_armstrng,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ! [C] :
( m1_armstrng(C,A,B)
<=> m1_subset_1(C,B) ) ) ).
fof(dt_l1_armstrng,axiom,
$true ).
fof(existence_l1_armstrng,axiom,
? [A] : l1_armstrng(A) ).
fof(abstractness_v2_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> ( v2_armstrng(A)
=> A = g1_armstrng(u1_armstrng(A),u2_armstrng(A),u3_armstrng(A)) ) ) ).
fof(reflexivity_r3_armstrng,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k4_armstrng(A))
& m1_subset_1(C,k4_armstrng(A)) )
=> r3_armstrng(A,B,B) ) ).
fof(dt_k1_armstrng,axiom,
! [A,B] :
( v1_relat_1(B)
=> m1_subset_1(k1_armstrng(A,B),k1_zfmisc_1(A)) ) ).
fof(dt_k2_armstrng,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> m1_subset_1(k2_armstrng(A),k1_zfmisc_1(A)) ) ).
fof(dt_k3_armstrng,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k6_margrel1))
& m1_subset_1(C,k4_finseq_2(A,k6_margrel1)) )
=> m2_finseq_2(k3_armstrng(A,B,C),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) ).
fof(commutativity_k3_armstrng,axiom,
! [A,B,C] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k4_finseq_2(A,k6_margrel1))
& m1_subset_1(C,k4_finseq_2(A,k6_margrel1)) )
=> k3_armstrng(A,B,C) = k3_armstrng(A,C,B) ) ).
fof(dt_k4_armstrng,axiom,
! [A] : m2_relset_1(k4_armstrng(A),k1_zfmisc_1(A),k1_zfmisc_1(A)) ).
fof(dt_k5_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> m2_relset_1(k5_armstrng(A),k1_zfmisc_1(u1_armstrng(A)),k1_zfmisc_1(u1_armstrng(A))) ) ).
fof(dt_k6_armstrng,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> m1_subset_1(k6_armstrng(A,B,C),k4_armstrng(A)) ) ).
fof(redefinition_k6_armstrng,axiom,
! [A,B,C] :
( ( m1_subset_1(B,k1_zfmisc_1(A))
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> k6_armstrng(A,B,C) = k4_tarski(B,C) ) ).
fof(dt_k7_armstrng,axiom,
! [A] : m2_relset_1(k7_armstrng(A),k4_armstrng(A),k4_armstrng(A)) ).
fof(dt_k8_armstrng,axiom,
! [A,B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> m2_relset_1(k8_armstrng(A,B),k1_zfmisc_1(A),k1_zfmisc_1(A)) ) ).
fof(dt_k9_armstrng,axiom,
! [A,B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> m1_subset_1(k9_armstrng(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(dt_k10_armstrng,axiom,
! [A,B] : m2_relset_1(k10_armstrng(A,B),k1_zfmisc_1(A),k1_zfmisc_1(A)) ).
fof(dt_k11_armstrng,axiom,
! [A,B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> m1_subset_1(k11_armstrng(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(dt_k12_armstrng,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ( v6_armstrng(k12_armstrng(A,B),A)
& m2_relset_1(k12_armstrng(A,B),k1_zfmisc_1(A),k1_zfmisc_1(A)) ) ) ).
fof(dt_k13_armstrng,axiom,
! [A,B] :
( m1_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> m1_subset_1(k13_armstrng(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(dt_k14_armstrng,axiom,
$true ).
fof(dt_u1_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> ( ~ v1_xboole_0(u1_armstrng(A))
& v1_finset_1(u1_armstrng(A)) ) ) ).
fof(dt_u2_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> ( v2_relat_1(u2_armstrng(A))
& m1_pboole(u2_armstrng(A),u1_armstrng(A)) ) ) ).
fof(dt_u3_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> m1_subset_1(u3_armstrng(A),k1_zfmisc_1(k4_card_3(u2_armstrng(A)))) ) ).
fof(dt_g1_armstrng,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k4_card_3(B))) )
=> ( v2_armstrng(g1_armstrng(A,B,C))
& l1_armstrng(g1_armstrng(A,B,C)) ) ) ).
fof(free_g1_armstrng,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A)
& v2_relat_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,k1_zfmisc_1(k4_card_3(B))) )
=> ! [D,E,F] :
( g1_armstrng(A,B,C) = g1_armstrng(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
fof(d9_armstrng,axiom,
! [A] :
( l1_armstrng(A)
=> k5_armstrng(A) = a_1_0_armstrng(A) ) ).
fof(d11_armstrng,axiom,
! [A] : k7_armstrng(A) = a_1_1_armstrng(A) ).
fof(t26_armstrng,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v6_armstrng(a_2_0_armstrng(A,B),A)
& m2_relset_1(a_2_0_armstrng(A,B),k1_zfmisc_1(A),k1_zfmisc_1(A)) ) ) ).
fof(t31_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C] :
( m2_relset_1(C,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ( ( v8_armstrng(B,A)
& v9_armstrng(B,A)
& v10_armstrng(B,A)
& C = a_2_1_armstrng(A,B) )
=> ( B = k8_armstrng(A,C)
& v6_armstrng(C,A)
& ! [D] :
( ( v6_armstrng(D,A)
& m2_relset_1(D,k1_zfmisc_1(A),k1_zfmisc_1(A)) )
=> ( B = k8_armstrng(A,D)
=> D = C ) ) ) ) ) ) ) ).
fof(t32_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( B = a_2_2_armstrng(A,C)
=> k2_xboole_0(k1_tarski(k4_tarski(C,A)),a_2_3_armstrng(A,C)) = k8_armstrng(A,B) ) ) ) ) ).
fof(d22_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> k9_armstrng(A,B) = a_2_4_armstrng(A,B) ) ).
fof(d23_armstrng,axiom,
! [A,B] : k10_armstrng(A,B) = a_2_5_armstrng(A,B) ).
fof(d24_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> k11_armstrng(A,B) = a_2_6_armstrng(A,B) ) ).
fof(t41_armstrng,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( C = k2_xboole_0(k1_tarski(A),a_2_7_armstrng(A,B))
=> ( v1_armstrng(C,A)
& v2_finsub_1(C) ) ) ) ) ).
fof(t42_armstrng,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_finset_1(A) )
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( B = a_2_8_armstrng(A,C)
=> k2_xboole_0(k1_tarski(A),a_2_9_armstrng(A,C)) = k9_armstrng(A,B) ) ) ) ) ).
fof(t43_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( B = a_2_2_armstrng(A,C)
=> k2_xboole_0(k1_tarski(A),a_2_10_armstrng(A,C)) = k9_armstrng(A,B) ) ) ) ) ).
fof(d26_armstrng,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ( r5_armstrng(A,B,C)
<=> ( r1_tarski(B,C)
& C = a_2_11_armstrng(A,B) ) ) ) ).
fof(d27_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> k13_armstrng(A,B) = a_2_12_armstrng(A,B) ) ).
fof(t49_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( B = a_2_2_armstrng(A,C)
=> k13_armstrng(A,B) = k1_tarski(C) ) ) ) ) ).
fof(t52_armstrng,axiom,
! [A] :
( v1_finset_1(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
=> ! [C] :
( ( v11_armstrng(B)
& C = a_2_13_armstrng(A,B) )
=> ( v1_xboole_0(B)
| B = k13_armstrng(A,k10_armstrng(A,C)) ) ) ) ) ).
fof(d32_armstrng,axiom,
! [A,B] :
( m2_relset_1(B,k1_zfmisc_1(A),k1_zfmisc_1(A))
=> k14_armstrng(A,B) = a_2_14_armstrng(A,B) ) ).
fof(fraenkel_a_1_0_armstrng,axiom,
! [A,B] :
( l1_armstrng(B)
=> ( r2_hidden(A,a_1_0_armstrng(B))
<=> ? [C,D] :
( m1_subset_1(C,k1_zfmisc_1(u1_armstrng(B)))
& m1_subset_1(D,k1_zfmisc_1(u1_armstrng(B)))
& A = k4_tarski(C,D)
& r2_armstrng(B,C,D) ) ) ) ).
fof(fraenkel_a_1_1_armstrng,axiom,
! [A,B] :
( r2_hidden(A,a_1_1_armstrng(B))
<=> ? [C,D] :
( m1_subset_1(C,k4_armstrng(B))
& m1_subset_1(D,k4_armstrng(B))
& A = k4_tarski(C,D)
& r3_armstrng(B,D,C) ) ) ).
fof(fraenkel_a_2_0_armstrng,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ( r2_hidden(A,a_2_0_armstrng(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k1_zfmisc_1(B))
& A = k6_armstrng(B,D,E)
& ( r1_tarski(C,D)
| r1_tarski(E,D) ) ) ) ) ).
fof(fraenkel_a_2_1_armstrng,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m2_relset_1(C,k1_zfmisc_1(B),k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_1_armstrng(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k1_zfmisc_1(B))
& A = k6_armstrng(B,D,E)
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
& ? [G] :
( m1_subset_1(G,k1_zfmisc_1(B))
& r3_armstrng(B,k6_armstrng(B,F,G),k6_armstrng(B,D,E))
& r2_hidden(k6_armstrng(B,F,G),C) ) ) ) ) ) ).
fof(fraenkel_a_2_2_armstrng,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_2_armstrng(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k1_zfmisc_1(B))
& A = k6_armstrng(B,D,E)
& ( r1_tarski(C,D)
| r1_tarski(E,D) ) ) ) ) ).
fof(fraenkel_a_2_3_armstrng,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_3_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = k6_armstrng(B,D,D)
& ~ r1_tarski(C,D) ) ) ) ).
fof(fraenkel_a_2_4_armstrng,axiom,
! [A,B,C] :
( m2_relset_1(C,k1_zfmisc_1(B),k1_zfmisc_1(B))
=> ( r2_hidden(A,a_2_4_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
& r4_armstrng(B,C,E,D) ) ) ) ) ).
fof(fraenkel_a_2_5_armstrng,axiom,
! [A,B,C] :
( r2_hidden(A,a_2_5_armstrng(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k1_zfmisc_1(B))
& A = k6_armstrng(B,D,E)
& ! [F] :
( ( r2_hidden(F,C)
& r1_tarski(D,F) )
=> r1_tarski(E,F) ) ) ) ).
fof(fraenkel_a_2_6_armstrng,axiom,
! [A,B,C] :
( m2_relset_1(C,k1_zfmisc_1(B),k1_zfmisc_1(B))
=> ( r2_hidden(A,a_2_6_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
=> ( ( r2_hidden(k6_armstrng(B,E,F),C)
& r1_tarski(E,D) )
=> r1_tarski(F,D) ) ) ) ) ) ) ).
fof(fraenkel_a_2_7_armstrng,axiom,
! [A,B,C] :
( m1_subset_1(C,k1_zfmisc_1(B))
=> ( r2_hidden(A,a_2_7_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& ~ r1_tarski(C,D) ) ) ) ).
fof(fraenkel_a_2_8_armstrng,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_8_armstrng(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k1_zfmisc_1(B))
& m1_subset_1(E,k1_zfmisc_1(B))
& A = k6_armstrng(B,D,E)
& ( r1_tarski(C,D)
| r1_tarski(E,D) ) ) ) ) ).
fof(fraenkel_a_2_9_armstrng,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_finset_1(B)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_9_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& ~ r1_tarski(C,D) ) ) ) ).
fof(fraenkel_a_2_10_armstrng,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_subset_1(C,k1_zfmisc_1(B)) )
=> ( r2_hidden(A,a_2_10_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& ~ r1_tarski(C,D) ) ) ) ).
fof(fraenkel_a_2_11_armstrng,axiom,
! [A,B,C] :
( r2_hidden(A,a_2_11_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(B)))
& A = k8_setfam_1(B,D)
& r1_tarski(D,C) ) ) ).
fof(fraenkel_a_2_12_armstrng,axiom,
! [A,B,C] :
( m2_relset_1(C,k1_zfmisc_1(B),k1_zfmisc_1(B))
=> ( r2_hidden(A,a_2_12_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& r2_hidden(k4_tarski(D,B),k8_armstrng(B,C)) ) ) ) ).
fof(fraenkel_a_2_13_armstrng,axiom,
! [A,B,C] :
( ( v1_finset_1(B)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
=> ( r2_hidden(A,a_2_13_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
=> ~ ( r2_hidden(E,C)
& r1_tarski(E,D) ) ) ) ) ) ).
fof(fraenkel_a_2_14_armstrng,axiom,
! [A,B,C] :
( m2_relset_1(C,k1_zfmisc_1(B),k1_zfmisc_1(B))
=> ( r2_hidden(A,a_2_14_armstrng(B,C))
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(B))
& A = D
& ? [E] :
( m1_subset_1(E,k1_zfmisc_1(B))
& ? [F] :
( m1_subset_1(F,k1_zfmisc_1(B))
& r2_hidden(k6_armstrng(B,E,F),C)
& r1_tarski(E,D)
& ~ r1_tarski(F,D) ) ) ) ) ) ).
%------------------------------------------------------------------------------