SET007 Axioms: SET007+409.ax
%------------------------------------------------------------------------------
% File : SET007+409 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Formalization of Simple Graphs
% 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 : sgraph1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 97 ( 31 unt; 0 def)
% Number of atoms : 382 ( 64 equ)
% Maximal formula atoms : 21 ( 3 avg)
% Number of connectives : 323 ( 38 ~; 7 |; 134 &)
% ( 24 <=>; 120 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 28 ( 26 usr; 1 prp; 0-5 aty)
% Number of functors : 42 ( 42 usr; 9 con; 0-4 aty)
% Number of variables : 229 ( 201 !; 28 ?)
% 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(fc1_sgraph1,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> v1_finset_1(k1_sgraph1(A,B)) ) ).
fof(cc1_sgraph1,axiom,
! [A] :
( v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( v1_xboole_0(B)
& v1_finset_1(B) ) ) ) ).
fof(rc1_sgraph1,axiom,
? [A] :
( l1_sgraph1(A)
& v1_sgraph1(A) ) ).
fof(fc2_sgraph1,axiom,
! [A] : ~ v1_xboole_0(k3_sgraph1(A)) ).
fof(t1_sgraph1,axiom,
$true ).
fof(t2_sgraph1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( r2_hidden(C,k1_sgraph1(A,B))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& C = D
& r1_xreal_0(A,D)
& r1_xreal_0(D,B) ) ) ) ) ).
fof(t3_sgraph1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k1_sgraph1(A,B))
<=> ( r1_xreal_0(A,C)
& r1_xreal_0(C,B) ) ) ) ) ) ).
fof(t4_sgraph1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k1_sgraph1(np__1,A) = k2_finseq_1(A) ) ).
fof(t5_sgraph1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> r1_tarski(k1_sgraph1(A,B),k2_finseq_1(B)) ) ) ) ).
fof(t6_sgraph1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(B,A)
=> r1_xboole_0(k2_finseq_1(A),k1_sgraph1(B,C)) ) ) ) ) ).
fof(t7_sgraph1,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_sgraph1(A,B) = k1_xboole_0 ) ) ) ).
fof(d2_sgraph1,axiom,
$true ).
fof(d3_sgraph1,axiom,
$true ).
fof(t8_sgraph1,axiom,
$true ).
fof(t9_sgraph1,axiom,
! [A,B] :
( r2_hidden(B,k2_sgraph1(A))
<=> ? [C] :
( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(A))
& B = C
& k4_card_1(C) = np__2 ) ) ).
fof(t10_sgraph1,axiom,
! [A,B] :
( r2_hidden(B,k2_sgraph1(A))
<=> ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(A))
& ? [C,D] :
( r2_hidden(C,A)
& r2_hidden(D,A)
& C != D
& B = k2_tarski(C,D) ) ) ) ).
fof(t11_sgraph1,axiom,
! [A] : r1_tarski(k2_sgraph1(A),k1_zfmisc_1(A)) ).
fof(t12_sgraph1,axiom,
! [A,B,C] :
( r2_hidden(k2_tarski(B,C),k2_sgraph1(A))
=> ( r2_hidden(B,A)
& r2_hidden(C,A)
& B != C ) ) ).
fof(t13_sgraph1,axiom,
k2_sgraph1(k1_xboole_0) = k1_xboole_0 ).
fof(t14_sgraph1,axiom,
! [A,B] :
( r1_tarski(A,B)
=> r1_tarski(k2_sgraph1(A),k2_sgraph1(B)) ) ).
fof(t15_sgraph1,axiom,
! [A] :
( v1_finset_1(A)
=> v1_finset_1(k2_sgraph1(A)) ) ).
fof(t16_sgraph1,axiom,
! [A] :
( ~ v1_realset1(A)
=> ~ v1_xboole_0(k2_sgraph1(A)) ) ).
fof(t17_sgraph1,axiom,
! [A] : k2_sgraph1(k1_tarski(A)) = k1_xboole_0 ).
fof(d5_sgraph1,axiom,
$true ).
fof(t18_sgraph1,axiom,
$true ).
fof(t19_sgraph1,axiom,
! [A] : r2_hidden(g1_sgraph1(k1_xboole_0,k1_subset_1(k2_sgraph1(k1_xboole_0))),k3_sgraph1(A)) ).
fof(d7_sgraph1,axiom,
! [A,B] :
( ( v1_sgraph1(B)
& l1_sgraph1(B) )
=> ( m1_sgraph1(B,A)
<=> m1_subset_1(B,k3_sgraph1(A)) ) ) ).
fof(t20_sgraph1,axiom,
$true ).
fof(t21_sgraph1,axiom,
! [A,B] :
( r2_hidden(B,k3_sgraph1(A))
<=> ? [C] :
( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(A))
& ? [D] :
( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
& B = g1_sgraph1(C,D) ) ) ) ).
fof(t22_sgraph1,axiom,
$true ).
fof(t23_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ( r1_tarski(u1_struct_0(B),A)
& r1_tarski(u1_sgraph1(B),k2_sgraph1(u1_struct_0(B))) ) ) ).
fof(t24_sgraph1,axiom,
$true ).
fof(t25_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C] :
~ ( r2_hidden(C,u1_sgraph1(B))
& ! [D,E] :
~ ( r2_hidden(D,u1_struct_0(B))
& r2_hidden(E,u1_struct_0(B))
& D != E
& C = k2_tarski(D,E) ) ) ) ).
fof(t26_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C,D] :
( r2_hidden(k2_tarski(C,D),u1_sgraph1(B))
=> ( r2_hidden(C,u1_struct_0(B))
& r2_hidden(D,u1_struct_0(B))
& C != D ) ) ) ).
fof(t27_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ( v1_finset_1(u1_struct_0(B))
& m1_subset_1(u1_struct_0(B),k1_zfmisc_1(A))
& v1_finset_1(u1_sgraph1(B))
& m1_subset_1(u1_sgraph1(B),k1_zfmisc_1(k2_sgraph1(u1_struct_0(B)))) ) ) ).
fof(d8_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C] :
( m1_sgraph1(C,A)
=> ( r1_sgraph1(A,B,C)
<=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& m2_relset_1(D,u1_struct_0(B),u1_struct_0(C))
& v3_funct_2(D,u1_struct_0(B),u1_struct_0(C))
& ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( r2_hidden(k2_tarski(E,F),u1_sgraph1(B))
<=> r2_hidden(k2_tarski(k1_funct_1(D,E),k1_funct_1(D,F)),u1_sgraph1(B)) ) ) ) ) ) ) ) ).
fof(t28_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ~ ( B != g1_sgraph1(k1_xboole_0,k1_subset_1(k2_sgraph1(k1_xboole_0)))
& ! [C,D] :
( m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
=> ~ ( ~ v1_xboole_0(C)
& B = g1_sgraph1(C,D) ) ) ) ) ).
fof(t29_sgraph1,axiom,
$true ).
fof(t30_sgraph1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k2_sgraph1(B)))
=> ! [D,E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(k2_sgraph1(k2_xboole_0(B,k1_tarski(D))))) )
=> ( ( r2_hidden(g1_sgraph1(B,C),k3_sgraph1(A))
& r2_hidden(D,A) )
=> ( r2_hidden(D,B)
| r2_hidden(g1_sgraph1(k2_xboole_0(B,k1_tarski(D)),E),k3_sgraph1(A)) ) ) ) ) ) ).
fof(t31_sgraph1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k2_sgraph1(B)))
=> ! [D,E] :
~ ( r2_hidden(D,B)
& r2_hidden(E,B)
& D != E
& r2_hidden(g1_sgraph1(B,C),k3_sgraph1(A))
& ! [F] :
( ( v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(k2_sgraph1(B))) )
=> ~ ( F = k2_xboole_0(C,k1_tarski(k2_tarski(D,E)))
& r2_hidden(g1_sgraph1(B,F),k3_sgraph1(A)) ) ) ) ) ) ).
fof(d9_sgraph1,axiom,
! [A,B] :
( r2_sgraph1(A,B)
<=> ( r2_hidden(g1_sgraph1(k1_xboole_0,k1_subset_1(k2_sgraph1(k1_xboole_0))),B)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
=> ! [E,F] :
( ( v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(k2_sgraph1(k2_xboole_0(C,k1_tarski(E))))) )
=> ( ( r2_hidden(g1_sgraph1(C,D),B)
& r2_hidden(E,A) )
=> ( r2_hidden(E,C)
| r2_hidden(g1_sgraph1(k2_xboole_0(C,k1_tarski(E)),F),B) ) ) ) ) )
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
=> ! [E,F] :
~ ( r2_hidden(g1_sgraph1(C,D),B)
& r2_hidden(E,C)
& r2_hidden(F,C)
& E != F
& ~ r2_hidden(k2_tarski(E,F),D)
& ! [G] :
( ( v1_finset_1(G)
& m1_subset_1(G,k1_zfmisc_1(k2_sgraph1(C))) )
=> ~ ( G = k2_xboole_0(D,k1_tarski(k2_tarski(E,F)))
& r2_hidden(g1_sgraph1(C,G),B) ) ) ) ) ) ) ) ).
fof(t32_sgraph1,axiom,
$true ).
fof(t33_sgraph1,axiom,
$true ).
fof(t34_sgraph1,axiom,
$true ).
fof(t35_sgraph1,axiom,
! [A] : r2_sgraph1(A,k3_sgraph1(A)) ).
fof(t36_sgraph1,axiom,
! [A,B] :
( r2_sgraph1(A,B)
=> r1_tarski(k3_sgraph1(A),B) ) ).
fof(t37_sgraph1,axiom,
! [A] :
( r2_sgraph1(A,k3_sgraph1(A))
& ! [B] :
( r2_sgraph1(A,B)
=> r1_tarski(k3_sgraph1(A),B) ) ) ).
fof(d10_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C] :
( m1_sgraph1(C,A)
=> ( m2_sgraph1(C,A,B)
<=> ( r1_tarski(u1_struct_0(C),u1_struct_0(B))
& r1_tarski(u1_sgraph1(C),u1_sgraph1(B)) ) ) ) ) ).
fof(d11_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C,D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( D = k4_sgraph1(A,B,C)
<=> ? [E] :
( v1_finset_1(E)
& ! [F] :
( r2_hidden(F,E)
<=> ( r2_hidden(F,u1_sgraph1(B))
& r2_hidden(C,F) ) )
& D = k4_card_1(E) ) ) ) ) ).
fof(t38_sgraph1,axiom,
$true ).
fof(t40_sgraph1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_sgraph1(B,A)
=> ! [C] :
~ ( r2_hidden(C,u1_struct_0(B))
& ! [D] :
( v1_finset_1(D)
=> ~ ( D = u1_struct_0(B)
& ~ r1_xreal_0(k4_card_1(D),k4_sgraph1(A,B,C)) ) ) ) ) ) ).
fof(t41_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C,D] :
~ ( r2_hidden(C,u1_struct_0(B))
& r2_hidden(D,u1_sgraph1(B))
& k4_sgraph1(A,B,C) = np__0
& r2_hidden(C,D) ) ) ).
fof(t42_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C,D] :
( v1_finset_1(D)
=> ( ( D = u1_struct_0(B)
& r2_hidden(C,D)
& k1_nat_1(np__1,k4_sgraph1(A,B,C)) = k4_card_1(D) )
=> ! [E] :
( m1_subset_1(E,D)
=> ~ ( C != E
& ! [F] :
~ ( r2_hidden(F,u1_sgraph1(B))
& F = k2_tarski(C,E) ) ) ) ) ) ) ).
fof(d12_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m2_finseq_1(E,u1_struct_0(B))
=> ( r3_sgraph1(A,B,C,D,E)
<=> ( k1_funct_1(E,np__1) = C
& k1_funct_1(E,k3_finseq_1(E)) = D
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,F)
=> ( r1_xreal_0(k3_finseq_1(E),F)
| r2_hidden(k2_tarski(k1_funct_1(E,F),k1_funct_1(E,k1_nat_1(F,np__1))),u1_sgraph1(B)) ) ) )
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ! [G] :
( m2_subset_1(G,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,F)
=> ( r1_xreal_0(k3_finseq_1(E),F)
| r1_xreal_0(G,F)
| r1_xreal_0(k3_finseq_1(E),G)
| ( k1_funct_1(E,F) != k1_funct_1(E,G)
& k2_tarski(k1_funct_1(E,F),k1_funct_1(E,k1_nat_1(F,np__1))) != k2_tarski(k1_funct_1(E,G),k1_funct_1(E,k1_nat_1(G,np__1))) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t43_sgraph1,axiom,
$true ).
fof(t44_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( r2_hidden(E,k5_sgraph1(A,B,C,D))
<=> ? [F] :
( m2_finseq_2(F,u1_struct_0(B),k3_finseq_2(u1_struct_0(B)))
& E = F
& r3_sgraph1(A,B,C,D,F) ) ) ) ) ) ).
fof(t45_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m2_finseq_2(E,u1_struct_0(B),k3_finseq_2(u1_struct_0(B)))
=> ( r3_sgraph1(A,B,C,D,E)
=> r2_hidden(E,k5_sgraph1(A,B,C,D)) ) ) ) ) ) ).
fof(d14_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C] :
( r4_sgraph1(A,B,C)
<=> ? [D] :
( m1_subset_1(D,u1_struct_0(B))
& r2_hidden(C,k5_sgraph1(A,B,D,D)) ) ) ) ).
fof(d15_sgraph1,axiom,
$true ).
fof(d18_sgraph1,axiom,
k8_sgraph1 = k7_sgraph1(np__3) ).
fof(t46_sgraph1,axiom,
? [A] :
( m1_subset_1(A,k1_zfmisc_1(k2_sgraph1(k2_finseq_1(np__3))))
& A = k4_setwiseo(k5_finsub_1(k5_numbers),k3_setwiseo(k5_numbers,np__1,np__2),k3_setwiseo(k5_numbers,np__2,np__3),k3_setwiseo(k5_numbers,np__3,np__1))
& k8_sgraph1 = g1_sgraph1(k2_finseq_1(np__3),A) ) ).
fof(t47_sgraph1,axiom,
( u1_struct_0(k8_sgraph1) = k2_finseq_1(np__3)
& u1_sgraph1(k8_sgraph1) = k4_setwiseo(k5_finsub_1(k5_numbers),k3_setwiseo(k5_numbers,np__1,np__2),k3_setwiseo(k5_numbers,np__2,np__3),k3_setwiseo(k5_numbers,np__3,np__1)) ) ).
fof(t48_sgraph1,axiom,
( r2_hidden(k3_setwiseo(k5_numbers,np__1,np__2),u1_sgraph1(k8_sgraph1))
& r2_hidden(k3_setwiseo(k5_numbers,np__2,np__3),u1_sgraph1(k8_sgraph1))
& r2_hidden(k3_setwiseo(k5_numbers,np__3,np__1),u1_sgraph1(k8_sgraph1)) ) ).
fof(t49_sgraph1,axiom,
r4_sgraph1(k5_numbers,k8_sgraph1,k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k8_finseq_1(k5_numbers,k12_finseq_1(k5_numbers,np__1),k12_finseq_1(k5_numbers,np__2)),k12_finseq_1(k5_numbers,np__3)),k12_finseq_1(k5_numbers,np__1))) ).
fof(s1_sgraph1,axiom,
( ( p1_s1_sgraph1(g1_sgraph1(k1_xboole_0,k1_subset_1(k2_sgraph1(k1_xboole_0))))
& ! [A] :
( m1_sgraph1(A,f1_s1_sgraph1)
=> ! [B] :
( ( r2_hidden(A,k3_sgraph1(f1_s1_sgraph1))
& p1_s1_sgraph1(A)
& r2_hidden(B,f1_s1_sgraph1) )
=> ( r2_hidden(B,u1_struct_0(A))
| p1_s1_sgraph1(g1_sgraph1(k2_xboole_0(u1_struct_0(A),k1_tarski(B)),k1_subset_1(k2_sgraph1(k2_xboole_0(u1_struct_0(A),k1_tarski(B)))))) ) ) )
& ! [A] :
( m1_sgraph1(A,f1_s1_sgraph1)
=> ! [B] :
~ ( p1_s1_sgraph1(A)
& r2_hidden(B,k2_sgraph1(u1_struct_0(A)))
& ~ r2_hidden(B,u1_sgraph1(A))
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k2_sgraph1(u1_struct_0(A))))
=> ~ ( C = k2_xboole_0(u1_sgraph1(A),k1_tarski(B))
& p1_s1_sgraph1(g1_sgraph1(u1_struct_0(A),C)) ) ) ) ) )
=> ! [A] :
( r2_hidden(A,k3_sgraph1(f1_s1_sgraph1))
=> p1_s1_sgraph1(A) ) ) ).
fof(dt_m1_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ( v1_sgraph1(B)
& l1_sgraph1(B) ) ) ).
fof(existence_m1_sgraph1,axiom,
! [A] :
? [B] : m1_sgraph1(B,A) ).
fof(dt_m2_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C] :
( m2_sgraph1(C,A,B)
=> m1_sgraph1(C,A) ) ) ).
fof(existence_m2_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ? [C] : m2_sgraph1(C,A,B) ) ).
fof(dt_l1_sgraph1,axiom,
! [A] :
( l1_sgraph1(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_sgraph1,axiom,
? [A] : l1_sgraph1(A) ).
fof(abstractness_v1_sgraph1,axiom,
! [A] :
( l1_sgraph1(A)
=> ( v1_sgraph1(A)
=> A = g1_sgraph1(u1_struct_0(A),u1_sgraph1(A)) ) ) ).
fof(dt_k1_sgraph1,axiom,
! [A,B] :
( ( v4_ordinal2(A)
& v4_ordinal2(B) )
=> m1_subset_1(k1_sgraph1(A,B),k1_zfmisc_1(k5_numbers)) ) ).
fof(dt_k2_sgraph1,axiom,
$true ).
fof(dt_k3_sgraph1,axiom,
$true ).
fof(dt_k4_sgraph1,axiom,
! [A,B,C] :
( m1_sgraph1(B,A)
=> m2_subset_1(k4_sgraph1(A,B,C),k1_numbers,k5_numbers) ) ).
fof(dt_k5_sgraph1,axiom,
! [A,B,C,D] :
( ( m1_sgraph1(B,A)
& m1_subset_1(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> m1_subset_1(k5_sgraph1(A,B,C,D),k1_zfmisc_1(k3_finseq_2(u1_struct_0(B)))) ) ).
fof(dt_k6_sgraph1,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> m1_sgraph1(k6_sgraph1(A,B),k5_numbers) ) ).
fof(dt_k7_sgraph1,axiom,
! [A] :
( m1_subset_1(A,k5_numbers)
=> m1_sgraph1(k7_sgraph1(A),k5_numbers) ) ).
fof(dt_k8_sgraph1,axiom,
m1_sgraph1(k8_sgraph1,k5_numbers) ).
fof(dt_u1_sgraph1,axiom,
! [A] :
( l1_sgraph1(A)
=> m1_subset_1(u1_sgraph1(A),k1_zfmisc_1(k2_sgraph1(u1_struct_0(A)))) ) ).
fof(dt_g1_sgraph1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k2_sgraph1(A)))
=> ( v1_sgraph1(g1_sgraph1(A,B))
& l1_sgraph1(g1_sgraph1(A,B)) ) ) ).
fof(free_g1_sgraph1,axiom,
! [A,B] :
( m1_subset_1(B,k1_zfmisc_1(k2_sgraph1(A)))
=> ! [C,D] :
( g1_sgraph1(A,B) = g1_sgraph1(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(d1_sgraph1,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> k1_sgraph1(A,B) = a_2_0_sgraph1(A,B) ) ) ).
fof(d4_sgraph1,axiom,
! [A] : k2_sgraph1(A) = a_1_0_sgraph1(A) ).
fof(d6_sgraph1,axiom,
! [A] : k3_sgraph1(A) = a_1_1_sgraph1(A) ).
fof(t39_sgraph1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_sgraph1(B,A)
=> ! [C] :
? [D] :
( v1_finset_1(D)
& D = a_3_0_sgraph1(A,B,C)
& k4_sgraph1(A,B,C) = k4_card_1(D) ) ) ) ).
fof(d13_sgraph1,axiom,
! [A,B] :
( m1_sgraph1(B,A)
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> k5_sgraph1(A,B,C,D) = a_4_0_sgraph1(A,B,C,D) ) ) ) ).
fof(d16_sgraph1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( m1_sgraph1(C,k5_numbers)
=> ( C = k6_sgraph1(A,B)
<=> ? [D] :
( m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(k2_finseq_1(k1_nat_1(B,A)))))
& D = a_2_1_sgraph1(A,B)
& C = g1_sgraph1(k2_finseq_1(k1_nat_1(B,A)),D) ) ) ) ) ) ).
fof(d17_sgraph1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m1_sgraph1(B,k5_numbers)
=> ( B = k7_sgraph1(A)
<=> ? [C] :
( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(k2_sgraph1(k2_finseq_1(A))))
& C = a_1_2_sgraph1(A)
& B = g1_sgraph1(k2_finseq_1(A),C) ) ) ) ) ).
fof(fraenkel_a_2_0_sgraph1,axiom,
! [A,B,C] :
( ( v4_ordinal2(B)
& v4_ordinal2(C) )
=> ( r2_hidden(A,a_2_0_sgraph1(B,C))
<=> ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& A = D
& r1_xreal_0(B,D)
& r1_xreal_0(D,C) ) ) ) ).
fof(fraenkel_a_1_0_sgraph1,axiom,
! [A,B] :
( r2_hidden(A,a_1_0_sgraph1(B))
<=> ? [C] :
( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(B))
& A = C
& k4_card_1(C) = np__2 ) ) ).
fof(fraenkel_a_1_1_sgraph1,axiom,
! [A,B] :
( r2_hidden(A,a_1_1_sgraph1(B))
<=> ? [C,D] :
( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(B))
& v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(k2_sgraph1(C)))
& A = g1_sgraph1(C,D) ) ) ).
fof(fraenkel_a_3_0_sgraph1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& m1_sgraph1(C,B) )
=> ( r2_hidden(A,a_3_0_sgraph1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = E
& r2_hidden(E,u1_struct_0(C))
& r2_hidden(k2_tarski(D,E),u1_sgraph1(C)) ) ) ) ).
fof(fraenkel_a_4_0_sgraph1,axiom,
! [A,B,C,D,E] :
( ( m1_sgraph1(C,B)
& m1_subset_1(D,u1_struct_0(C))
& m1_subset_1(E,u1_struct_0(C)) )
=> ( r2_hidden(A,a_4_0_sgraph1(B,C,D,E))
<=> ? [F] :
( m2_finseq_2(F,u1_struct_0(C),k3_finseq_2(u1_struct_0(C)))
& A = F
& r3_sgraph1(B,C,D,E,F) ) ) ) ).
fof(fraenkel_a_2_1_sgraph1,axiom,
! [A,B,C] :
( ( m2_subset_1(B,k1_numbers,k5_numbers)
& m2_subset_1(C,k1_numbers,k5_numbers) )
=> ( r2_hidden(A,a_2_1_sgraph1(B,C))
<=> ? [D,E] :
( m1_subset_1(D,k5_numbers)
& m1_subset_1(E,k5_numbers)
& A = k3_setwiseo(k5_numbers,D,E)
& r2_hidden(D,k2_finseq_1(C))
& r2_hidden(E,k1_sgraph1(k1_nat_1(C,np__1),k1_nat_1(C,B))) ) ) ) ).
fof(fraenkel_a_1_2_sgraph1,axiom,
! [A,B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(A,a_1_2_sgraph1(B))
<=> ? [C,D] :
( m1_subset_1(C,k5_numbers)
& m1_subset_1(D,k5_numbers)
& A = k3_setwiseo(k5_numbers,C,D)
& r2_hidden(C,k2_finseq_1(B))
& r2_hidden(D,k2_finseq_1(B))
& C != D ) ) ) ).
%------------------------------------------------------------------------------