SET007 Axioms: SET007+459.ax
%------------------------------------------------------------------------------
% File : SET007+459 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Correspondence Between Signatures and Graphs. Part I
% Version : [Urb08] axioms.
% English : Correspondence Between Monotonic Many Sorted Signatures and
% Well-Founded Graphs. Part I
% 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 : msscyc_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 51 ( 1 unt; 0 def)
% Number of atoms : 399 ( 30 equ)
% Maximal formula atoms : 16 ( 7 avg)
% Number of connectives : 419 ( 71 ~; 1 |; 208 &)
% ( 9 <=>; 130 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 49 ( 47 usr; 1 prp; 0-4 aty)
% Number of functors : 46 ( 46 usr; 5 con; 0-4 aty)
% Number of variables : 135 ( 121 !; 14 ?)
% 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_msscyc_1,axiom,
? [A] :
( l1_graph_1(A)
& v2_graph_1(A)
& v2_msscyc_1(A) ) ).
fof(rc2_msscyc_1,axiom,
? [A] :
( l1_graph_1(A)
& v1_graph_1(A)
& v2_graph_1(A)
& v5_graph_1(A)
& v6_graph_1(A)
& v7_graph_1(A)
& ~ v2_msscyc_1(A) ) ).
fof(fc1_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ~ v1_xboole_0(u2_graph_1(A)) ) ).
fof(rc3_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ? [B] :
( m1_graph_1(B,A)
& ~ v1_xboole_0(B)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_funct_1(B)
& v1_finset_1(B)
& v1_finseq_1(B)
& v8_graph_1(B,A) ) ) ).
fof(fc2_msscyc_1,axiom,
! [A,B] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A)
& ~ v1_xboole_0(B)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> ( ~ v1_xboole_0(k7_graph_2(A,B))
& v1_relat_1(k7_graph_2(A,B))
& v1_funct_1(k7_graph_2(A,B))
& v1_finset_1(k7_graph_2(A,B))
& v1_finseq_1(k7_graph_2(A,B)) ) ) ).
fof(cc1_msscyc_1,axiom,
! [A] :
( l1_graph_1(A)
=> ( ( v2_graph_1(A)
& v2_msscyc_1(A) )
=> ( v2_graph_1(A)
& v3_msscyc_1(A) ) ) ) ).
fof(cc2_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& v2_msscyc_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ( v1_xboole_0(B)
& v1_relat_1(B)
& v2_funct_1(B)
& v1_finset_1(B)
& v8_graph_1(B,A) ) ) ) ).
fof(cc3_msscyc_1,axiom,
! [A] :
( l1_graph_1(A)
=> ( ( v2_graph_1(A)
& v2_msscyc_1(A) )
=> ( v2_graph_1(A)
& v4_msscyc_1(A) ) ) ) ).
fof(cc4_msscyc_1,axiom,
! [A] :
( l1_graph_1(A)
=> ( ( v2_graph_1(A)
& ~ v4_msscyc_1(A) )
=> ( v2_graph_1(A)
& ~ v2_msscyc_1(A) ) ) ) ).
fof(rc4_msscyc_1,axiom,
? [A] :
( l1_graph_1(A)
& v2_graph_1(A)
& v4_msscyc_1(A) ) ).
fof(cc5_msscyc_1,axiom,
! [A] :
( l1_graph_1(A)
=> ( ( v2_graph_1(A)
& v4_msscyc_1(A) )
=> ( v2_graph_1(A)
& v3_msscyc_1(A) ) ) ) ).
fof(rc5_msscyc_1,axiom,
? [A] :
( l1_graph_1(A)
& v2_graph_1(A)
& ~ v2_msscyc_1(A)
& ~ v4_msscyc_1(A) ) ).
fof(rc6_msscyc_1,axiom,
? [A] :
( l1_graph_1(A)
& v2_graph_1(A)
& v3_msscyc_1(A) ) ).
fof(rc7_msscyc_1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v5_msualg_1(B,A)
& v3_msafree2(B,A)
& l3_msualg_1(B,A) )
=> ? [C] :
( m1_msafree(C,A,B)
& v1_relat_1(C)
& v2_relat_1(C)
& ~ v3_relat_1(C)
& v1_funct_1(C)
& v1_pre_circ(C,u1_struct_0(A)) ) ) ).
fof(fc3_msscyc_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A)
& v2_relat_1(B)
& v1_pre_circ(B,u1_struct_0(A))
& m1_pboole(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> ( ~ v1_xboole_0(k12_msafree(A,B,C))
& v1_finset_1(k12_msafree(A,B,C)) ) ) ).
fof(t1_msscyc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_finset_1(A) )
=> ( ! [B] :
( r2_hidden(B,k1_relat_1(A))
=> v1_finset_1(k1_funct_1(A,B)) )
=> v1_finset_1(k4_card_3(A)) ) ) ).
fof(d1_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ( m1_graph_1(B,A)
<=> ( m2_finseq_1(B,u2_graph_1(A))
& ? [C] :
( m2_finseq_1(C,u1_graph_1(A))
& r1_graph_2(A,C,B) ) ) ) ) ) ).
fof(t2_msscyc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B)
& v1_finseq_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ( r1_xreal_0(A,k3_finseq_1(B))
=> k1_graph_2(B,np__1,A) = k1_graph_2(k7_finseq_1(B,C),np__1,A) ) ) ) ) ).
fof(d2_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ( v1_msscyc_1(B,A)
<=> ? [C] :
( m2_finseq_1(C,u1_graph_1(A))
& r1_graph_2(A,C,B)
& k1_funct_1(C,np__1) = k1_funct_1(C,k3_finseq_1(C)) ) ) ) ) ).
fof(d3_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v2_msscyc_1(A)
<=> v1_xboole_0(u2_graph_1(A)) ) ) ).
fof(t3_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> r1_tarski(k2_xboole_0(k2_relat_1(u3_graph_1(A)),k2_relat_1(u4_graph_1(A))),u1_graph_1(A)) ) ).
fof(t4_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B,C] :
( m1_subset_1(C,u1_graph_1(A))
=> ! [D] :
( m1_subset_1(D,u1_graph_1(A))
=> ( ( C = k1_funct_1(u3_graph_1(A),B)
& D = k1_funct_1(u4_graph_1(A),B) )
=> r1_graph_2(A,k4_lang1(u1_graph_1(A),C,D),k9_finseq_1(B)) ) ) ) ) ).
fof(t5_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( r2_hidden(B,u2_graph_1(A))
=> ( v8_graph_1(k9_finseq_1(B),A)
& m1_graph_1(k9_finseq_1(B),A) ) ) ) ).
fof(t6_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( m1_graph_1(B,A)
=> ! [C] :
( m2_finseq_1(C,u1_graph_1(A))
=> ( ( v1_msscyc_1(B,A)
& r1_graph_2(A,C,B) )
=> k1_funct_1(C,np__1) = k1_funct_1(C,k3_finseq_1(C)) ) ) ) ) ).
fof(t7_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ! [B] :
( r2_hidden(B,u2_graph_1(A))
=> ! [C] :
( ( v8_graph_1(C,A)
& m1_graph_1(C,A) )
=> ( C = k9_finseq_1(B)
=> k7_graph_2(A,C) = k10_finseq_1(k1_funct_1(u3_graph_1(A),B),k1_funct_1(u4_graph_1(A),B)) ) ) ) ) ).
fof(t8_msscyc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> r1_xreal_0(k3_finseq_1(k1_graph_2(C,A,B)),k3_finseq_1(C)) ) ) ) ).
fof(t9_msscyc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v2_graph_1(C)
& ~ v2_msscyc_1(C)
& l1_graph_1(C) )
=> ! [D] :
( ( v8_graph_1(D,C)
& m1_graph_1(D,C) )
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,B)
& r1_xreal_0(B,k3_finseq_1(D)) )
=> ( v8_graph_1(k1_graph_2(D,A,B),C)
& m1_graph_1(k1_graph_2(D,A,B),C) ) ) ) ) ) ) ).
fof(t10_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> k3_finseq_1(k7_graph_2(A,B)) = k1_nat_1(k3_finseq_1(B),np__1) ) ) ).
fof(t11_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(C,k3_finseq_1(B)) )
=> ( k1_funct_1(k7_graph_2(A,B),C) = k1_funct_1(u3_graph_1(A),k1_funct_1(B,C))
& k1_funct_1(k7_graph_2(A,B),k1_nat_1(C,np__1)) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,C)) ) ) ) ) ) ).
fof(t12_msscyc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v1_relat_1(C)
& v1_funct_1(C)
& v1_finseq_1(C) )
=> ~ ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,B)
& r1_xreal_0(B,k3_finseq_1(C))
& v1_xboole_0(k1_graph_2(C,A,B)) ) ) ) ) ).
fof(t13_msscyc_1,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v2_graph_1(C)
& ~ v2_msscyc_1(C)
& l1_graph_1(C) )
=> ! [D] :
( ( v8_graph_1(D,C)
& m1_graph_1(D,C) )
=> ! [E] :
( ( v8_graph_1(E,C)
& m1_graph_1(E,C) )
=> ( ( r1_xreal_0(np__1,A)
& r1_xreal_0(A,B)
& r1_xreal_0(B,k3_finseq_1(D))
& E = k1_graph_2(D,A,B) )
=> k7_graph_2(C,E) = k2_graph_2(u1_graph_1(C),k7_graph_2(C,D),A,k1_nat_1(B,np__1)) ) ) ) ) ) ) ).
fof(t14_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> k1_funct_1(k7_graph_2(A,B),k1_nat_1(k3_finseq_1(B),np__1)) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,k3_finseq_1(B))) ) ) ).
fof(t15_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v8_graph_1(C,A)
& m1_graph_1(C,A) )
=> ( k1_funct_1(k7_graph_2(A,B),k1_nat_1(k3_finseq_1(B),np__1)) = k1_funct_1(k7_graph_2(A,C),np__1)
<=> ( ~ v1_xboole_0(k7_finseq_1(B,C))
& v8_graph_1(k7_finseq_1(B,C),A)
& m1_graph_1(k7_finseq_1(B,C),A) ) ) ) ) ) ).
fof(t16_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> ! [C] :
( ( ~ v1_xboole_0(C)
& v8_graph_1(C,A)
& m1_graph_1(C,A) )
=> ! [D] :
( ( ~ v1_xboole_0(D)
& v8_graph_1(D,A)
& m1_graph_1(D,A) )
=> ( B = k7_finseq_1(C,D)
=> ( k1_funct_1(k7_graph_2(A,B),np__1) = k1_funct_1(k7_graph_2(A,C),np__1)
& k1_funct_1(k7_graph_2(A,B),k1_nat_1(k3_finseq_1(B),np__1)) = k1_funct_1(k7_graph_2(A,D),k1_nat_1(k3_finseq_1(D),np__1)) ) ) ) ) ) ) ).
fof(t17_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> ( v1_msscyc_1(B,A)
=> k1_funct_1(k7_graph_2(A,B),np__1) = k1_funct_1(k7_graph_2(A,B),k1_nat_1(k3_finseq_1(B),np__1)) ) ) ) ).
fof(t18_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& ~ v2_msscyc_1(A)
& l1_graph_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> ( v1_msscyc_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ? [D] :
( v8_graph_1(D,A)
& m1_graph_1(D,A)
& k3_finseq_1(D) = C
& ~ v1_xboole_0(k7_finseq_1(D,B))
& v8_graph_1(k7_finseq_1(D,B),A)
& m1_graph_1(k7_finseq_1(D,B),A) ) ) ) ) ) ).
fof(d4_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v3_msscyc_1(A)
<=> ! [B] :
( ( v8_graph_1(B,A)
& m1_graph_1(B,A) )
=> ~ ( ~ v1_xboole_0(B)
& v1_msscyc_1(B,A) ) ) ) ) ).
fof(d5_msscyc_1,axiom,
! [A] :
( ( v2_graph_1(A)
& l1_graph_1(A) )
=> ( v4_msscyc_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_graph_1(A))
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& ! [D] :
( ( v8_graph_1(D,A)
& m1_graph_1(D,A) )
=> ( k1_funct_1(k7_graph_2(A,D),k1_nat_1(k3_finseq_1(D),np__1)) = B
=> ( v1_xboole_0(D)
| r1_xreal_0(k3_finseq_1(D),C) ) ) ) ) ) ) ) ).
fof(t19_msscyc_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v3_trees_2(A) )
=> ! [B] :
( m1_trees_1(B,k1_relat_1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> m1_trees_1(k16_finseq_1(k5_numbers,B,C),k1_relat_1(A)) ) ) ) ).
fof(t20_msscyc_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( m1_dtconstr(C,u1_struct_0(k5_msafree(A,B)),k5_trees_3(u1_struct_0(k5_msafree(A,B))),k1_msaterm(A,B))
=> ~ ( ~ v1_trees_9(C)
& ! [D] :
( m1_subset_1(D,u1_msualg_1(A))
=> k1_funct_1(C,k1_xboole_0) != k4_tarski(D,u1_struct_0(A)) ) ) ) ) ) ).
fof(t21_msscyc_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( l3_msualg_1(B,A)
=> ! [C] :
( m1_msafree(C,A,B)
=> ! [D] :
( m4_pboole(D,u1_struct_0(A),u4_msualg_1(A,B))
=> ( r2_pboole(u1_struct_0(A),C,D)
=> m1_msafree(D,A,B) ) ) ) ) ) ).
fof(t22_msscyc_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_msafree(C,A,B) )
=> ? [D] :
( m3_pboole(D,u1_struct_0(A),u4_msualg_1(A,k11_msafree(A,C)),u4_msualg_1(A,B))
& r2_msualg_3(A,k11_msafree(A,C),B,D) ) ) ) ) ).
fof(t23_msscyc_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( ( v2_relat_1(C)
& m1_msafree(C,A,B) )
=> ~ ( ~ v4_msafree2(B,A)
& v4_msafree2(k11_msafree(A,C),A) ) ) ) ) ).
fof(t24_msscyc_1,axiom,
$true ).
fof(t25_msscyc_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_msualg_1(A))
=> ( k8_funct_2(u1_msualg_1(A),k3_finseq_2(u1_struct_0(A)),u2_msualg_1(A),C) = k1_xboole_0
=> k1_relat_1(k5_msualg_1(A,C,B)) = k1_tarski(k1_xboole_0) ) ) ) ) ).
fof(t26_msscyc_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v5_msualg_1(B,A)
& l3_msualg_1(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( v5_msscyc_1(A)
=> v1_finset_1(k1_msualg_2(A,B,C)) ) ) ) ) ).
fof(t28_msscyc_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(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_msualg_1(A))
=> ! [F] :
( m1_subset_1(F,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),C))
=> ! [G] :
( m1_msaterm(G,A,B,k2_msaterm(A,B,E))
=> ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ! [I] :
( m1_subset_1(I,k1_funct_1(u4_msualg_1(A,k11_msafree(A,B)),D))
=> ~ ( F = k4_trees_4(k4_tarski(E,u1_struct_0(A)),G)
& r2_hidden(H,k4_finseq_1(G))
& I = k1_funct_1(G,H)
& r1_xreal_0(k9_msafree2(A,B,C,F),k9_msafree2(A,B,D,I)) ) ) ) ) ) ) ) ) ) ) ).
fof(d6_msscyc_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ( v5_msscyc_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> v1_finset_1(a_2_0_msscyc_1(A,B)) ) ) ) ).
fof(t27_msscyc_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v2_msualg_1(A)
& l1_msualg_1(A) )
=> ! [B] :
( ( v2_relat_1(B)
& m1_pboole(B,u1_struct_0(A)) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> a_3_0_msscyc_1(A,B,C) = k2_xboole_0(k12_msafree(A,B,C),k1_msualg_2(A,k11_msafree(A,B),C)) ) ) ) ).
fof(fraenkel_a_2_0_msscyc_1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& m1_subset_1(C,u1_struct_0(B)) )
=> ( r2_hidden(A,a_2_0_msscyc_1(B,C))
<=> ? [D] :
( m1_subset_1(D,u1_msualg_1(B))
& A = D
& k2_msualg_1(B,D) = C ) ) ) ).
fof(fraenkel_a_3_0_msscyc_1,axiom,
! [A,B,C,D] :
( ( ~ v3_struct_0(B)
& ~ v2_msualg_1(B)
& l1_msualg_1(B)
& v2_relat_1(C)
& m1_pboole(C,u1_struct_0(B))
& m1_subset_1(D,u1_struct_0(B)) )
=> ( r2_hidden(A,a_3_0_msscyc_1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,k1_funct_1(u4_msualg_1(B,k11_msafree(B,C)),D))
& A = E
& k9_msafree2(B,C,D,E) = np__0 ) ) ) ).
%------------------------------------------------------------------------------