SET007 Axioms: SET007+656.ax
%------------------------------------------------------------------------------
% File : SET007+656 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : On Segre's Product of Partial Line Spaces
% 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 : pencil_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 80 ( 0 unt; 0 def)
% Number of atoms : 552 ( 41 equ)
% Maximal formula atoms : 17 ( 6 avg)
% Number of connectives : 591 ( 119 ~; 2 |; 254 &)
% ( 30 <=>; 186 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 41 ( 40 usr; 0 prp; 1-3 aty)
% Number of functors : 30 ( 30 usr; 5 con; 0-4 aty)
% Number of variables : 216 ( 192 !; 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(rc1_pencil_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v1_pre_topc(A)
& ~ v3_pencil_1(A)
& ~ v4_pencil_1(A)
& v5_pencil_1(A)
& v6_pencil_1(A)
& ~ v7_pencil_1(A)
& v8_pencil_1(A) ) ).
fof(rc2_pencil_1,axiom,
? [A] :
( l1_pre_topc(A)
& ~ v3_struct_0(A)
& v1_pre_topc(A)
& ~ v3_pencil_1(A)
& ~ v4_pencil_1(A)
& v5_pencil_1(A)
& v6_pencil_1(A)
& v7_pencil_1(A)
& v8_pencil_1(A) ) ).
fof(fc1_pencil_1,axiom,
! [A] :
( ( ~ v3_pencil_1(A)
& l1_pre_topc(A) )
=> ~ v1_xboole_0(u1_pre_topc(A)) ) ).
fof(cc1_pencil_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v11_pencil_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_pralg_1(A) ) ) ).
fof(rc3_pencil_1,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v2_pralg_1(B)
& v11_pencil_1(B) ) ).
fof(rc4_pencil_1,axiom,
? [A] :
( v1_relat_1(A)
& v1_funct_1(A)
& v2_pralg_1(A)
& v11_pencil_1(A) ) ).
fof(cc2_pencil_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v14_pencil_1(A) )
=> ( v1_relat_1(A)
& v4_waybel_3(A) ) ) ).
fof(cc3_pencil_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v15_pencil_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v4_waybel_3(A)
& v2_pralg_1(A)
& v11_pencil_1(A) ) ) ).
fof(cc4_pencil_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v11_pencil_1(A)
& v15_pencil_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v2_pralg_1(A)
& v11_pencil_1(A)
& v12_pencil_1(A) ) ) ).
fof(cc5_pencil_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v11_pencil_1(A)
& v15_pencil_1(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v4_waybel_3(A)
& v2_pralg_1(A)
& v11_pencil_1(A)
& v14_pencil_1(A) ) ) ).
fof(rc5_pencil_1,axiom,
! [A] :
? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& v4_waybel_3(B)
& v2_pralg_1(B)
& v11_pencil_1(B)
& v12_pencil_1(B)
& v14_pencil_1(B)
& v15_pencil_1(B) ) ).
fof(fc2_pencil_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_relat_1(k1_pzfmisc1(A,B))
& v2_relat_1(k1_pzfmisc1(A,B))
& v1_funct_1(k1_pzfmisc1(A,B))
& v1_pre_circ(k1_pzfmisc1(A,B),A)
& v13_pencil_1(k1_pzfmisc1(A,B)) ) ) ).
fof(fc3_pencil_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& m1_pboole(B,A) )
=> ( v1_relat_1(k1_pzfmisc1(A,B))
& v2_relat_1(k1_pzfmisc1(A,B))
& v1_funct_1(k1_pzfmisc1(A,B))
& v1_pre_circ(k1_pzfmisc1(A,B),A)
& v13_pencil_1(k1_pzfmisc1(A,B))
& v16_pencil_1(k1_pzfmisc1(A,B),A) ) ) ).
fof(rc6_pencil_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v4_waybel_3(B)
& v2_pralg_1(B)
& m1_pboole(B,A) )
=> ? [C] :
( m4_pboole(C,A,k12_pralg_1(A,B))
& v1_relat_1(C)
& v2_relat_1(C)
& v1_funct_1(C)
& v13_pencil_1(C)
& v16_pencil_1(C,A) ) ) ).
fof(rc7_pencil_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v2_pralg_1(B)
& v14_pencil_1(B)
& m1_pboole(B,A) )
=> ? [C] :
( m4_pboole(C,A,k12_pralg_1(A,B))
& v1_relat_1(C)
& v2_relat_1(C)
& v1_funct_1(C)
& ~ v13_pencil_1(C)
& v16_pencil_1(C,A) ) ) ).
fof(rc8_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ? [B] :
( m1_pboole(B,A)
& v1_relat_1(B)
& v1_funct_1(B)
& ~ v13_pencil_1(B)
& v16_pencil_1(B,A) ) ) ).
fof(cc6_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( ( ~ v13_pencil_1(B)
& v16_pencil_1(B,A) )
=> v2_relat_1(B) ) ) ) ).
fof(fc4_pencil_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v13_pencil_1(B)
& v16_pencil_1(B,A)
& m1_pboole(B,A) )
=> ( ~ v1_xboole_0(k4_card_3(B))
& v1_fraenkel(k4_card_3(B))
& ~ v1_realset1(k4_card_3(B)) ) ) ).
fof(t1_pencil_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( ( k4_card_3(A) = k4_card_3(B)
& v2_relat_1(A) )
=> v2_relat_1(B) ) ) ) ).
fof(t2_pencil_1,axiom,
! [A] :
( r1_tarski(np__2,k1_card_1(A))
<=> ? [B,C] :
( r2_hidden(B,A)
& r2_hidden(C,A)
& B != C ) ) ).
fof(t3_pencil_1,axiom,
! [A] :
( r1_tarski(np__2,k1_card_1(A))
=> ! [B] :
? [C] :
( r2_hidden(C,A)
& B != C ) ) ).
fof(t4_pencil_1,axiom,
! [A] :
( r1_tarski(np__2,k1_card_1(A))
<=> ~ v1_realset1(A) ) ).
fof(t5_pencil_1,axiom,
! [A] :
( r1_tarski(np__3,k1_card_1(A))
<=> ? [B,C,D] :
( r2_hidden(B,A)
& r2_hidden(C,A)
& r2_hidden(D,A)
& B != C
& B != D
& C != D ) ) ).
fof(t6_pencil_1,axiom,
! [A] :
( r1_tarski(np__3,k1_card_1(A))
=> ! [B,C] :
? [D] :
( r2_hidden(D,A)
& B != D
& C != D ) ) ).
fof(d1_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_pencil_1(A,B,C)
<=> ~ ( B != C
& ! [D] :
( m1_subset_1(D,u1_pre_topc(A))
=> ~ r1_tarski(k2_tarski(B,C),D) ) ) ) ) ) ) ).
fof(d2_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_pencil_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_pre_topc(A))
=> ( r1_tarski(np__2,k1_card_1(k3_xboole_0(C,B)))
=> r1_tarski(C,B) ) ) ) ) ) ).
fof(d3_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v2_pencil_1(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r1_pencil_1(A,C,D) ) ) ) ) ) ) ).
fof(d4_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v3_pencil_1(A)
<=> v1_xboole_0(u1_pre_topc(A)) ) ) ).
fof(d5_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v4_pencil_1(A)
<=> m1_subset_1(u1_struct_0(A),u1_pre_topc(A)) ) ) ).
fof(d6_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v5_pencil_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_pre_topc(A))
=> r1_tarski(np__2,k1_card_1(B)) ) ) ) ).
fof(d7_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v6_pencil_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_pre_topc(A))
=> ! [C] :
( m1_subset_1(C,u1_pre_topc(A))
=> ( r1_tarski(np__2,k1_card_1(k3_xboole_0(B,C)))
=> B = C ) ) ) ) ) ).
fof(d8_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v7_pencil_1(A)
<=> ~ ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> r1_pencil_1(A,B,C) ) ) ) ) ).
fof(d9_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v8_pencil_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_pre_topc(A))
& r2_hidden(B,C) ) ) ) ) ).
fof(d10_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v9_pencil_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ? [D] :
( m2_finseq_1(D,u1_struct_0(A))
& B = k1_funct_1(D,np__1)
& C = k1_funct_1(D,k3_finseq_1(D))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,E)
=> ( r1_xreal_0(k3_finseq_1(D),E)
| ! [F] :
( m1_subset_1(F,u1_struct_0(A))
=> ! [G] :
( m1_subset_1(G,u1_struct_0(A))
=> ( ( F = k1_funct_1(D,E)
& G = k1_funct_1(D,k1_nat_1(E,np__1)) )
=> r1_pencil_1(A,F,G) ) ) ) ) ) ) ) ) ) ) ) ).
fof(d11_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ( v10_pencil_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( v1_pencil_1(C,A)
& v2_pencil_1(C,A)
& ! [D] :
( m2_finseq_1(D,k1_zfmisc_1(u1_struct_0(A)))
=> ~ ( C = k1_funct_1(D,np__1)
& r2_hidden(B,k1_funct_1(D,k3_finseq_1(D)))
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
=> ( r2_hidden(E,k2_relat_1(D))
=> ( v1_pencil_1(E,A)
& v2_pencil_1(E,A) ) ) )
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,E)
=> ( r1_xreal_0(k3_finseq_1(D),E)
| r1_tarski(np__2,k1_card_1(k3_xboole_0(k1_funct_1(D,E),k1_funct_1(D,k1_nat_1(E,np__1))))) ) ) ) ) ) ) ) ) ) ) ).
fof(d12_pencil_1,axiom,
! [A] :
( ( v8_pencil_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_pencil_1(A,B,C)
<=> ? [D] :
( m1_subset_1(D,u1_pre_topc(A))
& r1_tarski(k2_tarski(B,C),D) ) ) ) ) ) ).
fof(d13_pencil_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v11_pencil_1(A)
<=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> l1_pre_topc(B) ) ) ) ).
fof(d14_pencil_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v12_pencil_1(A)
<=> ! [B] :
( l1_pre_topc(B)
=> ~ ( r2_hidden(B,k2_relat_1(A))
& v3_pencil_1(B) ) ) ) ) ).
fof(d15_pencil_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v11_pencil_1(A) )
=> ( v12_pencil_1(A)
<=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> ( ~ v3_pencil_1(B)
& l1_pre_topc(B) ) ) ) ) ).
fof(d16_pencil_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v13_pencil_1(A)
<=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> v1_realset1(B) ) ) ) ).
fof(d17_pencil_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v14_pencil_1(A)
<=> ! [B] :
( l1_struct_0(B)
=> ~ ( r2_hidden(B,k2_relat_1(A))
& v3_realset2(B) ) ) ) ) ).
fof(d18_pencil_1,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v2_pralg_1(A) )
=> ( v14_pencil_1(A)
<=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> ( ~ v3_realset2(B)
& l1_struct_0(B) ) ) ) ) ).
fof(d19_pencil_1,axiom,
! [A] :
( v1_relat_1(A)
=> ( v15_pencil_1(A)
<=> ! [B] :
( r2_hidden(B,k2_relat_1(A))
=> ( ~ v3_struct_0(B)
& ~ v3_pencil_1(B)
& ~ v4_pencil_1(B)
& v5_pencil_1(B)
& v6_pencil_1(B)
& l1_pre_topc(B) ) ) ) ) ).
fof(d20_pencil_1,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v16_pencil_1(B,A)
<=> ? [C] :
( m1_subset_1(C,A)
& ! [D] :
( m1_subset_1(D,A)
=> ( C != D
=> ( ~ v1_xboole_0(k1_funct_1(B,D))
& v1_realset1(k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(t9_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ~ v1_realset1(D)
=> ~ v13_pencil_1(k2_polynom1(A,B,C,D)) ) ) ) ) ).
fof(t10_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ! [C,D] : v16_pencil_1(k2_polynom1(A,k1_pzfmisc1(A,B),C,D),A) ) ) ).
fof(t11_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v2_pralg_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m2_pboole(C,A,k12_pralg_1(A,B))
=> m4_pboole(k1_pzfmisc1(A,C),A,k12_pralg_1(A,B)) ) ) ) ).
fof(d21_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v13_pencil_1(B)
& v16_pencil_1(B,A)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( C = k3_pencil_1(A,B)
<=> ~ v1_realset1(k1_funct_1(B,C)) ) ) ) ) ).
fof(t12_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v13_pencil_1(B)
& v16_pencil_1(B,A)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,A)
=> ( C != k3_pencil_1(A,B)
=> ( ~ v1_xboole_0(k1_funct_1(B,C))
& v1_realset1(k1_funct_1(B,C)) ) ) ) ) ) ).
fof(t13_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_pboole(B,A)
=> ( r1_tarski(np__2,k1_card_1(k4_card_3(B)))
<=> ( v2_relat_1(B)
& ~ v13_pencil_1(B) ) ) ) ) ).
fof(d22_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(k4_card_3(k12_pralg_1(A,B)))))
=> ( C = k4_pencil_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v16_pencil_1(E,A)
& m4_pboole(E,A,k12_pralg_1(A,B))
& D = k4_card_3(E)
& ? [F] :
( m1_subset_1(F,A)
& m1_subset_1(k1_funct_1(E,F),u1_pre_topc(k1_pencil_1(A,B,F))) ) ) ) ) ) ) ) ).
fof(d23_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> k5_pencil_1(A,B) = g1_pre_topc(k4_card_3(k12_pralg_1(A,B)),k4_pencil_1(A,B)) ) ) ).
fof(t14_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k5_pencil_1(A,B)))
=> m1_pboole(C,A) ) ) ) ).
fof(t15_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> ~ ( ~ ! [C] :
( m1_subset_1(C,A)
=> v3_pencil_1(k1_pencil_1(A,B,C)) )
& v3_pencil_1(k5_pencil_1(A,B)) ) ) ) ).
fof(t16_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> ~ ( ! [C] :
( m1_subset_1(C,A)
=> ( ~ v4_pencil_1(k1_pencil_1(A,B,C))
& ~ ! [D] :
( m1_subset_1(D,A)
=> v3_pencil_1(k1_pencil_1(A,B,D)) ) ) )
& v4_pencil_1(k5_pencil_1(A,B)) ) ) ) ).
fof(t17_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v5_pencil_1(k1_pencil_1(A,B,C))
& ~ ! [D] :
( m1_subset_1(D,A)
=> v3_pencil_1(k1_pencil_1(A,B,D)) ) ) )
=> v5_pencil_1(k5_pencil_1(A,B)) ) ) ) ).
fof(t18_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_subset_1(C,A)
=> ( v6_pencil_1(k1_pencil_1(A,B,C))
& v5_pencil_1(k1_pencil_1(A,B,C))
& ~ ! [D] :
( m1_subset_1(D,A)
=> v3_pencil_1(k1_pencil_1(A,B,D)) ) ) )
=> v6_pencil_1(k5_pencil_1(A,B)) ) ) ) ).
fof(t19_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
=> ( v1_realset1(B)
=> ( v2_pencil_1(B,A)
& v1_pencil_1(B,A) ) ) ) ) ).
fof(t20_pencil_1,axiom,
! [A] :
( ( v6_pencil_1(A)
& l1_pre_topc(A) )
=> ! [B] :
( m1_subset_1(B,u1_pre_topc(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = B
=> v1_pencil_1(C,A) ) ) ) ) ).
fof(t21_pencil_1,axiom,
! [A] :
( l1_pre_topc(A)
=> ! [B] :
( m1_subset_1(B,u1_pre_topc(A))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
=> ( C = B
=> v2_pencil_1(C,A) ) ) ) ) ).
fof(t22_pencil_1,axiom,
! [A] :
( ( ~ v3_pencil_1(A)
& l1_pre_topc(A) )
=> v1_pencil_1(k2_pre_topc(A),A) ) ).
fof(t23_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v13_pencil_1(B)
& v16_pencil_1(B,A)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( ( r2_hidden(C,k4_card_3(B))
& r2_hidden(D,k4_card_3(B)) )
=> ! [E] :
( E != k3_pencil_1(A,B)
=> k1_funct_1(C,E) = k1_funct_1(D,E) ) ) ) ) ) ) ).
fof(t24_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_pre_topc(k6_pencil_1(A,B)))
<=> ? [D] :
( ~ v13_pencil_1(D)
& v16_pencil_1(D,A)
& m4_pboole(D,A,k12_pralg_1(A,B))
& C = k4_card_3(D)
& m1_subset_1(k1_funct_1(D,k3_pencil_1(A,D)),u1_pre_topc(k2_pencil_1(A,B,k3_pencil_1(A,D)))) ) ) ) ) ).
fof(t25_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_pboole(C,A)
=> ( m1_subset_1(C,u1_struct_0(k6_pencil_1(A,B)))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,u1_struct_0(k2_pencil_1(A,B,D)))
=> m1_subset_1(k2_polynom1(A,C,D,E),u1_struct_0(k6_pencil_1(A,B))) ) ) ) ) ) ) ).
fof(t26_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v13_pencil_1(B)
& v16_pencil_1(B,A)
& m1_pboole(B,A) )
=> ! [C] :
( ( ~ v13_pencil_1(C)
& v16_pencil_1(C,A)
& m1_pboole(C,A) )
=> ( r1_tarski(np__2,k1_card_1(k3_xboole_0(k4_card_3(B),k4_card_3(C))))
=> ( k3_pencil_1(A,B) = k3_pencil_1(A,C)
& ! [D] :
( D != k3_pencil_1(A,B)
=> k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ) ).
fof(t27_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v13_pencil_1(B)
& v16_pencil_1(B,A)
& m1_pboole(B,A) )
=> ! [C] :
( ~ v1_realset1(C)
=> ( v16_pencil_1(k2_polynom1(A,B,k3_pencil_1(A,B),C),A)
& ~ v13_pencil_1(k2_polynom1(A,B,k3_pencil_1(A,B),C)) ) ) ) ) ).
fof(t28_pencil_1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& ~ v3_pencil_1(A)
& v6_pencil_1(A)
& v8_pencil_1(A)
& l1_pre_topc(A) )
=> ( v10_pencil_1(A)
=> v9_pencil_1(A) ) ) ).
fof(t29_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v15_pencil_1(B)
& m1_pboole(B,A) )
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_pencil_1(A,B))))
=> ( ( ~ v1_realset1(C)
& v2_pencil_1(C,k6_pencil_1(A,B))
& v1_pencil_1(C,k6_pencil_1(A,B)) )
<=> ? [D] :
( ~ v13_pencil_1(D)
& v16_pencil_1(D,A)
& m4_pboole(D,A,k12_pralg_1(A,B))
& C = k4_card_3(D)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k2_pencil_1(A,B,k3_pencil_1(A,D)))))
=> ( E = k1_funct_1(D,k3_pencil_1(A,D))
=> ( v2_pencil_1(E,k2_pencil_1(A,B,k3_pencil_1(A,D)))
& v1_pencil_1(E,k2_pencil_1(A,B,k3_pencil_1(A,D))) ) ) ) ) ) ) ) ) ).
fof(dt_k1_pencil_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v11_pencil_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> l1_pre_topc(k1_pencil_1(A,B,C)) ) ).
fof(redefinition_k1_pencil_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v11_pencil_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> k1_pencil_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k2_pencil_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v15_pencil_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> ( ~ v3_struct_0(k2_pencil_1(A,B,C))
& ~ v3_pencil_1(k2_pencil_1(A,B,C))
& ~ v4_pencil_1(k2_pencil_1(A,B,C))
& v5_pencil_1(k2_pencil_1(A,B,C))
& v6_pencil_1(k2_pencil_1(A,B,C))
& l1_pre_topc(k2_pencil_1(A,B,C)) ) ) ).
fof(redefinition_k2_pencil_1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v15_pencil_1(B)
& m1_pboole(B,A)
& m1_subset_1(C,A) )
=> k2_pencil_1(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k3_pencil_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& ~ v13_pencil_1(B)
& v16_pencil_1(B,A)
& m1_pboole(B,A) )
=> m1_subset_1(k3_pencil_1(A,B),A) ) ).
fof(dt_k4_pencil_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> m1_subset_1(k4_pencil_1(A,B),k1_zfmisc_1(k1_zfmisc_1(k4_card_3(k12_pralg_1(A,B))))) ) ).
fof(dt_k5_pencil_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v4_waybel_3(B)
& v11_pencil_1(B)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k5_pencil_1(A,B))
& l1_pre_topc(k5_pencil_1(A,B)) ) ) ).
fof(dt_k6_pencil_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v15_pencil_1(B)
& m1_pboole(B,A) )
=> ( ~ v3_struct_0(k6_pencil_1(A,B))
& ~ v3_pencil_1(k6_pencil_1(A,B))
& ~ v4_pencil_1(k6_pencil_1(A,B))
& v5_pencil_1(k6_pencil_1(A,B))
& v6_pencil_1(k6_pencil_1(A,B))
& l1_pre_topc(k6_pencil_1(A,B)) ) ) ).
fof(redefinition_k6_pencil_1,axiom,
! [A,B] :
( ( ~ v1_xboole_0(A)
& v15_pencil_1(B)
& m1_pboole(B,A) )
=> k6_pencil_1(A,B) = k5_pencil_1(A,B) ) ).
fof(t7_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( r1_tarski(np__3,k1_card_1(A))
=> ! [B] :
( l1_pre_topc(B)
=> ( ( u1_struct_0(B) = A
& u1_pre_topc(B) = a_1_0_pencil_1(A) )
=> ( ~ v3_struct_0(B)
& ~ v3_pencil_1(B)
& ~ v4_pencil_1(B)
& ~ v7_pencil_1(B)
& v5_pencil_1(B)
& v6_pencil_1(B)
& v8_pencil_1(B) ) ) ) ) ) ).
fof(t8_pencil_1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ( r1_tarski(np__3,k1_card_1(A))
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(A))
=> ( k1_card_1(B) = np__2
=> ! [C] :
( l1_pre_topc(C)
=> ( ( u1_struct_0(C) = A
& u1_pre_topc(C) = k4_xboole_0(a_1_0_pencil_1(A),k1_tarski(B)) )
=> ( ~ v3_struct_0(C)
& ~ v3_pencil_1(C)
& ~ v4_pencil_1(C)
& v7_pencil_1(C)
& v5_pencil_1(C)
& v6_pencil_1(C)
& v8_pencil_1(C) ) ) ) ) ) ) ) ).
fof(fraenkel_a_1_0_pencil_1,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ( r2_hidden(A,a_1_0_pencil_1(B))
<=> ? [C] :
( m1_subset_1(C,k1_zfmisc_1(B))
& A = C
& np__2 = k1_card_1(C) ) ) ) ).
%------------------------------------------------------------------------------