SET007 Axioms: SET007+848.ax
%------------------------------------------------------------------------------
% File : SET007+848 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Axiomatization of Boolean Algebras Based on Sheffer Stroke
% 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 : sheffer1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 93 ( 7 unt; 0 def)
% Number of atoms : 700 ( 60 equ)
% Maximal formula atoms : 28 ( 7 avg)
% Number of connectives : 685 ( 78 ~; 0 |; 445 &)
% ( 15 <=>; 147 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 49 ( 48 usr; 0 prp; 1-3 aty)
% Number of functors : 29 ( 29 usr; 4 con; 0-6 aty)
% Number of variables : 177 ( 163 !; 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_sheffer1,axiom,
? [A] :
( l3_lattices(A)
& ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v10_lattices(A)
& v11_lattices(A)
& v12_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& v17_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A) ) ).
fof(cc1_sheffer1,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A) )
=> ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A) ) ) ) ).
fof(cc2_sheffer1,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A) )
=> ( ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v10_lattices(A)
& v11_lattices(A)
& v12_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& v17_lattices(A) ) ) ) ).
fof(rc2_sheffer1,axiom,
? [A] :
( l1_sheffer1(A)
& v6_sheffer1(A) ) ).
fof(rc3_sheffer1,axiom,
? [A] :
( l2_sheffer1(A)
& v7_sheffer1(A) ) ).
fof(rc4_sheffer1,axiom,
? [A] :
( l3_sheffer1(A)
& v8_sheffer1(A) ) ).
fof(rc5_sheffer1,axiom,
? [A] :
( l1_sheffer1(A)
& ~ v3_struct_0(A) ) ).
fof(rc6_sheffer1,axiom,
? [A] :
( l2_sheffer1(A)
& ~ v3_struct_0(A) ) ).
fof(rc7_sheffer1,axiom,
? [A] :
( l3_sheffer1(A)
& ~ v3_struct_0(A) ) ).
fof(cc3_sheffer1,axiom,
! [A] :
( l1_sheffer1(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v10_sheffer1(A)
& v11_sheffer1(A)
& v12_sheffer1(A) ) ) ) ).
fof(cc4_sheffer1,axiom,
! [A] :
( l2_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A) ) ) ) ).
fof(cc5_sheffer1,axiom,
! [A] :
( l1_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v6_lattices(A)
& v7_lattices(A) ) ) ) ).
fof(cc6_sheffer1,axiom,
! [A] :
( l3_lattices(A)
=> ( ( ~ v3_struct_0(A)
& v3_realset2(A) )
=> ( ~ v3_struct_0(A)
& v8_lattices(A)
& v9_lattices(A)
& v11_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& v17_lattices(A) ) ) ) ).
fof(fc1_sheffer1,axiom,
~ v3_struct_0(k4_sheffer1) ).
fof(fc2_sheffer1,axiom,
v3_realset2(k4_sheffer1) ).
fof(fc3_sheffer1,axiom,
( v7_robbins1(k4_sheffer1)
& v9_sheffer1(k4_sheffer1) ) ).
fof(rc8_sheffer1,axiom,
? [A] :
( l3_sheffer1(A)
& ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v6_lattices(A)
& v7_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v10_lattices(A)
& v11_lattices(A)
& v12_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v15_lattices(A)
& v16_lattices(A)
& v17_lattices(A)
& v4_robbins1(A)
& v5_robbins1(A)
& v6_robbins1(A)
& v7_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& v9_sheffer1(A)
& v10_sheffer1(A)
& v11_sheffer1(A)
& v12_sheffer1(A) ) ).
fof(t1_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v5_robbins1(A)
& l2_robbins1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k3_robbins1(A,k5_robbins1(A,B,C)) = k6_robbins1(A,k3_robbins1(A,B),k3_robbins1(A,C)) ) ) ) ).
fof(t2_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v5_lattices(A)
& v5_robbins1(A)
& l2_robbins1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k3_robbins1(A,k6_robbins1(A,B,C)) = k5_robbins1(A,k3_robbins1(A,B),k3_robbins1(A,C)) ) ) ) ).
fof(d1_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v1_sheffer1(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_lattices(A,B,C) = C
& k2_lattices(A,C,B) = C ) ) ) ) ) ).
fof(d2_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v1_sheffer1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B = k1_sheffer1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k2_lattices(A,B,C) = C
& k2_lattices(A,C,B) = C ) ) ) ) ) ) ).
fof(d3_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v2_sheffer1(A)
<=> ? [B] :
( m1_subset_1(B,u1_struct_0(A))
& ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_lattices(A,B,C) = C
& k1_lattices(A,C,B) = C ) ) ) ) ) ).
fof(d4_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v2_sheffer1(A)
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( B = k2_sheffer1(A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( k1_lattices(A,B,C) = C
& k1_lattices(A,C,B) = C ) ) ) ) ) ) ).
fof(d5_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v3_sheffer1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k1_lattices(A,B,k2_lattices(A,C,D)) = k2_lattices(A,k1_lattices(A,B,C),k1_lattices(A,B,D)) ) ) ) ) ) ).
fof(d6_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_sheffer1(A,B,C)
<=> ( k1_lattices(A,C,B) = k1_sheffer1(A)
& k1_lattices(A,B,C) = k1_sheffer1(A)
& k2_lattices(A,C,B) = k2_sheffer1(A)
& k2_lattices(A,B,C) = k2_sheffer1(A) ) ) ) ) ) ).
fof(d7_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( v4_sheffer1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ? [C] :
( m1_subset_1(C,u1_struct_0(A))
& r1_sheffer1(A,C,B) ) ) ) ) ).
fof(d8_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( ( v4_sheffer1(A)
& v11_lattices(A)
& v1_sheffer1(A)
& v6_lattices(A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( C = k3_sheffer1(A,B)
<=> r1_sheffer1(A,C,B) ) ) ) ) ) ).
fof(t3_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v1_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k3_lattices(A,B,k3_sheffer1(A,B)) = k1_sheffer1(A) ) ) ).
fof(t4_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v1_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k4_lattices(A,B,k3_sheffer1(A,B)) = k2_sheffer1(A) ) ) ).
fof(t5_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k3_lattices(A,B,k1_sheffer1(A)) = k1_sheffer1(A) ) ) ).
fof(t6_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k4_lattices(A,B,k2_sheffer1(A)) = k2_sheffer1(A) ) ) ).
fof(t7_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k4_lattices(A,k3_lattices(A,k3_lattices(A,B,C),D),B) = B ) ) ) ) ).
fof(t8_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v6_robbins1(A)
& v3_sheffer1(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k3_lattices(A,k4_lattices(A,k4_lattices(A,B,C),D),B) = B ) ) ) ) ).
fof(d9_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_lattices(A) )
=> ( v5_sheffer1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k2_lattices(A,B,B) = B ) ) ) ).
fof(t9_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> v5_sheffer1(A) ) ).
fof(t10_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> v6_robbins1(A) ) ).
fof(t11_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> v8_lattices(A) ) ).
fof(t12_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> v9_lattices(A) ) ).
fof(t13_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> v14_lattices(A) ) ).
fof(t14_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> v1_sheffer1(A) ) ).
fof(t15_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> v13_lattices(A) ) ).
fof(t16_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
=> v2_sheffer1(A) ) ).
fof(t17_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& l3_lattices(A) )
=> v5_lattices(A) ) ).
fof(t18_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v8_lattices(A)
& v9_lattices(A)
& v6_robbins1(A)
& v3_sheffer1(A)
& l3_lattices(A) )
=> v7_lattices(A) ) ).
fof(t19_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> k6_lattices(A) = k1_sheffer1(A) ) ).
fof(t20_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> k5_lattices(A) = k2_sheffer1(A) ) ).
fof(t21_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& v3_sheffer1(A)
& l3_lattices(A) )
=> k6_lattices(A) = k1_sheffer1(A) ) ).
fof(t22_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v11_lattices(A)
& v13_lattices(A)
& v14_lattices(A)
& v16_lattices(A)
& v17_lattices(A)
& v3_sheffer1(A)
& l3_lattices(A) )
=> k5_lattices(A) = k2_sheffer1(A) ) ).
fof(t23_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ( r1_sheffer1(A,B,C)
<=> r2_lattices(A,B,C) ) ) ) ) ).
fof(t24_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v6_robbins1(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& v4_sheffer1(A)
& l3_lattices(A) )
=> v16_lattices(A) ) ).
fof(t25_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& v1_sheffer1(A)
& v2_sheffer1(A)
& v3_sheffer1(A)
& l3_lattices(A) )
=> v4_sheffer1(A) ) ).
fof(t26_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> ( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) )
<=> ( v2_sheffer1(A)
& v1_sheffer1(A)
& v4_lattices(A)
& v6_lattices(A)
& v11_lattices(A)
& v3_sheffer1(A)
& v4_sheffer1(A) ) ) ) ).
fof(d10_sheffer1,axiom,
k4_sheffer1 = g3_sheffer1(k1_tarski(k1_xboole_0),k2_midsp_1,k2_midsp_1,k7_vectsp_2,k2_midsp_1) ).
fof(d11_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_sheffer1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_sheffer1(A,B,C) = k2_binop_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A),u1_sheffer1(A),B,C) ) ) ) ).
fof(d12_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_sheffer1(A) )
=> ( v9_sheffer1(A)
<=> ( ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k5_sheffer1(A,B,B) = k3_robbins1(A,B) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k1_lattices(A,B,C) = k5_sheffer1(A,k5_sheffer1(A,B,B),k5_sheffer1(A,C,C)) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k2_lattices(A,B,C) = k5_sheffer1(A,k5_sheffer1(A,B,C),k5_sheffer1(A,B,C)) ) )
& ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_sheffer1(A,B,C) = k1_lattices(A,k3_robbins1(A,B),k3_robbins1(A,C)) ) ) ) ) ) ).
fof(d13_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_sheffer1(A) )
=> ( v10_sheffer1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k5_sheffer1(A,k5_sheffer1(A,B,B),k5_sheffer1(A,B,B)) = B ) ) ) ).
fof(d14_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_sheffer1(A) )
=> ( v11_sheffer1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_sheffer1(A,B,k5_sheffer1(A,C,k5_sheffer1(A,C,C))) = k5_sheffer1(A,B,B) ) ) ) ) ).
fof(d15_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_sheffer1(A) )
=> ( v12_sheffer1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k5_sheffer1(A,k5_sheffer1(A,B,k5_sheffer1(A,C,D)),k5_sheffer1(A,B,k5_sheffer1(A,C,D))) = k5_sheffer1(A,k5_sheffer1(A,k5_sheffer1(A,C,C),B),k5_sheffer1(A,k5_sheffer1(A,D,D),B)) ) ) ) ) ) ).
fof(t27_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& v7_robbins1(A)
& v9_sheffer1(A)
& l3_sheffer1(A) )
=> v10_sheffer1(A) ) ).
fof(t28_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& v7_robbins1(A)
& v9_sheffer1(A)
& l3_sheffer1(A) )
=> v11_sheffer1(A) ) ).
fof(t29_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& v7_robbins1(A)
& v9_sheffer1(A)
& l3_sheffer1(A) )
=> v12_sheffer1(A) ) ).
fof(d16_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_sheffer1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k6_sheffer1(A,B) = k5_sheffer1(A,B,B) ) ) ).
fof(t30_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v12_sheffer1(A)
& l3_sheffer1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> k6_sheffer1(A,k5_sheffer1(A,B,k5_sheffer1(A,C,D))) = k5_sheffer1(A,k5_sheffer1(A,k6_sheffer1(A,C),B),k5_sheffer1(A,k6_sheffer1(A,D),B)) ) ) ) ) ).
fof(t31_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v10_sheffer1(A)
& l3_sheffer1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> B = k6_sheffer1(A,k6_sheffer1(A,B)) ) ) ).
fof(t32_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_sheffer1(A)
& v10_sheffer1(A)
& v11_sheffer1(A)
& v12_sheffer1(A)
& l3_sheffer1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_sheffer1(A,B,C) = k5_sheffer1(A,C,B) ) ) ) ).
fof(t33_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_sheffer1(A)
& v10_sheffer1(A)
& v11_sheffer1(A)
& v12_sheffer1(A)
& l3_sheffer1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> k5_sheffer1(A,B,k5_sheffer1(A,B,B)) = k5_sheffer1(A,C,k5_sheffer1(A,C,C)) ) ) ) ).
fof(t34_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_sheffer1(A)
& v10_sheffer1(A)
& v11_sheffer1(A)
& v12_sheffer1(A)
& l3_sheffer1(A) )
=> v4_lattices(A) ) ).
fof(t35_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_sheffer1(A)
& v10_sheffer1(A)
& v11_sheffer1(A)
& v12_sheffer1(A)
& l3_sheffer1(A) )
=> v6_lattices(A) ) ).
fof(t36_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_sheffer1(A)
& v10_sheffer1(A)
& v11_sheffer1(A)
& v12_sheffer1(A)
& l3_sheffer1(A) )
=> v11_lattices(A) ) ).
fof(t37_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_sheffer1(A)
& v10_sheffer1(A)
& v11_sheffer1(A)
& v12_sheffer1(A)
& l3_sheffer1(A) )
=> v3_sheffer1(A) ) ).
fof(t38_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v9_sheffer1(A)
& v10_sheffer1(A)
& v11_sheffer1(A)
& v12_sheffer1(A)
& l3_sheffer1(A) )
=> ( ~ v3_struct_0(A)
& v10_lattices(A)
& v17_lattices(A)
& l3_lattices(A) ) ) ).
fof(dt_l1_sheffer1,axiom,
! [A] :
( l1_sheffer1(A)
=> l1_struct_0(A) ) ).
fof(existence_l1_sheffer1,axiom,
? [A] : l1_sheffer1(A) ).
fof(dt_l2_sheffer1,axiom,
! [A] :
( l2_sheffer1(A)
=> ( l1_sheffer1(A)
& l3_lattices(A) ) ) ).
fof(existence_l2_sheffer1,axiom,
? [A] : l2_sheffer1(A) ).
fof(dt_l3_sheffer1,axiom,
! [A] :
( l3_sheffer1(A)
=> ( l1_sheffer1(A)
& l3_robbins1(A) ) ) ).
fof(existence_l3_sheffer1,axiom,
? [A] : l3_sheffer1(A) ).
fof(abstractness_v6_sheffer1,axiom,
! [A] :
( l1_sheffer1(A)
=> ( v6_sheffer1(A)
=> A = g1_sheffer1(u1_struct_0(A),u1_sheffer1(A)) ) ) ).
fof(abstractness_v7_sheffer1,axiom,
! [A] :
( l2_sheffer1(A)
=> ( v7_sheffer1(A)
=> A = g2_sheffer1(u1_struct_0(A),u2_lattices(A),u1_lattices(A),u1_sheffer1(A)) ) ) ).
fof(abstractness_v8_sheffer1,axiom,
! [A] :
( l3_sheffer1(A)
=> ( v8_sheffer1(A)
=> A = g3_sheffer1(u1_struct_0(A),u2_lattices(A),u1_lattices(A),u1_robbins1(A),u1_sheffer1(A)) ) ) ).
fof(dt_k1_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> m1_subset_1(k1_sheffer1(A),u1_struct_0(A)) ) ).
fof(dt_k2_sheffer1,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l3_lattices(A) )
=> m1_subset_1(k2_sheffer1(A),u1_struct_0(A)) ) ).
fof(dt_k3_sheffer1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l3_lattices(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k3_sheffer1(A,B),u1_struct_0(A)) ) ).
fof(dt_k4_sheffer1,axiom,
l3_sheffer1(k4_sheffer1) ).
fof(dt_k5_sheffer1,axiom,
! [A,B,C] :
( ( ~ v3_struct_0(A)
& l1_sheffer1(A)
& m1_subset_1(B,u1_struct_0(A))
& m1_subset_1(C,u1_struct_0(A)) )
=> m1_subset_1(k5_sheffer1(A,B,C),u1_struct_0(A)) ) ).
fof(dt_k6_sheffer1,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l1_sheffer1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> m1_subset_1(k6_sheffer1(A,B),u1_struct_0(A)) ) ).
fof(dt_u1_sheffer1,axiom,
! [A] :
( l1_sheffer1(A)
=> ( v1_funct_1(u1_sheffer1(A))
& v1_funct_2(u1_sheffer1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A))
& m2_relset_1(u1_sheffer1(A),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_struct_0(A)) ) ) ).
fof(dt_g1_sheffer1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ( v6_sheffer1(g1_sheffer1(A,B))
& l1_sheffer1(g1_sheffer1(A,B)) ) ) ).
fof(free_g1_sheffer1,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A) )
=> ! [C,D] :
( g1_sheffer1(A,B) = g1_sheffer1(C,D)
=> ( A = C
& B = D ) ) ) ).
fof(dt_g2_sheffer1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ( v7_sheffer1(g2_sheffer1(A,B,C,D))
& l2_sheffer1(g2_sheffer1(A,B,C,D)) ) ) ).
fof(free_g2_sheffer1,axiom,
! [A,B,C,D] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,A),A)
& m1_relset_1(D,k2_zfmisc_1(A,A),A) )
=> ! [E,F,G,H] :
( g2_sheffer1(A,B,C,D) = g2_sheffer1(E,F,G,H)
=> ( A = E
& B = F
& C = G
& D = H ) ) ) ).
fof(dt_g3_sheffer1,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& v1_funct_2(D,A,A)
& m1_relset_1(D,A,A)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,A),A)
& m1_relset_1(E,k2_zfmisc_1(A,A),A) )
=> ( v8_sheffer1(g3_sheffer1(A,B,C,D,E))
& l3_sheffer1(g3_sheffer1(A,B,C,D,E)) ) ) ).
fof(free_g3_sheffer1,axiom,
! [A,B,C,D,E] :
( ( v1_funct_1(B)
& v1_funct_2(B,k2_zfmisc_1(A,A),A)
& m1_relset_1(B,k2_zfmisc_1(A,A),A)
& v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(A,A),A)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& v1_funct_2(D,A,A)
& m1_relset_1(D,A,A)
& v1_funct_1(E)
& v1_funct_2(E,k2_zfmisc_1(A,A),A)
& m1_relset_1(E,k2_zfmisc_1(A,A),A) )
=> ! [F,G,H,I,J] :
( g3_sheffer1(A,B,C,D,E) = g3_sheffer1(F,G,H,I,J)
=> ( A = F
& B = G
& C = H
& D = I
& E = J ) ) ) ).
%------------------------------------------------------------------------------