SET007 Axioms: SET007+186.ax
%------------------------------------------------------------------------------
% File : SET007+186 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Definitions and Properties of Many Sorted Sets
% 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 : mboolean [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 43 ( 8 unt; 0 def)
% Number of atoms : 170 ( 5 equ)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 130 ( 3 ~; 2 |; 21 &)
% ( 8 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 14 ( 12 usr; 1 prp; 0-3 aty)
% Number of functors : 16 ( 16 usr; 1 con; 0-3 aty)
% Number of variables : 128 ( 127 !; 1 ?)
% 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_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ( v1_relat_1(k1_mboolean(A,B))
& v2_relat_1(k1_mboolean(A,B))
& v1_funct_1(k1_mboolean(A,B)) ) ) ).
fof(fc2_mboolean,axiom,
! [A] :
( v1_relat_1(k2_mboolean(A,k1_pboole(A)))
& v3_relat_1(k2_mboolean(A,k1_pboole(A)))
& v1_funct_1(k2_mboolean(A,k1_pboole(A))) ) ).
fof(d1_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( C = k1_mboolean(A,B)
<=> ! [D] :
( r2_hidden(D,A)
=> k1_funct_1(C,D) = k1_zfmisc_1(k1_funct_1(B,D)) ) ) ) ) ).
fof(t1_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r6_pboole(A,B,k1_mboolean(A,C))
<=> ! [D] :
( m1_pboole(D,A)
=> ( r1_pboole(A,D,B)
<=> r2_pboole(A,D,C) ) ) ) ) ) ).
fof(t2_mboolean,axiom,
! [A] : r6_pboole(A,k1_mboolean(A,k1_pboole(A)),k2_pre_circ(A,k1_tarski(k1_xboole_0))) ).
fof(t3_mboolean,axiom,
! [A,B] : r6_pboole(A,k1_mboolean(A,k2_pre_circ(A,B)),k2_pre_circ(A,k1_zfmisc_1(B))) ).
fof(t4_mboolean,axiom,
! [A,B] : r6_pboole(A,k1_mboolean(A,k2_pre_circ(A,k1_tarski(B))),k2_pre_circ(A,k2_tarski(k1_xboole_0,k1_tarski(B)))) ).
fof(t5_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> r2_pboole(A,k1_pboole(A),B) ) ).
fof(t6_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r2_pboole(A,B,C)
=> r2_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C)) ) ) ) ).
fof(t7_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> r2_pboole(A,k2_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C)),k1_mboolean(A,k2_pboole(A,B,C))) ) ) ).
fof(t8_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r6_pboole(A,k2_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C)),k1_mboolean(A,k2_pboole(A,B,C)))
=> ! [D] :
( r2_hidden(D,A)
=> r3_xboole_0(k1_funct_1(B,D),k1_funct_1(C,D)) ) ) ) ) ).
fof(t9_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> r6_pboole(A,k1_mboolean(A,k3_pboole(A,B,C)),k3_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C))) ) ) ).
fof(t10_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> r2_pboole(A,k1_mboolean(A,k4_pboole(A,B,C)),k2_pboole(A,k2_pre_circ(A,k1_tarski(k1_xboole_0)),k4_pboole(A,k1_mboolean(A,B),k1_mboolean(A,C)))) ) ) ).
fof(t11_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( r2_pboole(A,B,k4_pboole(A,C,D))
<=> ( r2_pboole(A,B,C)
& r5_pboole(A,B,D) ) ) ) ) ) ).
fof(t12_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> r2_pboole(A,k2_pboole(A,k1_mboolean(A,k4_pboole(A,B,C)),k1_mboolean(A,k4_pboole(A,C,B))),k1_mboolean(A,k5_pboole(A,B,C))) ) ) ).
fof(t13_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( r2_pboole(A,B,k5_pboole(A,C,D))
<=> ( r2_pboole(A,B,k2_pboole(A,C,D))
& r5_pboole(A,B,k3_pboole(A,C,D)) ) ) ) ) ) ).
fof(t14_mboolean,axiom,
$true ).
fof(t15_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( ( r2_pboole(A,B,C)
| r2_pboole(A,D,C) )
=> r2_pboole(A,k3_pboole(A,B,D),C) ) ) ) ) ).
fof(t16_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( r2_pboole(A,B,C)
=> r2_pboole(A,k4_pboole(A,B,D),C) ) ) ) ) ).
fof(t17_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( ( r2_pboole(A,B,C)
& r2_pboole(A,D,C) )
=> r2_pboole(A,k5_pboole(A,B,D),C) ) ) ) ) ).
fof(t18_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> r2_pboole(A,k11_pboole(A,B,C),k1_mboolean(A,k1_mboolean(A,k2_pboole(A,B,C)))) ) ) ).
fof(t19_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r2_pboole(A,B,C)
<=> r1_pboole(A,B,k1_mboolean(A,C)) ) ) ) ).
fof(t20_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> r2_pboole(A,k12_pboole(A,B,C),k1_mboolean(A,k11_pboole(A,B,C))) ) ) ).
fof(d2_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( C = k2_mboolean(A,B)
<=> ! [D] :
( r2_hidden(D,A)
=> k1_funct_1(C,D) = k3_tarski(k1_funct_1(B,D)) ) ) ) ) ).
fof(t21_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r1_pboole(A,B,k2_mboolean(A,C))
<=> ? [D] :
( m1_pboole(D,A)
& r1_pboole(A,B,D)
& r1_pboole(A,D,C) ) ) ) ) ).
fof(t22_mboolean,axiom,
! [A] : r6_pboole(A,k2_mboolean(A,k1_pboole(A)),k1_pboole(A)) ).
fof(t23_mboolean,axiom,
! [A,B] : r6_pboole(A,k2_mboolean(A,k2_pre_circ(A,B)),k2_pre_circ(A,k3_tarski(B))) ).
fof(t24_mboolean,axiom,
! [A,B] : r6_pboole(A,k2_mboolean(A,k2_pre_circ(A,k1_tarski(B))),k2_pre_circ(A,B)) ).
fof(t25_mboolean,axiom,
! [A,B,C] : r6_pboole(A,k2_mboolean(A,k2_pre_circ(A,k2_tarski(k1_tarski(B),k1_tarski(C)))),k2_pre_circ(A,k2_tarski(B,C))) ).
fof(t26_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r1_pboole(A,B,C)
=> r2_pboole(A,B,k2_mboolean(A,C)) ) ) ) ).
fof(t27_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r2_pboole(A,B,C)
=> r2_pboole(A,k2_mboolean(A,B),k2_mboolean(A,C)) ) ) ) ).
fof(t28_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> r6_pboole(A,k2_mboolean(A,k2_pboole(A,B,C)),k2_pboole(A,k2_mboolean(A,B),k2_mboolean(A,C))) ) ) ).
fof(t29_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> r2_pboole(A,k2_mboolean(A,k3_pboole(A,B,C)),k3_pboole(A,k2_mboolean(A,B),k2_mboolean(A,C))) ) ) ).
fof(t30_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> r6_pboole(A,k2_mboolean(A,k1_mboolean(A,B)),B) ) ).
fof(t31_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> r2_pboole(A,B,k1_mboolean(A,k2_mboolean(A,B))) ) ).
fof(t32_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( ( r2_pboole(A,k2_mboolean(A,B),C)
& r1_pboole(A,D,B) )
=> r2_pboole(A,D,C) ) ) ) ) ).
fof(t33_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ( ! [D] :
( m1_pboole(D,A)
=> ( r1_pboole(A,D,C)
=> r2_pboole(A,D,B) ) )
=> r2_pboole(A,k2_mboolean(A,C),B) ) ) ) ).
fof(t34_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( ( v2_relat_1(C)
& m1_pboole(C,A) )
=> ( ! [D] :
( m1_pboole(D,A)
=> ( r1_pboole(A,D,C)
=> r6_pboole(A,k3_pboole(A,D,B),k1_pboole(A)) ) )
=> r6_pboole(A,k3_pboole(A,k2_mboolean(A,C),B),k1_pboole(A)) ) ) ) ).
fof(t35_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( ( v2_relat_1(k2_pboole(A,B,C))
& ! [D] :
( m1_pboole(D,A)
=> ! [E] :
( m1_pboole(E,A)
=> ( ( r1_pboole(A,D,k2_pboole(A,B,C))
& r1_pboole(A,E,k2_pboole(A,B,C)) )
=> ( D = E
| r6_pboole(A,k3_pboole(A,D,E),k1_pboole(A)) ) ) ) ) )
=> r6_pboole(A,k2_mboolean(A,k3_pboole(A,B,C)),k3_pboole(A,k2_mboolean(A,B),k2_mboolean(A,C))) ) ) ) ).
fof(t36_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( ( v2_relat_1(D)
& m1_pboole(D,A) )
=> ( ( r2_pboole(A,C,k2_mboolean(A,k2_pboole(A,B,D)))
& ! [E] :
( m1_pboole(E,A)
=> ( r1_pboole(A,E,D)
=> r6_pboole(A,k3_pboole(A,E,C),k1_pboole(A)) ) ) )
=> r2_pboole(A,C,k2_mboolean(A,B)) ) ) ) ) ).
fof(t37_mboolean,axiom,
! [A,B] :
( ( v2_relat_1(B)
& v1_pre_circ(B,A)
& m1_pboole(B,A) )
=> ( ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ~ ( r1_pboole(A,C,B)
& r1_pboole(A,D,B)
& ~ r2_pboole(A,C,D)
& ~ r2_pboole(A,D,C) ) ) )
=> r1_pboole(A,k2_mboolean(A,B),B) ) ) ).
fof(dt_k1_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> m1_pboole(k1_mboolean(A,B),A) ) ).
fof(dt_k2_mboolean,axiom,
! [A,B] :
( m1_pboole(B,A)
=> m1_pboole(k2_mboolean(A,B),A) ) ).
%------------------------------------------------------------------------------