SET007 Axioms: SET007+592.ax
%------------------------------------------------------------------------------
% File : SET007+592 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Predicate Calculus for Boolean Valued Functions. Part II
% 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 : bvfunc_4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 28 ( 0 unt; 0 def)
% Number of atoms : 168 ( 34 equ)
% Maximal formula atoms : 10 ( 6 avg)
% Number of connectives : 168 ( 28 ~; 0 |; 9 &)
% ( 1 <=>; 130 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 9 ( 8 usr; 0 prp; 1-4 aty)
% Number of functors : 14 ( 14 usr; 1 con; 0-4 aty)
% Number of variables : 115 ( 115 !; 0 ?)
% 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(t1_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( r1_bvfunc_1(A,B,k14_bvfunc_1(A,C,D))
=> r1_bvfunc_1(A,k6_valuat_1(A,B,C),D) ) ) ) ) ) ).
fof(t2_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( r1_bvfunc_1(A,k6_valuat_1(A,B,C),D)
=> r1_bvfunc_1(A,B,k14_bvfunc_1(A,C,D)) ) ) ) ) ) ).
fof(t3_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k8_bvfunc_1(A,B,k6_valuat_1(A,B,C)) = B ) ) ) ).
fof(t4_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k6_valuat_1(A,B,k8_bvfunc_1(A,B,C)) = B ) ) ) ).
fof(t5_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k6_valuat_1(A,B,k5_valuat_1(A,B)) = k18_bvfunc_1(A) ) ) ).
fof(t6_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k8_bvfunc_1(A,B,k5_valuat_1(A,B)) = k19_bvfunc_1(A) ) ) ).
fof(t7_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k15_bvfunc_1(A,B,C) = k6_valuat_1(A,k14_bvfunc_1(A,B,C),k14_bvfunc_1(A,C,B)) ) ) ) ).
fof(t8_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k14_bvfunc_1(A,B,C) = k8_bvfunc_1(A,k5_valuat_1(A,B),C) ) ) ) ).
fof(t9_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> k9_bvfunc_1(A,B,C) = k8_bvfunc_1(A,k6_valuat_1(A,k5_valuat_1(A,B),C),k6_valuat_1(A,B,k5_valuat_1(A,C))) ) ) ) ).
fof(t10_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
<=> ( k14_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
& k14_bvfunc_1(A,C,B) = k19_bvfunc_1(A) ) ) ) ) ) ).
fof(t11_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
& k15_bvfunc_1(A,C,D) = k19_bvfunc_1(A) )
=> k15_bvfunc_1(A,B,D) = k19_bvfunc_1(A) ) ) ) ) ) ).
fof(t12_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
=> k15_bvfunc_1(A,k5_valuat_1(A,B),k5_valuat_1(A,C)) = k19_bvfunc_1(A) ) ) ) ) ).
fof(t13_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m2_fraenkel(E,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
& k15_bvfunc_1(A,D,E) = k19_bvfunc_1(A) )
=> k15_bvfunc_1(A,k6_valuat_1(A,B,D),k6_valuat_1(A,C,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) ).
fof(t14_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m2_fraenkel(E,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
& k15_bvfunc_1(A,D,E) = k19_bvfunc_1(A) )
=> k15_bvfunc_1(A,k14_bvfunc_1(A,B,D),k14_bvfunc_1(A,C,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) ).
fof(t15_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m2_fraenkel(E,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
& k15_bvfunc_1(A,D,E) = k19_bvfunc_1(A) )
=> k15_bvfunc_1(A,k8_bvfunc_1(A,B,D),k8_bvfunc_1(A,C,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) ).
fof(t16_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m2_fraenkel(D,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [E] :
( m2_fraenkel(E,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ( ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
& k15_bvfunc_1(A,D,E) = k19_bvfunc_1(A) )
=> k15_bvfunc_1(A,k15_bvfunc_1(A,B,D),k15_bvfunc_1(A,C,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) ).
fof(t17_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [E] :
( m1_eqrel_1(E,A)
=> k6_bvfunc_2(A,k15_bvfunc_1(A,B,C),D,E) = k6_valuat_1(A,k6_bvfunc_2(A,k14_bvfunc_1(A,B,C),D,E),k6_bvfunc_2(A,k14_bvfunc_1(A,C,B),D,E)) ) ) ) ) ) ).
fof(t18_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,B,C,D),k7_bvfunc_2(A,B,C,E)) ) ) ) ) ) ).
fof(t19_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( ( v2_bvfunc_2(D,A)
& r2_hidden(E,D)
& r3_bvfunc_2(A,C,D,E)
& k14_bvfunc_1(A,B,C) = k19_bvfunc_1(A) )
=> k14_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),C) = k19_bvfunc_1(A) ) ) ) ) ) ) ).
fof(t20_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( r3_bvfunc_2(A,B,C,D)
=> r1_bvfunc_1(A,k7_bvfunc_2(A,B,C,D),B) ) ) ) ) ) ).
fof(t21_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( r3_bvfunc_2(A,B,C,D)
=> r1_bvfunc_1(A,B,k6_bvfunc_2(A,B,C,D)) ) ) ) ) ) ).
fof(t22_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,B,C,E)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,B,C,D),k6_bvfunc_2(A,B,C,E)) ) ) ) ) ) ) ).
fof(t23_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [D] :
( m1_eqrel_1(D,A)
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( r3_bvfunc_2(A,B,C,D)
=> r1_bvfunc_1(A,k7_bvfunc_2(A,B,C,D),k7_bvfunc_2(A,B,C,E)) ) ) ) ) ) ) ).
fof(t24_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k15_bvfunc_1(A,B,C),D,E),k15_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),k6_bvfunc_2(A,C,D,E))) ) ) ) ) ) ).
fof(t25_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k6_valuat_1(A,B,C),D,E),k6_valuat_1(A,B,k6_bvfunc_2(A,C,D,E))) ) ) ) ) ) ).
fof(t26_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [E] :
( m1_eqrel_1(E,A)
=> r1_bvfunc_1(A,k14_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),C),k7_bvfunc_2(A,k14_bvfunc_1(A,B,C),D,E)) ) ) ) ) ) ).
fof(t27_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
=> k15_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),k6_bvfunc_2(A,C,D,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) ).
fof(t28_bvfunc_4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_fraenkel(B,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [C] :
( m2_fraenkel(C,A,k6_margrel1,k1_fraenkel(A,k6_margrel1))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [E] :
( m1_eqrel_1(E,A)
=> ( k15_bvfunc_1(A,B,C) = k19_bvfunc_1(A)
=> k15_bvfunc_1(A,k7_bvfunc_2(A,B,D,E),k7_bvfunc_2(A,C,D,E)) = k19_bvfunc_1(A) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------