SET007 Axioms: SET007+619.ax
%------------------------------------------------------------------------------
% File : SET007+619 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Predicate Calculus for Boolean Valued Functions. Part III
% 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 : bvfunc11 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 27 ( 2 unt; 0 def)
% Number of atoms : 165 ( 13 equ)
% Maximal formula atoms : 10 ( 6 avg)
% Number of connectives : 164 ( 26 ~; 3 |; 5 &)
% ( 0 <=>; 130 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 10 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-4 aty)
% Number of functors : 15 ( 15 usr; 1 con; 0-4 aty)
% Number of variables : 119 ( 119 !; 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_bvfunc11,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( r1_setfam_1(C,D)
=> r1_tarski(k22_bvfunc_1(A,B,C),k22_bvfunc_1(A,B,D)) ) ) ) ) ) ).
fof(t2_bvfunc11,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> r1_tarski(k22_bvfunc_1(A,B,C),k22_bvfunc_1(A,B,k3_partit1(A,C,D))) ) ) ) ) ).
fof(t3_bvfunc11,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> r1_tarski(k22_bvfunc_1(A,B,k2_partit1(A,C,D)),k22_bvfunc_1(A,B,C)) ) ) ) ) ).
fof(t4_bvfunc11,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_eqrel_1(C,A)
=> ( r1_tarski(k22_bvfunc_1(A,B,C),k22_bvfunc_1(A,B,k6_partit1(A)))
& r1_tarski(k22_bvfunc_1(A,B,k3_pua2mss1(A)),k22_bvfunc_1(A,B,C)) ) ) ) ) ).
fof(t5_bvfunc11,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( ( v2_bvfunc_2(B,A)
& B = k2_tarski(C,D) )
=> ( C = D
| ! [E,F] :
~ ( r2_hidden(E,C)
& r2_hidden(F,D)
& r1_xboole_0(E,F) ) ) ) ) ) ) ) ).
fof(t6_bvfunc11,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( ( v2_bvfunc_2(B,A)
& B = k2_tarski(C,D) )
=> ( C = D
| k2_bvfunc_2(A,B) = k2_partit1(A,C,D) ) ) ) ) ) ) ).
fof(t7_bvfunc11,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_bvfunc_2(A)))
=> ! [C] :
( m1_eqrel_1(C,A)
=> ! [D] :
( m1_eqrel_1(D,A)
=> ( B = k2_tarski(C,D)
=> ( C = D
| k5_bvfunc_2(A,C,B) = D ) ) ) ) ) ) ).
fof(t8_bvfunc11,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,B,C)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,B,D,E),k7_bvfunc_2(A,C,D,E)) ) ) ) ) ) ) ).
fof(t9_bvfunc11,axiom,
$true ).
fof(t10_bvfunc11,axiom,
$true ).
fof(t11_bvfunc11,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)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ) ).
fof(t12_bvfunc11,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,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ).
fof(t13_bvfunc11,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)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k6_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ) ).
fof(t14_bvfunc11,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,k7_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ).
fof(t15_bvfunc11,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,k5_valuat_1(A,k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)),k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,E),C,D)) ) ) ) ) ) ).
fof(t16_bvfunc11,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)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k7_bvfunc_2(A,k5_valuat_1(A,k6_bvfunc_2(A,B,C,D)),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,E),C,D)) ) ) ) ) ) ) ).
fof(t17_bvfunc11,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)
=> ( v2_bvfunc_2(C,A)
=> k5_valuat_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)) = k7_bvfunc_2(A,k5_valuat_1(A,k6_bvfunc_2(A,B,C,E)),C,D) ) ) ) ) ) ) ).
fof(t18_bvfunc11,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)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k6_bvfunc_2(A,k5_valuat_1(A,k6_bvfunc_2(A,B,C,D)),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,E),C,D)) ) ) ) ) ) ) ).
fof(t19_bvfunc11,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)
=> ( v2_bvfunc_2(C,A)
=> k5_valuat_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)) = k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,E),C,D) ) ) ) ) ) ) ).
fof(t20_bvfunc11,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)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k5_valuat_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)),k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,D),C,E)) ) ) ) ) ) ) ).
fof(t21_bvfunc11,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)
=> k5_valuat_1(A,k6_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E)) = k7_bvfunc_2(A,k6_bvfunc_2(A,k5_valuat_1(A,B),C,D),C,E) ) ) ) ) ) ).
fof(t22_bvfunc11,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)
=> k5_valuat_1(A,k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)) = k6_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,D),C,E) ) ) ) ) ) ).
fof(t23_bvfunc11,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)
=> k5_valuat_1(A,k6_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E)) = k7_bvfunc_2(A,k7_bvfunc_2(A,k5_valuat_1(A,B),C,D),C,E) ) ) ) ) ) ).
fof(t24_bvfunc11,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)
=> ( v2_bvfunc_2(C,A)
=> r1_bvfunc_1(A,k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,E),C,D)) ) ) ) ) ) ) ).
fof(t25_bvfunc11,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,k6_bvfunc_2(A,B,C,D),C,E),k6_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E)) ) ) ) ) ) ).
fof(t26_bvfunc11,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,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E)) ) ) ) ) ) ).
fof(t27_bvfunc11,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,k7_bvfunc_2(A,k6_bvfunc_2(A,B,C,D),C,E),k7_bvfunc_2(A,k7_bvfunc_2(A,B,C,D),C,E)) ) ) ) ) ) ).
%------------------------------------------------------------------------------