TPTP Problem File: SET948+1.p
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%------------------------------------------------------------------------------
% File : SET948+1 : TPTP v9.0.0. Released v3.2.0.
% Domain : Set theory
% Problem : Basic properties of sets, theorem 101
% Version : [Urb06] axioms : Especial.
% English : ((in(C,union(A,B)) & in(D,union(A,B)) ) => (C=D | disjoint(C,D)) )
% => union(intersection(A,B)) = intersection(union(A),union(B))
% Refs : [Byl90] Bylinski (1990), Some Basic Properties of Sets
% : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source : [Urb06]
% Names : zfmisc_1__t101_zfmisc_1 [Urb06]
% Status : Theorem
% Rating : 0.94 v9.0.0, 0.92 v8.2.0, 0.94 v7.4.0, 0.93 v7.3.0, 0.97 v7.2.0, 0.93 v7.1.0, 0.96 v7.0.0, 0.93 v6.4.0, 1.00 v3.2.0
% Syntax : Number of formulae : 19 ( 8 unt; 0 def)
% Number of atoms : 43 ( 10 equ)
% Maximal formula atoms : 5 ( 2 avg)
% Number of connectives : 34 ( 10 ~; 2 |; 7 &)
% ( 8 <=>; 7 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 5 ( 4 usr; 0 prp; 1-2 aty)
% Number of functors : 3 ( 3 usr; 0 con; 1-2 aty)
% Number of variables : 47 ( 43 !; 4 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
% library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
! [A,B] :
( in(A,B)
=> ~ in(B,A) ) ).
fof(commutativity_k2_xboole_0,axiom,
! [A,B] : set_union2(A,B) = set_union2(B,A) ).
fof(commutativity_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) ).
fof(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( subset(A,B)
& subset(B,A) ) ) ).
fof(d2_xboole_0,axiom,
! [A,B,C] :
( C = set_union2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
| in(D,B) ) ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( subset(A,B)
<=> ! [C] :
( in(C,A)
=> in(C,B) ) ) ).
fof(d3_xboole_0,axiom,
! [A,B,C] :
( C = set_intersection2(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,A)
& in(D,B) ) ) ) ).
fof(d4_tarski,axiom,
! [A,B] :
( B = union(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] :
( in(C,D)
& in(D,A) ) ) ) ).
fof(fc2_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(A,B)) ) ).
fof(fc3_xboole_0,axiom,
! [A,B] :
( ~ empty(A)
=> ~ empty(set_union2(B,A)) ) ).
fof(idempotence_k2_xboole_0,axiom,
! [A,B] : set_union2(A,A) = A ).
fof(idempotence_k3_xboole_0,axiom,
! [A,B] : set_intersection2(A,A) = A ).
fof(rc1_xboole_0,axiom,
? [A] : empty(A) ).
fof(rc2_xboole_0,axiom,
? [A] : ~ empty(A) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(symmetry_r1_xboole_0,axiom,
! [A,B] :
( disjoint(A,B)
=> disjoint(B,A) ) ).
fof(t101_zfmisc_1,conjecture,
! [A,B] :
( ! [C,D] :
( ( in(C,set_union2(A,B))
& in(D,set_union2(A,B)) )
=> ( C = D
| disjoint(C,D) ) )
=> union(set_intersection2(A,B)) = set_intersection2(union(A),union(B)) ) ).
fof(t4_xboole_0,axiom,
! [A,B] :
( ~ ( ~ disjoint(A,B)
& ! [C] : ~ in(C,set_intersection2(A,B)) )
& ~ ( ? [C] : in(C,set_intersection2(A,B))
& disjoint(A,B) ) ) ).
fof(t97_zfmisc_1,axiom,
! [A,B] : subset(union(set_intersection2(A,B)),set_intersection2(union(A),union(B))) ).
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