Let F be a nonempty family of sets with the following properties:
(a) If X∈F, then there are some Y∈F and Z∈F such that Y∩Z=∅ and Y∪Z=X.
(b) If X∈F, and Y∪Z=X,Y∩Z=∅, then either Y∈F or Z∈F.
Show that there is a decreasing sequence X0⊇X1⊇X2⊇... of sets Xn∈F such that n=0⋂∞Xn=∅.
F. Galvin