  Let F be a finite nonempty family of finite nonempty sets. We prove the
following: (i) F satisfies the condition of the title if and only if for every
pair of distinct subfamilies {A_1,...,A_r}, {B_1,...,B_s} of F, the union of
the A_i is different from the union of the B_i. (ii) If F satisfies the
condition of the title, then the number of subsets of the union of the members
of F containing at least one set of F is odd. We give two applications of these
results, one to number theory and one to commutative algebra.
