a note on the union-closed sets conjecture

A collection A of finite sets is closed under union if A, B ? A implies that A?B ? A. The Union-Closed Sets Conjecture states that if A is a union-closed collection of sets, containing at least one non-empty set, then there is an element which belongs to at least half of the sets in A. We show that if q is the minimum cardinality of ?A taken over all counterexamples A, then any counterexample A has cardinality at least 4q-1.