A size-width inequality for distributive lattices

We show that every collection ofw sets such that none contains any other generates at least 3w-2 sets under the operations of taking intersections and unions. In particular, we prove that if the finite distributive lattice ? contains an antichain of sizew, then |?| ≧3w, forw≠1, 2, 3, 6, where the minimal exceptional cases arise from the Boolean algebras ?_n withn=0, 1, 2, 3, 4 atoms.