The following conjecture of U Faigle and B Sands is proved: For every number R > 0 there exists a number n(R) such that if is a finite distributive lattice whose width w() (size of the largest antichain) is at least n(R), then || ⩾ Rw(). In words this says that as one considers increasingly large distributive lattices, the maximum sized antichain contains a vanishingly small proportion of the elements.