Irreducible uniform cover

A uniform cover is irreducible when it is not the disjoint union, as a multiset, of two nonempty uniform covers. There are only finitely many irreducible uniform covers of a fixed finite set: their multiplicity vectors form an antichain in ${\mathbb{N}}^{2^n}$, and Dickson lemma forbids an infinite antichain there.

Every subset of ${\mathbb{N}}^d$ has finitely many coordinatewise minimal elements. Equivalently, ${\mathbb{N}}^d$ has no infinite antichain in its coordinatewise partial order.