If is a partially ordered set such that every chain in has an upper bound, then has a maximal element.

Note that the empty chain in has an upper bound in if and only if is non-empty. Because this case is rather different from the case of non-empty chains, Zorn's Lemma is often stated in the following form: If is a non-empty partially ordered set such that every non-empty chain in has an upper bound, then has a maximal element.

In ZF, Zorn's Lemma is equivalent to the Axiom of Choice.