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

Note that the empty chain in $X$ has an upper bound in $X$ if and only if $X$ 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 $X$ is a non-empty partially ordered set such that every non-empty chain in $X$ has an upper bound, then $X$ has a maximal element. (In other words: Any non-empty inductively ordered set has a maximal element.)

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