Zorn's lemma

Let $(P,\le )$ be a nonempty partially ordered set. If every chain in a partial order $C\subseteq P$ has an upper bound in $P$, then $P$ has a maximal element. The hypothesis meansThe upper bound may depend on $C$ and need not belong to $C$. A least upper bound is not required. The conclusion asserts that some $m\in P$ satisfies $m\le x⟹x=m$; it does not assert a greatest element, which would satisfy $x\le m$ for every $x\in P$.^[ref]

Here a chain in a partial order may be the empty set. Every member of $P$ is an upper bound of the empty chain in a partial order, so including it in the hypothesis already forces $P$ to be nonempty. If “chain” means only nonempty chains in a partial order, the separate nonempty-$P$ assumption is essential: the empty poset otherwise satisfies the hypothesis vacuously and has no maximal element. These are two equivalent conventions for the theorem.

The strength of Zorn's lemma is that compatible partial constructions can have a completed, unextendible member without an explicit procedure for finding it. The next examples separate the upper bound calculation from the reason a maximal element has the desired property. All chains in a partial order must be covered, including uncountable ones, rather than just increasing sequences. The condition is sufficient, not necessary: a poset can have a maximal element and still have an unbounded chain in a partial order elsewhere.