If every chain has an upper bound, a maximal element exists.Suggest a slogan
Let \((P,\le)\) be a partially ordered set in which every chain has an upper bound in \(P\). Then \(P\) has a maximal element: an element \(m\in P\) such that \(m\le x\) with \(x\in P\) implies \(x=m\).
The empty chain is a chain, so the hypothesis forces \(P\) to be nonempty.
State Zorn's lemma.