Zorn's lemma

Zorn’s lemma: Let $(P,\le)$ be a partially ordered set . Suppose that for every chain $C\subseteq P$ (i.e. a subset that is totally ordered by $\le$), there exists an upper bound $u\in P$ of $C$. Then $P$ has a maximal element: an element $m\in P$ such that there is no $p\in P$ with $m<p$.

Over the usual background axioms of set theory, Zorn’s lemma is equivalent to the axiom of choice and to the well-ordering theorem .

Proof sketch (idea): Using choice, one builds an ascending chain by repeatedly extending it when possible; maximality is forced when the process cannot continue, and the chain hypothesis ensures the process has an upper bound that becomes maximal.