Zorn's lemma

Zorn’s lemma: Let $(P,\le)$ be a partially ordered set . If every chain $C\subseteq P$ has an upper bound in $P$, then $P$ has a maximal element.

Here a chain means a subset $C\subseteq P$ on which the restriction of $\le$ is a total order , and an element $m\in P$ is maximal if there is no $p\in P$ with $m<p$ (i.e., $m\le p$ and $m\ne p$). Over ZF, Zorn’s lemma is equivalent to the Axiom of Choice and to the well-ordering theorem .