Kuratowski's Lemma

Theorem
Let $\struct {S, \preceq}, S \ne \O$ be a non-empty ordered set.

Then every chain in $S$ is the subset of some maximal chain.

Also see

 * Zorn's Lemma
 * Kneser's Lemma
 * Tukey's Lemma
 * Hausdorff Maximal Principle