Kuratowski's Lemma

Theorem
Let $\left({S, \preceq}\right), S \ne \varnothing$ 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

He published this in 1922, thinking it little more than a corollary of the Hausdorff Maximal Principle.

In 1935, Max Zorn published a similar version, acknowledging Kuratowski's earlier work. This later version became the more famous one.