# Shortest proof of Kuratowski's lemma

I want a short proof of Kuratowski's lemma (about maximal chains).

Does proving Kuratowski's lemma need first prove Zorn lemma?

I am writing a book in which I claim that the reader needs to know only basic set theory in order to understand it. For this reason I need a short proof of Kuratowski's lemma, for the readers of my book.

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Well, the proof using ZL seems to be straightforward, see PlanetMath. –  Martin Sleziak Oct 18 '12 at 12:34
There are elementary proofs, but they all provide little insight. The best proof is IMHO by transfinite recursion: Add elements until you cannot add elements anymore without the set not being a chain.. –  Michael Greinecker Oct 18 '12 at 12:34
Zorn's lemma is basic set theory. –  Chris Eagle Oct 18 '12 at 12:36
There are easy, elementary proofs of (KL implies ZL) and (ZL implies KL). Thus, proving KL can't be significantly easier than proving ZL. –  Chris Eagle Oct 18 '12 at 12:39