[[Set theory MOC]]
# Zorn's lemma

**Zorn's lemma** is a proposition of set theory that is equivalent to the [[Axiom of Choice]] and the [[Well ordering principle]] over [[ZF]].[^2008]

> If $P$ is a [[Poset|partially ordered set]] in which every [[Poset#^chain|chain]] has an upper bound, then $P$ has a [[Poset#^maximal|maximal]] element. #m/thm/set/zfc

[^2008]: 2008\. [[Sources/@romanAdvancedLinearAlgebra2008|Advanced Linear Algebra]], p. 12

Zorn's lemma may be weakened to the [[Ultrafilter lemma]].

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