Wikipedia: Hausdorff maximality theorem

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The Hausdorff maximality theorem, defined by Felix Hausdorff in 1914 is an alternate formulation of Zorn's Lemma. It states that
#given a set A with a partial order named <, and two elements a and b
#given that a < b are incommensurable ie:
#*given that a < b is untrue
#*given that b < a is untrue

then A must have a subset named D with two elements d and e
such that e < d or d < e.

The Hausdorff maximality theorem, described by Felix Hausdorff in 1914, is an alternate formulation of Zorn's lemma and therefore also equivalent to the axiom of choice. It states that in a partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.