Hausdorff maximality theorem

The Hausdorff maximality theorem, defined by Felix Hausdorff in 1914 is an alternate formulation of Zorn's Lemma. It states that

1. given a set A with a partial order named <, and two elements a and b
2. given that a < b are incommensurable ie:
3. *given that a < b is untrue
4. *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.