Hausdorff's Maximal Principle is equivalent to Axiom of Choice

Theorem
Every ordered set has a maximal chain the axiom of choice holds.

Proof

 * Hausdorff Maximal Principle Implies Well-Ordering Theorem
 * Well-Ordering Theorem is Equivalent to Axiom of Choice
 * Hausdorff Maximal Principle, which is proven using the Axiom of Choice