Hausdorff's Maximal Principle is equivalent to Axiom of Choice

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

Proof

 * Axiom of Choice implies Hausdorff's Maximal Principle


 * Hausdorff's Maximal Principle implies Zermelo's Well-Ordering Theorem
 * Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice