Hausdorff Maximal Principle

Statement

Let $A$ be a set. Let $<:A\to A\to \mathrm{Prop}$ be a partial order, and let $B\subseteq A$ be a totally ordered subset of $A$. Then there exists a maximum totally ordered subset $C$ of $A$ with $B\subseteq C\subseteq A$. That is, there is no $D$ such that $C⊊D\subseteq A$ and $D$ is totally ordered.

Remarks

This theorem is equivalent to AC in ZF.