Ordering on Singleton is Total Ordering

Theorem
Let $S = \set s$ be a singleton.

Let $\RR$ be an ordering on $S$.

Then $\RR$ is a total ordering on $S$.

Proof
By definition of ordering, $\RR$ is a reflexive relation.

Hence from Reflexive Relation on Singleton is Well-Ordering:
 * $\struct {S, \RR}$ is a well-ordered set.

Hence by definition of well-ordered set:
 * $\RR$ is a total ordering on $S$.

Hence the result.