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 fortiori 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.

$\blacksquare$