Total Ordering is Total Relation

Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a total ordering.

Then $\RR$ is a total relation.

Proof

By definition of total ordering:

$\RR$ is a reflexive relation on the strength of being an ordering

$\RR$ is a connected relation on the strength of being a total ordering.

$\blacksquare$