Inverse of Non-Reflexive Relation is Non-Reflexive

Theorem
Let $\RR$ be a relation on a set $S$.

If $\RR$ is non-reflexive, then so is $\RR^{-1}$.

Proof
Let $\RR$ be non-reflexive.

Then:

Thus $\RR^{-1}$ is not antireflexive.

Also:

Thus $\RR^{-1}$ is not reflexive.

Hence the result, by definition of non-reflexive relation.