Inverse of Non-Reflexive Relation is Non-Reflexive

Theorem
Let $\mathcal R$ be a relation on a set $S$.

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

Proof
Let $\mathcal R$ be non-reflexive.

Then:

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

Also:

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

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