Inverse of Antisymmetric Relation is Antisymmetric

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

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

Proof
Let $\mathcal R$ be antisymmetric.

Then:
 * $\left({x, y}\right) \land \left({y, x}\right) \in \mathcal R \implies x = y$

It follows that:
 * $\left({y, x}\right) \land \left({x, y}\right)\in \mathcal R^{-1} \implies x = y$

Thus it follows that $\mathcal R^{-1}$ is also antisymmetric.