Reflexive Reduction of Antisymmetric Relation is Asymmetric

Theorem
Let $S$ be a set.

Let $\mathcal R$ be an Antisymmetric Relation on $S$.

Let $\mathcal R^\ne$ be the Reflexive Reduction of $\mathcal R$.

Then $\mathcal R^\ne$ is asymmetric.

Proof
Let $a, b \in S$.

Suppose for the sake of contradiction that
 * $a \mathrel{\mathcal R^\ne} b$ and $b \mathrel{\mathcal R^\ne} a$.

Then by the definition of reflexive reduction, $a \mathrel{\mathcal R} b$, $b \mathrel{\mathcal R} a$, and $a \ne b$.

But this contradicts the antisymmetry of $\mathcal R$.

Thus by the definition of an Asymmetric Relation, $\mathcal R^\ne$ is asymmetric.