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
Aiming for a contradiction, suppose that $\mathcal R^\ne$ is not asymmetric.

That is:
 * $\exists a, b \in S: a \mathrel{\mathcal R^\ne} b$ 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 definition, $\mathcal R^\ne$ is an asymmetric relation.