Asymmetric Relation is Antireflexive

Theorem
Every relation which is asymmetric is also antireflexive.

Proof
Let $\mathcal R$ be asymmetric.

Then, by definition, $\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \notin \mathcal R$.

Suppose $\left({x, x}\right) \in \mathcal R$. Then:

Thus $\mathcal R$ is antireflexive.