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.