Asymmetric Relation is Antisymmetric

Theorem
Every relation which is asymmetric is also antisymmetric.

Proof
Let $$\mathcal{R}$$ be asymmetric.

Then from the definition of asymmetric, $$\left({x, y}\right) \in \mathcal{R} \Longrightarrow \left({y, x}\right) \notin \mathcal{R}$$.

Thus $$\lnot \exists \left({x, y}\right) \in \mathcal{R}: \left({y, x}\right) \in \mathcal{R}$$.

Thus $$\left\{{\left({x, y}\right) \in \mathcal{R} \land \left({y, x}\right) \in \mathcal{R}}\right\} = \varnothing$$.

Thus $$\left({x, y}\right) \in \mathcal{R} \land \left({y, x}\right) \in \mathcal{R} \Longrightarrow x = y$$ is vacuously true.