Relation Asymmetric therefore Antisymmetric

Theorem

Every relation which is asymmetric is also antisymmetric.

Proof

Let $\mathcal R$ be asymmetric.

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

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

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

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