Asymmetric Relation is Antisymmetric

Theorem
Let $\mathcal R$ be an asymmetric relation.

Then $\mathcal R$ is also antisymmetric.

Proof
Let $\mathcal R$ be asymmetric.

Then from the definition of asymmetric:
 * $\tuple {x, y} \in \mathcal R \implies \tuple {y, x} \notin \mathcal R$

Thus:
 * $\neg \exists \tuple {x, y} \in \mathcal R: \tuple {y, x} \in \mathcal R$

Thus:
 * $\set {\tuple {x, y} \in \mathcal R \land \tuple {y, x} \in \mathcal R} = \O$

Thus:
 * $\tuple {x, y} \in \mathcal R \land \tuple {y, x} \in \mathcal R \implies x = y$

is vacuously true.