Asymmetric Relation is Antisymmetric

Theorem
Let $\RR$ be an asymmetric relation.

Then $\RR$ is also antisymmetric.

Proof
Let $\RR$ be asymmetric.

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

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

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

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

is vacuously true.