Equivalence of Definitions of Asymmetric Relation

Definition 1 implies Definition 2
Let $\RR$ be a relation which fulfills the condition:
 * $\left({x, y}\right) \in \RR \implies \left({y, x}\right) \notin \RR$

Then:

Hence $\RR$ is asymmetric by definition 2.

Definition 2 implies Definition 1
Let $\RR$ be a relation which fulfils the condition:
 * $\RR \cap \RR^{-1} = \O$

Then:

Hence $\RR$ is asymmetric by definition 1.