Equivalence of Definitions of Asymmetric Relation

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

Then:

Hence $\mathcal R$ is asymmetric by definition 2.

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

Then:

Hence $\mathcal R$ is asymmetric by definition 1.