Equivalence of Definitions of Symmetric Relation

Theorem
The following definitions of antisymmetric relation are equivalent:

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

Then:

Thus $\mathcal R^{-1} \subseteq \mathcal R$.

From Inverse Relation Equal iff Subset:
 * $\mathcal R^{-1} = \mathcal R$

Hence $\mathcal R$ is symmetric by the definition 2.

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

Thus:

Hence $\mathcal R$ is symmetric by the definition 1.