Equivalence of Definitions of Coreflexive Relation

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

Then:

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

Definition 2 implies Definition 1
Let $\mathcal R$ be a relation which fulfils the condition:
 * $\mathcal R \subseteq \Delta_S$

Then:

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