Symmetric Transitive and Serial Relation is Reflexive

Theorem
Let $\RR$ be a relation which is:
 * symmetric
 * transitive
 * serial.

Then $\RR$ is reflexive.

Thus such a relation is an equivalence.

Proof
Let $S$ be a set on which $\RR$ is a relation which is symmetric, transitive and serial.

As $\RR$ is symmetric:
 * $x \mathrel \RR y \implies y \mathrel \RR x$

As $\RR$ is transitive:
 * $x \mathrel \RR y \land y \mathrel \RR x \implies x \mathrel \RR x$

As $\RR$ is serial:
 * $\forall x \in S: \exists y \in S: x \mathrel \RR y$

Let $x \in S$.

Then

Thus:
 * $\forall x: x \mathrel \RR x$

and by definition $\RR$ is reflexive.

It follows by definition that such a relation is an equivalence relation.