Connected Equivalence Relation is Trivial

Theorem
Let $S$ be a set.

Let $\RR$ be a relation on $S$ which is both connected and an equivalence relation.

Then $\RR$ is the trivial relation on $S$.

Proof
By definition of equivalence relation, $\RR$ is an equivalence relation :


 * $\Delta_S \cup \RR^{-1} \cup \RR \circ \RR \subseteq \RR$

From Relation is Connected iff Union with Inverse and Diagonal is Trivial Relation:
 * $\Delta_S \cup \RR^{-1} \cup \RR = S \times S$

Hence the result.