Equivalence Class is Unique

Theorem
Let $\RR$ be an equivalence relation on $S$.

For each $x \in S$, the one and only one $\RR$-class to which $x$ belongs is $\eqclass x \RR$.

Proof
This follows directly from the Fundamental Theorem on Equivalence Relations: the set of $\RR$-classes forms a partition of $S$.