Equivalence Class is Unique

Theorem
Let $\mathcal R$ be an equivalence relation on $S$.

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

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