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 $$\left[\!\left[{x}\right]\!\right]_{\mathcal{R}}$$.

Proof
This follows directly from the fact that the set of $$\mathcal{R}$$-classes forms a partition of $$S$$.