Equivalence Class Equivalent Statements/1 iff 2

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

Let $x, y \in S$.


 * $(1): \quad x$ and $y$ are in the same $\RR$-class
 * $(2): \quad \eqclass x \RR = \eqclass y \RR$

Proof
By Equivalence Class is Unique:
 * $\eqclass x \RR$ is the unique $\RR$-class to which $x$ belongs

and:
 * $\eqclass y \RR$ is the unique $\RR$-class to which $y$ belongs.

As these are unique for each, they must be the same set.