Condition for Membership of Equivalence Class

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

Let $\eqclass x \RR$ denote the $\RR$-equivalence class of $x$.

Then:
 * $\forall y \in S: y \in \eqclass x \RR \iff \tuple {x, y} \in \RR$

Proof
From the definition of an equivalence class:


 * $\eqclass x \RR = \set {y \in S: \tuple {x, y} \in \RR}$

Let $y \in S$ such that $y \in \eqclass x \RR$.

Then by definition $\tuple {x, y} \in \RR$.

Similarly, let $\tuple {x, y} \in \RR$.

Again by definition, $y \in \eqclass x \RR$.