Condition for Membership of Equivalence Class

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

Let $\left[\!\left[{x}\right]\!\right]_\mathcal R$ denote the $\mathcal R$-equivalence class of $x$.

Then:
 * $\forall y \in S: y \in \left[\!\left[{x}\right]\!\right]_\mathcal R \iff \left({x, y}\right) \in \mathcal R$

Proof
From the definition of an equivalence class:


 * $\left[\!\left[{x}\right]\!\right]_\mathcal R = \left\{{y \in S: \left({x, y}\right) \in \mathcal R}\right\}$

Let $y \in S$ such that $y \in \left[\!\left[{x}\right]\!\right]_\mathcal R$.

Then by definition $\left({x, y}\right) \in \mathcal R$.

Similarly, let $\left({x, y}\right) \in \mathcal R$.

Again by definition, $y \in \left[\!\left[{x}\right]\!\right]_\mathcal R$.