Equivalence Class holds Equivalent Elements

Theorem
Let $$\mathcal{R}$$ be an equivalence relation on a set $$S$$. Then:

$$\left({x, y}\right) \in \mathcal{R} \iff \left[\left[{x}\right]\right]_{\mathcal{R}} = \left[\left[{y}\right]\right]_{\mathcal{R}}$$

Proof

 * First we prove that $$\left({x, y}\right) \in \mathcal{R} \Longrightarrow \left[\left[{x}\right]\right]_{\mathcal{R}} = \left[\left[{y}\right]\right]_{\mathcal{R}}$$.

Suppose $$\left({x, y}\right) \in \mathcal{R}: x, y \in S$$.

Now:

... so we have shown that $$\left({x, y}\right) \in \mathcal{R} \Longrightarrow \left[\left[{x}\right]\right]_{\mathcal{R}} = \left[\left[{y}\right]\right]_{\mathcal{R}}$$.


 * Next we prove that $$\left[\left[{x}\right]\right]_{\mathcal{R}} = \left[\left[{y}\right]\right]_{\mathcal{R}} \Longrightarrow \left({x, y}\right) \in \mathcal{R}$$:

... so we have shown that $$\left[\left[{x}\right]\right]_{\mathcal{R}} = \left[\left[{y}\right]\right]_{\mathcal{R}} \Longrightarrow \left({x, y}\right) \in \mathcal{R}$$.


 * Thus, we have:

So by Material Equivalence, $$\left({x, y}\right) \in \mathcal{R} \iff \left[\left[{x}\right]\right]_{\mathcal{R}} = \left[\left[{y}\right]\right]_{\mathcal{R}}$$.