# Equivalence Class of Element is Subset

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## Theorem

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

The $\mathcal R$-class of every element of $S$ is a subset of the set the element is in:

- $\forall x \in S: \eqclass x {\mathcal R} \subseteq S$

## Proof

\(\displaystyle y\) | \(\in\) | \(\displaystyle \eqclass x {\mathcal R}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \tuple {x, y}\) | \(\in\) | \(\displaystyle \mathcal R\) | Definition of Equivalence Class | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x\) | \(\in\) | \(\displaystyle S \land y \in S\) | Definition of Relation | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \eqclass x {\mathcal R}\) | \(\subseteq\) | \(\displaystyle S\) | Definition of Subset |

$\blacksquare$