# 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

$\blacksquare$