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

$\blacksquare$