Equivalence Class of Element is Subset

Theorem

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

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

$\forall x \in S: \eqclass x \RR \subseteq S$

Proof

\(\ds y\)

\(\in\)

\(\ds \eqclass x \RR\)

\(\ds \leadsto \ \ \)

\(\ds \tuple {x, y}\)

\(\in\)

\(\ds \RR\)

Definition of Equivalence Class

\(\ds \leadsto \ \ \)

\(\ds x\)

\(\in\)

\(\ds S \land y \in S\)

Definition of Relation

\(\ds \leadsto \ \ \)

\(\ds \eqclass x \RR\)

\(\subseteq\)

\(\ds S\)

Definition of Subset

$\blacksquare$