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:

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

$\ds y$

$\in$

$\ds \eqclass x \RR$

$\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

$\blacksquare$