# Equivalence Class is not Empty

## Theorem

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

Then no $\mathcal R$-class is empty.

## Proof

 $\, \displaystyle \forall \eqclass x {\mathcal R} \subseteq S: \exists x \in S: \,$ $\displaystyle x$ $\in$ $\displaystyle \eqclass x {\mathcal R}$ Definition of Equivalence Class $\displaystyle \leadsto \ \$ $\displaystyle \eqclass x {\mathcal R}$ $\ne$ $\displaystyle \O$ Definition of Empty Set

$\blacksquare$