Equivalence Class Equivalent Statements

Theorem
Let $\RR$ be an equivalence relation on $S$.

Let $x, y \in S$.


 * $(1): \quad x$ and $y$ are in the same $\RR$-class
 * $(2): \quad \eqclass x \RR = \eqclass y \RR$
 * $(3): \quad x \mathrel \RR y$
 * $(4): \quad x \in \eqclass y \RR$
 * $(5): \quad y \in \eqclass x \RR$
 * $(6): \quad \eqclass x \RR \cap \eqclass y \RR \ne \O$