Equivalence Class Equivalent Statements

Theorem
Let $\mathcal R$ be an equivalence on $S$.

Then $\forall x, y \in S$, the following statements are all equivalent:


 * $(1): \quad x$ and $y$ are in the same $\mathcal R$-class
 * $(2): \quad \left[\!\left[{x}\right]\!\right]_\mathcal R = \left[\!\left[{y}\right]\!\right]_\mathcal R$
 * $(3): \quad x \mathcal R y$
 * $(4): \quad x \in \left[\!\left[{y}\right]\!\right]_\mathcal R$
 * $(5): \quad y \in \left[\!\left[{x}\right]\!\right]_\mathcal R$
 * $(6): \quad \left[\!\left[{x}\right]\!\right]_\mathcal R \cap \left[\!\left[{y}\right]\!\right]_\mathcal R \ne \varnothing$

Proof

 * $(1)$ and $(2)$ are equivalent because, by Equivalence Class is Unique, $\left[\!\left[{x}\right]\!\right]_\mathcal R$ is the unique $\mathcal R$-class to which $x$ belongs, and $\left[\!\left[{y}\right]\!\right]_\mathcal R$ is the unique $\mathcal R$-class to which $y$ belongs. As these are unique for each, they must therefore be the same set.


 * $(2)$ is equivalent to $(3)$ by Equivalence Class holds Equivalent Elements.


 * $(3)$ is equivalent to $(4)$ by the definition of Equivalence Class.


 * $(3)$ is equivalent to $(5)$ through dint of the symmetry of $\mathcal R$ and the definition of Equivalence Class.


 * $(3)$ is equivalent to $(6)$ from Equivalence Classes are Disjoint.