Definition:Epsilon Relation

Definition
As $\in$ (the membership primitive) is technically not a class itself nor a relation, but a primitive, it is necessary to introduce a class which behaves identically to the membership relation $\in$ for sets. Currently, it is improper syntax to say, for example, that $\in$ is a strict well-ordering on the ordinals, so we introduce the $\in$-relation to be able to create analogous statements for equality. This will be referred to as the epsilon relation and will be denoted by $\Epsilon$.


 * $\Epsilon = \{ \left({ x, y }\right) : x \in y \}$. Thus, it is a relation, taking arguments from ordered pairs of sets $x$ and $y$.

The relation is a collection of ordered pairs such that $\left({ x, y }\right) \in \Epsilon \iff x \in y$. (Notice that this does not hold if $x$ and $y$ are proper classes).