Epsilon Relation is Proper

Theorem
Let $\mathbb U$ be the universal class.

Let $\Epsilon$ be the epsilon relation.

Then $\struct {\mathbb U, \Epsilon}$ is a proper relational structure.

Proof
Let $x \in \mathbb U$.

Then by the Axiom of Extension:
 * $x = \map {\Epsilon^{-1} } x$

where $\map {\Epsilon^{-1} } x$ denotes the preimage of $x$ under $\Epsilon$.

Since $x$ is a set, $\map {\prec^{-1} } x = x$ is a set.

As this holds for all $x \in \mathbb U$, $\struct {\mathbb U, \Epsilon}$ is a proper relational structure.