Epsilon Relation is Proper

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

Let $\Epsilon$ be the epsilon relation.

Then $\left({\mathbb U, \Epsilon}\right)$ is a proper relational structure.

Proof
Let $x \in \mathbb U$.

Then $x = \Epsilon^{-1} \left({ x }\right)$, the preimage of $x$ under $\Epsilon$, by the Axiom of Extension.

Since $x$ is a set, $\prec^{-1} \left({x}\right) = x$ is a set.

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