Epsilon Relation is Strictly Well-Founded

Theorem
Let $E$ denote the epsilon relation.

Then $E$ is a foundational relation on every class $A$:


 * $E Fr A$

Proof
By the axiom of foundation:


 * $\forall S: \left({\exists x: x \in S \implies \exists y \in S: \forall x \in S: \neg x \in y }\right)$

that is:
 * $\forall S: \left({S \ne \varnothing \implies \exists y \in S: \forall x \in S: \neg x \in y }\right)$

This holds for all sets whose construction is based on the Zermelo-Fraenkel axioms.

Therefore by the Axiom of Subsets it will hold for all subsets of $A$ that are sets.

Therefore, by the definition of a foundational relation, $E$ must be foundational for every class $A$, including proper classes.