Epsilon Relation is Strictly Well-Founded

Theorem

Let $\Epsilon$ denote the epsilon relation.

Then $\Epsilon$ is a strictly well-founded relation on every class $A$.

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$\forall S: \paren {\exists x: x \in S \implies \exists y \in S: \forall x \in S: \neg x \in y}$

$\forall S: \paren {S \ne \O \implies \exists y \in S: \forall x \in S: \neg x \in y}$

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

We can weaken the antecedent of the above statement with this statement:

$\forall S: \paren {\paren {S \ne \O \land S \subseteq A} \implies \exists y \in S: \forall x \in S: \neg x \in y}$

Note that this step does not require that $A$ be a set: it can be any class, even a proper class.

By definition, it follows that $\Epsilon$ is a strictly well-founded relation on every class $A$.