Epsilon is Foundational

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Theorem

Let $\Epsilon$ denote the epsilon relation.


Then $\Epsilon$ is a foundational relation on every class $A$.


Proof

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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, by Nonempty Class has Members:

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

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


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

$\forall S: \left({\left({S \ne \varnothing \land S \subseteq A}\right) \implies \exists y \in S: \forall x \in S: \neg x \in y}\right)$

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 foundational relation on every class $A$.