# Epsilon is Foundational

## Theorem

Let $\Epsilon$ denote the epsilon relation.

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

## Proof

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.

By the axiom of foundation:

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

That is, by Nonempty Class has Members:

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

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 foundational relation on every class $A$.