# 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

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Let $x \in \mathbb U$.

Then by the Axiom of Extension:

- $x = \Epsilon^{-1} \left({x}\right)$

where $\Epsilon^{-1} \left({x}\right)$ denotes the preimage of $x$ under $\Epsilon$.

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.

$\blacksquare$