# If Set Exists then Empty Set Exists

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## Theorem

If at least one set exists, then there exists an empty set.

## Proof

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Let $S$ be a set.

By the axiom of class comprehension, there is an empty class:

- $\O = \set { x : x \ne x }$

Since $x \in \O$ is never true, it follows vacuously that:

- $x \in \O \implies x \in S$

By the subclass definition:

- $\O \subseteq S$

By Subclass of Set is Set, $\O$ is a set.

$\blacksquare$

## Sources

- 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Separation Schema