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