Empty Mapping to Empty Set is Bijective

Theorem

Let $\nu: \O \to \O$ be an empty mapping.

Then $\nu$ is a bijection.

Proof 1

From Empty Mapping is Injective, $\nu$ is injective.

As the codomain of $\nu$ is empty, $\nu$ is vacuously surjective.

$\blacksquare$

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Proof 2

From Empty Mapping is Injective, $\nu$ is injective.

$\Box$

As the codomain of $\nu$ is empty, from Empty Mapping is Surjective iff Codomain is Empty, we have that $\nu$ is surjective.

$\Box$

Hence the result.

$\blacksquare$