Injection from Proper Class to Class

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Theorem

Let $A$ be a class.

Let $\mathrm P$ be a proper class.

Let $f: \mathrm P \to A$ be an injection.


Then $A$ is proper.


Proof

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Aiming for contradiction, suppose that $A$ is not proper.

Then it must be a set.


By Injection to Image is Bijection, it follows that the restriction $f \restriction_{\mathrm P \times f \left({\mathrm P}\right)}$ is a bijection.

By the corollary of Class is Proper iff Bijection from Class to Proper Class, $f \left({\mathrm P}\right)$ is proper.


But since $f \left({\mathrm P}\right) \subseteq A$, this contradicts Subclass of Set is Set.

And so by contradiction, $A$ cannot be a set.

Therefore $A$ is proper.

$\blacksquare$