Union of Small Classes is Small

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Theorem

Let $x$ and $y$ be small classes.

Then $x \cup y$ is also small.


Proof

Let $\mathscr M \left({A}\right)$ denote that $A$ is small.


\(\displaystyle \mathscr M \left({x}\right) \land \mathscr M \left({y}\right)\) \(\implies\) \(\displaystyle \mathscr M \left({\left\{{x, y}\right\}}\right)\) Axiom of Pairing
\(\displaystyle \) \(\implies\) \(\displaystyle \mathscr M \left({\bigcup \left\{{x, y}\right\}}\right)\) Axiom of Union
\(\displaystyle \) \(\implies\) \(\displaystyle \mathscr M \left({x \cup y}\right)\) Union of Doubleton

$\blacksquare$


Sources