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 $\map {\mathscr M} A$ denote that $A$ is small.


\(\displaystyle \map {\mathscr M} x \land \map {\mathscr M} y\) \(\leadsto\) \(\displaystyle \map {\mathscr M} {\set {x, y} }\) Axiom of Pairing
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map {\mathscr M} {\bigcup \set {x, y} }\) Axiom of Unions
\(\displaystyle \) \(\leadsto\) \(\displaystyle \map {\mathscr M} {x \cup y}\) Union of Doubleton

$\blacksquare$


Sources