# 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

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 5.8$