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.
\(\ds \map {\mathscr M} x \land \map {\mathscr M} y\) | \(\leadsto\) | \(\ds \map {\mathscr M} {\set {x, y} }\) | Axiom of Pairing | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \map {\mathscr M} {\bigcup \set {x, y} }\) | Axiom of Unions | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \map {\mathscr M} {x \cup y}\) | Union of Doubleton |
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.8$