Union of Subclass is Subclass of Union of Class
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Theorem
Let $A$ and $B$ be classes.
Let $\bigcup A$ and $\bigcup B$ denote the union of $A$ and union of $B$ respectively.
Let $A$ be a subclass of $B$:
- $A \subseteq B$
Then $\bigcup A$ is a subclass of $\bigcup B$:
- $\bigcup A \subseteq \bigcup B$
Proof
Let $x \in \bigcup A$.
Then:
- $\exists y \in A: x \in y$
But as $A \subseteq B$ it follows that $y \in B$.
That is:
- $\exists y \in B: x \in y$
That is:
- $x \in \bigcup B$
Hence the result by definition of subclass.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom: Exercise $5.2. \ \text {(a)}$