Union of Subclass is Subclass of Union of Class

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.