Union of Subclass is Subclass of Union of Class

Theorem
Let $A$ and $B$ be classes.

Let $\displaystyle \bigcup A$ and $\displaystyle \bigcup B$ denote the union of $A$ and union of $B$ respectively.

Let $A$ be a subclass of $B$:
 * $A \subseteq B$

Then $\displaystyle \bigcup A$ is a subset of $\displaystyle \bigcup B$:
 * $\displaystyle \bigcup A \subseteq \displaystyle \bigcup B$

Proof
By the axiom of unions, both $\displaystyle \bigcup A$ and $\displaystyle \bigcup B$ are sets.

Let $x \in \displaystyle \bigcup A$.

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:
 * $x \in \displaystyle \bigcup B$

Hence the result by definition of subset.