Union of Transitive Class is Subclass

Theorem
Let $A$ be a transitive class.

Let $\ds \bigcup A$ denote the union of $A$.

Then:
 * $\ds \bigcup A \subseteq A$

Proof
Let $A$ be transitive.

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

Then by definition:
 * $\exists y \in A: x \in y$

By definition of transitive class:
 * $x \in y \land y \in A \implies x \in A$

and so:
 * $x \in A$

Hence the result by definition of subclass.