Union of Transitive Class is Transitive

Theorem
Let $A$ be a class.

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

Let $A$ be transitive.

Then $\displaystyle \bigcup A$ is also transitive.

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

By Class is Transitive iff Union is Subset we have that
 * $\displaystyle \bigcup A \subseteq A$

Thus by definition of subclass:
 * $x \in A$

As $A$ is transitive:
 * $x \subseteq A$

Let $z \in x$.

As $x \subseteq A$, it follows by definition of subclass that:
 * $z \in A$

Thus we have that:
 * $\exists x \in A: z \in x$

and so by definition of union of class:
 * $z \in \displaystyle \bigcup A$

Thus we have that:
 * $z \in x \implies z \in \displaystyle \bigcup A$

and so by definition of subclass:
 * $x \subseteq \displaystyle \bigcup A$

Thus we have that:
 * $x \in \displaystyle \bigcup A \implies x \subseteq \displaystyle \bigcup A$

Hence $\displaystyle \bigcup A$ is a transitive class by definition.