Union of Class is Transitive if Every Element is Transitive/Proof

Proof
Let $A$ be such that every $y \in A$ is transitive.

Let $x \in \bigcup A$.

Then $x$ is an element of some element $y$ of $A$.

We have that $y$ is transitive.

Hence, by definition of transitive class:
 * $x \subseteq y$

Because $y \in A$, by definition of union of class:
 * $y \subseteq \bigcup A$

So:
 * $x \subseteq \bigcup A$

As this is true for all $x \in A$, it follows by definition that $\bigcup A$ is transitive.