Union of Class is Transitive if Every Element is Transitive

Theorem
Let $A$ be a class.

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

Let $A$ be such that every element of $A$ is transitive.

Then $\bigcup A$ is also transitive.

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 $A$.

By hypothesis, $y$ is transitive.

Hence, by definition of transitive, $x \subseteq y$.

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.