Union of Class is Subclass implies Class is Transitive

Theorem
Let $A$ be a class.

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

Let:
 * $\displaystyle \bigcup A \subseteq A$

Then $A$ is transitive.

Proof
Let $\displaystyle \bigcup A \subseteq A$.

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

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

By definition of subclass:
 * $x \in A$

Thus we have that:


 * $x \in y \land y \in A \implies x \in A$

It follows by definition that $A$ is a transitive class.