Union of Transitive Class is Transitive/Proof 2

Theorem

Let $A$ be a class.

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

Let $A$ be transitive.

Then $\bigcup A$ is also transitive.

Proof

Let $A$ be transitive.

Let $x \in \bigcup A$.

By Class is Transitive iff Union is Subclass we have that:

$\bigcup A \subseteq A$

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:

$z \in \bigcup A$

Thus we have that:

$z \in x \implies z \in \bigcup A$

and so by definition of subclass:

$x \subseteq \bigcup A$

Thus we have that:

$x \in \bigcup A \implies x \subseteq \bigcup A$

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

$\blacksquare$