# Union of Transitive Class is Subclass

## Theorem

Let $A$ be a transitive class.

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

Then:

$\ds \bigcup A \subseteq A$

## Proof

Let $A$ be transitive.

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

Then by definition:

$\exists y \in A: x \in y$

By definition of transitive class:

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

and so:

$x \in A$

Hence the result by definition of subclass.

$\blacksquare$