# Class is Transitive iff Union is Subset

## Theorem

A class $A$ is transitive if and only if:

$\displaystyle \bigcup A \subseteq A$

## Proof

### Necessary Condition

$A$ be transitive.

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

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.

$\Box$

### Sufficient Condition

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.

$\blacksquare$