Class is Transitive iff Union is Subclass

Theorem

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

$\bigcup A \subseteq A$

Proof

Necessary Condition

Let $A$ be transitive.

Let $x \in \bigcup A$.

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 $\bigcup A \subseteq A$.

Let $x \in \bigcup A$.

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.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 10$ Some useful facts about transitivity: Theorem $10.1$