Union of Class is Transitive if Every Element is Transitive

Theorem

Let $A$ be a class.

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

Let $A$ be such that every element of $A$ is transitive.

Then $\bigcup A$ is also transitive.

Proof

Let $A$ be such that every $y \in A$ is transitive.

Let $x \in \bigcup A$.

Then $x$ is an element of some element $y$ of $A$.

We have by hypothesis that $y$ is transitive.

Hence, by definition of transitive class:

$x \subseteq y$

Because $y \in A$, by definition of union of class:

$y \subseteq \bigcup A$

So:

$x \subseteq \bigcup A$

As this is true for all $x \in A$, it follows by definition that $\bigcup A$ is transitive.

$\blacksquare$

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: $\S 5$ The union axiom: Exercise $5.3. \ \text {(d)}$