Set is Transitive iff Subset of Power Set

Theorem

A set $a$ is transitive if and only if:

$a \subseteq \powerset a$

where $\powerset a$ denotes the power set of $a$.

Proof

Necessary Condition

Let $a$ be transitive.

Let $x \in a$.

By definition of transitive set:

$x \subseteq a$

Then by definition of power set:

$x \in \powerset a$

Hence, by definition of subset:

$a \subseteq \powerset a$

$\Box$

Sufficient Condition

Let $a \subseteq \powerset a$.

Let $x \in a$.

Then by definition of subset:

$x \in \powerset a$

By definition of power set:

$x \subseteq a$

As this is true for all $x \in a$, it follows by definition that $a$ 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: $\S 10$ Some useful facts about transitivity: Theorem $10.4$