Equivalent Statements for Transitive Set

Theorem

Let $a$ be a set.

The following statements are equivalent:

$(1): \quad \bigcup a \subseteq a$

$(2): \quad a \subseteq \powerset a$

$(3): \quad a$ is a transitive set

Proof

$(1) \iff (3)$

This is shown in Class is Transitive iff Union is Subclass.

$\Box$

$(2) \iff (3)$

This is shown in Set is Transitive iff Subset of Power Set.

$\blacksquare$

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: Remark