Equivalent Statements for Transitive Set

Theorem
Let $a$ be a set.


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


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


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

$(1) \iff (3)$
This is shown in Class is Transitive iff Union is Subclass.

$(2) \iff (3)$
This is shown in Set is Transitive iff Subset of Power Set.