Equivalent Statements for Transitive Set

Theorem
Let $A$ be a class.


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


 * $(2): \quad A \subseteq \powerset A$ is a transitive class


 * $(3): \quad A$ is a transitive class

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

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