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A set $A$ is transitive if it satisfies any of the following equivalent conditions:

  1. each element of an element of $A$ is an element of $A$
  2. each element of $A$ is a subset of $A$
  3. $\bigcup A\subseteq A$
  4. $A \subseteq \mathcal{P}(A)$

The word transitive is used because $a\in b\in A$ implying $a\in A$ reminds the transitivity in order relations. In fact, for ordinals, set membership will be an order relation