Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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Cantor's Attic (original site)
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A set is transitive if and only if all of its elements are subsets.

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

Properties of Transitive Sets

If $A$ is transitive, then if $x$ and $A$ are connected somehow by membership (that is, $x \in y \in z \ldots \in A$), then $x \in A$.

The intersection of two transitive sets is transitive.

In set theory, transitive sets play an important role in models of ZFC. See transitive ZFC model.