How does the axiom of extensionality determine when two sets are identical?

In set theory, the axiom of extensionality establishes that a set is determined entirely by its members. Under this principle, two sets are considered identical if and only if they contain precisely the same elements.123

This means that identity in set theory is defined by membership rather than by the specific conditions or properties used to describe or specify the sets. Even if two sets are defined by different criteria, they are regarded as the same set as long as their membership is identical. For example, this principle ensures that there is only one null set, as all sets with no members share the same content.3