Two sets with exactly the same members are equal. In a transitive model, all actual members of its sets are available internally, so ambient extensionality gives the internal axiom as well.
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The Axiom of Extensionality is a fundamental principle in set theory, specifically within the framework of Zermelo-Fraenkel set theory (ZF), which is one of the most common foundational systems for mathematics. The axiom states that two sets are considered to be equal if and only if they have the same elements.