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# State/Axiom of extensionality

Axiom: (Axiom of Extensionality) If $X$ and $Y$ are sets, then $X = Y$ if and only if for every set $z$, $z \in X$ if and only if $z \in Y$.

In other words, two sets are equal if they have the same elements (every element of one set is an element of the other).

This is one of the Zermelo-Fraenkel axioms of set theory.

## Logical Connections

This statement logically relies on the following definitions and statements:

The following statements and definitions rely on the material of this page: Def/Empty set, Def/Intersection, Def/Natural number, Def/Ordered pair, Def/Power set, Def/Union, and Def/Unordered pair

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