axiom of extensionality

If $X$ and $Y$ have the same elements, then $X=Y$.

The Axiom of Extensionality is one of the axioms of Zermelo-Fraenkel set theory.In symbols, it reads:

$$\forall u(u\in X\leftrightarrow u\in Y)\to X=Y.$$

Note that the converse,

$$X=Y\to \forall u(u\in X\leftrightarrow u\in Y)$$

is an axiom of the predicate calculus. Hence we have,

$$X=Y\leftrightarrow \forall u(u\in X\leftrightarrow u\in Y).$$

Therefore the Axiom of Extensionality expresses the most fundamental notion of a set: a set is determined by its elements.