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.

Title	axiom of extensionality
Canonical name	AxiomOfExtensionality
Date of creation	2013-03-22 13:42:40
Last modified on	2013-03-22 13:42:40
Owner	Sabean (2546)
Last modified by	Sabean (2546)
Numerical id	5
Author	Sabean (2546)
Entry type	Axiom
Classification	msc 03E30
Synonym	extensionality