One of the Zermelo-Fraenkel axioms of set theory, and the most obviously fundamental. It simply says that two sets are equal to each other if they have the same elements.

This is not entirely trivial. Two sets could be defined by different properties, for example the set of primes less than 5, and the set of solutions of x^2 - 5x + 6 = 0. But a set is an elementary entity defined purely by what members it has, so these are both the set {2, 3}.

Zermelo restricted this principle to sets but it may also be applied to classes generally, in which case it is not an axiom, but is called the Principle of Extensionality.

Formally: Ax Ay (x = y <=> Az (z in x <=> z in y))

ariels points out another one that having the same members would not necessarily define the same set if we didn't have this axiom. If sets came in flavours or colours, so that 2 could be red or blue, while still being 2. Then {red 2, red 3} and {blue 2, blue 3} might be different. This axiom says that if 2 is the same thing regardless of its colour, then these are both {2, 3} and the same set. Extensionality says that membership is the only way sets can be distinguished.

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