Axiom:Axiom of Extension

Axiom
Let $A$ and $B$ be sets.

The axiom of extension states that $A$ and $B$ are equal they contain the same elements.

This can be formulated as follows:

Formulation 2
In set theories that define $=$ instead of admitting it as a primitive, the axiom of extension can be formulated as:

The order of the elements in the sets is immaterial.

Hence a set is completely and uniquely determined by its elements.

Class Theory
The axiom of extension in the context of class theory has the same form:

Also see

 * Definition:Set Equality
 * Definition:Equals