Axiom:Axiom of Extension

Axiom
Two sets are equal iff they contain the same elements:


 * $\forall x: \left({x \in A \iff x \in B}\right) \iff A = B$

The order of the elements in the sets is immaterial.

Also defined as
For set theories that define $=$ instead of admitting it as a primitive, the Axiom of Extension becomes:


 * $\forall x: \left({ \left({ A = B \land A \in x }\right) \implies B \in x }\right)$

where $A$ and $B$ denote sets.

Also known as
Otherwise known as the Axiom of Extensionality or Axiom of Extent.

Also see

 * Definition:Set Equality
 * Definition:Equals