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

Alternate Axiom of Extensionality
For set theories that define $=$ instead of admitting it as a primitive, the Axiom of Extensionality becomes:


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

where $A$ and $B$ denote sets.

Also see

 * Set Equality
 * Equals