Axiom:Axiom of Extension

Axiom
Two sets are equal they contain the same elements:


 * $\forall x: \paren {x \in A \iff x \in B} \iff A = B$

The order of the elements in the sets is immaterial.

That is, a set is completely and uniquely determined by its elements.

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


 * $\forall x: \paren {\paren {A = B \land A \in x} \implies B \in x}$

where $A$ and $B$ denote sets.

Also known as
Otherwise known as the axiom of extensionality.

Also see

 * Definition:Set Equality
 * Definition:Equals