# Axiom:Axiom of Extension

## Axiom

Two sets are equal if and only if 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.

## Notes

This is the fundamental definition of what a set is: a set is determined by its elements.

## Also known as

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