---
tag: 0TB16T2C97
kind: statement
role: axiom
name: Axiom of extensionality
aliases: [extensionality]
msc: ["03E30"]
uses: [0SJ3RX9TXA, 0EZ758XE73, 0V6GAWTBYW, 0B88PQTJ66]
sources:
  - url: https://en.wikipedia.org/wiki/Axiom_of_extensionality
    locator: lead section
same_as:
  wikidata: Q740424
  wikipedia: Axiom of extensionality
---

## Statement

Let $A$ and $B$ be [[sets|0SJ3RX9TXA]].
If $A$ and $B$ have the same [[elements|0EZ758XE73]], then $A = B$:
$$\forall A \, \forall B \, \bigl( \forall x \, (x \in A \leftrightarrow x \in B) \to A = B \bigr).$$

## Remarks

It is one of the axioms of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].
The converse is a rule of logic: equal sets have the same elements.
In practice, $A = B$ is often proved with $A \subseteq B$ and $B \subseteq A$. See [[subset|0FE8Q5S42Y]].

## Explanations

### L1 human

A set is nothing more than its elements.
If two sets contain exactly the same things, they are the same set, however you describe them.
For example, $\{x \in \mathbb{R} \mid x^2 = 1\}$ and $\{-1, 1\}$ are equal.