Axiom of extensionality

Statement

Let $A$ and $B$ be sets. If $A$ and $B$ have the same elements, 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. 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.

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.