Definition:Set Equality/Definition 1

Definition
Two sets are equal if and only if they have the same elements.

This can be defined rigorously as:
 * $S = T \iff \paren {\forall x: x \in S \iff x \in T}$

where $S$ and $T$ are both sets.

Also see

 * Axiom:Axiom of Extension


 * Equivalence of Definitions of Set Equality