Definition:Set Equality

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

This can be defined rigorously as:

$$S = T \iff \left( {\forall x: x \in S \iff x \in T}\right)$$

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

Axiomatic Set Theory
The concept of set equality is axiomatised in the Axiom of Extension in Zermelo-Fraenkel set theory.