Definition:Set Equality/Definition 2

Definition
Let $S$ and $T$ be sets.

Then $S$ and $T$ are equal iff:
 * $S$ is a subset of $T$

and
 * $T$ is a subset of $S$

Notation
This can be denoted in several ways:
 * $S = T \iff \left({S \subseteq T}\right) \land \left({T \subseteq S}\right)$

or:
 * $S = T \iff \left({S \subseteq T}\right) \land \left({S \supseteq T}\right)$

or:
 * $S = T \iff S \subseteq T \subseteq S$

Thus the subset relation is antisymmetric.

Also see

 * Equivalence of Definitions of Set Equality