Definition:Set Equality/Definition 2

Definition

Let $S$ and $T$ be sets.

$S$ and $T$ are equal if and only if both:

$S$ is a subset of $T$

and

$T$ is a subset of $S$

Notation

This can be denoted in several ways:

$S = T \iff \paren {S \subseteq T} \land \paren {T \subseteq S}$

or:

$S = T \iff \paren {S \subseteq T} \land \paren {S \supseteq T}$

or:

$S = T \iff S \subseteq T \subseteq S$

Equality of Classes

In the context of class theory, the same definition applies.

Let $A$ and $B$ be classes.

$A$ and $B$ are equal, denoted $A = B$, if and only if:

$A \subseteq B$ and $B \subseteq A$

where $\subseteq$ denotes the subclass relation.