Definition:Set Equality/Definition 1

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Let $S$ and $T$ be sets.

$S$ and $T$ are equal if and only if they have the same elements:

$S = T \iff \paren {\forall x: x \in S \iff x \in T}$

Otherwise, $S$ and $T$ are distinct, or unequal.

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:

$\forall x: \paren {x \in A \iff x \in B}$

where $\in$ denotes class membership.

Also see