Definition 3.1.4 (Equality of sets). Two sets \(A\) and \(B\) are \(equal\), \(A=B\), iff every element of \(A\) is an element of \(B\) and vice versa. To put it another way, \(A=B\) if and only if every element \(x\) of \(A\) belongs also to \(B\), and every element \(y\) of \(B\) belongs also to \(A\).
$$ (\forall x, x \in A \implies x \in B) \land (\forall x, x \in B \implies x \in A) $$