Corollary 4.7.2.4. Let $S$ and $T$ be sets. Then $S$ and $T$ have the same cardinality if and only if there exists a bijection $S \xrightarrow {\sim } T$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Choose well-orderings $(S, \leq _ S)$ and $(T, \leq _ T)$ having order types $|S|$ and $|T|$, respectively. If $|S| = |T|$, then there is an isomorphism of linearly ordered sets $(S, \leq _ S) \simeq (T, \leq _{T} )$, and therefore a bijection $S \xrightarrow {\sim } T$. The converse follows from Proposition 4.7.2.3. $\square$