An ordinal between an ordinal and its cardinality

Question

Suppose $\alpha, \beta$ are ordinals (the first is not a cardinal) and $|\alpha| \subset \beta \subseteq \alpha$. Then why $|\alpha| \sim \beta?$

I know Cantor–Bernstein theorem does the job but how?

Well, there is injective function from $|\alpha|$ into $\beta$ .

How can I show that there is injective function from $\beta$ into $|\alpha|$?

Answer 1

Hint: There is an injective function from $\beta$ into $\alpha$, and $\alpha$ admits a bijection with $|\alpha|$.

Answer 2

Since $|\alpha|\sim\alpha$, so there is bijection $f:\alpha\to|\alpha|$. If $i:\beta\to\alpha$ is inclusion map, consider $f\circ i$.