How to show $A_1 \approx A_2 \iff \mathrm{card}(A_1)=\mathrm{card}(A_2)$

Question

For any set $A_1$ and $A_2$, let us define the relation $\approx$
if there exists a bijection between $A_1$ and $A_2$.
Then I want to show that $A_1 \approx A_2 \iff \mathrm{card}(A_1)=\mathrm{card}(A_2)$
where $\mathrm{card}$ means cardinal number.
I can use the fact : $A \preceq B$ and $B\preceq A \rightarrow A\approx B$ where $\preceq $ means there exists an one-to-one fucntion from $A$ to $B$.

Answer 1

HINT: It suffices to show that $\operatorname{card}(\alpha)=\alpha$ whenever $\alpha$ is a cardinal; and that $\approx$ is an equivalence relation.