$\operatorname{Card}(X) \leq\operatorname{Card}(Y)$ iff $\aleph (X) \leq \aleph(Y)$

Question

For any two sets $X$ and $Y$, we write $\operatorname{Card}(X)\leq\operatorname{Card}(Y)$ if an injection $X \rightarrow Y$ exist.

I have tried

Suppose $\aleph (X) \leq \aleph(Y)$, where $\aleph (X)$ is the initial ordinal of the equipotence class of $X$. Then $\aleph (X) \subset \aleph(Y) $. Thus $\aleph (X)$ is isomorphic to some ideal of $\aleph (Y)$. Further how to prove that $\operatorname{Card}(X)\leq\operatorname{Card}(Y)$

Please help me to solve further. Thank you

Answer 1

HINT: If $f\colon X\to Y$ is an injection, then there is an injection from $\aleph(X)$ into $\aleph(Y)$. Now use the fact that those are initial ordinals to conclude that $\aleph(X)\leq\aleph(Y)$.