$\omega_2$ is not the countable union of countable sets

Question

I'm not sure I quite understand the form of the proof in this post '$\omega_2$ is a not countable union of countable sets without AC' and similar ones. Is the idea to firstly show that there is an injection from the union into $\omega_1$, and then to show there is a surjection into $\omega_2$ to give a contradiction?

Answer 1

There is no surjection from $\omega_1$ onto $\omega_2$. So if a set can be mapped injectively into $\omega_1$, then it cannot be mapped surjectively onto $\omega_2$.