I was just reading a proof of the dimension theorem in Steven Roman's Advanced Linear Algebra. In addressing the cases of infinite bases, Roman proceeds to show that if $\mathcal{B}$ and $\mathcal{C}$ are bases of a space $V$, then $|\mathcal{B}|\leq |\mathcal{C}|$, working up to an application of the BSC-Theorem. Anyway, he uses the string
$$|\mathcal{B}|\leq\aleph_0|\mathcal{C}|=|\mathcal{C}|.$$
Sorry if it's an elementary question, but why does the equality follow? Here $\mathcal{C}$ is any infinite basis. Is it definition? I tried looking up multiplication of ordinals, but didn't find anything useful. Thanks.