Cardinality of a Grothendieck Universe Let us work in $ZFC+U$ where $U$ is the existence of a Grothendieck Universe. Let $\mathrm{On}(U) = \mathrm{On}\cap U$ denote the set of ordinals in $U$. How can I show that the cardinality of $U$ is equal to its set of ordinals, i.e. $|\mathrm{On}(U)| = |U|$?

It's slightly easier if you first prove that $U=V_\kappa$ for a strongly inaccessible $\kappa$.
Now you can, quite easily too, prove that for every $\alpha$, $|V_{\omega+\alpha}|=\beth_\alpha$. So $|V_\kappa|=\beth_\kappa$.
Finally, $\kappa\leq\beth_\kappa$, and by induction $\beth_\alpha<\kappa$ for $\alpha<\kappa$ (here the strong inaccessibility plays a crucial role), and therefore $\kappa=\beth_\kappa$.
