## How do we formally define the successor universe $\mathscr{U}^+$ of a universe $\mathscr{U}$ in $\mathsf{ZFC}$?

A set $$\mathscr{U}$$ is a universe if the following conditions are met:

• For any $$x \in \mathscr{U}$$ we have $$x \subseteq \mathscr{U}$$

• For any $$x,y \in \mathscr{U}$$ we have $$\{x,y\} \in \mathscr{U}$$,

• For any $$x \in \mathscr{U}$$ we have $$\mathcal{P}(x) \in \mathscr{U}$$,

• For any family $$(x_i)_{i \in I}$$ of elements $$x_i \in \mathscr{U}$$ indexed by an element $$I \in \mathscr{U}$$ we have $$\bigcup_{i \in I} x_i \in \mathscr{U}$$.

Grothendieck introduced an addition axiom $$\mathscr{U}$$A which says that every set $$x$$ is contained in some universe $$\mathscr{U}$$.

I’ve seen some authors use the concept of the successor universe $$\mathscr{U}^+$$ of a given universe $$\mathscr{U}$$. It is the smallest universe which contains $$\mathscr{U}$$. However, I’m not sure how to prove that such a thing exists in $$\mathsf{ZFC}$$. If we knew that for any two universes $$\mathscr{U}$$ and $$\mathscr{V}$$ we have either $$\mathscr{U} \in \mathscr{V}$$ or $$\mathscr{V} \in \mathscr{U}$$, it would be easy. But I’m not sure if we can prove that latter without showing first that universes are equivalent to $$V_\kappa$$ for inaccessible cardinals $$\kappa$$.