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$ .