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  • $\begingroup$ You are right. I apologize for my poor wording. I edited the title and the question. First of all, I wanted to write "construct" instead of "define". Second, I wanted to prove not the existence of a universe in $\mathsf{ZFC}$, but the implication "$\mathscr{U}$ exists $\Longrightarrow \mathscr{U}^+$ exists". $\endgroup$ Commented Nov 18, 2018 at 21:55
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    $\begingroup$ @Jxt921 That implication isn't provable in ZFC either - what is provable is that every universe has at most one successor, and that "every universe has a successor" is equivalent to "there is no largest inaccessible." $\endgroup$ Commented Nov 18, 2018 at 23:57