Is the following correct?


* If something is provable for small sets in ZFC with Axiom of Universes ("For any set x, there exists a universe U such that x∈U.") then it is provable for any sets in ZFC (without Axiom of Universes).


I want to work in ZFC with Axiom of Universes, but I wish my results be downgradable to plain ZFC. Help?



Downgrading from ZFC with universes to ZFC
No can do. ZFC+AU proves con(ZFC), ZFC doesn't.


