TITLE: Model of homotopy type theory in ZFC
QUESTION [8 upvotes]: There is a model of ZFC in homotopy type theory
Does exist a model of homotopy type theory in ZFC? 
Is there a proof of "equal logical expressivity" of these theories?
p.s. I use word "model" in common sense, because I don't know model theory

REPLY [8 votes]: Yes, as long as by "ZFC" you include its extension with some inaccessible cardinals, or (perhaps) are willing to play games with natural models.  In other words, the only difference in consistency strength is that "ZFC" by default doesn't include any universes, while "HoTT" by default includes countably many.