Russell's paradox
This argument is known as Russell's paradox. As a point of subtlety, the version of Russell's paradox we have presented here is actually a theorem of Zermelo; we can conclude from the contradiction obtained that we must reject the hypothesis that "R""U"∈"U", thus disproving the existence of a set containing all sets. This was possible because we have used restricted comprehension (as featured in ZFC) in the definition of "R""A" above, which in turn entailed that