We prove "There is no set that contains every set.".
For an arbitrary set U, we construct a set A not belonging to U.
Let A = {x∈U | x ∉ x }.
Then, x∈A <--> x∈U & x ∉ x (Axiom schema of specification).
Let x be A.
Then, A∈A <--> A∈U & A∉A.
If A∈U, then this reduces to
A∈A <--> A∉A...
I think that people get stuck on there narrow view of what something that exists has to be defined as. Where as omnipotence to a human seems to encompass everything you can "do" in the real known world rather then what you don't know that you can do outside of it. I hate to quote bible...