 Ackermann set theory actually proves that there are classes which have non-set classes as elements; the difference between sets and classes provably cannot be as in von Neumann-Gödel-Bernays class theory. A quick proof of this concerns ordinals. There is a proper class von Neumann ordinal \(\Omega\), the class of all set von Neumann ordinals. We can prove the existence of \(\Omega + 1\) using set comprehension: if \(\Omega\) were the last ordinal, then “\(x\) is a von Neumann ordinal with a successor” would be a predicate not mentioning sethood, with no parameters (so all parameters sets), and true only of sets. But this would make the class of all set ordinals a set, and the class of all set ordinals is \(\Omega\) itself, which would lead to the Burali-Forti paradox. So \(\Omega + 1\) must exist, and is a proper class with the proper class \(\Omega\) as an element.