 There is a meta-theorem of ZF called the Reflection Principle which asserts that any first-order assertion which is true of the universe \(V\) is also true of some set. This means that for any particular proof in ZF, there is a set \(M\) which might as well be the universe (because any proof uses only finitely many axioms). A suitable such set \(M\) can be construed as the universe of sets and the actual universe \(V\) can be construed as the universe of classes. The set \(M\) has the closure properties asserted in Elements and Subsets if it is a limit rank; it can be chosen to have as many of the closure properties asserted in Set Comprehension (translated into terms of \(M)\) as a proof in Ackermann set theory requires. This machinery is what is used to show that Ackermann set theory proves nothing about sets that ZF cannot prove: one translates a proof in Ackermann set theory into a proof in ZFC using the Reflection Principle.