 The Russell class can be shown to be the same size as the universe using Schröder-Bernstein: the injection from \(R\) into \(V\) is obvious, and \(V\) can be embedded into \(R\) using the map \(x \mapsto \{\{x\}, \varnothing \}\) (clearly no set \(\{\{x\}, \varnothing \}\) belongs to itself). So a class is proper iff it is the same size as the universe (limitation of size).