Let ZFC2 be the Second-Order formalisation of ZFC.
The Second-Order Axiom schema of Comprehension (part of the deductive system for SOL) says that for every formula (of SOL) there is a relation with the same extension (shapiro 1991).
If we formalise ZFC2, then the domain is all sets and the second-order quantifiers range over all the subsets of the domain. But then, what stops Russell's Paradox from arising?
I know it doesn't because ZFC2 is equivalent to Morse-Kelley set theory.