The Horatio Principle, SRH, Russell’s Paradox and Zermelo

ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC, objects like R are called proper classes. ZFC is silent about types, although some argue that Zermelo's axioms tacitly presuppose a background type theory.

In ZFC, given a set A, it is possible to define a set B that consists of exactly the sets in A that are not members of themselves. B cannot be in A by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything.

[http://en.wikipedia.org/wiki/Russell's_paradox]

This could well be the style of proof I’m looking for with the SRH. The above says that for a set A we can define B = {x in A & x not in x}. Thus if B is in A then there is Russell’s Paradox so B is not in A. This assumes the Zermelo axiom that existence can only be asserted for subsets of a given set, not for subsets of “all things”.