Mereological universalism and Platonism

1. If mereological universalism holds, then for any predicate F that is satisfied by an object, there is a mereological sum of all the objects that satisfy F.
2. If Platonism is true, then for any object x there is a property U_x such that (a) x has U_x, (b) nothing else has U_x, and (c) U_x does not have mereological parts.

Call any such a property an "identity of x". Condition (c) is probably satisfied by all properties, but I could imagine someone thinking that conjunctive properties are mereological sums of their conjuncts, so (c) would be needed to rule out conjunctive properties (and if V_x is a conjunctive property satisfying (a) and (b), then we can let U_x be the property of having V_x or being a unicorn, and this won't have mereological parts).

Assume Platonism and mereological universalism. For any predicate G(x), say that I_G(P) if and only if P is an identity of an object x such that G(x). For any object S, say that x∈S if and only if an identity of x is a mereological part of S. Then by (1) and (2), for any predicate G that has at least one satisfier there is an object M such that x∈M if and only if G(x). Just let M be any mereological sum of all the objects satisfying I_G (to get that if x∈M, then G(x), note that it follows from (c) that no identity can overlap more than one identity).
But now we have a contradiction. Let G(x) be the Russell predicate not(x∈x). Then G has at least one satisfier: e.g., you satisfy G. So there is an M such that x∈M if and only if G(x). So, M∈M if and only if not(M∈M), a contradiction.

So, our Platonist mereological universalist needs to restrict her universalism to concrete objects or something like that. Lewis's Parts of Classes is no doubt relevant, and what I said above may well overlap with Lewis (I haven't actually read that book--just ordered from Interlibrary Loan).