In the book<>written by Charles C.Pinter it said that in order to prove "the axiom of replacement together with strengthened version of axiom of choise imply axiom of limitation of size"(X is a proper class iff it is 1-1 correspondence with the universal sets V
In my perspective,I can't understand why it need the 'strong' version of axiom of choice.I think axiom of replacement with usual axiom of choice can imply VN sufficiently.Here is my proof:
1.It is obvious that X→V is injective(By definition V contains all the sets).Thus INJ holds
2.Now we shall show X→V is surjective:
For any Va∈V('a'is any ordinal number and Va is a set),by axiom of replacement it follows that there exists a set A⊂X which is A→Va is bijective.If such a set 'A'does not exist then X would be a set.A contradiction.Thus SUJ holds.
Therefore X→V is 1-1 correspondence.
I haven't use strong version of AC.Can someone tell me is there any mistake or logic lackage in my proof(especially for proving X to V is surjective) and give me an explicit correct proof for this exercise?Thanks!