I think your second set doesn't exist (exactly like the set of all sets doesn't exist) under the axiom of choice, for I can use the axiom of choice to select an

I think your second set doesn't exist (exactly like the set of all sets doesn't exist) under the axiom of choice, for I can use the axiom of choice to select an element, hence giving it a name :). I took some liberty with the allowable names, of course. You still have a point, though.