But that's perfectly valid. We have just considered that set, so obviously it's not an element of itself. I think maybe what you meant to say is after that: "No

But that's perfectly valid. We have just considered that set, so obviously it's not an element of itself. I think maybe what you meant to say is after that: "Now consider an arbitrary element in this set." That would present a problem. Of course so would showing it's non-empty, but I think there could be a way around that.