3. U{0} e {0}.
Indeed, for any ordinal x, we have U(x+) = x and x e x+.
I've been going through some old exam papers, and I came upon an interesting part of a question that I'm not sure how to solve. Throughout the question I've been proving things like is false, at most one of and is true, and so on. I've been asked whether any of the following can occur, using only axioms (ZF1)-(ZF4) and (ZF9). (Extensionality, null set, pairs, unions, foundation.)
1.
2.
3.
I think (1) is false, by extension I think (2) is false (but can't get my reasoning right in my head), and I'm not sure about (3).