
Ordinal Problem.
Hi, the question posed to me is as follows:
Prove or find a counterexample to:
1.$\displaystyle \alpha\cup\{\alpha \}$ is an ordinal $\displaystyle \Rightarrow \alpha$ is an ordinal
2.$\displaystyle \alpha\cup\bigcup{\alpha}$ is an ordinal $\displaystyle \Longleftrightarrow \alpha$ is an ordinal
I've found proofs to both of these questions but I'm not entirely sure that they are both correct. Can anyone see a way these implications do not hold?
Thank you.

a u {a} is an ordinal <> a is an ordinal.
a is an ordinal > a u Ua is an ordinal.
But it it does not hold for all a that
a u Ua is an ordinal > a is an ordinal.
You should find a simple example of an a such that a u Ua is an ordinal but a is not an ordinal.