This question has been bothering me for some while now and I still can't figure it out:

Prove that: (1) without the choice-axiom.

The way is defined in my courses is the following:

Let (where R is a wellorder on A).

Then ( is a ordinal and the ordertype of x)

I define: to be the set of all initial segments of A with respect to the wellorder R, wich is well-ordered by

Now is defined by not an injection?

In this case the (1) would become trivial, so I expect it's not. But Why?

It seems evidently true that and