Given $B \not= \emptyset$ a finite set of ordinals, prove that there is a maximal element in $B$ with respect to $\in$ , meaning $\exists A \in B \forall x \in B (x=A or x \in A)$
I think this is true because of the well-order of ordinals, but i don't know how to prove it, because for instance $B = \{ \emptyset, \omega\}$ so obviously $\omega$ is the maximal element in $B$ but $\omega$ is not finite, and every proof i thought of works only for finite ordinals.
Please help
Being an ordinal isn't particularly important:
This isn't too hard to prove by induction.
Since the ordinals are totally ordered by $\in$, $B$ (with the ordering given by $\in$) is a finite totally ordered set, so the theorem applies.
Incidentally, for this particular problem, there is a set-theoretic calculation of the maximum too: $\max(B) = \cup B$ (of course, you'll have to prove that $\cup B \in B$.