Assume that A is a set and define (as in Hartogs' theorem) $\alpha$ to be the set of ordinals dominated by A. Show that (i) $\alpha$ is a cardinal number, (ii) card A < $\alpha$ , and (iii) $\alpha$ is the least cardinal greater than card A.
This question is from Enderton textbook. Any help would be appreciated!
Edit: I have tried the following to think about (i), (ii) and (iii), although I don't think it's a meaningful progress. (i) $\alpha$ has to be an initial ordinal number for it to be a cardinal number. How do we show if $\alpha$ is an initial ordinal number? (ii) $\alpha$ is equinumerous to card($\alpha$). Since $\alpha$ is the set of ordinals dominated by A, $card (\alpha)\leq card A$.
I feel like this problem shouldn't be hard, but I'm stuck for some reason. I would appreciate any help on this problem and any suggestion as to how I could improve on thinking about those problems. (I don't think my understanding of axiom of choice is thorough - how would you advise me to study the axiom of choice?)