Dissecting a proof of the $\Delta$-system lemma (part II) This is part II of this question I asked yesterday. In the link you can find a proof of the $\Delta$-system lemma. In case 1 it uses the axiom of choice (correct me if I'm wrong). Now one can also prove the $\Delta$-system lemma differently, for example as follows:


I have two questions about it:
1) It seems to me that by using the ordinals to index the family of sets we have eliminated the axiom of choice from the proof. Have we or did we just use it a bit earlier in the proof where we index the family $B$?
2) But, more importantly, why is it ok to assume that $b \in B$ are subsets of $\omega_1$? In the theorem there is no such restriction. Can one just "wlog" this? The answer is probably yes since otherwise the proof would be wrong but I don't see how.
Thanks for your help!

You need some AC to prove the statement just for a family of pairs of $\omega_1$.
If $\omega_1$ is the union of a countable family $\{B_n:n \in \omega \}$ of countable sets (which is consistent with ZF!), then the family $\{\{n, \beta\}: n<\omega, \beta \in B_n-\omega \}$ does not contain an uncountable $\Delta$-system.
