I am writing an essay for which I need to prove that sufficiently many graphs of a certain type exist. Is it true that for any set of sets (or set of cardinals) $S$ a countable subset $C$ exists such that
$$\forall s \in S \exists c\in C[c>s]$$
if we speak about cardinals or
$$\forall s \in S \exists c\in C[|c|>|s|]$$
when speaking about sets of sets? does this require well ordering or any other kind of explanation?