I have been looking at certain collections of cardinals lately and am trying to figure out if they are sets or not.
Is it true, that if $P$ is some large cardinal property and there is some $\kappa_{\text{first}}$, for which this property holds, that the collection of cardinals $\mathcal{C} = \{\kappa \mid \kappa \text{ has not } P\}$ is a set? Can I argue that $\kappa_{\text{first}}$ must have greater cardinality than all elements of $\mathcal{C}$ and therefore contains $\mathcal{C}$ as a subset?
On the other hand, if I know that a collection of cardinals $\mathcal{D}$ contains cardinals of arbitrarily large cardinality, must $\mathcal{D}$ be a proper class? Is it sufficient to argue that if $\mathcal{D}$ was a set, then $\lambda = \sup{\mathcal{D}}^+$ would be a cardinal not in $\mathcal{D}$, but since $\mathcal{D}$ contains arbitrarily large cardinals, one of them would have cardinality greater than $\lambda$, which already yields a contradiction?