Maximum cardinality

Question

Let $X$ be some set and $P$ be some subset of ${\frak P}(X)$. We can define the smallest cardinal $\frak k$ such that any $A \in P$ has cardinal $\leq \frak k$. Indeed we can consider the set of all cardinals of elements of $P$ and pick the supremum. On the other hand, it is not clear to me that we can define the smallest cardinal $\frak k$ such that any $A \in P$ has cardinal $< \frak k$ (strictly less). Is it possible indeed?

Answer 1

In the context of $\sf ZFC$ the cardinals are well-ordered. It suffices to show that there is a cardinal which is greater than any of the cardinals of the sets in $P$, in which case there is a least such cardinal.

Since $A\in P$ means $A\subseteq X$, this means that all those cardinals are strictly less than $|X|^+$. Therefore, there is a least cardinal $\kappa$ such that $|A|<\kappa$ for all $A\in P$.