Let $S$ is a partition of a set $U$. Let $c$ is an ultrafilter on $U$.

Prove or disprove the following conjecture:

At least one of the following is true:

• $\exists D\in S, C\in c:C\subseteq D$ or
• $\exists C\in c\forall D\in S: \mathrm{card}(C\cap D)\le 1$.

Let $S$ is a partition of a set $U$. Let $c$ is an ultrafilter on $U$.

Prove or disprove this conjecture:

At least one of the following is true:

• $\exists D\in S, C\in c:C\subseteq D$ or
• $\exists C\in c\forall D\in S: \mathrm{card}(C\cap D)\le 1$.