Let $\displaystyle \{U_i : i \in I\}$ be a collection of sets, all containing a common element $\displaystyle x \in U_i$ for each $\displaystyle i \in I$.

Do we need the axiom of choice to show that $\displaystyle \displaystyle x \in \cap_{i\in I}U_i$?

The reason I ask is because wikipedia says that we need the axiom of choice to prove the Baire Category Theorem and in the proof it gives on the page, I can't see where this would be if it wasn't in the above.