My definition of Baire Space is the modern one
A topological space $X$ is a Baire Space iff the intersection of a countable family of open dense everywhere subsets of $X$ is dense everywhere.
There is an old definition that states
A topological space $X$ is a Baire Space iff every non empty open subset of $X$ is of second category.
I'm trying to prove that the old definition implies the modern one
My attempt: Suppose that $X$ is not a Baire Space in the modern sense, then there is a family $\{A_n\}_{n \in \mathbb{N}}$ of open everywhere dense subsets of $X$ and an open set $U \subseteq X$ such that $$U \cap \big( \bigcap_{n \in \mathbb{N}}A_n \big) = \emptyset,$$
so $$U = U - \big( \bigcap_{n \in \mathbb{N}}A_n \big) = \bigcup_{n \in \mathbb{N}} \big( U - A_n \big) $$
I'm tempted to think that for every $n \in \mathbb{N}$, $U - A_n $ is nowhere dense, which would give the result, but I'm not sure.
How do I see that $\mathrm{int}_X ( \mathrm{cl}_X (U-A_n)) = \emptyset???$
Thanks, any help would be appreciated.