This is another, shorter, proof for the fact that $MA_{{\aleph_{0}}}$ always holds.

Let $(P,\leq)$ be a partially ordered set and $\mathcal{D}$ be a collection of subsets of $P$. We remember that a filter $G$ on $(P,\leq)$ is $\mathcal{D}$-generic if $G\cap D\neq\varnothing$ for all $D\in\mathcal{D}$ which are dense in $(P,\leq)$. (In this context “dense” means: If $D$ is dense in $(P,\leq)$, then for every $p\in P$ there’s a $d\in D$ such that $d\leq p$.)

Let $(P,\leq)$ be a partially ordered set and $\mathcal{D}$ a countable collection of dense subsets of $P$. Then there exists a $\mathcal{D}$-generic filter $G$ on $P$. Moreover, it could be shown that for every $p\in P$ there’s such a $\mathcal{D}$-generic filter $G$ with $p\in G$.