This may be stupid question, but why is it that given a topological manifold $\displaystyle X$ and some atlas $\displaystyle \mathfrak{A}$ then their exists a unique $\displaystyle C^{\infty}$ structure $\displaystyle \mathfrak{A}^*$ on $\displaystyle X$ which contains $\displaystyle \mathfrak{A}$?

I can see why their exists some $\displaystyle C^{\infty}$ structure. Just define $\displaystyle \Omega$ to be the set of all atlases on $\displaystyle X$ containing $\displaystyle \mathfrak{A}$, order it in the natural way and apply Zorn's lemma. But, why is it unique? Is it because the way one constructs the ordering any two maximal atlases $\displaystyle \mathfrak{M},\mathfrak{N}$ would need to be comparable and thus equal?

Any help would be appreciated!