What does it mean for a smooth atlas to not be properly contained in any larger smooth atlas?

Question

In Lee's Introduction to Smooth Manifolds, he defines a smooth atlas as maximal if it is not properly contained in any larger smooth atlas. What does this mean? Any atlas covers the manifold so it can't be the domain that he's referring to. In the following example, also given by Lee but in a different context, which one is maximal?

$\mathscr{A}_1 = \{(\mathbb{R}^n, Id_{\mathbb{R}^n})\}$

$\mathscr{A}_2 = \{(B_1(x), Id_{B_1(x)}) : x \in \mathbb{R}^n \}$

Answer 1

Neither $\mathscr{A}_1$ nor $\mathscr{A}_2$ is maximal, since they are both properly contained in $\mathscr{A}_1 \cup \mathscr{A}_2$, and that union isn't maximal either. On the other hand, any smooth atlas is contained in a unique maximal atlas---just add in all of the compatible smooth charts.

Answer 2

The word "larger" here is redundant with "contained in" and contributes no additional meaning. "Contained in" means literally one atlas is a subset of the other, as a set of charts. So, an atlas $A$ is maximal if there does not exist any other atlas $B$ such that $A\subset B$.