If NP is not a proper subset of coNP, why does NP not equal coNP?
$\{2,3\}$ is not a proper subset of $\{3,4\}$, yet the two clearly are not equal.

Comparing sets is not like comparing numbers: two sets might not be comparable.

Additionally, NP is not a subset of coNP, or at least, it is not known that this is the case. You are either misreading the textbook, or your textbook is wrong, since proving that $NP \subseteq coNP$ would be a massive result.

$P\subseteq coNP$, perhaps that is what you actually read?