From Computational Complexity - A Modern Approach, it is claimed in a remark that the following holds true.
Remark 5.5
Note that $\mathbf\Sigma_1^p = \mathbf{NP}$ and $\mathbf\Pi_1^p = \mathbf{coNP}$. More generally, for every $i \geq 1$, $\mathbf\Pi_i^p = \mathbf{co\Sigma}_i^p = \{ \overline{L} : L \in \mathbf\Sigma_i^p\}$. Note also that $\mathbf\Sigma_i^p \subseteq \mathbf\Pi_{i+1}^p$, and so we can also define the polynomial hierarchy as $\cup_{i>0} \mathbf\Pi_i^p$.
However, I am having trouble understanding this. Similar claims are made in lecture notes found on the web, but none of them give a proof since they are all left as exercise. Would it be possible to prove this using NTM instead of verifiers?