## Would \$\Sigma_i^P \neq \Pi_i^P\$ imply that polynomial hierachy cannot collapse to the \$i\$-th level?

If $$\Sigma_i^P = \Pi_i^P$$, then it follows that the polynomial hierarchy collapses to the $$i$$-th level.

What about the case $$\Sigma_i^P \neq \Pi_i^P$$? For example, consider the case of $$NP \neq coNP$$. As far as I understand, this would imply the polynomial hierarchy cannot collapse to the first level, since if $$PH =NP$$, then in particular, $$coNP \subseteq NP$$, which means $$NP = coNP$$. Can we expand this idea to proof the general case: $$\Sigma_i^P \neq \Pi_i^P$$ implies $$PH$$ cannot collapse to $$i$$-th level?