Let $A$ be a non-empty set of regular cardinals such that $\vert A\vert <\text{min}\ A$, and $\{\nu_i\mid i<i_0\}\subseteq \text{pcf}\ A$ be a strict increasing sequence having limit length $i_0$.Then $\mathcal{J}_{<\nu_{i_0}:=\text{sup}\{\nu_i\mid i<i_0\}}A=\bigcup\{\mathcal{J}_{<\nu_i}A\mid i<i_0\}$ hence there is $\nu\in\text{pcf}\ A$ such that $\nu\geq\nu_{i_0}$. 

But in this case does $\nu_{i_0}\text{is regular}\Rightarrow\nu_{i_0}\in\text{pcf}\ A$ hold? or is there a counter-example in higher axioms than ZFC? I see when the special case, $A$ is an interval, this holds true.

[When $A$ denotes a set of regulars, $\mathcal{J}_{<\nu}A:=\{B\subseteq A\mid (\forall D\ \text{ultrafilter on}\ A), B\in D\Rightarrow \text{cf}\ \prod A/{D}<\nu\}$.]