Let $\mathcal{C}$ be a chain of subspaces of a Banach space $\mathcal{X}$. For each $\mathcal{Y}\in\mathcal{C}$, define its immediate predecessor \begin{equation*}\mathcal{Y}_{-}=\bigvee\{\mathcal{Z}\in\mathcal{C}:\mathcal{Z}\subset \mathcal{Y},\mathcal{Z}\neq\mathcal{Y}\}.\end{equation*}
I encounter the following statement in one paper:
The condition $\mathcal{Y}_{-}=\mathcal{Y}$ implies that $\mathcal{C}$ contains elements having infinite codimension in $\mathcal{X}$.
I tried to prove this statement but do not actually know where to start. Can someone give some hint?
Thanks!