Steinberg representation for $\mathrm{GL}(2)$ is irreducible

Question

Let $(\pi, V) = \mathcal{B}(\chi_{1}, \chi_{2})$ be a principal series representation of $\mathrm{GL}(2, K)$ ($K$ is a local field), where $\chi_{1}, \chi_{2}$ are characters of $K^{\times}$. If $(\chi_{1}\chi_{2}^{-1})(y) = |y|^{-1}$, the representation is reducible and it has a 1-dimensional subspace spanned by $f_{0}(g) = \chi(\det (g))$, where $\chi = (\chi_{1}\chi_{2})^{1/2}$. I want to show that the quotient representation $(\pi, V/V')$ where $V' = \mathbb{C}f_{0}$ is irreducible, which is so-called (twisted) Steinberg representation. This is an exercise 4.5.2 in Bump, but actually I have no idea how to show this. Can we use Jacquet functor in this case? Then how?