Exercise :
For the Simple Linear Model $\mathbb E[y_x] = b_0 + b_1x$, prove that for a newly given $x_0$ and $y_{x_0}$ a new observation while $\hat{y_{x_0}}$ its point estimate, it is : $$\operatorname{cov}(y_{x_0},\hat{y_{x_0}}) = 0$$
Question :
It is :
$$\operatorname{cov}(y_{x_0},\hat{y_{x_0}}) = \mathbb E[(y_{x_0}-\mathbb E[y_{x_0}])(\hat{y_{x_0}}-\mathbb E[\hat{y_{x_0}}])]$$
but how would one continue to prove the expression asked from that point on ?