conditional distribution multivariate gaussian generalized inverse

given $$\begin{bmatrix}X\\Y\end{bmatrix}\sim N\left(\begin{bmatrix}\mu_X\\\mu_Y\end{bmatrix};\begin{bmatrix}\Sigma_{XX}&\Sigma_{XY}\\\Sigma_{YX}&\Sigma_{YY}\end{bmatrix}\right)$$ How can I prove that $$\Sigma_{YX}-\Sigma_{YX}\Sigma_{XX}^{-}\Sigma_{XX}=0$$, where $$\Sigma_{XX}^{-}$$ is the reflexive g-inverse, i.e. $$\Sigma_{XX}\Sigma_{XX}^{-}\Sigma_{XX}=\Sigma_{XX}$$.

The bigger picture is that I am trying to find en expression for the conditional distribution $$Y|X=x$$ when $$\Sigma_{XX}$$ is singular. My approach is to prove that $$X$$ and $$Y-\Sigma_{YX}\Sigma_{XX}^{-}X$$ are independent. Once this is proved the rest is trivial. However, I got stuck with the problem above.