conditional multi-variate normal when the condition is a range

Question

Suppose $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \sim N(\mu, \Sigma)$, where $x_1, X_2$ are vectors.

I understand the formula for the conditional distribution $P(X_1|X_2)$, but does anyone know how I can get $P(X_1 | X_2>z)$?

Or more generally: $$ P\left(X_1 | \begin{bmatrix} x_{21} > z_1 \\ x_{22} > z_2 \\ x_{23}>z_3 \\... \end{bmatrix} \right) $$