Consider a random vector of dimension $5\times 1$, $X\equiv (X_1, X_2, X_3, X_4, X_5)$, with joint CDF denoted by $G$.
Consider the following probability $$ p\equiv Pr(X_1\geq a_1,X_2\geq a_2,X_3\geq a_3,X_4\geq a_4,X_5\geq a_5 ), $$ where $(a_1,_2,a_3,a_4,a_5)\in \mathbb{R}^5$.
I would like to write $p$ using the CDF of $X$, ($G$ introduced above). Could you advise on how to do that?
My thoughts: At the moment, I have been able to derive the formula above for a vector with at most 3 components. For example, when $X$ is scalar $$ Pr(X_1\geq a_1)=P_{X_1}([a_1, \infty])=1-G(a_1) $$
When $X$ is bivariate $$ Pr(X_1\geq a_1, X_2\geq a_2)=P_{X_1,X_2}([a_1, \infty]\times [a_2, \infty])=1+G(a_1,a_2)-G(\infty, a_2)-G(a_1, \infty) $$ (see also this question).
When $X$ is trivariate $$ Pr(X_1\geq a_1, X_2\geq a_2, X_3\geq a_3)=P_{X_1,X_2,X_3}([a_1, \infty]\times [a_2, \infty]\times [a_3, \infty])=1-G(a_1,a_2,a_3)+G(\infty, a_2, a_3)+G(a_1, \infty, a_3)-G(\infty, \infty, a_3)+G(a_1, a_2, \infty)-G(\infty, a_2, \infty)-G(a_1, \infty, \infty) $$ How do I extend this to the $5$-variate case?