Proof of Expected value by Abel's Summation

Question

Using Abel's summation lemma, prove that: $$ \mathbb{E}[X] = \int_0^\infty (1-F(x)) \ dx, $$ where $F(x)$ is the CDF of a random variable $X$

I do know how to prove this equation by another method. But using Abel's summation is definitely out of my zone. Please give me some advices.