I am confused by a step in the proof of the Hoeffding identity as provided in the book by Denuit, Dhaene, Goovaerts and Kaas. The following is a screen shot of the beginning of the proof:
I am confused about the transition from the second last to last step here. If $X$ was a non-negative random variable, then it follows that $$ \mathbb{E}X = \mathbb{E}\left ( \int_0^{\infty} 1_{u \le X} du \right ) $$ but the result here seems to hold for non-negative random variables. A similar question was asked here, but the accepted solution is just for the non-negative case, unless I have missed something. I'm looking for a clear justification of the last step highlighted in red.
This rest on the fact that $$ \int_{-\infty}^{+\infty}\left(\mathbf 1\{u\leqslant X_1\}-\mathbf 1\{u\leqslant X_2\}\right)du=X_1-X_2, $$ which can be seen by dealing with the cases $X_1\leqslant X_2$ and $X_2\lt X_1$.