# Is the joint density the same if the inequality sign is flipped?

I know that $$F(x,y)= \mathbb P [X \leq x, Y \leq y] = \int^x_{-\infty}\int^y_{-\infty}f_{X,Y}(u,v)dvdu$$, is differentiated with respect to $$x$$ and $$y$$ to calculate the joint density $$f_{X,Y}(x,y)$$. I have seen in a few places that $$\partial^2_{xy}\mathbb P [X \leq x, Y \leq y] = \partial^2_{xy}\mathbb P [X > x, Y \leq y]$$, so that either expression can be used to calculate the joint density. I can't seem to verify if this is true or not.

I have the following relation: $$\mathbb P [X > x, Y \leq y] + \mathbb P [X \leq x, Y \leq y] = \mathbb P [Y \leq y]$$ from the law of total probability, and so calculating just $$\partial_x$$ of this expression gives:

$$\partial_x \mathbb P [X > x, Y \leq y] + \partial_x \mathbb P [X \leq x, Y \leq y] = 0$$. And so $$\partial^2_{xy} \mathbb P [X > x, Y \leq y] + \partial^2_{xy} \mathbb P [X \leq x, Y \leq y] = 0$$, which implies $$\partial^2_{xy}\mathbb P [X \leq x, Y \leq y] = -\partial^2_{xy}\mathbb P [X > x, Y \leq y]$$.

This seems to say that the densities from either expression gives different results. Does my logic here make sense? Any help would be appreciated. Thanks!