Let $(X,Y)$ have joint density function $f$ and joint distribution function $F$. Suppose that $$f(x_1,y_1)f(x_2,y_2)\leq f(x_1,y_2)f(x_2,y_1),$$ holds for $x_1\leq a\leq x_2$ and $y_1\leq b\leq y_2$. How to show that $$F(a,b)\leq F_X(a)F_Y(b)?$$ Here $F_X$ and $F_Y$ are the cumulative distribution functions of $X$ and $Y$ respectively.
My approach was this: $$ \begin{align}(F(a,b))^2 &= \left(\int_{-\infty}^a\int_{-\infty}^b f(x_1,y_1)dx_1dy_1\right)\left(\int_{-\infty}^a\int_{-\infty}^b f(x_2,y_2)dx_2dy_2\right) \\ &= \int_{-\infty}^a\int_{-\infty}^b\int_{-\infty}^a\int_{-\infty}^b f(x_1,y_1)f(x_2,y_2)dx_1dy_1dx_2dy_2 \\ &\leq\int_{-\infty}^a\int_{-\infty}^b\int_{-\infty}^a\int_{-\infty}^b f(x_1,y_2)f(x_2,y_1)dx_1dy_1dx_2dy_2. \\ \end{align} $$ But this leads to nowhere.