I came across this question in a review of an old exam I took.  I didn't get the answer correctly then, and I'm struggling to figure the answer out now.  Can anyone help me reason through this?

If $F_X(z) > F_Y (z)$ for all $z\in \mathbb{R}$ then $P(X < Y ) > 0$?

Here is what I attempted:

I figured I might be able to approach this by proving this through contradiction.  I started by assuming $P(X<Y)=0$.  Then, 

\begin{eqnarray*}
F_{X}(z)=P(X\le z) & = & P(X\le z,X<Y)+P(X\le z,X\ge Y)\\
 & = & 0+P(X\le z,X\ge Y)
\end{eqnarray*}

Can anyone help from here?

Thanks.