automatons: the intersection of non-CFL with the regular language is not $ phi $. Why?

I used the logic that, since regular languages ​​are subsets of CFLs, CFLs and non-CFLs are different, those that are not CFLs and regular languages ​​must be different. But this is not true. Can someone please explain?

The real question was:

If L is not CFL and F is finite language. Show that L F is not CFL.