A curious inequality

I first saw this in UW's challenge of the week (if I remember correctly), where they used to post a nice math puzzle every week and give the winners (drawn at random from the correct solutions) a gift certificate to Baskin Robbins.

[That has now been discontinued and I believe the pages also have been taken down.]

Anyway, here is the puzzle.

If $x,y \gt 0$ are real numbers, show that

$$x^y + y^x \gt 1$$