Whew ! I finally spotted the mistake !
You're taking z>0, while, since the plane x+y+z=1 intersects the xy-plane, there are some negative z that satisfy the conditions !
Actually, we have z=1-x-y>1-2=-1, because x+y<2 (large inequalities, but I'm too lazy to do the latex)
So try it with the boundaries for z : -1 and 1-x-y
(I haven't done the calculations yet)