Finding $P[X+Y > 1, X > 1]$

I’m trying to solve the following problem: i have two independent exponentially distributed r.v. $X$ and $Y$ both with $lambda = 1$. I want to know the probability $P[X+Y > 1, X > 1]$. Since they are independent, i wrote the joint p.d.f. as $f_{X,Y}(x,y) = f_X(x) cdot f_Y(y) = e^{-x}cdot e^{-y}$. Then i’ve tried to solve
$$
int_{1}^{infty} int_{1 – x}^{infty} e^{-x}cdot e^{-y}, dy, dx
$$
so,
$$
int_{1}^{infty} e^{-x}cdot e^{x – 1} dx = e^{-1} cdot int_{1}^{infty} dx = infty
$$
but i get as result $infty$. What am i doing wrong? I think the integral is correctly soved so the initial fomula of the joint may be wrong…