[math-fun] vector domination probability problem (insanely annoying)

I stumbled across this apparently true fact: Choose N random i.i.d. real numbers (from some fixed probability density) then sort them; result is (a,b,c,d,...,z). Do it again independently, result is (A,B,C,D,...,Z). Then what is the chance that a<A, b<B, c<C, ..., and z<Z simultaneously? ("vector domination") My original incorrect answer was 2^(-N). Complete with wrong proof. My new correct answer is 1/(N+1). I proved this for N=1,2,3,4,5,6. It presumably is true for N=7,8,9... too. But what is the proof? Surely something this simple must have a simple proof?