In 2012, the amount of nuclear weapons globally is approx. 23,375. This is not enough to destroy the world, but enough to cause a nuclear winter, mass starvation, and radioactive fall- out particles.
Such nuclear fallout would force humanity into self sufficient shelters, or face mass starvation, without the ability to use sunlight to grow food alternative sources of food would need to be used, dung beetles, worms, mushrooms, sunless means of food production, or survival via stored cans, flour, and so forth.
99% of the population would die within the following decades, crops once used widely would no longer have viable seed stock, and transportation networks would be destroyed.
Inbreeding may become a post apocalytipc issue depending on the amount of humans surviving, while not a strictly extinction level event, life will never be the same