Stanislav Petrov is a rather famous person (of course only on Lesswrong, not in the real world).
But there is another Russian who saved the world: Vasili Alexandrovich Arkhipov.
On this day in 1962, at the height of the Cuban Missile Crisis Vasili Arkhipov prevented the launch of a nuclear torpedo and thus a possible nuclear war.
It's strange that Petrov attracts much more attention than Arkhipov. E.g. googling "Stanislav Petrov" produces 101.000 results, "Vasili Arkhipov" only 9.040 results. By contrast searching for "Britney Spears" generates about 295.000.000 results. Sorta depressing.
Anyway, let this day be the Vasili Arkhipov Day.
We do have enough nuclear weapons to kill all humans, and there is some chance of them all simultaneously being detonated - so I would not endorse the first clause. The second clause is not the only alternative, though: consider also the case where the probability of disaster per unit time is small.
It's not clear this is the case, actually, despite it being a part of the common culture. Nuclear winter style scenarios are possible, but even then it's not clear that all humans would die.