As many have explained, they did calculations. But similar to the fears around LHC creating a black hole and dooming the planet not so long ago, there is one further reason why it is particularly unlikely:
The surrounding cosmos has done way worse than our little nukes. Remember the Chelyabinsk event, a ~10 meter meteorite exploding over Russia in 2013? That cute thing had about 7 to 8 times the explosive yield of the first three nukes (trinity, little boy, fat man) combined.
By meteorite standards, this was a firecracker at best. The legendary dinosaur killer was many **many** times more powerful. If our puny little bombs could set the atmosphere ablaze in nuclear fire, those things would have done so multiple timers over. Or rather every decade.
Same with the aforementioned black holes, each day cosmic rays hit Earth at energies current particle physicists can only dream of.
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