And a 100 billion dollar line of credit from the treasury. The FDIC describes itself as being “ backed by the full faith and credit of the United States government.” If there was a systemic issue, the government would likely need to intervene.
The Fed can do that any time they want by buying bank debt with newly created dollars, which effectively socializes the risk in the banking system, so it doesn't fail, hopefully at something resembling market rates. The Fed then runs the risk that they don't get paid back instead of an individual bank, but the modern Fed cannot fail, that sort of thing can only affect the currency as a whole, by weakening it generally speaking.