Is the Hadith, "He who has the equivalent of a mustard seed's weight of faith in his heart shall not enter Hellfire," authentic, and can a heart ever be devoid of a mustard seed's weight of faith?

"He who has an atom's weight of pride in his heart shall not enter Paradise," and "He who has an atom's weight of faith in his heart shall not remain eternally in Hellfire." The intended meaning of faith here is believing in the existence of Allah and the truthfulness of the message of Muhammad, peace be upon him, and this meaning is not found in the hearts of disbelievers.