The phrase "arrogant fool" refers to someone who is both full of themselves and lacking intelligence. It describes a person who thinks highly of themselves but often makes foolish or ignorant decisions. Full definition
Atheists are arrogant fools who believe in themselves a bit more than is warranted. (religion.blogs.cnn.com)
He has been exposed as a deluded, arrogant fool on many occasions. (justarsenal.com)
And only arrogant fools respond to the pain of their neighbors the way that Job's friends responded to his — with elaborate theological explanations, blame, and calls for repentance. (rachelheldevans.com)