What sin was Brad Pitt guilty of in the movie "Se7en"?

In the 1995 movie "Se7en", one of the key characters who is a New York police detective David Mills (Brad Pitt) is guilty of the sin of wrath. Mills is working with detective William Somerset (Morgan Freeman). Just before Somerset is about to retire, the two men must track down a serial killer Kevin Spacey as John Doe. Doe is a guy who is using the seven deadly sins (cardinal vices within Christian teachings) as a motif for murder.

At the end of the movie, Mills shoots Doe. This act serves as Doe's last plot. It represents wrath. Doe had stated that he envied Mills and his wife Tracy (Gwyneth Paltrow). Because Doe has killed Mills' pregnant wife, Mills can't stop himself. He is a cop out of control; he kills Doe. Police converge at the deadly scene and take the devastated Mills away. He is full of wrath; he showed an uncontrollable anger and complete abhorrence towards John Doe.