Wordorigins.org

No, it’s not a doggie dog world. The phrase is dog eat dog, a reference to ruthless competition. It is a translation of the Latin proverb, canis caninam non est, dog will not eat dog. English use of the phrase dates to the late 18th century. From the Times (London) of 19 June 1789:

As it is an established fact, that sharper will not rob sharper, nor dog eat dog.

(Source: ADS-L)