I'll go pedantic, but ない is technically not an irregular negative form of ある. The negative form of ある is the regular あらぬ/あらない, but it has been substituted with ない, which is an entirely different word. You'll often hear あらへん in Kansai dialect, which is derived from ある.

Edit: there are idioms that use あらず which is also a standard negative form of ある, like なきにしもあらず.