No. Some ACs are used as mood stabilizers.
It's one of those logic/sub-group things: For example.... An AP is a mood stabilizer. But a mood stabilizer is not necessarily an AP.

Uhhh, I guess it'd be simplest to say....

Mood stabilizer is the big umbrella.

It contains various classifications of meds, such as:
AED/AC
AP
AAP

Then of course, each of these groups is comprised of the various individual meds.

I had to go double-check on Lithium. It is, of course just thought of as a mood stabilizer, and it is, but its subgroup is anti-psychotic.

Haha, I hope this is making sense (if not, I can make a flow chart or something ). It's coming up on 6 am, so I might not be the best judge of comprehensible explanation atm.