It depends on where you are going. For Caribbean, Central and South America I can almost always find a AA flight that's significantly cheaper in F and all on mainline (via DFW or MIA).

From what I've seen neither UA nor AA is interested in a domestic fare war. Even if they wanted to they are constrained by gates from the days the MAC was a little too close to NW.

On the other hand I've also felt UA dropped the ball on aligning EWR traffic to go after 1-stop TATL traffic. There was really good connections from the NW/CO days, but then they got a little too focused on funneling people to ORD post merger (a bit like DL and ATL). The fact that AF is now running A340s for CDG/MSP route shows the TATL demand was there and left on the table IMHO.