Hang on, supplementary point from the antipodean upstart
Lets not forget that the TCAS is tracking targets outside the range that it displays. The 2700 thing is to reduce display clutter.
Lets say that one is climbing to F060 at 3000 fpm. The TCAS is watching targets up to 7200'/9700' above depending on software status.
Your autopilot plans (roughly) to start the pitch over to capture F060 at a maximum 5700', six seconds (again, roughly) before it would shoot through F060 if that altitude were not armed for capture. The autothrottles are also perhaps rolling back to keep the speed limit for the terminal control area.
Worst case then is another fellow descending at 3000fpm rate to F070 in your vicinity. Will he not also commence a pitch up at at least 6 seconds before to capture that altitude, and will his TCAS not be tracking you?
Can someone perhaps resolve this as 'so many seconds to closest point of approach' for RA? I cannot remember off the top of my head what CPA triggers an RA. Maybe our answer lies there.