An aircraft with TCAS can receive an RA in respect of non-TCAS equipped traffic. All that is required is for that traffic to have a working mode C transponder. (And be on a conflicting flight path, of course.) I've seen this, twice. (Works a treat, by the way.)
Regarding the OP, I doubt that the Boeing received an RA in respect of the approach underneath you, unless he was closing on you, and closer to your altitude than 500 feet, and within about a mile. (Or, 20 seconds to point of closest passing, to be more accurate.)
TCAS does not measure "separation" in terms of feet or miles. If that was the case, there would be either numerous false alarms in the circuit area, or inadequate protection at cruise speed. It measures the projected hazard in terms of closure rate based on time.