B737-800 Captain reported an NMAC during initial approach with another aircraft that had entered Class B airspace without a clearance.

Narrative

Being vectored onto a downwind for a visual to [Runway] XXR; I noticed TCAS traffic pop up ahead on the display. Immediately after that ATC instructed us to climb to 5;000 to avoid the unknown traffic. I could see the traffic at this point. The automation was slow to climb; so I disconnected the autopilot to help expedite. ATC then instructed us to turn left to 140 for traffic avoidance. Just as I started the turn--or continued it; really--we got an aural traffic alert followed immediately by an RA. All of this happened very quickly. The RA commanded a descent; which was unexpected. But the FO noticed immediately and verbalized it. I disconnected the auto throttle and pushed the nose down; but overcoming the climb momentum was difficult and took a lot of effort. The traffic passed in front of us and to our right; quite closely; apparently climbing. ATC was advised we were complying with an RA. On the 140 heading at about 3;700 feet we were clear of the conflict and ATC had us maintain 4;000 and turn back to 170. The controller informed us the conflict aircraft had entered Class B without clearance and was being tracked. We continued vectors for a visual to XXR and landed normally with no further incident. I believe the vector to 140 ATC gave us made the difference.Cause - Aircraft entering ZZZ Class B airspace without clearance.At the time we were in Class B airspace and slowing to configure at this point was not necessary; but had I been at a slower speed we would have had more time; although we were converging. Also; I turned the autopilot off and pushed the thrust manually because I sensed we were going to have to maneuver; but we were not technically in an RA. I should have turned the auto throttle off then also; which would have helped some in the reversal from climb to descent; maybe.