Pilot reported a NMAC with another aircraft that ATC did not see or call as traffic.

Narrative

We were on an IFR flight plan climbing out of SLE on an assigned heading of 360 and talking to Seattle Center; which was moderately busy. The Controller cleared us up to 5;000 ft.; and shortly after gave us a left turn to 280 'for traffic.' We made the turn and as we rolled out I noticed a target on the MFD that was moving perpendicular to us coming directly at our 9 o'clock. I zoomed in the MFD and noted that the traffic was around 2-3 miles away and level at 4;500 ft. I believe we were climbing through 3;300 at that point; and I expected ATC to give us another turn or stop our climb - that never happened. Between 3;800-3;900 ft. I told the student to stop his climb and I also pushed on the yoke to help him level off. Almost immediately after stopping the climb; the TIS gave us the audible traffic alert 'traffic; 9 o'clock; high.' We never got a visual on the traffic as it was above our wing. Once the frequency had an opening; I called with 'Stopping our climb for traffic avoidance.' The Controller came back and sounded a little confused; and said we could continue climb at pilot's discretion. A few seconds later the Controller asked if we received an 'RA' for that traffic...I responded that we did get an audible alert from our ADS-B just after we stopped our climb. The Controller came back and apologized for her mistake; saying that she didn't see the other traffic and the scope did not give her an audible alert for the potential conflict (as it should have). The Controller cleared us for a climb that could have put us right into another aircraft; and their equipment malfunctioned and failed to alert them of the conflict. The flight continued with out any other safety issues.