Flight Crew Air Transport Pilot (ATP)
ATC Issue All Types; Conflict NMAC; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter VFR In IMC
Flight Crew Took Evasive Action
Human Factors
During climbout from CVG; we were given a climb to 13;000 feet and cleared to WADAL intersection. Passing out of 9;000 feet (climbing at 1500 feet/min) we were told to contact Indy Center on 135.57. We checked in with Indy Center (135.57) and stated we were climbing to 13;000 feet. Passing through about 10;000 feet; the TCAS gave an aural TA warning and the display popped up on the MFD.  It showed an aircraft 800 feet above us descending at our 12 O'clock position towards us (about 6 miles out). At the same time; Indy center issued a turn 15 degrees to the left. Simultaneously we got a descending RA (indicating 2;000 feet/min down). I commanded my OE student to immediately disconnected the A/P comply with the RA; which he did. As we were descending per the RA; we visually acquired the aircraft as it pop out of the clouds at about 200 feet above us. It passed off our right side and directly over us. This appeared to be low wing aircraft; single engine type. This aircraft was squawking 1200 and appeared to be VFR.  They nor Indy Center were not controlling this aircraft at the time. I did speak with the Indy Center manager afterwards; he was unaware at that point what had transpired.Cause - VFR aircraft not controlled by ATC and flying in the clouds without an IFR clearance.Suggestions - I learned the importance of always complying with TCAS instructions in an expeditious manner. Had we not done this and utilized our training; it may have ended poorly.
Air carrier Captain reported an NMAC with a VFR small aircraft during departure from CVG airport.