FLC OF AN MLG ACR ACFT CLBED ABOVE ASSIGNED ALT IN RESPONSE TO A TCASII RA.

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WE WERE ON A VECTOR OF 260 DEGS TO INTERCEPT THE CVG 040 DEG RADIAL LEVEL AT FL220. AN ACR LGT WAS LEVELING AT FL210. HE WAS AT OUR 1 O'CLOCK LOW; SO WE DID NOT HAVE HIM VISUALLY. WE HAD HIM ON TCASII AT 3-4 MI. WE THEN GOT A TA FOLLOWED SHORTLY BY AN RA WITH A COMMAND TO CLB 1500 FPM. WE COMPLIED UNTIL THE FO GOT A QUICK VISUAL OF THE ACFT. WE TOPPED OUT AT 22400 FT AND QUICKLY RETURNED TO FL220. WE IMMEDIATELY CONTACTED CTR TO SEE IF THEY SHOWED ANY CONFLICT. THEY SAID THAT IT WAS NOT A PROB AND THERE HAD BEEN NO CONFLICT. THE TFC WAS IN A POS THAT WAS DIFFICULT TO SEE SO WE HAD TO DEPEND ON THE TCASII UNTIL WE COULD VISUALLY CONFIRM NO CONFLICT. THE TFC WAS MOST PROBABLY CLBING AT A RAPID RATE JUST PRIOR TO LEVEL OFF. ALSO; WE HAD BEEN DISPATCHED WITH ONE AIR CONDITIONING PACK INOP AND THEREFORE WE WERE AT FL220 AND 325 KIAS. I (CAPT) WAS GETTING THE ATIS AND NEVER HEARD THE 'TFC' AURAL ON TCASII AS IT IS ONLY HEARD OVER A SEPARATE COCKPIT SPEAKER AND NOT OVER THE HEADPHONES THAT I AM REQUIRED TO WEAR.