B737-700 CAPT INSISTED ON USING THE APU AFTER IT WAS MEL'ED INOP. THE FIRE WARNING SYS DID NOT TEST PROPERLY.

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WE JUST RECEIVED THE PLANE THAT FLEW IN. AFTER MY OUTSIDE PREFLT; I STARTED MY COCKPIT PREFLT. I QUICKLY DISCOVERED THE APU SQUIB WAS NOT TESTING. I SHOWED THE CAPT. HE IMMEDIATELY CALLED THE COMPANY AND HAD IT MEL'ED. SO WE HAD THEM CONNECT THE GND PWR AND TURNED OFF THE APU; BUT THE ENG TURNED VERY SLOWLY; AND WE ABORTED THE START. CAPT TESTED THE APU SQUIB AGAIN; AND NOW IT TESTED PERFECT; OVER AND OVER AGAIN. CAPT WANTED TO START THE ENGS WITH THE APU. I SAID 'YOU CAN'T; IT IS MEL'ED!' HE SAID 'IT IS TESTING FINE' AND CRANKED THE APU. I WAS PRETTY UPSET AND THE CAPT KNEW IT. BUT HE THOUGHT IT WAS THE CORRECT THING TO DO. I WAS OVERRULED. WE BEGAN TO START THE FIRST ENG; AND IT TURNED SLOWLY AGAIN. WE BOTH LOOKED AT BLEED SOURCES; AND THE CAPT HAD LEFT THE WING HEAT SWITCH ON; FROM HIS FLOW. (I WISH I WOULD HAVE CAUGHT IT.) HE TURNED OFF THE WING HEAT SWITCH; AND STARTED FINE; BOTH ENGS. REST OF TRIP WENT FINE. BUT WHEN WE GOT CLOSE TO OUR DEST; I REMINDED THE CAPT THAT WHEN HE CALLS IN RANGE; TO 'MAKE SURE WE GET 'AIR AND ELECTRICS' AS PER THE MEL.' CAPT OVERRULED AGAIN; AND MENTIONED IT WAS STILL TESTING FINE. WE DISCUSSED THIS A LOT FOR THE NEXT 2 DAYS. HE DID NOT AGREE WITH MY INTERP ON MEL'S. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE FO RPTED HE HAD TO FLY WITH THE CAPT FOR 2 MORE DAYS. AFTER THE INCIDENT THE RELATIONS BTWN THE CAPT AND FO WERE CHILLY AT BEST. THE ACR DOES NOT HAVE A WAY TO ENFORCE CRM. THE FO DID NOT RPT THE INCIDENT TO ANYONE.