A B737 FLIGHT CREW DEPARTING LAS WAS ADVISED BY TOWER AFTER TAKEOFF THEY DEPARTED WITHOUT CLEARANCE; BUT A CHECK OF THE TAPES VERIFIED THEY WERE IN FACT CLEARED FOR TAKEOFF.

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WE TAXIED OUT TO RWY 7L FOR AN A8 INTXN DEP. COMPANY WAS HOLDING SHORT AND AS WE NEARED THE RWY; TWR CLRED HIM FOR TKOF. WE CONTINUED TO TAXI AND JUST AS WE GOT TO THE HOLD SHORT; TWR CLRED US FOR TKOF. WE GOT AIRBORNE AND TWR ASKED US IF WE HAD A VISUAL ON OUR COMPANY. HE THEN SAID THAT WE HAD NOT RECEIVED CLRNC TO TAKE OFF. WE SAID THAT WE HAD AND I STATED THAT; 'YOU CAN CHK THE TAPES.' WE MAINTAINED VISUAL WITH COMPANY AND SWITCHED TO DEP. THE REST OF THE FLT WAS UNEVENTFUL. WHILE TAXIING IN; GND ADVISED THAT MCCARRAN TWR WANTED US TO CALL THEM. I PHONED THE TWR SUPVR AND HE HAD CHKED THE TAPES AND TOLD US THAT WE DID IN FACT HAVE CLRNC FOR TKOF. HE SAID THAT NO FURTHER ACTION WAS NECESSARY FOR US. HOWEVER; I INFORMED HIM THAT WE WOULD BE SUBMITTING A SAFETY RPT. HE THEN ASKED FOR MY NAME AND I REPLIED THAT HE COULD CALL COMPANY OPS IF HE NEEDED THAT INFO. THAT SEEMED TO END THE CONVERSATION. SLOW DOWN WHEN ISSUING CLRNCS! IF YOU ARE TRYING TO MAKE UP FOR THE MASSIVE DELAYS OUT OF THE ARPT BY RUSHING TO IMPROVE YOUR DEP RATE AT NIGHT; THEN IT COULD LEAD TO A MISTAKE.