A MEDIUM TRANSPORT AIRCRAFT CREW WAS CLEARED INTO POSITION AND HOLD LAX 25L BUT MISTAKENLY TOOK OFF.

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LAX RWY 25L XE02Z. SECOND FLT LEG OF THE DAY; CREW HAD ONLY BEEN ON DUTY SINCE XA30Z; WX 8 SM. WE WERE CLRED TO TAXI TO RWY 25L VIA TXWY A. UPON TAXIING TO RWY 25L; GND CTL INSTRUCTED US TO COME UP ON TWR FREQ AROUND TXWY AF. SHORTLY AFTER SWITCHING TO TWR FREQ; TWR ASKED 'ACFT X; WILL YOU BE READY AT THE END?' 'YES; ACFT X WILL BE READY AT THE END.' ABOUT 100 FT BEFORE REACHING RWY 25L; TFC IS A 6 MI FINAL CLRED TO POS. THIS WAS RECEIVED BY THE FLT CREW AS 'ACFT X TFC IS A 6 MI FINAL CLRED FOR TKOF.' LINE-UP ITEMS WERE COMPLETED AS WE ROLLED OUT ONTO THE RWY. THIS INCLUDED THE GND SPOILERS AND LNDG LIGHTS; BOTH OF WHICH WE NEVER TURN ON UNLESS A CLRED FOR TKOF IS RECEIVED. I ALSO LOOKED DOWN THE AISLE TO MAKE SURE THAT EVERYONE WAS SEATED AND READY TO GO. (THERE WERE NO ARRS OR DEPS ON RWY 25L DURING THE FEW MINS BEFORE THIS OCCURRENCE.) AS WE ROLLED DOWN THE RWY; I HEARD THE CTLR CLEAR AN ACFT TO LAND OR TAXI R; BUT WE HAD ALREADY REACHED V1. I HEARD AS WE ROTATED THE CTLR START TO ADDRESS US BUT THEN PAUSED AND SAID 'ACFT X...I HAVE A NUMBER FOR YOU TO COPY...ADVISE WHEN READY.' THE CAPT SAID TO ME WE WERE CLRED TO TAKE OFF RIGHT? BUT WE WERE ALREADY AIRBORNE. NO RA'S WERE DETECTED. I FEEL THERE WERE 4 BIG ERRORS HERE: 1) THE WORD