B777 DEP FROM JFK EXPERIENCED TCASII TA WITH OPPOSITE DIRECTION TFC AT 6000 FT; ATC ISSUED A TURN TO THE CONFLICTING ACFT.

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WE WERE CLRED TO CLB AND MAINTAIN 11000 FT MSL; ON AN ASSIGNED HDG OF 050 DEGS. IMMEDIATELY AFTER ROLLING OUT ON HDG 050 DEGS; THE TCASII GAVE US A 'TFC; TFC' AND I OBSERVED TFC AT OUR ALT (6000 FT) OPPOSITE DIRECTION; CLOSING FAST. ATC IMMEDIATELY INSTRUCTED US TO 'TURN R IMMEDIATELY TO HDG 100 DEGS;' AND SUBSEQUENTLY ATC INSTRUCTED OUR OPPOSITE DIRECTION TO TURN IMMEDIATELY 45 DEGS R. AS THE TFC PASSED US; THE TCASII SHOWED HIM TO BE LESS THAN 2 MI; STILL AT OUR ALT. THE OPPOSITE DIRECTION TFC WAS A LEARJET. WE DID EVERYTHING WE WERE INSTRUCTED TO DO; AND DID ABSOLUTELY NOTHING WRONG. NEITHER DID THE LEARJET. THIS IS A SIMPLE CASE OF ATC LOSING SEPARATION DURING HIS HANDLING OF OUR FLT. BEFORE HANDING US OFF TO THE NEXT SECTOR; THE CTLR THANKED BOTH OUR FLT AND THE LEARJET FOR 'OUR HELP.' ALLOWING US TO CLB THROUGH AN ALT OCCUPIED BY OPPOSITE DIRECTION TFC CONSTITUTES A DANGEROUS SAFETY HAZARD. IT IS FURTHER COMPOUNDED BY TURNING US INTO THE TFC SO AS TO CAUSE US TO BE WITHIN MERE SECONDS OF COLLISION.