EK require you to provide a letter stating you have terminated your employment with your previous employer only after you accept an offer of employment with them. ie. Prior to to commencing your training.
Unless EK have an understanding with a specific airline, such as the Air NZ thing.
I don't think EK has any such understandings with any OZ airlines, so you would have to provide proof of your resignation from your current job prior to starting at EK. All that is required is your letter of termination of employment.