Not to be anal, and I know it really doesn't have anything to do with the slip-up, but Jasmine was not his wife in that scene. Will Smith's buddy makes a big deal out of it just a few scenes later, telling him that he'll never fly for Nasa if he marries a stripper.
It would probably take an AF vet like me to spot it, but the real slip up was not the collar, but the epaulets. One seems to pop in and out from under the collar at will a couple times during the closeups while he is saying goodbye to his girl friend Jasmine. Of course she's such a doll it's hard to pay attention to him!!!!!!