Puzzle #234: Different Birthdays

Two people have different birthdays. With each expressed as MM-DD, all eight digits are different. If this is possible, what are the two birthdays? If not, prove that it is not possible.

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, November 29, 2017 at noon Pacific Time. I will post the answer shortly after.