The sum of opposite numbers equals zero (the additive identity)
5 + -5 = 0 so the opposite of 5 is -5
The absolute value of a number is the positive distance it lies from zero on a number line.
The absolute value of 5 is 5, so Mike is not correct in his second statement.
Strangely if you swapped the two numbers then both parts would be correct:
The opposite of -5 is 5 and the absolute value of -5 is 5.