I take a slightly different approach, based on the apparent actual
meanings of the words for which "AM" and "PM are abbreviations.
It seems intuitively obvious to me that between 11:59 Ante-Meridiem and
12:01 Post-Meridiem must lie 12:00 Meridiem. (Though 12:00:01 - one
second later - would be Post-Meridiem again.)
Similarly, though less an "obvious necessity" consequence, between 11:59
Post-Meridiem and 12:01 Ante-Meridiem lies 12:00 Midnight. (I understand
"meridiem" to be derived from a word which would have literally meant
"mid-day".)