Here's a similar test:
($me->age!=23 || $me->age!=57)
In English that would be "my age is not twenty-three or my age is not fifty-seven".
That statement is true if either of its parts is true (because they're joined by OR). If I'm 57 then I'm not 23. if I'm 23 then I'm not 57. And if I'm 4 then I'm neither. In all three cases the statement "my age is not twenty-three or my age is not fifty-seven" is true.
In short:
!(A && B) ==> (!A || !B)
If they're not both true then one of them must be false.
It goes the other way, too:
!(A || B) ==> (!A && !B)
If neither of them are true then they must both be false.