Write a function/program that accepts a string of lower/uppercase letters [A-Za-z] as input, that checks whether the occuring letters are unique and in alphabetical order (ignoring lower and uppercase) or not. The output must be truthy if they are unique and in alphabetical order and falsy if not.
Here some testcases
a true abcdefGHIjklmnopqrSTUVWXyz true aa false puz true puzz false puzZ false puZ true PuZ true pzu false pzU false abcdABCD false dcba false
If you want, run your program on all words of a wordlist like this one and and post some interesting ones =).
Lowest number of bytes wins.