05AB1E, 1017 bytes
Σ7ÝRAS«Σ•_ćúθœιu0•hSk}ÙQ Explanation:
Σ } # Sort by k # the index of each char of the input in... 7ÝR •_ćúθœιu0•hS # the numbers 7 -> [7...0,9,8,A...F] AS # then the alphabet « Ù # concatenated Ù remove duplicates # remove duplicates Q # and check that it's not changed