Wash and dry all vegetables thoroughly.
Shred the green cabbage and purple cabbage using a sharp knife or a food processor.
Grate the carrots using a box grater or food processor.
In a large mixing bowl, combine the shredded green cabbage, purple cabbage, and grated carrots.
In a separate bowl, mix together the mayonnaise, apple cider vinegar, sugar, and Dijon mustard until smooth.
Pour the dressing over the vegetable mixture.
Toss the vegetables and dressing together until evenly coated.
Season with salt and pepper to taste.
Cover the bowl with plastic wrap and refrigerate for at least 30 minutes.
Give the coleslaw a final toss before serving.