Using a large sharp knife, slice the cabbages into 1/4-inch-thick slices. Lay the slices flat and finely chop them, first sideways, then up and down. You should have about 5 cups green cabbage and 1 cup purple. Toss the green and purple cabbages and the carrot in a large bowl.
For the dressing: In another bowl, mix the vinegar, sugar, onion, green pepper, oil, celery seeds, garlic, dry mustard, salt, and pepper, stirring until the sugar is dissolved.
With a measuring cup, transfer half of the dressing to the bowl with the cabbage and carrots. Toss with clean hands or a large spatula to combine. Keep adding the dressing, ¼ cup at a time, being careful not to drench the cabbage. You may not need all of the dressing; any leftover can be saved in a covered container in the refrigerator for up to 5 days and used as a dressing for salad or to marinate vegetables.
Cover and refrigerate for several hours, until chilled. The slaw keeps, refrigerated, for 5 days.