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MB3 154b: Putting Dishes in a Dishwasher and Immersion in a Mikvah on Shabbos (323:6-7)
Washing dishes if having dirty dishes lying around bothers you. Putting dishes into a dishwasher on Shabbos vis-a-vis hachana. Immersing a utensil in a mikvah on Shabbos. Going to the mikvah on Shabbos.