fly the coop

fly the coop AND fly the nest

(to leave somewhere, especially to leave your home for the first time in order to live away from the family ) {mainly US} — оторваться от дома; покинуть родительское гнездо; быть отрезанным ломтем

Example: The last of our kids has finally flown the coop so we have the whole house to ourselves.

Example: The latest Social Trends survey found grown men were increasingly reluctant to fly the nest.Nearly a third of men aged between 20 and 35 are living with their parents, compared with only one in four in 1977/8. news.bbc.co.uk)

Related vocabulary:
fly / leave the nest
1 (of a young bird) to become able to fly and leave its nest
2 (informal) (of somebody's child) to leave home (our parents' home) and live somewhere else
[отрезанный ломоть]
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A coop is a place where chickens are kept.