[{"id":1,"active":true,"key":[]},{"id":2,"active":true,"key":[{"code":"fake_code","ids":[""],"labels":["d"]}]},{"id":3,"active":true,"key":[{"code":"fake_code","ids":[""],"labels":["a","b","c"]}]}]

db.collection.find({key:{$elemMatch:{labels:{$in:["a","b","c","d"]}}}},{_id:0,id:1})