[{"userName":"4567","library":[{"serialNumber":"Book_1"},{"serialNumber":"Book_2"},{"serialNumber":"Book_4"}]},{"userName":"123","library":[{"serialNumber":"Book_2"}]},{"userName":"13526","library":[{"serialNumber":"Book_7"},{"serialNumber":"Book_6"},{"serialNumber":"Book_5"},{"serialNumber":"Book_4"},{"serialNumber":"Book_3"},{"serialNumber":"Book_5"}]},{"userName":"0000","library":[]},{"userName":"0000","library":null},{"userName":"00001111"},{"userName":"11110000","library":[{"serialNumber":"Book_0"},{"serialNumber":"Book_0"},{"serialNumber":"Book_1"},{"serialNumber":"Book_1"},{"serialNumber":"Book_2"}]},{"userName":"888999","library":[{"serialNumber":"Book_0"},{"serialNumber":"Book_0"}]}]

db.collection.aggregate([{$match:{$expr:{$and:[{$eq:[{$type:"$library"},"array"]},{$ne:["$library",[]]}]}}},{$addFields:{allUnique:{$eq:[{$size:{"$setUnion":["$library.serialNumber",[]]}},{$size:"$library"}]}}},{$match:{allUnique:false}},{$project:{userName:1,_id:0}}])