ContainsOnly
[$list_1$, $list_2$]ContainsOnly[{b, a, a}, {a, b, c}]
The first list contains elements not present in the second list:
ContainsOnly[{b, a, d}, {a, b, c}]
ContainsOnly[{}, {a, b, c}]
Use Equal as the comparison function to have numerical tolerance:
ContainsOnly[{a, 1.0}, {1, a, b}, {SameTest -> Equal}]