How do you output boolean if column containing lists have elements from another larger list?

I have a column where each row contains a list of strings of varying lengths. I need to create a new column that has a list of booleans (equivalent to the original list) of whether or not each element is found in ANOTHER (larger) list.

This is what I am doing and well, it clearly does not work. I based it off of this question:
How to return list of booleans to see if elements of one list in another list

data = [
    [1, ["cat", "cat", "mouse"]],
    [2, ["dog", "horse"]],
    [3, ["cat"]],
    [
        4,
        np.nan,
    ],
]

df = pd.DataFrame(data, columns=["ID", "list"])
df

main_list = ["cat", "dog", "mouse", "pig", "cow"]

df["contains_item_from_list"] = df["list"].apply(
    (lambda x: [x in main_list for x in b])
)

desired output:

ID     list          contains_item_from_list
1  [cat,cat,mouse]      [True, True, True]
2  [dog,horse]          [True, False]
3  [cat]                [True]
4   NaN                 [False]

>Solution :

You can also apply a function that iterates over each list in list. This should be faster than exploding the column:

main_set = set(main_list)
df["contains_item_from_list"] = df['list'].apply(lambda x: [w in main_set for w in x] if isinstance(x, list) else [x in main_set])

Output:

   ID               list contains_item_from_list
0   1  [cat, cat, mouse]      [True, True, True]
1   2       [dog, horse]           [True, False]
2   3              [cat]                  [True]
3   4                NaN                 [False]