Create new variable based on another variable that contrains character strings of numbers?

I have a character variable that contains entries representing diagnoses. I would like to create a variable to indicate whether a specific diagnosis is present.

For example, in the below data frame I have a variable called Diagnoses. This variable contains strings of numbers, whereby the numbers represent different diagnoses (e.g., 1 = diabetes, 2 = depression). I want to code a new variable to indicate whether the character variable contains a number I am interested in (e.g., 2 = depression).

Diagnoses   Relevant_Diagnoses 
1,3,5,7,8      0
1,2,3,4        1
2,3            1  
3,5,8          0

I've got the below code, but am aware I need to change the last ifesle statement so that it checks whether 2 is included in the list (rather than ==2), but I'm not sure how. What is the best way to go about this?

df$Relevant_Diagnoses <- ifelse(is.na(df$Diagnoses), NA, ifelse(df$Diagnoses == 2, 1, 0)) 

Many thanks for your help!