How to implement conditional probability distribution on set-valued Random variables
Keeping it to discrete space, the joint probability of an event can be interpreted as the fraction of the size of this subset compared to the whole state space.
In case of events, being sets themselves, the definition remains the same.
$$P(X_i,Y_j) = \frac{|\{(x,y) \in  (X_i,Y_j)|\}} {|\Omega|} $$
$|\Omega|$ is just the set of all combinations of single events (x,y).
Let us now check your formula with the following example:
$$X_0 = Y_1 = \{0\}, ~~ X_1 = Y_0 = \{1\}$$
Then the total space space $X \times Y$,
$$ \Omega  = \{(0,0),(0,1),(1,0),(1,1)\} $$
And for example:
$$ P(X_0,Y_0) = P( (0,0) ) = \frac{1}{4} $$
Your formula however implies,
$$ | (X_i \cap Y_i) \cap  (X_j \cap Y_j) | = 0 $$
as $X_i \cap Y_i$ is empty per contruction in the example.