This doesn't have all the symbols and l want to verify if my reasoning is sound. It is also one half of the iff proof.
Given : The union of X and Y is a subspace of V.
I have to prove that either X is contained in Y or Y is contained in X.
Reasoning :
Suppose there exists w belonging to X which doesn't belong to Y and there exists v belonging to Y which doesn't belong to X. Both w,v belong to the union of X and Y . Let z= w+v. z doesn't belong to the union of X and Y. Hence, the union of X and Y is not a subspace. This is a contradiction. Therefore, all w belonging to X, must also belong to Y or all v belonging to Y must belong to X.
I am not sure about z not belonging to the union in all cases but l can think of some particular examples.
I think this prove is very weak if not wrong and l will check better ones.