If P(X>Y)=m and P(X>k)=m then E(Y) =k

Question

I feel like the statement would only be true if the median of Y was equal to the mean of Y ie Y is symmetrical. Because the probability only cares about the median and expected value cares only median. That is, i believe If P(X>Y)=m and P(X>k)=m then median(Y) =k is definitely true but the title is only true when median=mean iff distribution of Y is symmetrical.