Probability Question from Actuarial exam: claims for wind damage

Hi,
Attached is a similar problem to the last one I posted (about iid). The solution is included and I was hoping that someone can please explain this to be. I almost have it but not totally. Thanks!!

Since we want the expected value of the largest of the three claims and the claims are independent is this the logic of the start of the solution?

Let Y = the largest value of the 3 claims, X₁, X₂, and X₃

For Y = y, we want \(\displaystyle F_{Y}(y)=p(y>X_{1} \cap y>X_{2} \cap y>X_{3})= p(X_{1}<y \cap X_{2}<y \cap X_{3}<y)=p(X_{1})p(X_{2})p(X_{3})=[p(X_{1})]^3\)

What I have above is not 100% obtained from the solution I provided, but just after reading the solution once and thinking about it a bit. My logic lies heavily on the statement \(\displaystyle F_{Y}(y)=p(y>X_{1} \cap y>X_{2} \cap y>X_{3})\)

Is this the correct thinking?