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EX = integral from 0 to infinity of 1 - F(x) dx

Say this integral can be:

EX = integral from 0 to infinity of P(X > x), then

EX = integral from 0 to infinity of integral from x to infinity of f(y)dydx

= integral from 0 to infinity of integral from 0 to y of dxf(y)dy = EX

Can somebody explain me how and why the last integral is included?

Thanks.