If [tex]U\subseteq V[/tex], then either [tex]U=V[/tex] or [tex]U[/tex] is contained entirely in [tex]V[/tex].
In the first case, we get [tex]\mathbb P(U)=\mathbb P(V)[/tex].
In the second case, we have [tex]\mathbb P(U)=\mathbb P(U\cap V)=\mathbb P(U\mid V)\cdot\mathbb P(V)[/tex], or [tex]\mathbb P(U\mid V)=\dfrac{\mathbb P(U)}{\mathbb P(V)}[/tex]. The LHS must be between 0 and 1, and this can only occur as long as [tex]\mathbb P(U)\le\mathbb P(V)[/tex].
