HMMT 2010 T4

Let

$$\begin{gathered} e^{x}+e^{y}=A \\ x e^{x}+y e^{y}=B \\ x^{2} e^{x}+y^{2} e^{y}=C \\ x^{3} e^{x}+y^{3} e^{y}=D \\ x^{4} e^{x}+y^{4} e^{y}=E . \end{gathered}$$

Prove that if $A, B, C$, and $D$ are all rational, then so is $E$.