Conditional expectation of two exponential random variables

$X$ and $Y$ are two independent exponential random variables:

$P(X \geq x)=e^{-\lambda_1 x}$ for every $x>0$;

$P(Y \geq y)=e^{-\lambda_2 y}$ for every $y>0$;

How to calculate the expectation of $X$ under the condition that $X<Y$?

i.e. How to calculate $E(X|X<Y)$?

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I'm also wondering why I am wrong:

$E(X)=E(X|X<Y)*P(X<Y)+E(X|X \geq Y)*P(X \geq Y)$

$E(X)=1/\lambda_1$

$P(X<Y)=\lambda_1/(\lambda_1+\lambda_2)$

$E(X|X \geq Y)=E(X)+E(Y)=1/\lambda_1+1/\lambda_2$

$P(X \geq Y)=\lambda_2/(\lambda_1+\lambda_2)$

However, the result of $E(X|X<Y)$ is $0$.

===I see $E(X|X \geq Y)$ should be $E(X)+E(Y|Y<X)$...