A simple question about conditional expectation

Question

If I got

$$E\left(\min\left(X,Y\right)\right)$$

Why is it equal to

$$E\left(\min\left(X,Y\right)\right)=E\left(\min\left(X,Y\right)\mid\min\left(X,Y\right)=X\right)P\left(X\le Y\right)+E\left(\min\left(X,Y\right)\mid\min\left(X,Y\right)=Y\right)P\left(X>Y\right)$$

Answer 1

Let $A$ be the event $)X \leq Y$ and $B=(X>Y)$. Then $E(X\wedge Y)=E((X\wedge Y) I_A)+E((X\wedge Y) I_B)=E(X\wedge Y : A) P(A)+E(X\wedge Y : B) P(B)$.

Note: $E(X:A)$ is defined as $\frac 1 {P(A)} EXI_A$.