Consider two random variables $X_1=\min (W_1, W_2)$ and $ X_2=\min (W_3, W_4),$ where $W_1$, $W_2$,$W_3$ and $W_4$ are positive, identically distributed random variables. While $W_1$, $W_2$ are independent $W_3$, $W_4$ are correlated. We assume that the range of both $X_1$ and $X_2$ is from $0$ to $1$. Is it true that
$$\Pr \{X_2 \leq x \} \leq \Pr \{X_1 \leq x \},~~~ \forall x?$$
If yes, how to prove it? I would be grateful if any pointers to existing literature are given. Thanks!