This is an extension of my previous question from the post
Consider the two random variables $X_1$ and $X_2$ defined via non-negative random variables $Y_1$ and $Y_2$ as
$ X_1=\min(Y_1,Y_2)\\$ where $Y_1$ and $Y_2$ are i.i.d and
$ X_2=\min(Y_1,Y_2)$ - where $Y_1$ and $Y_2$ are correlated.
The question is can we say $X_1 \leq X_2$ always? How to prove it?