Does $\{\omega \in \Omega: X(\omega) \leq t\} =\{\omega \in \Omega: \delta(\omega) =1\}\cap \{\omega \in \Omega: T(\omega) \leq t\}$ hold?
Solution 1:
$Y(\omega)<X (\omega)\leq t$ then $X(\omega) \leq t$ but $\delta(\omega)=0$.
For an explicit counter-example consider constant r.v.'s $X=2, Y=1, t=2$.