Let $\{X_t\}_t$ be a sequence of discrete random variables and assume that
\begin{equation}
\\Pr\big(X_t=x \, | \, X_{1},\dots, X_{t-1}\big) \geq \nu_x \quad \text{ for all } t=1,2,\dots
\end{equation} 


Consider a sequence of binary random variables $\{Y_t\}_t$ such that If $Y_t=1$, then $X_t=x$, i.e.,  $Y_t\leq 1\{X_t=x\}$, for all $t=1,2,\dots$

Could you help me to show that
\begin{equation}
\\Pr\big(X_t=x \, | \, Y_{1},\dots, Y_{t-1}\big) \geq   \nu_x  \quad \text{ for all } t=1,2,\dots \quad ?
\end{equation}