The title may sound strange. Sorry for that but the question is short and easy to understand.
I have a set $A \in \mathcal{F}_t$ where $(\mathcal{F}_t)$ is a filtration on some probability space. For some fixed $t \in \mathbb{R}_+$, it holds that $A\subseteq \{\tau \geq t\}$ where $\tau$ is a stopping time.
What I want to show is the following. For all $s: 0 \leq s < t$, it holds that $A \cap \{\tau \leq s\} \in \mathcal{F}_s$.
Since $s < t$, I have that $\{\tau \geq t\} \subseteq\{\tau \geq s\}$. Therefore, $A\subseteq \{\tau \geq s\}$. Then I write $A \cap \{\tau < s\} \subseteq \{\tau \geq s\} \cap \{\tau < s\} =\emptyset$. Hence $A \cap \{\tau < s\} =\emptyset$. This gives $A \cap \{\tau \leq s\} = A \cap \{\tau = s\}$. I don't know if this intersection lies in $\mathcal{F}_s$ but this is where I am stuck.
Note that
$$A \cap \{\tau \leq s\} \subseteq \{\tau \geq t\} \cap \{\tau \leq s\} = \{t \leq \tau \leq s\} = \emptyset \in \mathcal{F}_s$$
for any $s<t$.