Inequality between stopping times in continuous time

Question

Let $(\mathcal{F}_t)_{t\geq0}$ be a filtration and let $\sigma$ and $\tau$ be $(\mathcal{F}_t)_{t\geq0}$-stopping times. I would like to show that the events $\{\tau\leq\sigma\}$, $\{\tau<\sigma\}$ and $\{\tau=\sigma\}$ are elements of $\mathcal{F}_{\sigma}\cap\mathcal{F}_\tau=\mathcal{F}_{\sigma\land\tau}$. I can do this is discrete time, but I get stuck in the case of continuous time. So far I have reasoned that since $$ \mathcal{F}_{\sigma\land\tau}=\{A\in\mathcal{F}\mid A\cap\{\sigma\land\tau\leq t\}\in\mathcal{F}_t\;\forall t\geq 0\}, $$ I must e.g. show that $\{\tau<\sigma\}\cap\{\sigma\land\tau\leq t\}\in\mathcal{F}_t$ for all $t\geq0$. I have $$ \{\tau<\sigma\}\cap\{\sigma\land\tau\leq t\}=\{\tau<\sigma\}\cap\{\tau\leq t\}, $$ but I am not seeing how to proceed further. I would very much appreciate a hint from someone. Thank you in advance.