Let $\sigma,\tau$ be stopping times. $\{\sigma<\tau\}\in\mathcal{F}_{\sigma}$?

Question

Consider a filtered prob space $(\Omega,\mathcal{F},(\mathcal{F}_s)_{s\in [0,T]},\mathbb{P})$ satisfying the usual conditions. Let $\sigma,\tau$ be stopping times. Is it possible to show that $$\{\sigma<\tau\}\in\mathcal{F}_{\sigma}?$$ I tried the following: $$\{\sigma<\tau\}=\bigcup_{q\in[0,T]\cap\mathbb{Q}}\{\sigma<q\}\cap\{q<\tau\}$$ Now, fix $s\in[0,T]$. Consider $$A_s:=\{\sigma<q\}\cap\{q<\tau\}\cap\{\sigma\leq s\}$$ and try to show that $A_s\in\mathcal{F}_s$. Then the claim follows by the definition of $\mathcal{F}_{\sigma}$. If $q\leq s$, then $$A_s=\{\sigma<q\}\cap\{q<\tau\}\in\mathcal{F}_q\subset\mathcal{F}_s.$$ If $q>s$, then $$A_s=\{\sigma\leq s\}\cap\{\tau>q\}$$ and i got stuck.