I am having a difficult time understanding a certain equivalence regarding the maximum of two stopping times. The context is showing, for two stopping times $\tau_1$ and $\tau_2$, that $\tau_1 \vee \tau_2$ is a stopping time as well.
I understand that this amounts to showing that the random variable $\tau_1 \vee \tau_2$ is $\mathcal{F}_t$-measurable. It is also clear intuitively speaking. The trouble I have is with the intersection in the following $$\{\tau_1 \vee \tau_2 \le t\} = \{\tau_1 \le t\} \cap \{\tau_2 \le t\}$$
My understanding seems to be wrong, for which I kindly ask for clarification and guidance for, but I would expect (for $\tau_1 \le \tau_2)$ that $\{\tau_1 \le t\} \subseteq \{\tau_2 \le t\}$ to be true -- which is to say, a larger stopping ''accommodates'' more elements in $\Omega$. Under this reasoning, I should expect $$\{\tau_1 \vee \tau_2 \le t\} = \{\tau_1 \le t\} \cup \{\tau_2 \le t\}$$
For reference, I will direct you to theorems 28 and 38. This is one such example, but several sources seem to verify the intersection, rather than the union, in the equivalence above.
Any intuition and guidance are certainly appreciated!