I want to prove that for stopping times $S,T$ : $$ \{ \min(S,t) < \min(T,t) \} \in \mathcal{F}_t$$
I want to prove implication for both directions ( I took it from the lecture notes of the course I study):
A random time $T$ is an $\mathbb{F}^+$ -stopping time iff for all $t>0$ one has $\{ T <t \} \in \mathcal{F}_t$.
I have deleted my useless old ideas. I would like to apologise for mess in comments and confusion I did to saz and Ansel B.
Below, I am trying to write down rigorously without missing steps what Ansel B said about the problem number 1.
Firstly, to prove that $ \{T \leq t \} \in \mathcal{F}_T $ I want to see that: $$ \{T \leq t \} \in \{F \in \mathcal{F}_\infty | F \cap \{T \leq t\} \in \mathcal{F}_t, \forall t > 0 \}$$
Is it trivial, or I should say something on that? My explanation to this step is: as $\mathcal{F}_\infty = \sigma(\mathcal{F}_t, t \geq 0 )$ we indeed have $F$ inside of even $\{T \leq t\}$ as $ \mathcal{F}_\infty$ is created by $\mathcal{F}_t$-s.
By analogy, we say that $ \{ \min(T,t) \leq t \} \in \mathcal{F}_{\min(T,t)} $.
Moreover $\mathcal{F}_{\min(T,t)} \subset \mathcal{F}_{t} $. ( Is it trivial?)
Similarly with $ \{ \min(S,t) \leq t \} \in \mathcal{F}_{\min(S,t)} \subset \mathcal{F}_{t}$.
Finally 2 subsets of $\mathcal{F}_{t}$: $ \{ \min(S,t) \leq t \} $ and $ \{ \min(T,t) \leq t \} $ together lead to 4 cases:
$$\{S \leq T\}, \{t \leq T\}, \{S \leq t\}, \{t \leq t\}$$ where first case is by definition correct, as we choose from the beginning $ S \leq T$. Second is impossible, Third is definion of stopping time $S$, hence is always in $\mathcal{F}_{t}$. Fourth is trivial case and also in $\mathcal{F}_{t}$.
Therefore we can conclude that $ \{ \min(S,t) < \min(T,t) \} \in \mathcal{F}_t$.
First note that if $S$ and $T$ are stopping times then so is $\min(S,T)$ as $\{\min(S,T)>t\}=\{S>t\}\cap\{T>t\}\in\mathcal{F}_t$. Also it is very easily seen that any stopping time $T$ is $\mathcal{F}_T$ measurable. And for a fixed $t$, $t$ is trivally a stopping time, thus $\min(T,t)$ is $\mathcal{F}_{\min(T,t)}\subset\mathcal{F}_t$-measurable, similarly with $\min(S,t)$, so $\{\min(S,t)<\min(T,t)\}\in\mathcal{F}_t$.