Let $x \in (\{a\} \cup [t, \infty)) \cap (\{b\} \cup[s, \infty))$. Show that $x \in \{x\} \cup [\max(t,s), \infty)$.
Show that $x \in \{x\} \cup [\max(t,s), \infty)$.
$x$ is always an element of $\{x\}$, and since $\{x\}\subseteq \{x\}\cup B$, it is also an element of $\{x\}\cup B$, no matter what $B$ is.