Prove that $(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$. Logically, the following proposition makes sense:

$(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$

Or, in english, if sets $S$ and $T$ share no elements, and the union of the sets is equal to one of the sets, then the other set must be empty.

However, what I'm struggling with is how to translate this english into mathematical symbols.

Would anyone know how to solve this with mathematical symbols?
$$ (S \cap T = \varnothing)\tag{1}$$
$$(S \cup T = T) \tag{2}$$

Now:
Starting with $(2)$
$$S \cup T = T \implies S \subseteq T \implies S \cap T = S$$

Since we have $(1): S\cap T = \varnothing$, then $$S\cap T = S = \varnothing$$
Informally, assume that S is non-empty. Then $S \cup T$ is not T, which contradicts the assumption.