I wish to write
Let $S$ be a nonempty set and let $T$ be a collection of subsets of $S$. If $A \cap B \neq \emptyset$ for all pairs $A,B$ of elements of $T$, then there exists an element $x \in S$ such that $x \in C$ for all $C \in T$
symbolically. My attempt is as follows:
Let $S \neq \emptyset, T = \{ S_1,S_2,\dots,S_n \} \ni S_i \subseteq S.$ For $\forall (i,j) \in \{ (i,j) \in \mathbb{Z} : 1 \le i,j,\le n \land i \neq j \}, S_i \cap S_j \neq \emptyset \implies \exists x \in S, \forall C \in T, x \in C.$
I hope that I did it correctly. Please let me know what you think.