Does $X$ have a lower bound?

Question

Let $P=\{A \subseteq \mathbb N: A$ is non-empty, finite and has an even number of elements$\}$. Consider $X=\{A \in P : 1 \in A\}$. Does $X$ have a lower bound in $(P,\subseteq)$?

I think that there is a lower bound.

Answer 1

Hint: consider $\{1,2\}$ and $\{1,3\}$.