(related to Proposition: Intersection of a Set With Another Set is Subset of This Set)

- Let $A$ and $B$ be sets.
- Let $A\cap B$ be the intersection of $A$ and $B$.
- We will prove $A\cap B\subseteq A$. The proof for $A\cap B\subseteq B$ is similar.

- Let $x\in A\cap B$.

- By the definition of intersection, it is true that $x\in A\wedge x\in B.$
- By the truth table of conjunction "and" ("$\wedge$") it must be true that $x\in A$.

- Since $x\in A$ for all $x\in A\cap B$, then by definition of subset $A\cap B\subseteq A$.∎

**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016