Prove $A \cap B \subseteq A $ I am learning from Apostle's Calculus Vol.1 and could really do with some verification. Question 12(sect. 12.5): prove $A \cap B \subseteq A $ Proof: Let $x \in A \cap B$ if $x \in A$ and $x \in B$. Thus, $A \cap B \subseteq A $ if $A \subseteq B$ or $A \subset B$

Your first sentence introduces an element $x$, which is not mentioned in the second one. It doesn't make sense.

Simply say that if $x\in A\cap B$, then, by the definition of $A\cap B$, $x\in A$. Since thus occurs for each $x\in A$, $A\cap B\subseteq A$.