Prove that if A is in the set of B and A is in the set of C then A is in the set of B n C

Question

anonymous · Answer

$$A \subseteq B$$ and $$A \subseteq C$$ then $$A \subseteq BnC$$

anonymous · Answer

let $x\in A$ then since $A \subseteq B$ we know $x\in B$
similarly since $x\in A$ and $A \subseteq C$ we have $x\in C$
since $x$ is an arbitrary element of A and is in both B and C, we know $x\in B\cap C$

anonymous · Answer

sorry last line should read 
since $x$ is an arbitrary element of A we know $A \subseteq B\cap C$

anonymous · Answer

Proof. Suppose $$A \subseteq B$$ and $$A \subseteq C$$ Further suppose that there  exists an arbitrary x in A: $$\exists x \in A$$ By how subsets are defined, $$x \in B$$ and $$x \in C$$Thus, $$x \subseteq BnC$$Again using the definition of a subset (since x is in A but also BnC), $$A \subseteq BnC$$

anonymous · Answer

Note that my second to last step is incorrect. It should read: $$x \in BnC$$This is because x is not necessarily a set and so may not be a subset.

anonymous · Answer

Yeah, I figured that it was a type. Thank you very much.

anonymous · Answer

typo*