MHB KRISTINE's question at Yahoo Answers (Set inclusion)

[Fernando Revilla]

Here is a link to the question:

How do I prove If A subset (B union C) and (A intersect B)=empty set, then A subset C? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello KRISTINE,

By hypothesis, $(i)\;A\subset B\cup C\quad(ii)\;A\cap B=\emptyset$

If $x\in A$, then (by $(i)$) $x\in B$ or $x\in C$.

Suppose $x\in B$. Then, (by $(ii)$) $x\not \in A$ which contradicts the hypothesis $x\in A$. So, necessarily $x\in C$. We have proven $A\subset C$.