Can Disjoint Sets Help Prove Subset Relations in Set Theory?

[eddiep1993]

I am currently reading vellemans how to prove it for the purpose of being able to construct a proof on my own. I would like to carry on this knowledge to also help me out with spivaks calculus So the problem is:

Prove that if a and b\c are disjoint, then a$\bigcap$b$\subseteq$c.
1.goal: a$\bigcap$b\c=∅ → a$\bigcap$b$\subseteq$c

2. Givens:a$\bigcap$b\c=∅ Goal:a$\bigcap$b$\subseteq$c

3.Givens:a$\bigcap$b\c=∅ x$\in$a x$\in$b Goal: x$\in$c
This is as far as I got. I haveE no idea where to go from here and I feel the solution is starting right at me. I think it might have something to do with the fact that a and b\c are disjoints. This might be in the wrong section but I don't no where else to put it

 

[tiny-tim]

hi eddiep1993! welcome to pf!

… I think it might have something to do with the fact that a and b\c are disjoints.

that's right

if x ε a, then x not ε in b/c …

carry on from there ​

 

[haruspex]

3.Givens:a$\bigcap$b\c=∅ x$\in$a x$\in$b Goal: x$\in$c

Maybe: a$\bigcap$b\c=∅ x$\in$a x$\in$b x$\notin$c Goal: contradiction