Disproving the Statement: A Contradiction in Set Theory | Proof Help

[INdeWATERS]

I am having issues with a proof, as follows.
*U = universal set , P(U) = power set of a universal set

For all sets A, B, C ∈ P(U), if A ⊆ C and B ⊆ C, then A ⊆ B or B ⊆ A.

I am pretty sure the statement is false and so I have to disprove it, i.e. prove the negation. I am stuck on how to negate. My attempts are as follows...

(1) There exist sets A, B, C ∈ P(U) such that A ⊆ C or B ⊆ C and A ⊄ B and B ⊄ A.
(2) There exist sets A, B, C ∈ P(U) such that if A ⊆ C or B ⊆ C then A ⊄ B and B ⊄ A.

Would the contrapositive of the statement be easier to work with??
For all sets A, B, C ∈ P(U), if A ⊆ B or B ⊆ A then, A ⊆ C and B ⊆ C.

Thank you for your time and help!

 

[mathman]

Take a pair of sets that are not empty and not overlapping and call them A and B. Let C = A∪B. Then obviously both A and B are contained in C, but neither contains the other.

 

[arsenal997]

It is really easy to disprove that with an example:

If you have a set as follows, A={a,b,c}, then the power set will be the next,

P(A)={empty,A,{a,b},{b,c},{a,c}}, then if you define B=A, C={b,c}, and D={a,c}, then $$C\subsetB$$ and $$D\subsetB$$ but C is not a subset of D, neither D is a subset of C.
q.e.d.

With that example the proff is done