Prove or disprove:  if A is a subset of B and B is not a subset of C, then A is not a subset of C 
Prove or disprove:  if A is a subset of B and B is not a subset of C, then A is not a subset of C.

I know it is false for the counter example:
A = {1, 2}
B = {1, 2, 3, 4}
C = {1, 2, 6, 5}
How can I prove that mathematically?
 A: You just did; it's called a Counterexample.
You simply note that these particular $A$, $B$, and $C$ satisfy the hypotheses ("$A$ is a subset of $B$ and $B$ is not a subset of $C$"), but fail to satisfy the proposed conclusion ("$A$ is not a subset of $C$"). So the proposed implication cannot always hold. This disproves the assertion.
A: There exist an element x that is a member of set A and since set A is a subset of set B, then that element x is also a member of set B. 
Since set B is not a subset of set C that there does not exist an element x which is a member of set C, therefore element x is not in set C
Therefore element x is a member of set A but not an element in set C therefore set A is not a subset of set C.
A: A is  the subset of B given:
B is not the subset of C given:
Let x be an arbitrary element such that x belongs to A than we have to show that x doesn't belong to C
=>xEA (:. given)
=>xEB  (A is subset of B)
=>x doesn't belong to C  (B is not the subset of C)
but x is an arbitraray element of A hence proved that A is not the subset of C
A: You can say that if $A$ is included in $B \cap C$ then it is obviously a subset of $B$ and $C$ by definition of the intersection. 
But $B$ need not to be a subset of $C$ for $B \cap C$ to be non empty hence for $A$ to be non empty. Your example illustrate this quite well
