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?
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 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