IF A-B ⊆ C Then A-C ⊆ B?

[@JacobViscusi]

An alternate non contradiction based proof: Suppose A, B, and C satisfy the premise A-B \subseteq C. Consider an arbitrary x in A-C, so that by definition, x is in A but is not in C. Since x is not in C, x can not be in any subset of C, so in particular, x is not in A-B. Recalling that x is in A and pairing with the new fact that x is not in A-B, it must be the case that x is in B. Since x was arbitrary, we have shown A-C \subseteq B.

[@RelativelyPowder]

yes this proof is also valid, i like the approach of "if an element is in 𝐴−𝐶, then it must belong to 𝐵 because it cannot be in 𝐴−𝐵 (as 𝐴−𝐵⊆𝐶)". Very unique logic!