Proving a set statement

Question

$A, B, C$ are subsets of a set $U$:

$A ⊆ B → A ∩ B$ $\nsubseteq$ $C$ $(1)$

$A ⊆ B ∨ A ⊆ C$ $(2)$

$A ∩ C ⊆ B$ $(3)$

I have to prove that this is valid:

$A ∩ B$ $\nsubseteq$ $C$ $(4)$

It is recommended to use this in our proof:

$X ⊆ Y ↔ X ∩ Y = X$ $(5)$

Should this be solved by mathematical induction or somehow else? I don't know what to do about that. Sorry for bad English. That is not my first language.

Answer 1

I'd solve it by cases on (2). If $A \subseteq B$, you're done by (1).

Suppose now that $A \subseteq C$ (6).

By (6) we can say $A \cap C = A$.

So by (3): $A \cap C = A \subseteq B$.

So $A \subseteq B$ holds.

By (1): $A \cap B \nsubseteq C$.