Assume that $A,B , A\cap B$ Ordinals, and we want to prove that following can't be $A\cap B \in A $ and $A\cap B \in B$ using just the definition of Ordinals and that $A \in B <=> A \subset B ,or , A=B$
Any ideas ?!
I did arrive that $A - B \not= \emptyset$ and $B-A \not= \emptyset$
stuck here !!??
If $A \cap B \in A$ and $A \cap B \in B$ then $A \cap B \in A \cap B$. This contradicts the axiom of foundation.