Equivalency of two definitions of WARP (Weak Axiom of Revealed Preference)

Question

I have two definitions for WARP as follows. How can I prove they are equivalent?

First Definition:

$C(A) \cap B \neq \emptyset \Rightarrow C(B) \cap A \subset C(A)$

Second definition from "Microeconomics Theory" (by Mas-Colell, Whinston, and Creen):

The choice structure $(\beta, C(.))$ satisfies the Weak Axiom of Revealed Preference if the following property holds:

If for some $B\in\beta$ with $x, y \in B$ we have $x \in C(B)$, then for any $B' \in \beta$ with $x, y \in B'$ and $y \in C(B')$, we must also have $x \in C(B')$.

Answer 1

Suppose that the condition in the first definition holds, $x\in C(B)$, $y\in C(B')$, and $x,y\in B\cap B'$. Then $y\in C(B')\cap B$, so $x\in C(B)\cap B'\subseteq C(B')$. Thus, the condition in the first definition implies the one in the second.

Now assume the condition in the second definition, and suppose that $C(A)\cap B\ne\varnothing$. Let $y\in C(A)\cap B$, and let $x$ be any element of $C(B)\cap A$; we must show that $x\in C(A)$. Clearly $x,y\in A\cap B$, $x\in C(B)$, and $y\in C(A)$, so by hypothesis $x\in C(A)$. Thus, $C(B)\cap A\subseteq C(A)$, and the condition in the second definition implies the one in the first.