Showing that intersections are not defined

Question

I'm checking to see why intersections are not defined when looking at the class $A$ defined by:

$$ A = ON \cup [ON]^2\;,$$

where $ON$ is the class of ordinals and $[ON]^2$ the class of unordered pairs of distinct ordinals. Intersections are defined in $A$ if for any $x , y \in A$, we can find a $z \in A$ for which $\forall u \in A[u \in z \leftrightarrow u \in x \wedge u \in y].$

Here's what I've been thinking so far:

I have to find some $x, y \in A$ with the intersection of $x$ and $y$ not being in $A$. If I'm taking $x$ and $y$ in $A$, then they have to either be ordinals, or ordered pairs of ordinals. Since the intersection of two ordinals is an ordinal, I can't choose both $x$ and $y$ to be ordinals. I'm guessing I have to let $x$ be an ordinal and $y$ be an ordered pair.

This is where I'm getting stuck. I've tried several choices for $x$ and $y$, with $x$ being an ordinal, and $y$ being an ordered pair of ordinals, but I'm not getting the result I want.

Can anyone help? Thanks in advance!

Answer 1

HINT: What is $2\cap\{1,2\}$? I’ve left the answer spoiler-protected.

$2\cap\{1,2\}=\{0,1\}\cap\{1,2\}=\{1\}\notin A$.