I am trying to prove the following:
If $m,n\in \omega$, then $m\oplus n<\omega$
My proof is as follows
Say wolog $m\in n$
Let $$f:(\{0\}×m)\cup (\{1\}×n)\rightarrow 2×\omega$$ $$f(t, p)=<t,p>$$
$f$ is 1-1 and not onto since there is no element in the domain that maps to $<1,n>$
Thus
$$m \oplus n < |2×\omega|=|\omega|=\omega$$
Is this correct? Is there a more efficient way?