On page 222 in Enderton's Elements of Set Theory, there's a remark which is then justified as an exercise:
Show that for any order types $\rho$ and $\sigma$ there exist structures $\langle A,R\rangle$ and $\langle B,S\rangle$ of types $\rho$ and $\sigma$, respectively, such that $A\cap B=\varnothing$.
Essentially the isomorphism type of a partially ordered set $\langle A,<\rangle$ is the set of all partially ordered sets which are order isomorphic to $\langle A,<\rangle$ and has the least rank.
Enderton claims that we can find disjoint members from any pair of isomorphism types. I find it hard to prove this (which is a foundational for defining order type arithmetic).
NOTE: the isomorphic equivalence class is too large to be a set but Enderton used Scott's trick to make a set. Now Enderton is working with it for isomorphism type arithmetic, but the detail about finding disjoint members from any two iso-types is too elusive for me.
Can I get insight as to the proof?