Proof of isomorphism between sum of ordinals

Question

I want to prove that the sum of two ordinals $\alpha$, $\beta$ is isomorphic to the corresponding sum $\alpha \oplus \beta$ of two well-ordered sets, where i'm using this definition of sum of two well-ordered sets: $(A,<_1),(B,<_2)$, $A \oplus B$ is $(A \ \times \{0\} \cup B \ \times \{1\},<)$ with $(a,0)<(a',0)$ iff $a <_1 a'$; $(b,1)<(b',1)$ iff $b <_2 b'$ and $(a,0)<(b,1)$ for all $a,b$. To prove the above result, should i act by transfinite induction? Or could i find an explicit bijection? For example this: $\phi: \alpha +\beta \to \alpha \oplus \beta$ where if $\gamma \in \alpha+\beta, \gamma < \alpha$ $\phi(\gamma)=(\gamma,0)$ and if $\gamma \in \alpha+\beta, \gamma \geq \alpha$ so $\gamma= \alpha + \mu$ and $\phi(\gamma)=(\mu,1)?$