Many years ago Staszek Radziszowski and I published a very elementary lemma that covers this case and many similar cases.
Let $(X,\le)$ be a partially ordered set and let $\varPhi$ be a family of functions from $X$ to $X$. Suppose that, for $x,x'\in X$ and $\phi\in\varPhi$ we have $\phi(x)\le x$ and $x\le x'\implies \phi(x)\le\phi(x')$. Call $x\in X$ $\varPhi$-stable if $\phi(x)=x$ for all $\phi\in\varPhi$. Let $\varPhi^*(x)$ be the closure of $\lbrace x\rbrace$ under $\varPhi$.
Lemma. For each $x\in X$, $\varPhi^*(x)$ contains at most one $\varPhi$-stable element.
To apply it to this case, $X$ is the set of all (labelled) graphs on $n$ vertices and $x\le x'$ means that $x$ is a super-graph of $x'$. There is a function for each pair of vertices $i,j$ which makes them adjacent if their degree sum is at least $n$ and does nothing otherwise. There is at least one $\varPhi$-stable graph (add edges until you can't add more) so by the lemma it is unique.
See Lemma 2 of this paper.