Set Theory and Beyond

I defined the ordered pair as $$⟨x,y⟩:=\{\{x\},\{x,y\}\}.$$ I might've used the notation $(x,y)$ in some places by accident, but let's be particularly careful of our notation in this post. Having defined an ordered pair $⟨x,y⟩$, we might wonder what an ordered triple $⟨x,y,z⟩$ would be. One way to approach this is to define $n$-tuple by recursion on $n$. So we say: $$⟨x_0⟩:=x_0,$$ $$⟨x_0,\dots ,x_n⟩:=⟨⟨x_0,\dots ,x_{n-1}⟩,x_n⟩.$$ In this definition, we know what is a 1-tuple and we can easily see that a 2-tuple exactly corresponds to an ordered pair. From there, we can see that $⟨x,y,z⟩$ is actually $⟨⟨x,y⟩,z⟩$, and so on. However, there is a subtle issue here and it has to do with the rigor that I'm suppressing from my posts. Very briefly, our definition would here go something like: "given sets $x_0,\dots ,x_{n-1}$,...