I want to say formally that $C_0$, ..., $C_{n+1}$ are distinct. Two ways seem to be
(1) $\qquad\qquad\qquad\qquad (\forall x,y \in \Bbb N)(x \ne y \to C_x \ne C_y)$,
(2) $\qquad\qquad\qquad\qquad C_{0 \le i \le n} \space \ne \space C_{i+(1 \le j \le n+1)}$.
Are (1) and (2) interchangable, assuming (the likely obvious) that $i, j, n \in \Bbb N$? And is there a better way?
If you really want to say it symbolically (keep in mind that in most cases it's better to just write precisely in natural language), a useful notation is indexed conjunctions, the $\wedge$-analogue of $\sum$: we write "$\bigwedge_{i\in I}\alpha_i$" to denote that $\alpha_i$ is true for each $i\in I$. (Similarly, we have $\bigvee$ for disjunctions.)
And we can further fold abbreviations into the subscript, just as we do for $\sum$; so the proposition $$\bigwedge_{0\le i<j\le n+1}C_i\not=C_j$$ does the job. (Of course there's an implicit assumption that $i,j$ range over naturals, but this is standard here.)