Theorem fz0dif1

Description: Split the first element of a finite set of sequential nonnegative integers. (Contributed by AV, 12-Sep-2025)

	Ref	Expression
Assertion	fz0dif1	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots N\right)\setminus \left\{0\right\}=\left(1\dots N\right)$$

Proof

Step	Hyp	Ref	Expression
1		elnn0uz	$$⊢N\in {\mathbb{N}}_0\leftrightarrow N\in {\mathbb{Z}}_{\ge 0}$$
2		fzdif1	$$⊢N\in {\mathbb{Z}}_{\ge 0}\to \left(0\dots N\right)\setminus \left\{0\right\}=\left(0+1\dots N\right)$$
3	1 2	sylbi	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots N\right)\setminus \left\{0\right\}=\left(0+1\dots N\right)$$
4		0p1e1	$$⊢0+1=1$$
5	4	oveq1i	$$⊢\left(0+1\dots N\right)=\left(1\dots N\right)$$
6	3 5	eqtrdi	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots N\right)\setminus \left\{0\right\}=\left(1\dots N\right)$$