Theorem fz0sn0fz1

Description: A finite set of sequential nonnegative integers is the union of the singleton containing 0 and a finite set of sequential positive integers. (Contributed by AV, 20-Mar-2021)

	Ref	Expression
Assertion	fz0sn0fz1	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots N\right)=\left\{0\right\}\cup \left(1\dots N\right)$$

Proof

Step	Hyp	Ref	Expression
1		0elfz	$$⊢N\in {\mathbb{N}}_0\to 0\in \left(0\dots N\right)$$
2		fzsplit	$$⊢0\in \left(0\dots N\right)\to \left(0\dots N\right)=\left(0\dots 0\right)\cup \left(0+1\dots N\right)$$
3		0p1e1	$$⊢0+1=1$$
4	3	oveq1i	$$⊢\left(0+1\dots N\right)=\left(1\dots N\right)$$
5	4	uneq2i	$$⊢\left(0\dots 0\right)\cup \left(0+1\dots N\right)=\left(0\dots 0\right)\cup \left(1\dots N\right)$$
6	2 5	eqtrdi	$$⊢0\in \left(0\dots N\right)\to \left(0\dots N\right)=\left(0\dots 0\right)\cup \left(1\dots N\right)$$
7	1 6	syl	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots N\right)=\left(0\dots 0\right)\cup \left(1\dots N\right)$$
8		0z	$$⊢0\in \mathbb{Z}$$
9		fzsn	$$⊢0\in \mathbb{Z}\to \left(0\dots 0\right)=\left\{0\right\}$$
10	8 9	mp1i	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots 0\right)=\left\{0\right\}$$
11	10	uneq1d	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots 0\right)\cup \left(1\dots N\right)=\left\{0\right\}\cup \left(1\dots N\right)$$
12	7 11	eqtrd	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots N\right)=\left\{0\right\}\cup \left(1\dots N\right)$$