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 e. NN0 -> ( 0 ... N ) = ( { 0 } u. ( 1 ... N ) ) )

Proof

Step	Hyp	Ref	Expression
1		0elfz	|- ( N e. NN0 -> 0 e. ( 0 ... N ) )
2		fzsplit	|- ( 0 e. ( 0 ... N ) -> ( 0 ... N ) = ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... N ) ) )
3		0p1e1	|- ( 0 + 1 ) = 1
4	3	oveq1i	|- ( ( 0 + 1 ) ... N ) = ( 1 ... N )
5	4	uneq2i	|- ( ( 0 ... 0 ) u. ( ( 0 + 1 ) ... N ) ) = ( ( 0 ... 0 ) u. ( 1 ... N ) )
6	2 5	eqtrdi	|- ( 0 e. ( 0 ... N ) -> ( 0 ... N ) = ( ( 0 ... 0 ) u. ( 1 ... N ) ) )
7	1 6	syl	|- ( N e. NN0 -> ( 0 ... N ) = ( ( 0 ... 0 ) u. ( 1 ... N ) ) )
8		0z	|- 0 e. ZZ
9		fzsn	|- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
10	8 9	mp1i	|- ( N e. NN0 -> ( 0 ... 0 ) = { 0 } )
11	10	uneq1d	|- ( N e. NN0 -> ( ( 0 ... 0 ) u. ( 1 ... N ) ) = ( { 0 } u. ( 1 ... N ) ) )
12	7 11	eqtrd	|- ( N e. NN0 -> ( 0 ... N ) = ( { 0 } u. ( 1 ... N ) ) )