Theorem 0elfz

Description: 0 is an element of a finite set of sequential nonnegative integers with a nonnegative integer as upper bound. (Contributed by AV, 6-Apr-2018)

	Ref	Expression
Assertion	0elfz	|- ( N e. NN0 -> 0 e. ( 0 ... N ) )

Proof

Step	Hyp	Ref	Expression
1		0nn0	|- 0 e. NN0
2	1	a1i	|- ( N e. NN0 -> 0 e. NN0 )
3		id	|- ( N e. NN0 -> N e. NN0 )
4		nn0ge0	|- ( N e. NN0 -> 0 <_ N )
5		elfz2nn0	|- ( 0 e. ( 0 ... N ) <-> ( 0 e. NN0 /\ N e. NN0 /\ 0 <_ N ) )
6	2 3 4 5	syl3anbrc	|- ( N e. NN0 -> 0 e. ( 0 ... N ) )