Theorem fzo1lb

Description: 1 is the left endpoint of a half-open integer range based at 1 iff the right endpoint is an integer greater than 1. (Contributed by AV, 4-Sep-2025)

	Ref	Expression
Assertion	fzo1lb	|- ( 1 e. ( 1 ..^ N ) <-> N e. ( ZZ>= ` 2 ) )

Proof

Step	Hyp	Ref	Expression
1		1z	|- 1 e. ZZ
2		3anass	|- ( ( 1 e. ZZ /\ N e. ZZ /\ 1 < N ) <-> ( 1 e. ZZ /\ ( N e. ZZ /\ 1 < N ) ) )
3	1 2	mpbiran	|- ( ( 1 e. ZZ /\ N e. ZZ /\ 1 < N ) <-> ( N e. ZZ /\ 1 < N ) )
4		fzolb	|- ( 1 e. ( 1 ..^ N ) <-> ( 1 e. ZZ /\ N e. ZZ /\ 1 < N ) )
5		eluz2b1	|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. ZZ /\ 1 < N ) )
6	3 4 5	3bitr4i	|- ( 1 e. ( 1 ..^ N ) <-> N e. ( ZZ>= ` 2 ) )