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 ∈ ( 1 ..^ 𝑁 ) ↔ 𝑁 ∈ ( ℤ≥ ‘ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		1z	⊢ 1 ∈ ℤ
2		3anass	⊢ ( ( 1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 1 < 𝑁 ) ↔ ( 1 ∈ ℤ ∧ ( 𝑁 ∈ ℤ ∧ 1 < 𝑁 ) ) )
3	1 2	mpbiran	⊢ ( ( 1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 1 < 𝑁 ) ↔ ( 𝑁 ∈ ℤ ∧ 1 < 𝑁 ) )
4		fzolb	⊢ ( 1 ∈ ( 1 ..^ 𝑁 ) ↔ ( 1 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 1 < 𝑁 ) )
5		eluz2b1	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 𝑁 ∈ ℤ ∧ 1 < 𝑁 ) )
6	3 4 5	3bitr4i	⊢ ( 1 ∈ ( 1 ..^ 𝑁 ) ↔ 𝑁 ∈ ( ℤ≥ ‘ 2 ) )