Theorem 1elfzo1

Description: 1 is in a half-open range of positive integers iff its upper bound is greater than 1. (Contributed by AV, 22-Nov-2022)

	Ref	Expression
Assertion	1elfzo1	⊢ ( 1 ∈ ( 1 ..^ 𝑀 ) ↔ ( 𝑀 ∈ ℕ ∧ 1 < 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		elfzo1	⊢ ( 1 ∈ ( 1 ..^ 𝑀 ) ↔ ( 1 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 1 < 𝑀 ) )
2		1nn	⊢ 1 ∈ ℕ
3		3anass	⊢ ( ( 1 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 1 < 𝑀 ) ↔ ( 1 ∈ ℕ ∧ ( 𝑀 ∈ ℕ ∧ 1 < 𝑀 ) ) )
4	2 3	mpbiran	⊢ ( ( 1 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 1 < 𝑀 ) ↔ ( 𝑀 ∈ ℕ ∧ 1 < 𝑀 ) )
5	1 4	bitri	⊢ ( 1 ∈ ( 1 ..^ 𝑀 ) ↔ ( 𝑀 ∈ ℕ ∧ 1 < 𝑀 ) )