Theorem elfzuz

Description: A member of a finite set of sequential integers belongs to an upper set of integers. (Contributed by NM, 17-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	elfzuz	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		elfzuzb	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) )
2	1	simplbi	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) )