Theorem eluzfz

Description: Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005) (Revised by Mario Carneiro, 28-Apr-2015)

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

Proof

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