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Metamath Proof Explorer

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 ) )