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Metamath Proof Explorer

Theorem elfz4

Description: Membership in a finite set of sequential integers. (Contributed by NM, 21-Jul-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion elfz4 ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ) ∧ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) → 𝐾 ∈ ( 𝑀 ... 𝑁 ) )

Proof

Step Hyp Ref Expression
1 elfz2 ⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ) ∧ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) )
2 1 biimpri ⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ) ∧ ( 𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁 ) ) → 𝐾 ∈ ( 𝑀 ... 𝑁 ) )