Metamath Proof Explorer


Theorem elfzel1

Description: Membership in a finite set of sequential integer implies the lower bound is an integer. (Contributed by NM, 6-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion elfzel1 ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑀 ∈ ℤ )

Proof

Step Hyp Ref Expression
1 elfzuz ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝐾 ∈ ( ℤ𝑀 ) )
2 eluzel2 ( 𝐾 ∈ ( ℤ𝑀 ) → 𝑀 ∈ ℤ )
3 1 2 syl ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑀 ∈ ℤ )