Metamath Proof Explorer


Theorem elfzle3

Description: Membership in a finite set of sequential integer implies the bounds are comparable. (Contributed by NM, 18-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion elfzle3 ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑀𝑁 )

Proof

Step Hyp Ref Expression
1 elfzuz2 ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ𝑀 ) )
2 eluzle ( 𝑁 ∈ ( ℤ𝑀 ) → 𝑀𝑁 )
3 1 2 syl ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑀𝑁 )