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 ) → 𝑀 ≤ 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		elfzuz2	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2		eluzle	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ≤ 𝑁 )
3	1 2	syl	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑀 ≤ 𝑁 )