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 ∈ ( ℤ≥ ‘ 𝑀 ) )
2		eluzel2	⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ )
3	1 2	syl	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑀 ∈ ℤ )