Theorem eluzfz1

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

	Ref	Expression
Assertion	eluzfz1	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		eluzel2	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ )
2		uzid	⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) )
3	1 2	syl	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) )
4		eluzfz	⊢ ( ( 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) )
5	3 4	mpancom	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) )