Theorem elfzuz2

Description: Implication of membership in a finite set of sequential integers. (Contributed by NM, 20-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	elfzuz2	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		elfzuzb	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) )
2		eqid	⊢ ( ℤ≥ ‘ 𝑀 ) = ( ℤ≥ ‘ 𝑀 )
3	2	uztrn2	⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
4	1 3	sylbi	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )