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Metamath Proof Explorer

Theorem elfzuz3

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

		Ref	Expression
	Assertion	elfzuz3	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ_≥ ‘ 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		elfzuzb	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝐾 ∈ ( ℤ_≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ( ℤ_≥ ‘ 𝐾 ) ) )
2	1	simprbi	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ_≥ ‘ 𝐾 ) )