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Metamath Proof Explorer

Theorem elfzubelfz

Description: If there is a member in a finite set of sequential integers, the upper bound is also a member of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-May-2018)

		Ref	Expression
	Assertion	elfzubelfz	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		elfzuz2	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( ℤ_≥ ‘ 𝑀 ) )
2		eluzfz2	⊢ ( 𝑁 ∈ ( ℤ_≥ ‘ 𝑀 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) )
3	1 2	syl	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) )