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Metamath Proof Explorer

Theorem elfzle1

Description: A member of a finite set of sequential integer is greater than or equal to the lower bound. (Contributed by NM, 6-Sep-2005) (Revised by Mario Carneiro, 28-Apr-2015)

		Ref	Expression
	Assertion	elfzle1	⊢ ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑀 ≤ 𝐾 )

Proof

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