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 ∈ ( ℤ𝑀 ) )
2 eluzfz2 ( 𝑁 ∈ ( ℤ𝑀 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) )
3 1 2 syl ( 𝐾 ∈ ( 𝑀 ... 𝑁 ) → 𝑁 ∈ ( 𝑀 ... 𝑁 ) )