This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem fzss1

Description: Subset relationship for finite sets of sequential integers. (Contributed by NM, 28-Sep-2005) (Proof shortened by Mario Carneiro, 28-Apr-2015)

Ref Expression
Assertion fzss1 ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 ... 𝑁 ) ⊆ ( 𝑀 ... 𝑁 ) )

Proof

Step Hyp Ref Expression
1 elfzuz ⊢ ( 𝑘 ∈ ( 𝐾 ... 𝑁 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝐾 ) )
2 id ⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) )
3 uztrn ⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝐾 ) ∧ 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) )
4 1 2 3 syl2anr ⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑘 ∈ ( 𝐾 ... 𝑁 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) )
5 elfzuz3 ⊢ ( 𝑘 ∈ ( 𝐾 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑘 ) )
6 5 adantl ⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑘 ∈ ( 𝐾 ... 𝑁 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑘 ) )
7 elfzuzb ⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑘 ) ) )
8 4 6 7 sylanbrc ⊢ ( ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑘 ∈ ( 𝐾 ... 𝑁 ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) )
9 8 ex ⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑘 ∈ ( 𝐾 ... 𝑁 ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) )
10 9 ssrdv ⊢ ( 𝐾 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝐾 ... 𝑁 ) ⊆ ( 𝑀 ... 𝑁 ) )