Theorem fzss2

Description: Subset relationship for finite sets of sequential integers. (Contributed by NM, 4-Oct-2005) (Revised by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Assertion	fzss2	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) → ( 𝑀 ... 𝐾 ) ⊆ ( 𝑀 ... 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		elfzuz	⊢ ( 𝑘 ∈ ( 𝑀 ... 𝐾 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) )
2	1	adantl	⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝐾 ) ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) )
3		elfzuz3	⊢ ( 𝑘 ∈ ( 𝑀 ... 𝐾 ) → 𝐾 ∈ ( ℤ≥ ‘ 𝑘 ) )
4		uztrn	⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ∧ 𝐾 ∈ ( ℤ≥ ‘ 𝑘 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑘 ) )
5	3 4	sylan2	⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝐾 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑘 ) )
6		elfzuzb	⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝑁 ∈ ( ℤ≥ ‘ 𝑘 ) ) )
7	2 5 6	sylanbrc	⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝐾 ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) )
8	7	ex	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) → ( 𝑘 ∈ ( 𝑀 ... 𝐾 ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) )
9	8	ssrdv	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) → ( 𝑀 ... 𝐾 ) ⊆ ( 𝑀 ... 𝑁 ) )