Theorem fzp1ss

Description: Subset relationship for finite sets of sequential integers. (Contributed by NM, 26-Jul-2005) (Revised by Mario Carneiro, 28-Apr-2015)

	Ref	Expression
Assertion	fzp1ss	⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 + 1 ) ... 𝑁 ) ⊆ ( 𝑀 ... 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		uzid	⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) )
2		peano2uz	⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) )
3		fzss1	⊢ ( ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝑀 + 1 ) ... 𝑁 ) ⊆ ( 𝑀 ... 𝑁 ) )
4	1 2 3	3syl	⊢ ( 𝑀 ∈ ℤ → ( ( 𝑀 + 1 ) ... 𝑁 ) ⊆ ( 𝑀 ... 𝑁 ) )