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 ) ) |
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 ) ) |