Description: A positive integer decreased by 1 is greater than or equal to 0. (Contributed by AV, 30-Oct-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | nnm1ge0 | ⊢ ( 𝑁 ∈ ℕ → 0 ≤ ( 𝑁 − 1 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nngt0 | ⊢ ( 𝑁 ∈ ℕ → 0 < 𝑁 ) | |
2 | 0z | ⊢ 0 ∈ ℤ | |
3 | nnz | ⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ ) | |
4 | zltlem1 | ⊢ ( ( 0 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 0 < 𝑁 ↔ 0 ≤ ( 𝑁 − 1 ) ) ) | |
5 | 2 3 4 | sylancr | ⊢ ( 𝑁 ∈ ℕ → ( 0 < 𝑁 ↔ 0 ≤ ( 𝑁 − 1 ) ) ) |
6 | 1 5 | mpbid | ⊢ ( 𝑁 ∈ ℕ → 0 ≤ ( 𝑁 − 1 ) ) |