Metamath Proof Explorer
Description: Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004)
|
|
Ref |
Expression |
|
Assertion |
nn0leltp1 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 ≤ 𝑁 ↔ 𝑀 < ( 𝑁 + 1 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
nn0z |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ ) |
2 |
|
nn0z |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ ) |
3 |
|
zleltp1 |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ 𝑀 < ( 𝑁 + 1 ) ) ) |
4 |
1 2 3
|
syl2an |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 ≤ 𝑁 ↔ 𝑀 < ( 𝑁 + 1 ) ) ) |