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Metamath Proof Explorer

Theorem nn0ltp1le

Description: Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Dec-2002) (Proof shortened by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion nn0ltp1le ⊢ M ∈ ℕ 0 ∧ N ∈ ℕ 0 → M < N ↔ M + 1 ≤ N

Proof

Step Hyp Ref Expression
1 nn0z ⊢ M ∈ ℕ 0 → M ∈ ℤ
2 nn0z ⊢ N ∈ ℕ 0 → N ∈ ℤ
3 zltp1le ⊢ M ∈ ℤ ∧ N ∈ ℤ → M < N ↔ M + 1 ≤ N
4 1 2 3 syl2an ⊢ M ∈ ℕ 0 ∧ N ∈ ℕ 0 → M < N ↔ M + 1 ≤ N