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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 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 < 𝑁 ↔ ( 𝑀 + 1 ) ≤ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 nn0z ⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ )
2 nn0z ⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ )
3 zltp1le ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 < 𝑁 ↔ ( 𝑀 + 1 ) ≤ 𝑁 ) )
4 1 2 3 syl2an ⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 < 𝑁 ↔ ( 𝑀 + 1 ) ≤ 𝑁 ) )