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Metamath Proof Explorer

Theorem nn0ltlem1

Description: Nonnegative integer ordering relation. (Contributed by NM, 10-May-2004) (Proof shortened by Mario Carneiro, 16-May-2014)

Ref Expression
Assertion nn0ltlem1 ( ( 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 zltlem1 ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 < 𝑁 ↔ 𝑀 ≤ ( 𝑁 − 1 ) ) )
4 1 2 3 syl2an ⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 < 𝑁 ↔ 𝑀 ≤ ( 𝑁 − 1 ) ) )