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 ) ) )