Theorem nnltlem1

Description: Positive integer ordering relation. (Contributed by NM, 21-Jun-2005)

	Ref	Expression
Assertion	nnltlem1	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 < 𝑁 ↔ 𝑀 ≤ ( 𝑁 − 1 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nnz	⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℤ )
2		nnz	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )
3		zltlem1	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 < 𝑁 ↔ 𝑀 ≤ ( 𝑁 − 1 ) ) )
4	1 2 3	syl2an	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 𝑀 < 𝑁 ↔ 𝑀 ≤ ( 𝑁 − 1 ) ) )