Theorem nn0lem1lt

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

	Ref	Expression
Assertion	nn0lem1lt	⊢ ( ( 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		zlem1lt	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 − 1 ) < 𝑁 ) )
4	1 2 3	syl2an	⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑀 − 1 ) < 𝑁 ) )