Metamath Proof Explorer


Theorem nn0leltp1

Description: Nonnegative integer ordering relation. (Contributed by Raph Levien, 10-Apr-2004)

Ref Expression
Assertion nn0leltp1 ( ( 𝑀 ∈ ℕ0𝑁 ∈ ℕ0 ) → ( 𝑀𝑁𝑀 < ( 𝑁 + 1 ) ) )

Proof

Step Hyp Ref Expression
1 nn0z ( 𝑀 ∈ ℕ0𝑀 ∈ ℤ )
2 nn0z ( 𝑁 ∈ ℕ0𝑁 ∈ ℤ )
3 zleltp1 ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀𝑁𝑀 < ( 𝑁 + 1 ) ) )
4 1 2 3 syl2an ( ( 𝑀 ∈ ℕ0𝑁 ∈ ℕ0 ) → ( 𝑀𝑁𝑀 < ( 𝑁 + 1 ) ) )