This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem nn0leltp1

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

Ref Expression
Assertion nn0leltp1 ( ( 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 zleltp1 ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 ↔ 𝑀 < ( 𝑁 + 1 ) ) )
4 1 2 3 syl2an ⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 ≤ 𝑁 ↔ 𝑀 < ( 𝑁 + 1 ) ) )