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Metamath Proof Explorer

Theorem xnn0ge0

Description: An extended nonnegative integer is greater than or equal to 0. (Contributed by Alexander van der Vekens, 6-Jan-2018) (Revised by AV, 10-Dec-2020)

		Ref	Expression
	Assertion	xnn0ge0	⊢ ( 𝑁 ∈ ℕ₀^* → 0 ≤ 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		elxnn0	⊢ ( 𝑁 ∈ ℕ₀^* ↔ ( 𝑁 ∈ ℕ₀ ∨ 𝑁 = +∞ ) )
2		nn0ge0	⊢ ( 𝑁 ∈ ℕ₀ → 0 ≤ 𝑁 )
3		0lepnf	⊢ 0 ≤ +∞
4		breq2	⊢ ( 𝑁 = +∞ → ( 0 ≤ 𝑁 ↔ 0 ≤ +∞ ) )
5	3 4	mpbiri	⊢ ( 𝑁 = +∞ → 0 ≤ 𝑁 )
6	2 5	jaoi	⊢ ( ( 𝑁 ∈ ℕ₀ ∨ 𝑁 = +∞ ) → 0 ≤ 𝑁 )
7	1 6	sylbi	⊢ ( 𝑁 ∈ ℕ₀^* → 0 ≤ 𝑁 )