Theorem elrege0

Description: The predicate "is a nonnegative real". (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Mario Carneiro, 18-Jun-2014)

	Ref	Expression
Assertion	elrege0	⊢ ( 𝐴 ∈ ( 0 [,) +∞ ) ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		elicopnf	⊢ ( 0 ∈ ℝ → ( 𝐴 ∈ ( 0 [,) +∞ ) ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ ( 0 [,) +∞ ) ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) )