Theorem elxrge02

Description: Elementhood in the set of nonnegative extended reals. (Contributed by Thierry Arnoux, 18-Dec-2016)

	Ref	Expression
Assertion	elxrge02	⊢ ( 𝐴 ∈ ( 0 [,] +∞ ) ↔ ( 𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞ ) )

Proof

Step	Hyp	Ref	Expression
1		0xr	⊢ 0 ∈ ℝ*
2		pnfxr	⊢ +∞ ∈ ℝ*
3		0lepnf	⊢ 0 ≤ +∞
4		eliccioo	⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞ ) → ( 𝐴 ∈ ( 0 [,] +∞ ) ↔ ( 𝐴 = 0 ∨ 𝐴 ∈ ( 0 (,) +∞ ) ∨ 𝐴 = +∞ ) ) )
5	1 2 3 4	mp3an	⊢ ( 𝐴 ∈ ( 0 [,] +∞ ) ↔ ( 𝐴 = 0 ∨ 𝐴 ∈ ( 0 (,) +∞ ) ∨ 𝐴 = +∞ ) )
6		biid	⊢ ( 𝐴 = 0 ↔ 𝐴 = 0 )
7		ioorp	⊢ ( 0 (,) +∞ ) = ℝ+
8	7	eleq2i	⊢ ( 𝐴 ∈ ( 0 (,) +∞ ) ↔ 𝐴 ∈ ℝ+ )
9		biid	⊢ ( 𝐴 = +∞ ↔ 𝐴 = +∞ )
10	6 8 9	3orbi123i	⊢ ( ( 𝐴 = 0 ∨ 𝐴 ∈ ( 0 (,) +∞ ) ∨ 𝐴 = +∞ ) ↔ ( 𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞ ) )
11	5 10	bitri	⊢ ( 𝐴 ∈ ( 0 [,] +∞ ) ↔ ( 𝐴 = 0 ∨ 𝐴 ∈ ℝ+ ∨ 𝐴 = +∞ ) )