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Theorem xrnepnf 13060

Description: An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xrnepnf	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))

**Proof of Theorem xrnepnf**
Step	Hyp	Ref	Expression
1		pm5.61 1008	. 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞))
2		elxr 13058	. . . 4 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
3		df-3or 1093	. . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞))
4		or32 931	. . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞))
5	2, 3, 4	3bitri 298	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞))
6		df-ne 2935	. . 3 ⊢ (𝐴 ≠ +∞ ↔ ¬ 𝐴 = +∞)
7	5, 6	anbi12i 634	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞))
8		renepnf 11184	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
9		mnfnepnf 11192	. . . . . 6 ⊢ -∞ ≠ +∞
10		neeq1 2996	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞))
11	9, 10	mpbiri 259	. . . . 5 ⊢ (𝐴 = -∞ → 𝐴 ≠ +∞)
12	8, 11	jaoi 863	. . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → 𝐴 ≠ +∞)
13	12	neneqd 2939	. . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → ¬ 𝐴 = +∞)
14	13	pm4.71i 564	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞))
15	1, 7, 14	3bitr4i 304	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 207 ∧ wa 396 ∨ wo 853 ∨ w3o 1091 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ℝcr 11028 +∞cpnf 11167 -∞cmnf 11168 ℝ^*cxr 11169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-ext 2711 ax-sep 5218 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-ex 1787 df-sb 2074 df-clab 2718 df-cleq 2731 df-clel 2814 df-ne 2935 df-nel 3039 df-rab 3392 df-v 3433 df-un 3888 df-in 3890 df-ss 3900 df-pw 4531 df-sn 4556 df-pr 4558 df-uni 4839 df-pnf 11172 df-mnf 11173 df-xr 11174

This theorem is referenced by: xaddnepnf 13180 xlt2addrd 32851 xrlexaddrp 45797 xrnpnfmnf 45917