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Theorem xrnepnf 12514

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		pm5.61 997	. 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞))
2		elxr 12512	. . . 4 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
3		df-3or 1084	. . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞))
4		or32 922	. . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞))
5	2, 3, 4	3bitri 299	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞))
6		df-ne 3017	. . 3 ⊢ (𝐴 ≠ +∞ ↔ ¬ 𝐴 = +∞)
7	5, 6	anbi12i 628	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞))
8		renepnf 10689	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
9		mnfnepnf 10697	. . . . . 6 ⊢ -∞ ≠ +∞
10		neeq1 3078	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞))
11	9, 10	mpbiri 260	. . . . 5 ⊢ (𝐴 = -∞ → 𝐴 ≠ +∞)
12	8, 11	jaoi 853	. . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → 𝐴 ≠ +∞)
13	12	neneqd 3021	. . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → ¬ 𝐴 = +∞)
14	13	pm4.71i 562	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞))
15	1, 7, 14	3bitr4i 305	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ℝcr 10536 +∞cpnf 10672 -∞cmnf 10673 ℝ^*cxr 10674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-pw 4541 df-sn 4568 df-pr 4570 df-uni 4839 df-pnf 10677 df-mnf 10678 df-xr 10679

This theorem is referenced by: xaddnepnf 12631 xlt2addrd 30482 xrlexaddrp 41640 xrnpnfmnf 41771