xrnemnf - Metamath Proof Explorer

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Theorem xrnemnf 13168

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

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

**Proof of Theorem xrnemnf**
Step	Hyp	Ref	Expression
1		pm5.61 1016	. 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞))
2		elxr 13167	. . . 4 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
3		df-3or 1104	. . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞))
4	2, 3	bitri 278	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞))
5		df-ne 2956	. . 3 ⊢ (𝐴 ≠ -∞ ↔ ¬ 𝐴 = -∞)
6	4, 5	anbi12i 640	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞))
7		renemnf 11282	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
8		pnfnemnf 11288	. . . . . 6 ⊢ +∞ ≠ -∞
9		neeq1 3017	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞))
10	8, 9	mpbiri 261	. . . . 5 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞)
11	7, 10	jaoi 871	. . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → 𝐴 ≠ -∞)
12	11	neneqd 2960	. . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → ¬ 𝐴 = -∞)
13	12	pm4.71i 569	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞))
14	1, 6, 13	3bitr4i 306	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ℝcr 11123 +∞cpnf 11264 -∞cmnf 11265 ℝ^*cxr 11266

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11269 df-mnf 11270 df-xr 11271

This theorem is used by: xaddnemnf 13288 xaddass 13301 xlesubadd 13315 xrge0nre 13506 xblss2ps 24627 xblss2 24628 nmoix 24955 nmoleub 24957 blcvx 25024 xrge0tsms 25061 metdstri 25078 nmoleub2lem 25342 xrge0tsmsd 33513 esumcvgre 34601 icorempo 38105 xrnmnfpnf 45917 xrred 46194

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