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Mirrors > Home > ILE Home > Th. List > xrnepnf | GIF version |
Description: An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xrnepnf | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.61 783 | . 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞)) | |
2 | elxr 9556 | . . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
3 | df-3or 963 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) | |
4 | or32 759 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞)) | |
5 | 2, 3, 4 | 3bitri 205 | . . 3 ⊢ (𝐴 ∈ ℝ* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞)) |
6 | df-ne 2307 | . . 3 ⊢ (𝐴 ≠ +∞ ↔ ¬ 𝐴 = +∞) | |
7 | 5, 6 | anbi12i 455 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞)) |
8 | renepnf 7806 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
9 | mnfnepnf 7814 | . . . . . 6 ⊢ -∞ ≠ +∞ | |
10 | neeq1 2319 | . . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞)) | |
11 | 9, 10 | mpbiri 167 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 ≠ +∞) |
12 | 8, 11 | jaoi 705 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → 𝐴 ≠ +∞) |
13 | 12 | neneqd 2327 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → ¬ 𝐴 = +∞) |
14 | 13 | pm4.71i 388 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞)) |
15 | 1, 7, 14 | 3bitr4i 211 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∨ wo 697 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 ≠ wne 2306 ℝcr 7612 +∞cpnf 7790 -∞cmnf 7791 ℝ*cxr 7792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-un 4350 ax-cnex 7704 ax-resscn 7705 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-rex 2420 df-rab 2423 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-pnf 7795 df-mnf 7796 df-xr 7797 |
This theorem is referenced by: xaddnepnf 9634 |
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