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Mirrors > Home > MPE Home > Th. List > xrnepnf | Structured version Visualization version 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 1002 | . 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞)) | |
2 | elxr 13156 | . . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
3 | df-3or 1087 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) | |
4 | or32 925 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞)) | |
5 | 2, 3, 4 | 3bitri 297 | . . 3 ⊢ (𝐴 ∈ ℝ* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞)) |
6 | df-ne 2939 | . . 3 ⊢ (𝐴 ≠ +∞ ↔ ¬ 𝐴 = +∞) | |
7 | 5, 6 | anbi12i 628 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞)) |
8 | renepnf 11307 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
9 | mnfnepnf 11315 | . . . . . 6 ⊢ -∞ ≠ +∞ | |
10 | neeq1 3001 | . . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞)) | |
11 | 9, 10 | mpbiri 258 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 ≠ +∞) |
12 | 8, 11 | jaoi 857 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → 𝐴 ≠ +∞) |
13 | 12 | neneqd 2943 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → ¬ 𝐴 = +∞) |
14 | 13 | pm4.71i 559 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞)) |
15 | 1, 7, 14 | 3bitr4i 303 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ℝcr 11152 +∞cpnf 11290 -∞cmnf 11291 ℝ*cxr 11292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-nel 3045 df-rab 3434 df-v 3480 df-un 3968 df-in 3970 df-ss 3980 df-pw 4607 df-sn 4632 df-pr 4634 df-uni 4913 df-pnf 11295 df-mnf 11296 df-xr 11297 |
This theorem is referenced by: xaddnepnf 13276 xlt2addrd 32769 xrlexaddrp 45302 xrnpnfmnf 45425 |
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