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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 13159 | . . . 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 2940 | . . 3 ⊢ (𝐴 ≠ +∞ ↔ ¬ 𝐴 = +∞) | |
| 7 | 5, 6 | anbi12i 628 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞)) | 
| 8 | renepnf 11310 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
| 9 | mnfnepnf 11318 | . . . . . 6 ⊢ -∞ ≠ +∞ | |
| 10 | neeq1 3002 | . . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞)) | |
| 11 | 9, 10 | mpbiri 258 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 ≠ +∞) | 
| 12 | 8, 11 | jaoi 857 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → 𝐴 ≠ +∞) | 
| 13 | 12 | neneqd 2944 | . . 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 1539 ∈ wcel 2107 ≠ wne 2939 ℝcr 11155 +∞cpnf 11293 -∞cmnf 11294 ℝ*cxr 11295 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-nel 3046 df-rab 3436 df-v 3481 df-un 3955 df-in 3957 df-ss 3967 df-pw 4601 df-sn 4626 df-pr 4628 df-uni 4907 df-pnf 11298 df-mnf 11299 df-xr 11300 | 
| This theorem is referenced by: xaddnepnf 13280 xlt2addrd 32763 xrlexaddrp 45368 xrnpnfmnf 45490 | 
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