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Mirrors > Home > ILE Home > Th. List > xrnemnf | GIF version |
Description: An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xrnemnf | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.61 740 | . 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞)) | |
2 | elxr 8850 | . . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
3 | df-3or 920 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) | |
4 | 2, 3 | bitri 182 | . . 3 ⊢ (𝐴 ∈ ℝ* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) |
5 | df-ne 2246 | . . 3 ⊢ (𝐴 ≠ -∞ ↔ ¬ 𝐴 = -∞) | |
6 | 4, 5 | anbi12i 447 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞)) |
7 | renemnf 7167 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
8 | pnfnemnf 8851 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
9 | neeq1 2258 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
10 | 8, 9 | mpbiri 166 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
11 | 7, 10 | jaoi 668 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → 𝐴 ≠ -∞) |
12 | 11 | neneqd 2266 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → ¬ 𝐴 = -∞) |
13 | 12 | pm4.71i 383 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞)) |
14 | 1, 6, 13 | 3bitr4i 210 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 102 ↔ wb 103 ∨ wo 661 ∨ w3o 918 = wceq 1284 ∈ wcel 1433 ≠ wne 2245 ℝcr 6980 +∞cpnf 7150 -∞cmnf 7151 ℝ*cxr 7152 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-pnf 7155 df-mnf 7156 df-xr 7157 |
This theorem is referenced by: (None) |
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