Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 | pm5.61 708 | . 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞)) | |
2 | elxr 8696 | . . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
3 | df-3or 886 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) | |
4 | 2, 3 | bitri 173 | . . 3 ⊢ (𝐴 ∈ ℝ* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) |
5 | df-ne 2206 | . . 3 ⊢ (𝐴 ≠ -∞ ↔ ¬ 𝐴 = -∞) | |
6 | 4, 5 | anbi12i 433 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞)) |
7 | renemnf 7074 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
8 | pnfnemnf 8697 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
9 | neeq1 2218 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
10 | 8, 9 | mpbiri 157 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
11 | 7, 10 | jaoi 636 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → 𝐴 ≠ -∞) |
12 | 11 | neneqd 2226 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → ¬ 𝐴 = -∞) |
13 | 12 | pm4.71i 371 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞)) |
14 | 1, 6, 13 | 3bitr4i 201 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ↔ wb 98 ∨ wo 629 ∨ w3o 884 = wceq 1243 ∈ wcel 1393 ≠ wne 2204 ℝcr 6888 +∞cpnf 7057 -∞cmnf 7058 ℝ*cxr 7059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-pnf 7062 df-mnf 7063 df-xr 7064 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |