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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 | pm5.61 784 | . 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞)) | |
2 | elxr 9703 | . . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
3 | df-3or 968 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) | |
4 | 2, 3 | bitri 183 | . . 3 ⊢ (𝐴 ∈ ℝ* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) |
5 | df-ne 2335 | . . 3 ⊢ (𝐴 ≠ -∞ ↔ ¬ 𝐴 = -∞) | |
6 | 4, 5 | anbi12i 456 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = -∞)) |
7 | renemnf 7938 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
8 | pnfnemnf 7944 | . . . . . 6 ⊢ +∞ ≠ -∞ | |
9 | neeq1 2347 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 ≠ -∞ ↔ +∞ ≠ -∞)) | |
10 | 8, 9 | mpbiri 167 | . . . . 5 ⊢ (𝐴 = +∞ → 𝐴 ≠ -∞) |
11 | 7, 10 | jaoi 706 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → 𝐴 ≠ -∞) |
12 | 11 | neneqd 2355 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → ¬ 𝐴 = -∞) |
13 | 12 | pm4.71i 389 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞)) |
14 | 1, 6, 13 | 3bitr4i 211 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∨ w3o 966 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 ℝcr 7743 +∞cpnf 7921 -∞cmnf 7922 ℝ*cxr 7923 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-pnf 7926 df-mnf 7927 df-xr 7928 |
This theorem is referenced by: xaddf 9771 xaddval 9772 xaddnemnf 9784 xaddass 9796 xlesubadd 9810 xblss2ps 12951 xblss2 12952 |
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