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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 | pm5.61 997 | . 2 ⊢ ((((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞)) | |
2 | elxr 12505 | . . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
3 | df-3or 1084 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞)) | |
4 | or32 922 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞) ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞)) | |
5 | 2, 3, 4 | 3bitri 299 | . . 3 ⊢ (𝐴 ∈ ℝ* ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞)) |
6 | df-ne 3017 | . . 3 ⊢ (𝐴 ≠ +∞ ↔ ¬ 𝐴 = +∞) | |
7 | 5, 6 | anbi12i 628 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∨ 𝐴 = +∞) ∧ ¬ 𝐴 = +∞)) |
8 | renepnf 10683 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
9 | mnfnepnf 10691 | . . . . . 6 ⊢ -∞ ≠ +∞ | |
10 | neeq1 3078 | . . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 ≠ +∞ ↔ -∞ ≠ +∞)) | |
11 | 9, 10 | mpbiri 260 | . . . . 5 ⊢ (𝐴 = -∞ → 𝐴 ≠ +∞) |
12 | 8, 11 | jaoi 853 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → 𝐴 ≠ +∞) |
13 | 12 | neneqd 3021 | . . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → ¬ 𝐴 = +∞) |
14 | 13 | pm4.71i 562 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ↔ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ ¬ 𝐴 = +∞)) |
15 | 1, 7, 14 | 3bitr4i 305 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ℝcr 10530 +∞cpnf 10666 -∞cmnf 10667 ℝ*cxr 10668 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-pw 4540 df-sn 4561 df-pr 4563 df-uni 4832 df-pnf 10671 df-mnf 10672 df-xr 10673 |
This theorem is referenced by: xaddnepnf 12624 xlt2addrd 30476 xrlexaddrp 41613 xrnpnfmnf 41744 |
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