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Mirrors > Home > ILE Home > Th. List > xrrebnd | GIF version |
Description: An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
xrrebnd | ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9684 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | id 19 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ) | |
3 | mnflt 9691 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
4 | ltpnf 9688 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞) | |
5 | 3, 4 | jca 304 | . . . 4 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
6 | 2, 5 | 2thd 174 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
7 | renepnf 7926 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
8 | 7 | necon2bi 2382 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℝ) |
9 | pnfxr 7931 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
10 | xrltnr 9687 | . . . . . . 7 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ ¬ +∞ < +∞ |
12 | breq1 3969 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞)) | |
13 | 11, 12 | mtbiri 665 | . . . . 5 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞) |
14 | 13 | intnand 917 | . . . 4 ⊢ (𝐴 = +∞ → ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
15 | 8, 14 | 2falsed 692 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
16 | renemnf 7927 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
17 | 16 | necon2bi 2382 | . . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
18 | mnfxr 7935 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
19 | xrltnr 9687 | . . . . . . 7 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 ⊢ ¬ -∞ < -∞ |
21 | breq2 3970 | . . . . . 6 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
22 | 20, 21 | mtbiri 665 | . . . . 5 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
23 | 22 | intnanrd 918 | . . . 4 ⊢ (𝐴 = -∞ → ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
24 | 17, 23 | 2falsed 692 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
25 | 6, 15, 24 | 3jaoi 1285 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
26 | 1, 25 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ w3o 962 = wceq 1335 ∈ wcel 2128 class class class wbr 3966 ℝcr 7732 +∞cpnf 7910 -∞cmnf 7911 ℝ*cxr 7912 < clt 7913 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-pre-ltirr 7845 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-xp 4593 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 |
This theorem is referenced by: xrre 9725 xrre2 9726 xrre3 9727 elioc2 9841 elico2 9842 elicc2 9843 xblpnfps 12840 xblpnf 12841 |
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