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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 9733 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | id 19 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ) | |
3 | mnflt 9740 | . . . . 5 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
4 | ltpnf 9737 | . . . . 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 7967 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
8 | 7 | necon2bi 2395 | . . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 ∈ ℝ) |
9 | pnfxr 7972 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
10 | xrltnr 9736 | . . . . . . 7 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ ¬ +∞ < +∞ |
12 | breq1 3992 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞)) | |
13 | 11, 12 | mtbiri 670 | . . . . 5 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞) |
14 | 13 | intnand 926 | . . . 4 ⊢ (𝐴 = +∞ → ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
15 | 8, 14 | 2falsed 697 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
16 | renemnf 7968 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
17 | 16 | necon2bi 2395 | . . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
18 | mnfxr 7976 | . . . . . . 7 ⊢ -∞ ∈ ℝ* | |
19 | xrltnr 9736 | . . . . . . 7 ⊢ (-∞ ∈ ℝ* → ¬ -∞ < -∞) | |
20 | 18, 19 | ax-mp 5 | . . . . . 6 ⊢ ¬ -∞ < -∞ |
21 | breq2 3993 | . . . . . 6 ⊢ (𝐴 = -∞ → (-∞ < 𝐴 ↔ -∞ < -∞)) | |
22 | 20, 21 | mtbiri 670 | . . . . 5 ⊢ (𝐴 = -∞ → ¬ -∞ < 𝐴) |
23 | 22 | intnanrd 927 | . . . 4 ⊢ (𝐴 = -∞ → ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)) |
24 | 17, 23 | 2falsed 697 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
25 | 6, 15, 24 | 3jaoi 1298 | . 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 972 = wceq 1348 ∈ wcel 2141 class class class wbr 3989 ℝcr 7773 +∞cpnf 7951 -∞cmnf 7952 ℝ*cxr 7953 < clt 7954 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-pre-ltirr 7886 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 |
This theorem is referenced by: xrre 9777 xrre2 9778 xrre3 9779 elioc2 9893 elico2 9894 elicc2 9895 xblpnfps 13192 xblpnf 13193 |
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