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Mirrors > Home > ILE Home > Th. List > xrre | GIF version |
Description: A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.) |
Ref | Expression |
---|---|
xrre | ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprl 521 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → -∞ < 𝐴) | |
2 | ltpnf 9687 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞) | |
3 | 2 | adantl 275 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞) |
4 | rexr 7923 | . . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
5 | pnfxr 7930 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
6 | xrlelttr 9710 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞)) | |
7 | 5, 6 | mp3an3 1308 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞)) |
8 | 4, 7 | sylan2 284 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞)) |
9 | 3, 8 | mpan2d 425 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → 𝐴 < +∞)) |
10 | 9 | imp 123 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → 𝐴 < +∞) |
11 | 10 | adantrl 470 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 < +∞) |
12 | xrrebnd 9723 | . . 3 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) | |
13 | 12 | ad2antrr 480 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞))) |
14 | 1, 11, 13 | mpbir2and 929 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2128 class class class wbr 3965 ℝcr 7731 +∞cpnf 7909 -∞cmnf 7910 ℝ*cxr 7911 < clt 7912 ≤ cle 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 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 |
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 3773 df-br 3966 df-opab 4026 df-po 4256 df-iso 4257 df-xp 4592 df-cnv 4594 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 |
This theorem is referenced by: xrrege0 9729 tgioo 12946 |
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