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Mirrors > Home > ILE Home > Th. List > ioomax | GIF version |
Description: The open interval from minus to plus infinity. (Contributed by NM, 6-Feb-2007.) |
Ref | Expression |
---|---|
ioomax | ⊢ (-∞(,)+∞) = ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 7822 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 7818 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | iooval2 9698 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)}) | |
4 | 1, 2, 3 | mp2an 422 | . 2 ⊢ (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
5 | rabid2 2607 | . . 3 ⊢ (ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} ↔ ∀𝑥 ∈ ℝ (-∞ < 𝑥 ∧ 𝑥 < +∞)) | |
6 | mnflt 9569 | . . . 4 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥) | |
7 | ltpnf 9567 | . . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
8 | 6, 7 | jca 304 | . . 3 ⊢ (𝑥 ∈ ℝ → (-∞ < 𝑥 ∧ 𝑥 < +∞)) |
9 | 5, 8 | mprgbir 2490 | . 2 ⊢ ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
10 | 4, 9 | eqtr4i 2163 | 1 ⊢ (-∞(,)+∞) = ℝ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∈ wcel 1480 {crab 2420 class class class wbr 3929 (class class class)co 5774 ℝcr 7619 +∞cpnf 7797 -∞cmnf 7798 ℝ*cxr 7799 < clt 7800 (,)cioo 9671 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-ioo 9675 |
This theorem is referenced by: unirnioo 9756 blssioo 12714 |
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