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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 7542 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 7538 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | iooval2 9331 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)}) | |
4 | 1, 2, 3 | mp2an 417 | . 2 ⊢ (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
5 | rabid2 2543 | . . 3 ⊢ (ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} ↔ ∀𝑥 ∈ ℝ (-∞ < 𝑥 ∧ 𝑥 < +∞)) | |
6 | mnflt 9251 | . . . 4 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥) | |
7 | ltpnf 9249 | . . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
8 | 6, 7 | jca 300 | . . 3 ⊢ (𝑥 ∈ ℝ → (-∞ < 𝑥 ∧ 𝑥 < +∞)) |
9 | 5, 8 | mprgbir 2433 | . 2 ⊢ ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
10 | 4, 9 | eqtr4i 2111 | 1 ⊢ (-∞(,)+∞) = ℝ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1289 ∈ wcel 1438 {crab 2363 class class class wbr 3845 (class class class)co 5652 ℝcr 7347 +∞cpnf 7517 -∞cmnf 7518 ℝ*cxr 7519 < clt 7520 (,)cioo 9304 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-cnex 7434 ax-resscn 7435 ax-pre-ltirr 7455 ax-pre-ltwlin 7456 ax-pre-lttrn 7457 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-po 4123 df-iso 4124 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 df-ioo 9308 |
This theorem is referenced by: unirnioo 9389 |
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