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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 8045 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 8041 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | iooval2 9947 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)}) | |
4 | 1, 2, 3 | mp2an 426 | . 2 ⊢ (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
5 | rabid2 2667 | . . 3 ⊢ (ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} ↔ ∀𝑥 ∈ ℝ (-∞ < 𝑥 ∧ 𝑥 < +∞)) | |
6 | mnflt 9815 | . . . 4 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥) | |
7 | ltpnf 9812 | . . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
8 | 6, 7 | jca 306 | . . 3 ⊢ (𝑥 ∈ ℝ → (-∞ < 𝑥 ∧ 𝑥 < +∞)) |
9 | 5, 8 | mprgbir 2548 | . 2 ⊢ ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
10 | 4, 9 | eqtr4i 2213 | 1 ⊢ (-∞(,)+∞) = ℝ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ∈ wcel 2160 {crab 2472 class class class wbr 4018 (class class class)co 5897 ℝcr 7841 +∞cpnf 8020 -∞cmnf 8021 ℝ*cxr 8022 < clt 8023 (,)cioo 9920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-ioo 9924 |
This theorem is referenced by: unirnioo 10005 blssioo 14522 |
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