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Mirrors > Home > MPE Home > Th. List > ioomax | Structured version Visualization version 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 11222 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 11219 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | iooval2 13308 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)}) | |
4 | 1, 2, 3 | mp2an 691 | . 2 ⊢ (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
5 | rabid2 3438 | . . 3 ⊢ (ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} ↔ ∀𝑥 ∈ ℝ (-∞ < 𝑥 ∧ 𝑥 < +∞)) | |
6 | mnflt 13054 | . . . 4 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥) | |
7 | ltpnf 13051 | . . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
8 | 6, 7 | jca 513 | . . 3 ⊢ (𝑥 ∈ ℝ → (-∞ < 𝑥 ∧ 𝑥 < +∞)) |
9 | 5, 8 | mprgbir 3068 | . 2 ⊢ ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
10 | 4, 9 | eqtr4i 2763 | 1 ⊢ (-∞(,)+∞) = ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 class class class wbr 5111 (class class class)co 7363 ℝcr 11060 +∞cpnf 11196 -∞cmnf 11197 ℝ*cxr 11198 < clt 11199 (,)cioo 13275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-pre-lttri 11135 ax-pre-lttrn 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-id 5537 df-po 5551 df-so 5552 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7927 df-2nd 7928 df-er 8656 df-en 8892 df-dom 8893 df-sdom 8894 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-ioo 13279 |
This theorem is referenced by: unirnioo 13377 resup 13783 reordt 22607 icopnfcld 24169 iocmnfcld 24170 blssioo 24196 reconnlem1 24227 ioombl1 24964 ioombl 24967 mbfdm 25028 ismbf 25030 ismbf2d 25042 ismbf3d 25056 tpr2rico 32583 esumcvgsum 32777 itgexpif 33309 retopsconn 33931 asindmre 36235 itgsubsticclem 44318 |
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