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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 10690 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | pnfxr 10687 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | iooval2 12763 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)}) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (-∞(,)+∞) = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
5 | rabid2 3380 | . . 3 ⊢ (ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} ↔ ∀𝑥 ∈ ℝ (-∞ < 𝑥 ∧ 𝑥 < +∞)) | |
6 | mnflt 12510 | . . . 4 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥) | |
7 | ltpnf 12507 | . . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
8 | 6, 7 | jca 514 | . . 3 ⊢ (𝑥 ∈ ℝ → (-∞ < 𝑥 ∧ 𝑥 < +∞)) |
9 | 5, 8 | mprgbir 3151 | . 2 ⊢ ℝ = {𝑥 ∈ ℝ ∣ (-∞ < 𝑥 ∧ 𝑥 < +∞)} |
10 | 4, 9 | eqtr4i 2845 | 1 ⊢ (-∞(,)+∞) = ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1531 ∈ wcel 2108 {crab 3140 class class class wbr 5057 (class class class)co 7148 ℝcr 10528 +∞cpnf 10664 -∞cmnf 10665 ℝ*cxr 10666 < clt 10667 (,)cioo 12730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-pre-lttri 10603 ax-pre-lttrn 10604 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7681 df-2nd 7682 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-ioo 12734 |
This theorem is referenced by: unirnioo 12829 resup 13227 reordt 21818 icopnfcld 23368 iocmnfcld 23369 blssioo 23395 reconnlem1 23426 ioombl1 24155 ioombl 24158 mbfdm 24219 ismbf 24221 ismbf2d 24233 ismbf3d 24247 tpr2rico 31148 esumcvgsum 31340 itgexpif 31870 retopsconn 32489 asindmre 34969 itgsubsticclem 42249 |
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