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Mirrors > Home > MPE Home > Th. List > ioopos | Structured version Visualization version GIF version |
Description: The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007.) |
Ref | Expression |
---|---|
ioopos | ⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 11257 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | pnfxr 11264 | . . 3 ⊢ +∞ ∈ ℝ* | |
3 | iooval2 13353 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (0(,)+∞) = {𝑥 ∈ ℝ ∣ (0 < 𝑥 ∧ 𝑥 < +∞)}) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ (0 < 𝑥 ∧ 𝑥 < +∞)} |
5 | ltpnf 13096 | . . . 4 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
6 | 5 | biantrud 532 | . . 3 ⊢ (𝑥 ∈ ℝ → (0 < 𝑥 ↔ (0 < 𝑥 ∧ 𝑥 < +∞))) |
7 | 6 | rabbiia 3436 | . 2 ⊢ {𝑥 ∈ ℝ ∣ 0 < 𝑥} = {𝑥 ∈ ℝ ∣ (0 < 𝑥 ∧ 𝑥 < +∞)} |
8 | 4, 7 | eqtr4i 2763 | 1 ⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3432 class class class wbr 5147 (class class class)co 7405 ℝcr 11105 0cc0 11106 +∞cpnf 11241 ℝ*cxr 11243 < clt 11244 (,)cioo 13320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-addrcl 11167 ax-rnegex 11177 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-ioo 13324 |
This theorem is referenced by: ioorp 13398 repos 13419 |
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