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Mirrors > Home > MPE Home > Th. List > ioorp | Structured version Visualization version GIF version |
Description: The set of positive reals expressed as an open interval. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
Ref | Expression |
---|---|
ioorp | ⊢ (0(,)+∞) = ℝ+ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioopos 13156 | . 2 ⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
2 | df-rp 12731 | . 2 ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
3 | 1, 2 | eqtr4i 2769 | 1 ⊢ (0(,)+∞) = ℝ+ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 {crab 3068 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 0cc0 10871 +∞cpnf 11006 < clt 11009 ℝ+crp 12730 (,)cioo 13079 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-addrcl 10932 ax-rnegex 10942 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-rp 12731 df-ioo 13083 |
This theorem is referenced by: rpsup 13586 advlog 25809 advlogexp 25810 logccv 25818 cxpcn3 25901 loglesqrt 25911 rlimcnp 26115 rlimcnp2 26116 divsqrtsumlem 26129 amgmlem 26139 logfacbnd3 26371 logexprlim 26373 dchrisum0lem2a 26665 logdivsum 26681 log2sumbnd 26692 elxrge02 31206 xrge0iifcnv 31883 xrge0iifiso 31885 xrge0iifhom 31887 xrge0mulc1cn 31891 esumdivc 32051 signsply0 32530 rpsqrtcn 32573 logdivsqrle 32630 itg2gt0cn 35832 dvasin 35861 hoicvrrex 44094 amgmwlem 46506 |
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