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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 12898 | . 2 ⊢ (0(,)+∞) = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
2 | df-rp 12473 | . 2 ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
3 | 1, 2 | eqtr4i 2764 | 1 ⊢ (0(,)+∞) = ℝ+ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {crab 3057 class class class wbr 5030 (class class class)co 7170 ℝcr 10614 0cc0 10615 +∞cpnf 10750 < clt 10753 ℝ+crp 12472 (,)cioo 12821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-addrcl 10676 ax-rnegex 10686 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-rp 12473 df-ioo 12825 |
This theorem is referenced by: rpsup 13325 advlog 25397 advlogexp 25398 logccv 25406 cxpcn3 25489 loglesqrt 25499 rlimcnp 25703 rlimcnp2 25704 divsqrtsumlem 25717 amgmlem 25727 logfacbnd3 25959 logexprlim 25961 dchrisum0lem2a 26253 logdivsum 26269 log2sumbnd 26280 elxrge02 30781 xrge0iifcnv 31455 xrge0iifiso 31457 xrge0iifhom 31459 xrge0mulc1cn 31463 esumdivc 31621 signsply0 32100 rpsqrtcn 32143 logdivsqrle 32200 itg2gt0cn 35455 dvasin 35484 hoicvrrex 43636 amgmwlem 45959 |
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