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Mirrors > Home > MPE Home > Th. List > iooretop | Structured version Unicode version |
Description: Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) |
Ref | Expression |
---|---|
iooretop |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retopbas 20481 |
. . 3
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2 | bastg 18713 |
. . 3
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3 | 1, 2 | ax-mp 5 |
. 2
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4 | ioorebas 11512 |
. 2
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5 | 3, 4 | sselii 3464 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-cnex 9453 ax-resscn 9454 ax-1cn 9455 ax-icn 9456 ax-addcl 9457 ax-addrcl 9458 ax-mulcl 9459 ax-mulrcl 9460 ax-mulcom 9461 ax-addass 9462 ax-mulass 9463 ax-distr 9464 ax-i2m1 9465 ax-1ne0 9466 ax-1rid 9467 ax-rnegex 9468 ax-rrecex 9469 ax-cnre 9470 ax-pre-lttri 9471 ax-pre-lttrn 9472 ax-pre-ltadd 9473 ax-pre-mulgt0 9474 ax-pre-sup 9475 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-tp 3993 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-tr 4497 df-eprel 4743 df-id 4747 df-po 4752 df-so 4753 df-fr 4790 df-we 4792 df-ord 4833 df-on 4834 df-lim 4835 df-suc 4836 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-om 6590 df-1st 6690 df-2nd 6691 df-recs 6945 df-rdg 6979 df-er 7214 df-en 7424 df-dom 7425 df-sdom 7426 df-sup 7806 df-pnf 9535 df-mnf 9536 df-xr 9537 df-ltxr 9538 df-le 9539 df-sub 9712 df-neg 9713 df-div 10109 df-nn 10438 df-n0 10695 df-z 10762 df-uz 10977 df-q 11069 df-ioo 11419 df-topgen 14505 df-bases 18647 |
This theorem is referenced by: icccld 20488 icopnfcld 20489 iocmnfcld 20490 zcld 20532 iccntr 20540 reconnlem1 20545 reconnlem2 20546 icoopnst 20653 iocopnst 20654 dvlip 21608 dvlipcn 21609 dvivthlem1 21623 dvne0 21626 lhop2 21630 lhop 21631 dvfsumle 21636 dvfsumabs 21638 dvfsumlem2 21642 ftc1 21657 dvloglem 22236 advlog 22242 advlogexp 22243 cxpcn3 22329 loglesqr 22339 log2sumbnd 22936 dya2iocbrsiga 26857 dya2icobrsiga 26858 lgamgulmlem2 27183 dvtanlem 28612 ftc1cnnc 28637 areacirclem1 28655 rfcnpre1 29912 rfcnpre2 29924 itgsin0pilem1 29961 |
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