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Mirrors > Home > MPE Home > Th. List > iooretop | Structured version Visualization version GIF version |
Description: Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) |
Ref | Expression |
---|---|
iooretop | ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retopbas 23072 | . . 3 ⊢ ran (,) ∈ TopBases | |
2 | bastg 21278 | . . 3 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
4 | ioorebas 12655 | . 2 ⊢ (𝐴(,)𝐵) ∈ ran (,) | |
5 | 3, 4 | sselii 3855 | 1 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2050 ⊆ wss 3829 ran crn 5408 ‘cfv 6188 (class class class)co 6976 (,)cioo 12554 topGenctg 16567 TopBasesctb 21257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-sup 8701 df-inf 8702 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-n0 11708 df-z 11794 df-uz 12059 df-q 12163 df-ioo 12558 df-topgen 16573 df-bases 21258 |
This theorem is referenced by: icccld 23078 icopnfcld 23079 iocmnfcld 23080 zcld 23124 iccntr 23132 reconnlem1 23137 reconnlem2 23138 icoopnst 23246 iocopnst 23247 dvlip 24293 dvlipcn 24294 dvivthlem1 24308 dvne0 24311 lhop2 24315 lhop 24316 dvfsumle 24321 dvfsumabs 24323 dvfsumlem2 24327 ftc1 24342 dvloglem 24932 advlog 24938 advlogexp 24939 cxpcn3 25030 loglesqrt 25040 lgamgulmlem2 25309 log2sumbnd 25822 dya2iocbrsiga 31184 dya2icobrsiga 31185 poimir 34372 ftc1cnnc 34413 areacirclem1 34429 rfcnpre1 40701 rfcnpre2 40713 ioontr 41224 iocopn 41233 icoopn 41238 islptre 41337 limciccioolb 41339 limcicciooub 41355 limcresiooub 41360 limcresioolb 41361 icccncfext 41606 itgsin0pilem1 41671 itgsbtaddcnst 41703 dirkercncflem2 41826 dirkercncflem3 41827 dirkercncflem4 41828 fourierdlem28 41857 fourierdlem32 41861 fourierdlem33 41862 fourierdlem48 41876 fourierdlem49 41877 fourierdlem56 41884 fourierdlem57 41885 fourierdlem59 41887 fourierdlem60 41888 fourierdlem61 41889 fourierdlem62 41890 fourierdlem68 41896 fourierdlem72 41900 fourierdlem73 41901 fouriersw 41953 iooborel 42071 |
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