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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 23369 | . . 3 ⊢ ran (,) ∈ TopBases | |
2 | bastg 21574 | . . 3 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
4 | ioorebas 12840 | . 2 ⊢ (𝐴(,)𝐵) ∈ ran (,) | |
5 | 3, 4 | sselii 3964 | 1 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 ⊆ wss 3936 ran crn 5556 ‘cfv 6355 (class class class)co 7156 (,)cioo 12739 topGenctg 16711 TopBasesctb 21553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-ioo 12743 df-topgen 16717 df-bases 21554 |
This theorem is referenced by: icccld 23375 icopnfcld 23376 iocmnfcld 23377 zcld 23421 iccntr 23429 reconnlem1 23434 reconnlem2 23435 icoopnst 23543 iocopnst 23544 dvlip 24590 dvlipcn 24591 dvivthlem1 24605 dvne0 24608 lhop2 24612 lhop 24613 dvfsumle 24618 dvfsumabs 24620 dvfsumlem2 24624 ftc1 24639 dvloglem 25231 advlog 25237 advlogexp 25238 cxpcn3 25329 loglesqrt 25339 lgamgulmlem2 25607 log2sumbnd 26120 dya2iocbrsiga 31533 dya2icobrsiga 31534 poimir 34940 ftc1cnnc 34981 areacirclem1 34997 rfcnpre1 41296 rfcnpre2 41308 ioontr 41807 iocopn 41816 icoopn 41821 islptre 41920 limciccioolb 41922 limcicciooub 41938 limcresiooub 41943 limcresioolb 41944 icccncfext 42190 itgsin0pilem1 42255 itgsbtaddcnst 42287 dirkercncflem2 42409 dirkercncflem3 42410 dirkercncflem4 42411 fourierdlem28 42440 fourierdlem32 42444 fourierdlem33 42445 fourierdlem48 42459 fourierdlem49 42460 fourierdlem56 42467 fourierdlem57 42468 fourierdlem59 42470 fourierdlem60 42471 fourierdlem61 42472 fourierdlem62 42473 fourierdlem68 42479 fourierdlem72 42483 fourierdlem73 42484 fouriersw 42536 iooborel 42654 |
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