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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 24781 | . . 3 ⊢ ran (,) ∈ TopBases | |
| 2 | bastg 22973 | . . 3 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
| 3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
| 4 | ioorebas 13491 | . 2 ⊢ (𝐴(,)𝐵) ∈ ran (,) | |
| 5 | 3, 4 | sselii 3980 | 1 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ⊆ wss 3951 ran crn 5686 ‘cfv 6561 (class class class)co 7431 (,)cioo 13387 topGenctg 17482 TopBasesctb 22952 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-ioo 13391 df-topgen 17488 df-bases 22953 |
| This theorem is referenced by: icccld 24787 icopnfcld 24788 iocmnfcld 24789 zcld 24835 iccntr 24843 reconnlem1 24848 reconnlem2 24849 icoopnst 24969 iocopnst 24970 dvlip 26032 dvlipcn 26033 dvivthlem1 26047 dvne0 26050 lhop2 26054 lhop 26055 dvfsumle 26060 dvfsumleOLD 26061 dvfsumabs 26063 dvfsumlem2 26067 dvfsumlem2OLD 26068 ftc1 26083 dvloglem 26690 advlog 26696 advlogexp 26697 cxpcn3 26791 loglesqrt 26804 lgamgulmlem2 27073 log2sumbnd 27588 dya2iocbrsiga 34277 dya2icobrsiga 34278 poimir 37660 ftc1cnnc 37699 areacirclem1 37715 dvrelog3 42066 aks4d1p1p6 42074 redvmptabs 42390 rfcnpre1 45024 rfcnpre2 45036 ioontr 45524 iocopn 45533 icoopn 45538 islptre 45634 limciccioolb 45636 limcicciooub 45652 limcresiooub 45657 limcresioolb 45658 icccncfext 45902 itgsin0pilem1 45965 itgsbtaddcnst 45997 dirkercncflem2 46119 dirkercncflem3 46120 dirkercncflem4 46121 fourierdlem28 46150 fourierdlem32 46154 fourierdlem33 46155 fourierdlem48 46169 fourierdlem49 46170 fourierdlem56 46177 fourierdlem57 46178 fourierdlem59 46180 fourierdlem60 46181 fourierdlem61 46182 fourierdlem62 46183 fourierdlem68 46189 fourierdlem72 46193 fourierdlem73 46194 fouriersw 46246 iooborel 46366 iooii 48815 i0oii 48817 io1ii 48818 |
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