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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 24721 | . . 3 ⊢ ran (,) ∈ TopBases | |
2 | bastg 22913 | . . 3 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ran (,) ⊆ (topGen‘ran (,)) |
4 | ioorebas 13463 | . 2 ⊢ (𝐴(,)𝐵) ∈ ran (,) | |
5 | 3, 4 | sselii 3973 | 1 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2098 ⊆ wss 3944 ran crn 5679 ‘cfv 6549 (class class class)co 7419 (,)cioo 13359 topGenctg 17422 TopBasesctb 22892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-ioo 13363 df-topgen 17428 df-bases 22893 |
This theorem is referenced by: icccld 24727 icopnfcld 24728 iocmnfcld 24729 zcld 24773 iccntr 24781 reconnlem1 24786 reconnlem2 24787 icoopnst 24907 iocopnst 24908 dvlip 25970 dvlipcn 25971 dvivthlem1 25985 dvne0 25988 lhop2 25992 lhop 25993 dvfsumle 25998 dvfsumleOLD 25999 dvfsumabs 26001 dvfsumlem2 26005 dvfsumlem2OLD 26006 ftc1 26021 dvloglem 26627 advlog 26633 advlogexp 26634 cxpcn3 26728 loglesqrt 26738 lgamgulmlem2 27007 log2sumbnd 27522 dya2iocbrsiga 34026 dya2icobrsiga 34027 poimir 37257 ftc1cnnc 37296 areacirclem1 37312 dvrelog3 41668 aks4d1p1p6 41676 rfcnpre1 44523 rfcnpre2 44535 ioontr 45034 iocopn 45043 icoopn 45048 islptre 45145 limciccioolb 45147 limcicciooub 45163 limcresiooub 45168 limcresioolb 45169 icccncfext 45413 itgsin0pilem1 45476 itgsbtaddcnst 45508 dirkercncflem2 45630 dirkercncflem3 45631 dirkercncflem4 45632 fourierdlem28 45661 fourierdlem32 45665 fourierdlem33 45666 fourierdlem48 45680 fourierdlem49 45681 fourierdlem56 45688 fourierdlem57 45689 fourierdlem59 45691 fourierdlem60 45692 fourierdlem61 45693 fourierdlem62 45694 fourierdlem68 45700 fourierdlem72 45704 fourierdlem73 45705 fouriersw 45757 iooborel 45877 iooii 48122 i0oii 48124 io1ii 48125 |
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