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Mathbox for Zhi Wang |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iscnrm4 | Structured version Visualization version GIF version |
Description: A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.) |
Ref | Expression |
---|---|
iscnrm4 | β’ (π½ β CNrm β (π½ β Top β§ βπ β π« βͺ π½βπ‘ β π« βͺ π½(((π β© ((clsβπ½)βπ‘)) = β β§ (((clsβπ½)βπ ) β© π‘) = β ) β βπ β ((neiβπ½)βπ )βπ β ((neiβπ½)βπ‘)(π β© π) = β ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscnrm3 47841 | . 2 β’ (π½ β CNrm β (π½ β Top β§ βπ β π« βͺ π½βπ‘ β π« βͺ π½(((π β© ((clsβπ½)βπ‘)) = β β§ (((clsβπ½)βπ ) β© π‘) = β ) β βπ β π½ βπ β π½ (π β π β§ π‘ β π β§ (π β© π) = β )))) | |
2 | id 22 | . . . . . 6 β’ (π½ β Top β π½ β Top) | |
3 | 2 | sepnsepo 47812 | . . . . 5 β’ (π½ β Top β (βπ β ((neiβπ½)βπ )βπ β ((neiβπ½)βπ‘)(π β© π) = β β βπ β π½ βπ β π½ (π β π β§ π‘ β π β§ (π β© π) = β ))) |
4 | 3 | imbi2d 340 | . . . 4 β’ (π½ β Top β ((((π β© ((clsβπ½)βπ‘)) = β β§ (((clsβπ½)βπ ) β© π‘) = β ) β βπ β ((neiβπ½)βπ )βπ β ((neiβπ½)βπ‘)(π β© π) = β ) β (((π β© ((clsβπ½)βπ‘)) = β β§ (((clsβπ½)βπ ) β© π‘) = β ) β βπ β π½ βπ β π½ (π β π β§ π‘ β π β§ (π β© π) = β )))) |
5 | 4 | 2ralbidv 3212 | . . 3 β’ (π½ β Top β (βπ β π« βͺ π½βπ‘ β π« βͺ π½(((π β© ((clsβπ½)βπ‘)) = β β§ (((clsβπ½)βπ ) β© π‘) = β ) β βπ β ((neiβπ½)βπ )βπ β ((neiβπ½)βπ‘)(π β© π) = β ) β βπ β π« βͺ π½βπ‘ β π« βͺ π½(((π β© ((clsβπ½)βπ‘)) = β β§ (((clsβπ½)βπ ) β© π‘) = β ) β βπ β π½ βπ β π½ (π β π β§ π‘ β π β§ (π β© π) = β )))) |
6 | 5 | pm5.32i 574 | . 2 β’ ((π½ β Top β§ βπ β π« βͺ π½βπ‘ β π« βͺ π½(((π β© ((clsβπ½)βπ‘)) = β β§ (((clsβπ½)βπ ) β© π‘) = β ) β βπ β ((neiβπ½)βπ )βπ β ((neiβπ½)βπ‘)(π β© π) = β )) β (π½ β Top β§ βπ β π« βͺ π½βπ‘ β π« βͺ π½(((π β© ((clsβπ½)βπ‘)) = β β§ (((clsβπ½)βπ ) β© π‘) = β ) β βπ β π½ βπ β π½ (π β π β§ π‘ β π β§ (π β© π) = β )))) |
7 | 1, 6 | bitr4i 278 | 1 β’ (π½ β CNrm β (π½ β Top β§ βπ β π« βͺ π½βπ‘ β π« βͺ π½(((π β© ((clsβπ½)βπ‘)) = β β§ (((clsβπ½)βπ ) β© π‘) = β ) β βπ β ((neiβπ½)βπ )βπ β ((neiβπ½)βπ‘)(π β© π) = β ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 βwrex 3064 β© cin 3942 β wss 3943 β c0 4317 π« cpw 4597 βͺ cuni 4902 βcfv 6536 Topctop 22745 clsccl 22872 neicnei 22951 CNrmccnrm 23165 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-en 8939 df-fin 8942 df-fi 9405 df-rest 17374 df-topgen 17395 df-top 22746 df-topon 22763 df-bases 22799 df-cld 22873 df-cls 22875 df-nei 22952 df-nrm 23171 df-cnrm 23172 |
This theorem is referenced by: (None) |
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