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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 47585 | . 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 47556 | . . . . 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 341 | . . . 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 3219 | . . 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 576 | . 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3062 βwrex 3071 β© cin 3948 β wss 3949 β c0 4323 π« cpw 4603 βͺ cuni 4909 βcfv 6544 Topctop 22395 clsccl 22522 neicnei 22601 CNrmccnrm 22815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-en 8940 df-fin 8943 df-fi 9406 df-rest 17368 df-topgen 17389 df-top 22396 df-topon 22413 df-bases 22449 df-cld 22523 df-cls 22525 df-nei 22602 df-nrm 22821 df-cnrm 22822 |
This theorem is referenced by: (None) |
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