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Mirrors > Home > MPE Home > Th. List > Mathboxes > isnrm4 | Structured version Visualization version GIF version |
Description: A topological space is normal iff any two disjoint closed sets are separated by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.) |
Ref | Expression |
---|---|
isnrm4 | β’ (π½ β Nrm β (π½ β Top β§ βπ β (Clsdβπ½)βπ β (Clsdβπ½)((π β© π) = β β βπ₯ β ((neiβπ½)βπ)βπ¦ β ((neiβπ½)βπ)(π₯ β© π¦) = β ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnrm3 23183 | . 2 β’ (π½ β Nrm β (π½ β Top β§ βπ β (Clsdβπ½)βπ β (Clsdβπ½)((π 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 47720 | . . . . 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© π) = β β βπ₯ β ((neiβπ½)βπ)βπ¦ β ((neiβπ½)βπ)(π₯ β© π¦) = β ) β ((π 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 3217 | . . 3 β’ (π½ β Top β (βπ β (Clsdβπ½)βπ β (Clsdβπ½)((π β© π) = β β βπ₯ β ((neiβπ½)βπ)βπ¦ β ((neiβπ½)βπ)(π₯ β© π¦) = β ) β βπ β (Clsdβπ½)βπ β (Clsdβπ½)((π 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 (Clsdβπ½)βπ β (Clsdβπ½)((π β© π) = β β βπ₯ β ((neiβπ½)βπ)βπ¦ β ((neiβπ½)βπ)(π₯ β© π¦) = β )) β (π½ β Top β§ βπ β (Clsdβπ½)βπ β (Clsdβπ½)((π 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 Nrm β (π½ β Top β§ βπ β (Clsdβπ½)βπ β (Clsdβπ½)((π 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 1086 = wceq 1540 β wcel 2105 βwral 3060 βwrex 3069 β© cin 3947 β wss 3948 β c0 4322 βcfv 6543 Topctop 22715 Clsdccld 22840 neicnei 22921 Nrmcnrm 23134 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-top 22716 df-cld 22843 df-cls 22845 df-nei 22922 df-nrm 23141 |
This theorem is referenced by: dfnrm3 47729 |
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