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| Mirrors > Home > MPE Home > Th. List > isopn3 | Structured version Visualization version GIF version | ||
| Description: A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
| Ref | Expression |
|---|---|
| clscld.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| isopn3 | β’ ((π½ β Top β§ π β π) β (π β π½ β ((intβπ½)βπ) = π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clscld.1 | . . . . 5 β’ π = βͺ π½ | |
| 2 | 1 | ntrval 22540 | . . . 4 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| 3 | inss2 4230 | . . . . . . . 8 β’ (π½ β© π« π) β π« π | |
| 4 | 3 | unissi 4918 | . . . . . . 7 β’ βͺ (π½ β© π« π) β βͺ π« π |
| 5 | unipw 5451 | . . . . . . 7 β’ βͺ π« π = π | |
| 6 | 4, 5 | sseqtri 4019 | . . . . . 6 β’ βͺ (π½ β© π« π) β π |
| 7 | 6 | a1i 11 | . . . . 5 β’ (π β π½ β βͺ (π½ β© π« π) β π) |
| 8 | id 22 | . . . . . . 7 β’ (π β π½ β π β π½) | |
| 9 | pwidg 4623 | . . . . . . 7 β’ (π β π½ β π β π« π) | |
| 10 | 8, 9 | elind 4195 | . . . . . 6 β’ (π β π½ β π β (π½ β© π« π)) |
| 11 | elssuni 4942 | . . . . . 6 β’ (π β (π½ β© π« π) β π β βͺ (π½ β© π« π)) | |
| 12 | 10, 11 | syl 17 | . . . . 5 β’ (π β π½ β π β βͺ (π½ β© π« π)) |
| 13 | 7, 12 | eqssd 4000 | . . . 4 β’ (π β π½ β βͺ (π½ β© π« π) = π) |
| 14 | 2, 13 | sylan9eq 2793 | . . 3 β’ (((π½ β Top β§ π β π) β§ π β π½) β ((intβπ½)βπ) = π) |
| 15 | 14 | ex 414 | . 2 β’ ((π½ β Top β§ π β π) β (π β π½ β ((intβπ½)βπ) = π)) |
| 16 | 1 | ntropn 22553 | . . 3 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) β π½) |
| 17 | eleq1 2822 | . . 3 β’ (((intβπ½)βπ) = π β (((intβπ½)βπ) β π½ β π β π½)) | |
| 18 | 16, 17 | syl5ibcom 244 | . 2 β’ ((π½ β Top β§ π β π) β (((intβπ½)βπ) = π β π β π½)) |
| 19 | 15, 18 | impbid 211 | 1 β’ ((π½ β Top β§ π β π) β (π β π½ β ((intβπ½)βπ) = π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β© cin 3948 β wss 3949 π« cpw 4603 βͺ cuni 4909 βcfv 6544 Topctop 22395 intcnt 22521 |
| 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-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-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 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-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-top 22396 df-ntr 22524 |
| This theorem is referenced by: ntridm 22572 ntrtop 22574 ntr0 22585 isopn3i 22586 opnnei 22624 cnntr 22779 llycmpkgen2 23054 dvnres 25448 dvcnvre 25536 taylthlem2 25886 ulmdvlem3 25914 abelth 25953 opnbnd 35210 ioontr 44224 cncfuni 44602 fperdvper 44635 dirkercncflem3 44821 dirkercncflem4 44822 fourierdlem58 44880 fourierdlem73 44895 |
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