Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > isopn3 | 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 13695 | . . . 4 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| 3 | inss2 3358 | . . . . . . . 8 β’ (π½ β© π« π) β π« π | |
| 4 | 3 | unissi 3834 | . . . . . . 7 β’ βͺ (π½ β© π« π) β βͺ π« π |
| 5 | unipw 4219 | . . . . . . 7 β’ βͺ π« π = π | |
| 6 | 4, 5 | sseqtri 3191 | . . . . . 6 β’ βͺ (π½ β© π« π) β π |
| 7 | 6 | a1i 9 | . . . . 5 β’ (π β π½ β βͺ (π½ β© π« π) β π) |
| 8 | id 19 | . . . . . . 7 β’ (π β π½ β π β π½) | |
| 9 | pwidg 3591 | . . . . . . 7 β’ (π β π½ β π β π« π) | |
| 10 | 8, 9 | elind 3322 | . . . . . 6 β’ (π β π½ β π β (π½ β© π« π)) |
| 11 | elssuni 3839 | . . . . . 6 β’ (π β (π½ β© π« π) β π β βͺ (π½ β© π« π)) | |
| 12 | 10, 11 | syl 14 | . . . . 5 β’ (π β π½ β π β βͺ (π½ β© π« π)) |
| 13 | 7, 12 | eqssd 3174 | . . . 4 β’ (π β π½ β βͺ (π½ β© π« π) = π) |
| 14 | 2, 13 | sylan9eq 2230 | . . 3 β’ (((π½ β Top β§ π β π) β§ π β π½) β ((intβπ½)βπ) = π) |
| 15 | 14 | ex 115 | . 2 β’ ((π½ β Top β§ π β π) β (π β π½ β ((intβπ½)βπ) = π)) |
| 16 | 1 | ntropn 13702 | . . 3 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) β π½) |
| 17 | eleq1 2240 | . . 3 β’ (((intβπ½)βπ) = π β (((intβπ½)βπ) β π½ β π β π½)) | |
| 18 | 16, 17 | syl5ibcom 155 | . 2 β’ ((π½ β Top β§ π β π) β (((intβπ½)βπ) = π β π β π½)) |
| 19 | 15, 18 | impbid 129 | 1 β’ ((π½ β Top β§ π β π) β (π β π½ β ((intβπ½)βπ) = π)) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β wb 105 = wceq 1353 β wcel 2148 β© cin 3130 β wss 3131 π« cpw 3577 βͺ cuni 3811 βcfv 5218 Topctop 13582 intcnt 13678 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-top 13583 df-ntr 13681 |
| This theorem is referenced by: ntridm 13711 ntrtop 13713 ntr0 13719 isopn3i 13720 cnntr 13810 |
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