Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > ntrss2 | Structured version Visualization version GIF version | ||
| Description: A subset includes its interior. (Contributed by NM, 3-Oct-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
| Ref | Expression |
|---|---|
| clscld.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| ntrss2 | β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clscld.1 | . . 3 β’ π = βͺ π½ | |
| 2 | 1 | ntrval 22531 | . 2 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| 3 | inss2 4228 | . . . 4 β’ (π½ β© π« π) β π« π | |
| 4 | 3 | unissi 4916 | . . 3 β’ βͺ (π½ β© π« π) β βͺ π« π |
| 5 | unipw 5449 | . . 3 β’ βͺ π« π = π | |
| 6 | 4, 5 | sseqtri 4017 | . 2 β’ βͺ (π½ β© π« π) β π |
| 7 | 2, 6 | eqsstrdi 4035 | 1 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β© cin 3946 β wss 3947 π« cpw 4601 βͺ cuni 4907 βcfv 6540 Topctop 22386 intcnt 22512 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-top 22387 df-ntr 22515 |
| This theorem is referenced by: ntrin 22556 neiint 22599 opnnei 22615 topssnei 22619 maxlp 22642 restntr 22677 iscnp4 22758 cnntri 22766 cnntr 22770 cnprest 22784 llycmpkgen2 23045 xkococnlem 23154 flimopn 23470 fclsneii 23512 fcfnei 23530 subgntr 23602 iccntr 24328 rectbntr0 24339 bcthlem5 24836 limcflf 25389 dvbss 25409 perfdvf 25411 dvreslem 25417 dvcnp2 25428 dvnres 25439 dvaddbr 25446 dvcmulf 25453 dvmptres2 25470 dvmptcmul 25472 dvmptntr 25479 dvcnvre 25527 taylthlem1 25876 taylthlem2 25877 ulmdvlem3 25905 lgamucov2 26532 ubthlem1 30110 kur14lem6 34190 cvmlift2lem12 34293 gg-dvcnp2 35162 opnbnd 35198 opnregcld 35203 cldregopn 35204 dvresntr 44620 |
| Copyright terms: Public domain | W3C validator |