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| 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 22970 | . 2 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| 3 | inss2 4229 | . . . 4 β’ (π½ β© π« π) β π« π | |
| 4 | 3 | unissi 4917 | . . 3 β’ βͺ (π½ β© π« π) β βͺ π« π |
| 5 | unipw 5451 | . . 3 β’ βͺ π« π = π | |
| 6 | 4, 5 | sseqtri 4014 | . 2 β’ βͺ (π½ β© π« π) β π |
| 7 | 2, 6 | eqsstrdi 4032 | 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 394 = wceq 1533 β wcel 2098 β© cin 3944 β wss 3945 π« cpw 4603 βͺ cuni 4908 βcfv 6547 Topctop 22825 intcnt 22951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 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 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-top 22826 df-ntr 22954 |
| This theorem is referenced by: ntrin 22995 neiint 23038 opnnei 23054 topssnei 23058 maxlp 23081 restntr 23116 iscnp4 23197 cnntri 23205 cnntr 23209 cnprest 23223 llycmpkgen2 23484 xkococnlem 23593 flimopn 23909 fclsneii 23951 fcfnei 23969 subgntr 24041 iccntr 24767 rectbntr0 24778 bcthlem5 25286 limcflf 25840 dvbss 25860 perfdvf 25862 dvreslem 25868 dvcnp2 25879 dvcnp2OLD 25880 dvnres 25891 dvaddbr 25898 dvcmulf 25906 dvmptres2 25924 dvmptcmul 25926 dvmptntr 25933 dvcnvre 25982 taylthlem1 26338 taylthlem2 26339 taylthlem2OLD 26340 ulmdvlem3 26368 lgamucov2 27001 ubthlem1 30736 kur14lem6 34891 cvmlift2lem12 34994 opnbnd 35879 opnregcld 35884 cldregopn 35885 dvresntr 45369 |
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