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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 22927 | . 2 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| 3 | inss2 4225 | . . . 4 β’ (π½ β© π« π) β π« π | |
| 4 | 3 | unissi 4912 | . . 3 β’ βͺ (π½ β© π« π) β βͺ π« π |
| 5 | unipw 5446 | . . 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 395 = wceq 1534 β wcel 2099 β© cin 3943 β wss 3944 π« cpw 4598 βͺ cuni 4903 βcfv 6542 Topctop 22782 intcnt 22908 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-top 22783 df-ntr 22911 |
| This theorem is referenced by: ntrin 22952 neiint 22995 opnnei 23011 topssnei 23015 maxlp 23038 restntr 23073 iscnp4 23154 cnntri 23162 cnntr 23166 cnprest 23180 llycmpkgen2 23441 xkococnlem 23550 flimopn 23866 fclsneii 23908 fcfnei 23926 subgntr 23998 iccntr 24724 rectbntr0 24735 bcthlem5 25243 limcflf 25797 dvbss 25817 perfdvf 25819 dvreslem 25825 dvcnp2 25836 dvcnp2OLD 25837 dvnres 25848 dvaddbr 25855 dvcmulf 25863 dvmptres2 25881 dvmptcmul 25883 dvmptntr 25890 dvcnvre 25939 taylthlem1 26295 taylthlem2 26296 taylthlem2OLD 26297 ulmdvlem3 26325 lgamucov2 26958 ubthlem1 30667 kur14lem6 34757 cvmlift2lem12 34860 opnbnd 35745 opnregcld 35750 cldregopn 35751 dvresntr 45229 |
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