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| Mirrors > Home > MPE Home > Th. List > ssntr | Structured version Visualization version GIF version | ||
| Description: An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.) |
| Ref | Expression |
|---|---|
| clscld.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| ssntr | β’ (((π½ β Top β§ π β π) β§ (π β π½ β§ π β π)) β π β ((intβπ½)βπ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elin 3964 | . . . . 5 β’ (π β (π½ β© π« π) β (π β π½ β§ π β π« π)) | |
| 2 | elpwg 4605 | . . . . . 6 β’ (π β π½ β (π β π« π β π β π)) | |
| 3 | 2 | pm5.32i 574 | . . . . 5 β’ ((π β π½ β§ π β π« π) β (π β π½ β§ π β π)) |
| 4 | 1, 3 | bitr2i 276 | . . . 4 β’ ((π β π½ β§ π β π) β π β (π½ β© π« π)) |
| 5 | elssuni 4941 | . . . 4 β’ (π β (π½ β© π« π) β π β βͺ (π½ β© π« π)) | |
| 6 | 4, 5 | sylbi 216 | . . 3 β’ ((π β π½ β§ π β π) β π β βͺ (π½ β© π« π)) |
| 7 | 6 | adantl 481 | . 2 β’ (((π½ β Top β§ π β π) β§ (π β π½ β§ π β π)) β π β βͺ (π½ β© π« π)) |
| 8 | clscld.1 | . . . 4 β’ π = βͺ π½ | |
| 9 | 8 | ntrval 22860 | . . 3 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| 10 | 9 | adantr 480 | . 2 β’ (((π½ β Top β§ π β π) β§ (π β π½ β§ π β π)) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| 11 | 7, 10 | sseqtrrd 4023 | 1 β’ (((π½ β Top β§ π β π) β§ (π β π½ β§ π β π)) β π β ((intβπ½)βπ)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1540 β wcel 2105 β© cin 3947 β wss 3948 π« cpw 4602 βͺ cuni 4908 βcfv 6543 Topctop 22715 intcnt 22841 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-top 22716 df-ntr 22844 |
| This theorem is referenced by: ntrin 22885 neiint 22928 restntr 23006 cnntri 23095 xkococnlem 23483 iccntr 24657 bcthlem5 25176 ftc1 25897 lgamucov 26883 cvmlift2lem12 34769 cvmlift3lem7 34780 opnregcld 35679 ftc1cnnc 37024 |
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