Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > ssntr | 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 3320 | . . . . 5 β’ (π β (π½ β© π« π) β (π β π½ β§ π β π« π)) | |
| 2 | elpwg 3585 | . . . . . 6 β’ (π β π½ β (π β π« π β π β π)) | |
| 3 | 2 | pm5.32i 454 | . . . . 5 β’ ((π β π½ β§ π β π« π) β (π β π½ β§ π β π)) |
| 4 | 1, 3 | bitr2i 185 | . . . 4 β’ ((π β π½ β§ π β π) β π β (π½ β© π« π)) |
| 5 | elssuni 3839 | . . . 4 β’ (π β (π½ β© π« π) β π β βͺ (π½ β© π« π)) | |
| 6 | 4, 5 | sylbi 121 | . . 3 β’ ((π β π½ β§ π β π) β π β βͺ (π½ β© π« π)) |
| 7 | 6 | adantl 277 | . 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 13695 | . . 3 β’ ((π½ β Top β§ π β π) β ((intβπ½)βπ) = βͺ (π½ β© π« π)) |
| 10 | 9 | adantr 276 | . 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 3196 | 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 set class |
| Syntax hints: β wi 4 β§ wa 104 = 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: ntrin 13709 neiint 13730 cnntri 13809 |
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