Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > opnneiss | GIF version | ||
| Description: An open set is a neighborhood of any of its subsets. (Contributed by NM, 13-Feb-2007.) |
| Ref | Expression |
|---|---|
| opnneiss | β’ ((π½ β Top β§ π β π½ β§ π β π) β π β ((neiβπ½)βπ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp3 999 | . 2 β’ ((π½ β Top β§ π β π½ β§ π β π) β π β π) | |
| 2 | eqid 2177 | . . . . 5 β’ βͺ π½ = βͺ π½ | |
| 3 | 2 | eltopss 13649 | . . . 4 β’ ((π½ β Top β§ π β π½) β π β βͺ π½) |
| 4 | sstr 3165 | . . . . 5 β’ ((π β π β§ π β βͺ π½) β π β βͺ π½) | |
| 5 | 4 | ancoms 268 | . . . 4 β’ ((π β βͺ π½ β§ π β π) β π β βͺ π½) |
| 6 | 3, 5 | stoic3 1431 | . . 3 β’ ((π½ β Top β§ π β π½ β§ π β π) β π β βͺ π½) |
| 7 | 2 | opnneissb 13795 | . . 3 β’ ((π½ β Top β§ π β π½ β§ π β βͺ π½) β (π β π β π β ((neiβπ½)βπ))) |
| 8 | 6, 7 | syld3an3 1283 | . 2 β’ ((π½ β Top β§ π β π½ β§ π β π) β (π β π β π β ((neiβπ½)βπ))) |
| 9 | 1, 8 | mpbid 147 | 1 β’ ((π½ β Top β§ π β π½ β§ π β π) β π β ((neiβπ½)βπ)) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β wb 105 β§ w3a 978 β wcel 2148 β wss 3131 βͺ cuni 3811 βcfv 5218 Topctop 13637 neicnei 13778 |
| 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-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
| 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 13638 df-nei 13779 |
| This theorem is referenced by: opnneip 13799 tpnei 13800 topssnei 13802 opnneiid 13804 neissex 13805 |
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