Nearby theorems
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| Mirrors > Home > MPE Home > Th. List > neii1 | Structured version Visualization version GIF version | ||
| Description: A neighborhood is included in the topology's base set. (Contributed by NM, 12-Feb-2007.) |
| Ref | Expression |
|---|---|
| neifval.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| neii1 | β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β π β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | neifval.1 | . . 3 β’ π = βͺ π½ | |
| 2 | 1 | neiss2 22596 | . 2 β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β π β π) |
| 3 | 1 | isnei 22598 | . . . 4 β’ ((π½ β Top β§ π β π) β (π β ((neiβπ½)βπ) β (π β π β§ βπ β π½ (π β π β§ π β π)))) |
| 4 | simpl 483 | . . . 4 β’ ((π β π β§ βπ β π½ (π β π β§ π β π)) β π β π) | |
| 5 | 3, 4 | syl6bi 252 | . . 3 β’ ((π½ β Top β§ π β π) β (π β ((neiβπ½)βπ) β π β π)) |
| 6 | 5 | impancom 452 | . 2 β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β (π β π β π β π)) |
| 7 | 2, 6 | mpd 15 | 1 β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β π β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwrex 3070 β wss 3947 βͺ cuni 4907 βcfv 6540 Topctop 22386 neicnei 22592 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-top 22387 df-nei 22593 |
| This theorem is referenced by: neisspw 22602 neiss 22604 opnnei 22615 neiuni 22617 topssnei 22619 innei 22620 neissex 22622 iscnp4 22758 llycmpkgen2 23045 neitx 23102 flimopn 23470 flfnei 23486 fclsneii 23512 fcfnei 23530 cnextcn 23562 limcflf 25389 cvmlift2lem1 34281 neiin 35205 neibastop2 35234 cnneiima 47502 |
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