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 22605 | . 2 β’ ((π½ β Top β§ π β ((neiβπ½)βπ)) β π β π) |
| 3 | 1 | isnei 22607 | . . . 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 484 | . . . 4 β’ ((π β π β§ βπ β π½ (π β π β§ π β π)) β π β π) | |
| 5 | 3, 4 | syl6bi 253 | . . 3 β’ ((π½ β Top β§ π β π) β (π β ((neiβπ½)βπ) β π β π)) |
| 6 | 5 | impancom 453 | . 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 397 = wceq 1542 β wcel 2107 βwrex 3071 β wss 3949 βͺ cuni 4909 βcfv 6544 Topctop 22395 neicnei 22601 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-top 22396 df-nei 22602 |
| This theorem is referenced by: neisspw 22611 neiss 22613 opnnei 22624 neiuni 22626 topssnei 22628 innei 22629 neissex 22631 iscnp4 22767 llycmpkgen2 23054 neitx 23111 flimopn 23479 flfnei 23495 fclsneii 23521 fcfnei 23539 cnextcn 23571 limcflf 25398 cvmlift2lem1 34293 neiin 35217 neibastop2 35246 cnneiima 47549 |
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