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| Mirrors > Home > MPE Home > Th. List > isneip | Structured version Visualization version GIF version | ||
| Description: The predicate "the class π is a neighborhood of point π". (Contributed by NM, 26-Feb-2007.) |
| Ref | Expression |
|---|---|
| neifval.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| isneip | β’ ((π½ β Top β§ π β π) β (π β ((neiβπ½)β{π}) β (π β π β§ βπ β π½ (π β π β§ π β π)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snssi 4812 | . . 3 β’ (π β π β {π} β π) | |
| 2 | neifval.1 | . . . 4 β’ π = βͺ π½ | |
| 3 | 2 | isnei 22829 | . . 3 β’ ((π½ β Top β§ {π} β π) β (π β ((neiβπ½)β{π}) β (π β π β§ βπ β π½ ({π} β π β§ π β π)))) |
| 4 | 1, 3 | sylan2 591 | . 2 β’ ((π½ β Top β§ π β π) β (π β ((neiβπ½)β{π}) β (π β π β§ βπ β π½ ({π} β π β§ π β π)))) |
| 5 | snssg 4788 | . . . . . 6 β’ (π β π β (π β π β {π} β π)) | |
| 6 | 5 | anbi1d 628 | . . . . 5 β’ (π β π β ((π β π β§ π β π) β ({π} β π β§ π β π))) |
| 7 | 6 | rexbidv 3176 | . . . 4 β’ (π β π β (βπ β π½ (π β π β§ π β π) β βπ β π½ ({π} β π β§ π β π))) |
| 8 | 7 | anbi2d 627 | . . 3 β’ (π β π β ((π β π β§ βπ β π½ (π β π β§ π β π)) β (π β π β§ βπ β π½ ({π} β π β§ π β π)))) |
| 9 | 8 | adantl 480 | . 2 β’ ((π½ β Top β§ π β π) β ((π β π β§ βπ β π½ (π β π β§ π β π)) β (π β π β§ βπ β π½ ({π} β π β§ π β π)))) |
| 10 | 4, 9 | bitr4d 281 | 1 β’ ((π½ β Top β§ π β π) β (π β ((neiβπ½)β{π}) β (π β π β§ βπ β π½ (π β π β§ π β π)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1539 β wcel 2104 βwrex 3068 β wss 3949 {csn 4629 βͺ cuni 4909 βcfv 6544 Topctop 22617 neicnei 22823 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 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 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 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 22618 df-nei 22824 |
| This theorem is referenced by: neips 22839 neindisj 22843 neindisj2 22849 neiptopnei 22858 cnpnei 22990 fbflim2 23703 cnpflf2 23726 neibl 24232 neibastop2 35551 neibastop3 35552 |
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