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| Mirrors > Home > MPE Home > Th. List > islp2 | Structured version Visualization version GIF version | ||
| Description: The predicate "π is a limit point of π " in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.) |
| Ref | Expression |
|---|---|
| lpfval.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| islp2 | β’ ((π½ β Top β§ π β π β§ π β π) β (π β ((limPtβπ½)βπ) β βπ β ((neiβπ½)β{π})(π β© (π β {π})) β β )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lpfval.1 | . . . 4 β’ π = βͺ π½ | |
| 2 | 1 | islp 22864 | . . 3 β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)βπ) β π β ((clsβπ½)β(π β {π})))) |
| 3 | 2 | 3adant3 1130 | . 2 β’ ((π½ β Top β§ π β π β§ π β π) β (π β ((limPtβπ½)βπ) β π β ((clsβπ½)β(π β {π})))) |
| 4 | ssdifss 4134 | . . 3 β’ (π β π β (π β {π}) β π) | |
| 5 | 1 | neindisj2 22847 | . . 3 β’ ((π½ β Top β§ (π β {π}) β π β§ π β π) β (π β ((clsβπ½)β(π β {π})) β βπ β ((neiβπ½)β{π})(π β© (π β {π})) β β )) |
| 6 | 4, 5 | syl3an2 1162 | . 2 β’ ((π½ β Top β§ π β π β§ π β π) β (π β ((clsβπ½)β(π β {π})) β βπ β ((neiβπ½)β{π})(π β© (π β {π})) β β )) |
| 7 | 3, 6 | bitrd 278 | 1 β’ ((π½ β Top β§ π β π β§ π β π) β (π β ((limPtβπ½)βπ) β βπ β ((neiβπ½)β{π})(π β© (π β {π})) β β )) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ w3a 1085 = wceq 1539 β wcel 2104 β wne 2938 βwral 3059 β cdif 3944 β© cin 3946 β wss 3947 β c0 4321 {csn 4627 βͺ cuni 4907 βcfv 6542 Topctop 22615 clsccl 22742 neicnei 22821 limPtclp 22858 |
| 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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
| 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 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-int 4950 df-iun 4998 df-iin 4999 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-top 22616 df-cld 22743 df-ntr 22744 df-cls 22745 df-nei 22822 df-lp 22860 |
| This theorem is referenced by: clslp 22872 lpbl 24232 reperflem 24554 islptre 44633 islpcn 44653 |
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