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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 22543 | . . 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 1132 | . 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 4115 | . . 3 β’ (π β π β (π β {π}) β π) | |
| 5 | 1 | neindisj2 22526 | . . 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 1164 | . 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 1087 = wceq 1541 β wcel 2106 β wne 2939 βwral 3060 β cdif 3925 β© cin 3927 β wss 3928 β c0 4302 {csn 4606 βͺ cuni 4885 βcfv 6516 Topctop 22294 clsccl 22421 neicnei 22500 limPtclp 22537 |
| 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 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
| 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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-iin 4977 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-top 22295 df-cld 22422 df-ntr 22423 df-cls 22424 df-nei 22501 df-lp 22539 |
| This theorem is referenced by: clslp 22551 lpbl 23911 reperflem 24233 islptre 44013 islpcn 44033 |
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