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| Mirrors > Home > MPE Home > Th. List > islp | Structured version Visualization version GIF version | ||
| Description: The predicate "the class π is a limit point of π". (Contributed by NM, 10-Feb-2007.) |
| Ref | Expression |
|---|---|
| lpfval.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| islp | β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)βπ) β π β ((clsβπ½)β(π β {π})))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lpfval.1 | . . . 4 β’ π = βͺ π½ | |
| 2 | 1 | lpval 22650 | . . 3 β’ ((π½ β Top β§ π β π) β ((limPtβπ½)βπ) = {π₯ β£ π₯ β ((clsβπ½)β(π β {π₯}))}) |
| 3 | 2 | eleq2d 2819 | . 2 β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)βπ) β π β {π₯ β£ π₯ β ((clsβπ½)β(π β {π₯}))})) |
| 4 | id 22 | . . 3 β’ (π β ((clsβπ½)β(π β {π})) β π β ((clsβπ½)β(π β {π}))) | |
| 5 | id 22 | . . . 4 β’ (π₯ = π β π₯ = π) | |
| 6 | sneq 4638 | . . . . . 6 β’ (π₯ = π β {π₯} = {π}) | |
| 7 | 6 | difeq2d 4122 | . . . . 5 β’ (π₯ = π β (π β {π₯}) = (π β {π})) |
| 8 | 7 | fveq2d 6895 | . . . 4 β’ (π₯ = π β ((clsβπ½)β(π β {π₯})) = ((clsβπ½)β(π β {π}))) |
| 9 | 5, 8 | eleq12d 2827 | . . 3 β’ (π₯ = π β (π₯ β ((clsβπ½)β(π β {π₯})) β π β ((clsβπ½)β(π β {π})))) |
| 10 | 4, 9 | elab3 3676 | . 2 β’ (π β {π₯ β£ π₯ β ((clsβπ½)β(π β {π₯}))} β π β ((clsβπ½)β(π β {π}))) |
| 11 | 3, 10 | bitrdi 286 | 1 β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)βπ) β π β ((clsβπ½)β(π β {π})))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1541 β wcel 2106 {cab 2709 β cdif 3945 β wss 3948 {csn 4628 βͺ cuni 4908 βcfv 6543 Topctop 22402 clsccl 22529 limPtclp 22645 |
| 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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 |
| 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-top 22403 df-cld 22530 df-cls 22532 df-lp 22647 |
| This theorem is referenced by: lpdifsn 22654 lpss3 22655 islp2 22656 islp3 22657 maxlp 22658 restlp 22694 lpcls 22875 limcnlp 25402 limcflflem 25404 |
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