Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > lpdifsn | Structured version Visualization version GIF version | ||
| Description: π is a limit point of π iff it is a limit point of π β {π}. (Contributed by Mario Carneiro, 25-Dec-2016.) |
| Ref | Expression |
|---|---|
| lpfval.1 | β’ π = βͺ π½ |
| Ref | Expression |
|---|---|
| lpdifsn | β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)βπ) β π β ((limPtβπ½)β(π β {π})))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lpfval.1 | . . 3 β’ π = βͺ π½ | |
| 2 | 1 | islp 22865 | . 2 β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)βπ) β π β ((clsβπ½)β(π β {π})))) |
| 3 | ssdifss 4135 | . . . 4 β’ (π β π β (π β {π}) β π) | |
| 4 | 1 | islp 22865 | . . . 4 β’ ((π½ β Top β§ (π β {π}) β π) β (π β ((limPtβπ½)β(π β {π})) β π β ((clsβπ½)β((π β {π}) β {π})))) |
| 5 | 3, 4 | sylan2 592 | . . 3 β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)β(π β {π})) β π β ((clsβπ½)β((π β {π}) β {π})))) |
| 6 | difabs 4293 | . . . . 5 β’ ((π β {π}) β {π}) = (π β {π}) | |
| 7 | 6 | fveq2i 6894 | . . . 4 β’ ((clsβπ½)β((π β {π}) β {π})) = ((clsβπ½)β(π β {π})) |
| 8 | 7 | eleq2i 2824 | . . 3 β’ (π β ((clsβπ½)β((π β {π}) β {π})) β π β ((clsβπ½)β(π β {π}))) |
| 9 | 5, 8 | bitrdi 287 | . 2 β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)β(π β {π})) β π β ((clsβπ½)β(π β {π})))) |
| 10 | 2, 9 | bitr4d 282 | 1 β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)βπ) β π β ((limPtβπ½)β(π β {π})))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 β cdif 3945 β wss 3948 {csn 4628 βͺ cuni 4908 βcfv 6543 Topctop 22616 clsccl 22743 limPtclp 22859 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3376 df-rab 3432 df-v 3475 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 22617 df-cld 22744 df-cls 22746 df-lp 22861 |
| This theorem is referenced by: perfdvf 25653 limcrecl 44644 |
| Copyright terms: Public domain | W3C validator |