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Mirrors > Home > MPE Home > Th. List > topsn | Structured version Visualization version GIF version |
Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4893). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
topsn | β’ (π½ β (TopOnβ{π΄}) β π½ = π« {π΄}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topgele 22756 | . . 3 β’ (π½ β (TopOnβ{π΄}) β ({β , {π΄}} β π½ β§ π½ β π« {π΄})) | |
2 | 1 | simprd 495 | . 2 β’ (π½ β (TopOnβ{π΄}) β π½ β π« {π΄}) |
3 | pwsn 4893 | . . 3 β’ π« {π΄} = {β , {π΄}} | |
4 | 1 | simpld 494 | . . 3 β’ (π½ β (TopOnβ{π΄}) β {β , {π΄}} β π½) |
5 | 3, 4 | eqsstrid 4023 | . 2 β’ (π½ β (TopOnβ{π΄}) β π« {π΄} β π½) |
6 | 2, 5 | eqssd 3992 | 1 β’ (π½ β (TopOnβ{π΄}) β π½ = π« {π΄}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3941 β c0 4315 π« cpw 4595 {csn 4621 {cpr 4623 βcfv 6534 TopOnctopon 22736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-top 22720 df-topon 22737 |
This theorem is referenced by: restsn2 22999 rrxtopn0 45519 |
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