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Mirrors > Home > MPE Home > Th. List > topgele | Structured version Visualization version GIF version |
Description: The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
topgele | β’ (π½ β (TopOnβπ) β ({β , π} β π½ β§ π½ β π« π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 22835 | . . . 4 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | 0opn 22826 | . . . 4 β’ (π½ β Top β β β π½) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π½ β (TopOnβπ) β β β π½) |
4 | toponmax 22848 | . . 3 β’ (π½ β (TopOnβπ) β π β π½) | |
5 | 3, 4 | prssd 4830 | . 2 β’ (π½ β (TopOnβπ) β {β , π} β π½) |
6 | toponuni 22836 | . . . 4 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
7 | eqimss2 4041 | . . . 4 β’ (π = βͺ π½ β βͺ π½ β π) | |
8 | 6, 7 | syl 17 | . . 3 β’ (π½ β (TopOnβπ) β βͺ π½ β π) |
9 | sspwuni 5107 | . . 3 β’ (π½ β π« π β βͺ π½ β π) | |
10 | 8, 9 | sylibr 233 | . 2 β’ (π½ β (TopOnβπ) β π½ β π« π) |
11 | 5, 10 | jca 510 | 1 β’ (π½ β (TopOnβπ) β ({β , π} β π½ β§ π½ β π« π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wss 3949 β c0 4326 π« cpw 4606 {cpr 4634 βͺ cuni 4912 βcfv 6553 Topctop 22815 TopOnctopon 22832 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-top 22816 df-topon 22833 |
This theorem is referenced by: topsn 22853 txindis 23558 dissneqlem 36852 ntrf2 43585 |
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