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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 22766 | . . . 4 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | 0opn 22757 | . . . 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 22779 | . . 3 β’ (π½ β (TopOnβπ) β π β π½) | |
5 | 3, 4 | prssd 4820 | . 2 β’ (π½ β (TopOnβπ) β {β , π} β π½) |
6 | toponuni 22767 | . . . 4 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
7 | eqimss2 4036 | . . . 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 5096 | . . 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 511 | 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 395 = wceq 1533 β wcel 2098 β wss 3943 β c0 4317 π« cpw 4597 {cpr 4625 βͺ cuni 4902 βcfv 6536 Topctop 22746 TopOnctopon 22763 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-top 22747 df-topon 22764 |
This theorem is referenced by: topsn 22784 txindis 23489 dissneqlem 36728 ntrf2 43432 |
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