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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 22278 | . . . 4 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | 0opn 22269 | . . . 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 22291 | . . 3 β’ (π½ β (TopOnβπ) β π β π½) | |
5 | 3, 4 | prssd 4783 | . 2 β’ (π½ β (TopOnβπ) β {β , π} β π½) |
6 | toponuni 22279 | . . . 4 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
7 | eqimss2 4002 | . . . 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 5061 | . . 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 513 | 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 397 = wceq 1542 β wcel 2107 β wss 3911 β c0 4283 π« cpw 4561 {cpr 4589 βͺ cuni 4866 βcfv 6497 Topctop 22258 TopOnctopon 22275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-top 22259 df-topon 22276 |
This theorem is referenced by: topsn 22296 txindis 23001 dissneqlem 35857 ntrf2 42484 |
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