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Mirrors > Home > ILE Home > Th. List > topgele | 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 13654 | . . . 4 β’ (π½ β (TopOnβπ) β π½ β Top) | |
2 | 0opn 13646 | . . . 4 β’ (π½ β Top β β β π½) | |
3 | 1, 2 | syl 14 | . . 3 β’ (π½ β (TopOnβπ) β β β π½) |
4 | toponmax 13665 | . . 3 β’ (π½ β (TopOnβπ) β π β π½) | |
5 | 0ex 4132 | . . . 4 β’ β β V | |
6 | prssg 3751 | . . . 4 β’ ((β β V β§ π β π½) β ((β β π½ β§ π β π½) β {β , π} β π½)) | |
7 | 5, 4, 6 | sylancr 414 | . . 3 β’ (π½ β (TopOnβπ) β ((β β π½ β§ π β π½) β {β , π} β π½)) |
8 | 3, 4, 7 | mpbi2and 943 | . 2 β’ (π½ β (TopOnβπ) β {β , π} β π½) |
9 | toponuni 13655 | . . . 4 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
10 | eqimss2 3212 | . . . 4 β’ (π = βͺ π½ β βͺ π½ β π) | |
11 | 9, 10 | syl 14 | . . 3 β’ (π½ β (TopOnβπ) β βͺ π½ β π) |
12 | sspwuni 3973 | . . 3 β’ (π½ β π« π β βͺ π½ β π) | |
13 | 11, 12 | sylibr 134 | . 2 β’ (π½ β (TopOnβπ) β π½ β π« π) |
14 | 8, 13 | jca 306 | 1 β’ (π½ β (TopOnβπ) β ({β , π} β π½ β§ π½ β π« π)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 = wceq 1353 β wcel 2148 Vcvv 2739 β wss 3131 β c0 3424 π« cpw 3577 {cpr 3595 βͺ cuni 3811 βcfv 5218 Topctop 13637 TopOnctopon 13650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-top 13638 df-topon 13651 |
This theorem is referenced by: (None) |
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