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Mirrors > Home > MPE Home > Th. List > tgss | Structured version Visualization version GIF version |
Description: Subset relation for generated topologies. (Contributed by NM, 7-May-2007.) |
Ref | Expression |
---|---|
tgss | β’ ((πΆ β π β§ π΅ β πΆ) β (topGenβπ΅) β (topGenβπΆ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrin 4229 | . . . . . 6 β’ (π΅ β πΆ β (π΅ β© π« π₯) β (πΆ β© π« π₯)) | |
2 | 1 | unissd 4913 | . . . . 5 β’ (π΅ β πΆ β βͺ (π΅ β© π« π₯) β βͺ (πΆ β© π« π₯)) |
3 | sstr2 3985 | . . . . 5 β’ (π₯ β βͺ (π΅ β© π« π₯) β (βͺ (π΅ β© π« π₯) β βͺ (πΆ β© π« π₯) β π₯ β βͺ (πΆ β© π« π₯))) | |
4 | 2, 3 | syl5com 31 | . . . 4 β’ (π΅ β πΆ β (π₯ β βͺ (π΅ β© π« π₯) β π₯ β βͺ (πΆ β© π« π₯))) |
5 | 4 | adantl 481 | . . 3 β’ ((πΆ β π β§ π΅ β πΆ) β (π₯ β βͺ (π΅ β© π« π₯) β π₯ β βͺ (πΆ β© π« π₯))) |
6 | ssexg 5317 | . . . . 5 β’ ((π΅ β πΆ β§ πΆ β π) β π΅ β V) | |
7 | 6 | ancoms 458 | . . . 4 β’ ((πΆ β π β§ π΅ β πΆ) β π΅ β V) |
8 | eltg 22853 | . . . 4 β’ (π΅ β V β (π₯ β (topGenβπ΅) β π₯ β βͺ (π΅ β© π« π₯))) | |
9 | 7, 8 | syl 17 | . . 3 β’ ((πΆ β π β§ π΅ β πΆ) β (π₯ β (topGenβπ΅) β π₯ β βͺ (π΅ β© π« π₯))) |
10 | eltg 22853 | . . . 4 β’ (πΆ β π β (π₯ β (topGenβπΆ) β π₯ β βͺ (πΆ β© π« π₯))) | |
11 | 10 | adantr 480 | . . 3 β’ ((πΆ β π β§ π΅ β πΆ) β (π₯ β (topGenβπΆ) β π₯ β βͺ (πΆ β© π« π₯))) |
12 | 5, 9, 11 | 3imtr4d 294 | . 2 β’ ((πΆ β π β§ π΅ β πΆ) β (π₯ β (topGenβπ΅) β π₯ β (topGenβπΆ))) |
13 | 12 | ssrdv 3984 | 1 β’ ((πΆ β π β§ π΅ β πΆ) β (topGenβπ΅) β (topGenβπΆ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β wcel 2099 Vcvv 3470 β© cin 3944 β wss 3945 π« cpw 4598 βͺ cuni 4903 βcfv 6542 topGenctg 17412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-topgen 17418 |
This theorem is referenced by: tgidm 22876 tgss3 22882 basgen 22884 2basgen 22886 tgfiss 22887 bastop1 22889 lecldbas 23116 txss12 23502 xrtgioo 24715 |
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