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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 4194 | . . . . . 6 β’ (π΅ β πΆ β (π΅ β© π« π₯) β (πΆ β© π« π₯)) | |
2 | 1 | unissd 4876 | . . . . 5 β’ (π΅ β πΆ β βͺ (π΅ β© π« π₯) β βͺ (πΆ β© π« π₯)) |
3 | sstr2 3952 | . . . . 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 483 | . . 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 5281 | . . . . 5 β’ ((π΅ β πΆ β§ πΆ β π) β π΅ β V) | |
7 | 6 | ancoms 460 | . . . 4 β’ ((πΆ β π β§ π΅ β πΆ) β π΅ β V) |
8 | eltg 22323 | . . . 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 22323 | . . . 4 β’ (πΆ β π β (π₯ β (topGenβπΆ) β π₯ β βͺ (πΆ β© π« π₯))) | |
11 | 10 | adantr 482 | . . 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 3951 | 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 397 β wcel 2107 Vcvv 3444 β© cin 3910 β wss 3911 π« cpw 4561 βͺ cuni 4866 βcfv 6497 topGenctg 17324 |
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-topgen 17330 |
This theorem is referenced by: tgidm 22346 tgss3 22352 basgen 22354 2basgen 22356 tgfiss 22357 bastop1 22359 lecldbas 22586 txss12 22972 xrtgioo 24185 |
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