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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 4226 | . . . . . 6 β’ (π΅ β πΆ β (π΅ β© π« π₯) β (πΆ β© π« π₯)) | |
2 | 1 | unissd 4910 | . . . . 5 β’ (π΅ β πΆ β βͺ (π΅ β© π« π₯) β βͺ (πΆ β© π« π₯)) |
3 | sstr2 3982 | . . . . 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 5314 | . . . . 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 22784 | . . . 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 22784 | . . . 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 3981 | 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 2098 Vcvv 3466 β© cin 3940 β wss 3941 π« cpw 4595 βͺ cuni 4900 βcfv 6534 topGenctg 17384 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-topgen 17390 |
This theorem is referenced by: tgidm 22807 tgss3 22813 basgen 22815 2basgen 22817 tgfiss 22818 bastop1 22820 lecldbas 23047 txss12 23433 xrtgioo 24646 |
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