![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eltg3i | Structured version Visualization version GIF version |
Description: The union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
eltg3i | β’ ((π΅ β π β§ π΄ β π΅) β βͺ π΄ β (topGenβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . 4 β’ ((π΅ β π β§ π΄ β π΅) β π΄ β π΅) | |
2 | pwuni 4948 | . . . 4 β’ π΄ β π« βͺ π΄ | |
3 | ssin 4231 | . . . 4 β’ ((π΄ β π΅ β§ π΄ β π« βͺ π΄) β π΄ β (π΅ β© π« βͺ π΄)) | |
4 | 1, 2, 3 | sylanblc 588 | . . 3 β’ ((π΅ β π β§ π΄ β π΅) β π΄ β (π΅ β© π« βͺ π΄)) |
5 | 4 | unissd 4918 | . 2 β’ ((π΅ β π β§ π΄ β π΅) β βͺ π΄ β βͺ (π΅ β© π« βͺ π΄)) |
6 | eltg 22873 | . . 3 β’ (π΅ β π β (βͺ π΄ β (topGenβπ΅) β βͺ π΄ β βͺ (π΅ β© π« βͺ π΄))) | |
7 | 6 | adantr 480 | . 2 β’ ((π΅ β π β§ π΄ β π΅) β (βͺ π΄ β (topGenβπ΅) β βͺ π΄ β βͺ (π΅ β© π« βͺ π΄))) |
8 | 5, 7 | mpbird 257 | 1 β’ ((π΅ β π β§ π΄ β π΅) β βͺ π΄ β (topGenβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β wcel 2099 β© cin 3946 β wss 3947 π« cpw 4603 βͺ cuni 4908 βcfv 6548 topGenctg 17419 |
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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-topgen 17425 |
This theorem is referenced by: eltg3 22878 tgiun 22895 tgidm 22896 tgrest 23076 leordtval2 23129 fnemeet1 35850 fnejoin2 35853 ontgval 35915 |
Copyright terms: Public domain | W3C validator |