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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 486 | . . . 4 β’ ((π΅ β π β§ π΄ β π΅) β π΄ β π΅) | |
2 | pwuni 4907 | . . . 4 β’ π΄ β π« βͺ π΄ | |
3 | ssin 4191 | . . . 4 β’ ((π΄ β π΅ β§ π΄ β π« βͺ π΄) β π΄ β (π΅ β© π« βͺ π΄)) | |
4 | 1, 2, 3 | sylanblc 590 | . . 3 β’ ((π΅ β π β§ π΄ β π΅) β π΄ β (π΅ β© π« βͺ π΄)) |
5 | 4 | unissd 4876 | . 2 β’ ((π΅ β π β§ π΄ β π΅) β βͺ π΄ β βͺ (π΅ β© π« βͺ π΄)) |
6 | eltg 22323 | . . 3 β’ (π΅ β π β (βͺ π΄ β (topGenβπ΅) β βͺ π΄ β βͺ (π΅ β© π« βͺ π΄))) | |
7 | 6 | adantr 482 | . 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 397 β wcel 2107 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: eltg3 22328 tgiun 22345 tgidm 22346 tgrest 22526 leordtval2 22579 fnemeet1 34884 fnejoin2 34887 ontgval 34949 |
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