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Mirrors > Home > MPE Home > Th. List > eltg2b | Structured version Visualization version GIF version |
Description: Membership in a topology generated by a basis. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
eltg2b | β’ (π΅ β π β (π΄ β (topGenβπ΅) β βπ₯ β π΄ βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eltg2 22874 | . 2 β’ (π΅ β π β (π΄ β (topGenβπ΅) β (π΄ β βͺ π΅ β§ βπ₯ β π΄ βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄)))) | |
2 | simpl 482 | . . . . . . 7 β’ ((π₯ β π¦ β§ π¦ β π΄) β π₯ β π¦) | |
3 | 2 | reximi 3081 | . . . . . 6 β’ (βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄) β βπ¦ β π΅ π₯ β π¦) |
4 | eluni2 4912 | . . . . . 6 β’ (π₯ β βͺ π΅ β βπ¦ β π΅ π₯ β π¦) | |
5 | 3, 4 | sylibr 233 | . . . . 5 β’ (βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄) β π₯ β βͺ π΅) |
6 | 5 | ralimi 3080 | . . . 4 β’ (βπ₯ β π΄ βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄) β βπ₯ β π΄ π₯ β βͺ π΅) |
7 | dfss3 3968 | . . . 4 β’ (π΄ β βͺ π΅ β βπ₯ β π΄ π₯ β βͺ π΅) | |
8 | 6, 7 | sylibr 233 | . . 3 β’ (βπ₯ β π΄ βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄) β π΄ β βͺ π΅) |
9 | 8 | pm4.71ri 560 | . 2 β’ (βπ₯ β π΄ βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄) β (π΄ β βͺ π΅ β§ βπ₯ β π΄ βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄))) |
10 | 1, 9 | bitr4di 289 | 1 β’ (π΅ β π β (π΄ β (topGenβπ΅) β βπ₯ β π΄ βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β wcel 2099 βwral 3058 βwrex 3067 β wss 3947 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: tg2 22881 tgcl 22885 eltop2 22891 tgss2 22903 basgen2 22905 2ndc1stc 23368 eltx 23485 tgqioo 24729 isfne2 35826 |
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