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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 22812 | . 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 3078 | . . . . . 6 β’ (βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄) β βπ¦ β π΅ π₯ β π¦) |
4 | eluni2 4906 | . . . . . 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 3077 | . . . 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 3965 | . . . 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 2098 βwral 3055 βwrex 3064 β wss 3943 βͺ cuni 4902 βcfv 6536 topGenctg 17390 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-topgen 17396 |
This theorem is referenced by: tg2 22819 tgcl 22823 eltop2 22829 tgss2 22841 basgen2 22843 2ndc1stc 23306 eltx 23423 tgqioo 24667 isfne2 35735 |
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