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Mirrors > Home > MPE Home > Th. List > bastg | Structured version Visualization version GIF version |
Description: A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
bastg | β’ (π΅ β π β π΅ β (topGenβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . . . . 6 β’ ((π΅ β π β§ π₯ β π΅) β π₯ β π΅) | |
2 | vex 3479 | . . . . . . . 8 β’ π₯ β V | |
3 | 2 | pwid 4625 | . . . . . . 7 β’ π₯ β π« π₯ |
4 | 3 | a1i 11 | . . . . . 6 β’ ((π΅ β π β§ π₯ β π΅) β π₯ β π« π₯) |
5 | 1, 4 | elind 4195 | . . . . 5 β’ ((π΅ β π β§ π₯ β π΅) β π₯ β (π΅ β© π« π₯)) |
6 | elssuni 4942 | . . . . 5 β’ (π₯ β (π΅ β© π« π₯) β π₯ β βͺ (π΅ β© π« π₯)) | |
7 | 5, 6 | syl 17 | . . . 4 β’ ((π΅ β π β§ π₯ β π΅) β π₯ β βͺ (π΅ β© π« π₯)) |
8 | 7 | ex 414 | . . 3 β’ (π΅ β π β (π₯ β π΅ β π₯ β βͺ (π΅ β© π« π₯))) |
9 | eltg 22460 | . . 3 β’ (π΅ β π β (π₯ β (topGenβπ΅) β π₯ β βͺ (π΅ β© π« π₯))) | |
10 | 8, 9 | sylibrd 259 | . 2 β’ (π΅ β π β (π₯ β π΅ β π₯ β (topGenβπ΅))) |
11 | 10 | ssrdv 3989 | 1 β’ (π΅ β π β π΅ β (topGenβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 β© cin 3948 β wss 3949 π« cpw 4603 βͺ cuni 4909 βcfv 6544 topGenctg 17383 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-topgen 17389 |
This theorem is referenced by: unitg 22470 tgclb 22473 tgtop 22476 tgidm 22483 tgss3 22489 bastop2 22497 elcls3 22587 ordtopn1 22698 ordtopn2 22699 leordtval2 22716 iocpnfordt 22719 icomnfordt 22720 iooordt 22721 tgcn 22756 tgcnp 22757 tgcmp 22905 2ndcsb 22953 2ndc1stc 22955 2ndcctbss 22959 2ndcomap 22962 ptopn 23087 xkoopn 23093 txopn 23106 txbasval 23110 ptpjcn 23115 flftg 23500 alexsubb 23550 blssopn 24004 iooretop 24282 bndth 24474 ovolicc2 25039 cncombf 25175 cnmbf 25176 ordtconnlem1 32904 elmbfmvol2 33266 dya2icoseg2 33277 iccllysconn 34241 rellysconn 34242 topjoin 35250 fnemeet2 35252 fnejoin1 35253 ontgval 35316 mblfinlem3 36527 mblfinlem4 36528 ismblfin 36529 cnambfre 36536 kelac2 41807 |
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