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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 22324 | . 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 484 | . . . . . . 7 β’ ((π₯ β π¦ β§ π¦ β π΄) β π₯ β π¦) | |
3 | 2 | reximi 3084 | . . . . . 6 β’ (βπ¦ β π΅ (π₯ β π¦ β§ π¦ β π΄) β βπ¦ β π΅ π₯ β π¦) |
4 | eluni2 4870 | . . . . . 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 3083 | . . . 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 3933 | . . . 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 562 | . 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 397 β wcel 2107 βwral 3061 βwrex 3070 β wss 3911 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: tg2 22331 tgcl 22335 eltop2 22341 tgss2 22353 basgen2 22355 2ndc1stc 22818 eltx 22935 tgqioo 24179 isfne2 34860 |
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