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Mirrors > Home > ILE Home > Th. List > eltg3 | Unicode version |
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Jim Kingdon, 4-Mar-2023.) |
Ref | Expression |
---|---|
eltg3 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-topgen 12703 |
. . . . . . 7
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2 | 1 | funmpt2 5255 |
. . . . . 6
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3 | funrel 5233 |
. . . . . 6
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4 | 2, 3 | ax-mp 5 |
. . . . 5
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5 | relelfvdm 5547 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 4, 5 | mpan 424 |
. . . 4
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7 | inex1g 4139 |
. . . 4
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8 | 6, 7 | syl 14 |
. . 3
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9 | eltg4i 13486 |
. . 3
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10 | inss1 3355 |
. . . . . . 7
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11 | sseq1 3178 |
. . . . . . 7
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12 | 10, 11 | mpbiri 168 |
. . . . . 6
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13 | 12 | biantrurd 305 |
. . . . 5
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14 | unieq 3818 |
. . . . . 6
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15 | 14 | eqeq2d 2189 |
. . . . 5
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16 | 13, 15 | bitr3d 190 |
. . . 4
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17 | 16 | spcegv 2825 |
. . 3
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18 | 8, 9, 17 | sylc 62 |
. 2
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19 | eltg3i 13487 |
. . . . 5
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20 | eleq1 2240 |
. . . . 5
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21 | 19, 20 | syl5ibrcom 157 |
. . . 4
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22 | 21 | expimpd 363 |
. . 3
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23 | 22 | exlimdv 1819 |
. 2
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24 | 18, 23 | impbid2 143 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-topgen 12703 |
This theorem is referenced by: tgval3 13489 tgtop 13499 eltop3 13502 tgidm 13505 bastop1 13514 tgrest 13600 tgcn 13639 txbasval 13698 |
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