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Mirrors > Home > ILE Home > Th. List > eltg | Unicode version |
Description: Membership in a topology generated by a basis. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
eltg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgval 11916 |
. . 3
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2 | 1 | eleq2d 2164 |
. 2
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3 | elex 2644 |
. . . 4
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4 | 3 | adantl 272 |
. . 3
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5 | inex1g 3996 |
. . . . . 6
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6 | uniexg 4290 |
. . . . . 6
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7 | 5, 6 | syl 14 |
. . . . 5
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8 | ssexg 3999 |
. . . . 5
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9 | 7, 8 | sylan2 281 |
. . . 4
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10 | 9 | ancoms 265 |
. . 3
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11 | id 19 |
. . . . 5
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12 | pweq 3452 |
. . . . . . 7
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13 | 12 | ineq2d 3216 |
. . . . . 6
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14 | 13 | unieqd 3686 |
. . . . 5
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15 | 11, 14 | sseq12d 3070 |
. . . 4
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16 | 15 | elabg 2775 |
. . 3
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17 | 4, 10, 16 | pm5.21nd 866 |
. 2
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18 | 2, 17 | bitrd 187 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-topgen 11840 |
This theorem is referenced by: eltg4i 11922 eltg3i 11923 bastg 11928 tgss 11930 eltop 11936 |
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