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Mirrors > Home > MPE Home > Th. List > eltg2 | Structured version Unicode version |
Description: Membership in a topology generated by a basis. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
eltg2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgval2 18696 |
. . 3
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2 | 1 | eleq2d 2524 |
. 2
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3 | elex 3087 |
. . . 4
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4 | 3 | adantl 466 |
. . 3
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5 | uniexg 6490 |
. . . . . 6
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6 | ssexg 4549 |
. . . . . 6
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7 | 5, 6 | sylan2 474 |
. . . . 5
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8 | 7 | ancoms 453 |
. . . 4
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9 | 8 | adantrr 716 |
. . 3
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10 | sseq1 3488 |
. . . . 5
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11 | sseq2 3489 |
. . . . . . . 8
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12 | 11 | anbi2d 703 |
. . . . . . 7
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13 | 12 | rexbidv 2868 |
. . . . . 6
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14 | 13 | raleqbi1dv 3031 |
. . . . 5
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15 | 10, 14 | anbi12d 710 |
. . . 4
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16 | 15 | elabg 3214 |
. . 3
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17 | 4, 9, 16 | pm5.21nd 893 |
. 2
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18 | 2, 17 | bitrd 253 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-iota 5492 df-fun 5531 df-fv 5537 df-topgen 14504 |
This theorem is referenced by: eltg2b 18699 tg1 18704 tgcl 18709 elmopn 20152 metutopOLD 20292 psmetutop 20293 |
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