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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgval 12145 | . . 3 | |
2 | 1 | eleq2d 2187 | . 2 |
3 | elex 2671 | . . . 4 | |
4 | 3 | adantl 275 | . . 3 |
5 | inex1g 4034 | . . . . . 6 | |
6 | uniexg 4331 | . . . . . 6 | |
7 | 5, 6 | syl 14 | . . . . 5 |
8 | ssexg 4037 | . . . . 5 | |
9 | 7, 8 | sylan2 284 | . . . 4 |
10 | 9 | ancoms 266 | . . 3 |
11 | id 19 | . . . . 5 | |
12 | pweq 3483 | . . . . . . 7 | |
13 | 12 | ineq2d 3247 | . . . . . 6 |
14 | 13 | unieqd 3717 | . . . . 5 |
15 | 11, 14 | sseq12d 3098 | . . . 4 |
16 | 15 | elabg 2803 | . . 3 |
17 | 4, 10, 16 | pm5.21nd 886 | . 2 |
18 | 2, 17 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1316 wcel 1465 cab 2103 cvv 2660 cin 3040 wss 3041 cpw 3480 cuni 3706 cfv 5093 ctg 12062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-topgen 12068 |
This theorem is referenced by: eltg4i 12151 eltg3i 12152 bastg 12157 tgss 12159 eltop 12165 |
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