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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-topgen 12068 | . . . . . . 7 | |
2 | 1 | funmpt2 5132 | . . . . . 6 |
3 | funrel 5110 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | relelfvdm 5421 | . . . . 5 | |
6 | 4, 5 | mpan 420 | . . . 4 |
7 | inex1g 4034 | . . . 4 | |
8 | 6, 7 | syl 14 | . . 3 |
9 | eltg4i 12151 | . . 3 | |
10 | inss1 3266 | . . . . . . 7 | |
11 | sseq1 3090 | . . . . . . 7 | |
12 | 10, 11 | mpbiri 167 | . . . . . 6 |
13 | 12 | biantrurd 303 | . . . . 5 |
14 | unieq 3715 | . . . . . 6 | |
15 | 14 | eqeq2d 2129 | . . . . 5 |
16 | 13, 15 | bitr3d 189 | . . . 4 |
17 | 16 | spcegv 2748 | . . 3 |
18 | 8, 9, 17 | sylc 62 | . 2 |
19 | eltg3i 12152 | . . . . 5 | |
20 | eleq1 2180 | . . . . 5 | |
21 | 19, 20 | syl5ibrcom 156 | . . . 4 |
22 | 21 | expimpd 360 | . . 3 |
23 | 22 | exlimdv 1775 | . 2 |
24 | 18, 23 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 cab 2103 cvv 2660 cin 3040 wss 3041 cpw 3480 cuni 3706 cdm 4509 wrel 4514 wfun 5087 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: tgval3 12154 tgtop 12164 eltop3 12167 tgidm 12170 bastop1 12179 tgrest 12265 tgcn 12304 txbasval 12363 |
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