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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 12141 | . . . . . . 7 | |
2 | 1 | funmpt2 5162 | . . . . . 6 |
3 | funrel 5140 | . . . . . 6 | |
4 | 2, 3 | ax-mp 5 | . . . . 5 |
5 | relelfvdm 5453 | . . . . 5 | |
6 | 4, 5 | mpan 420 | . . . 4 |
7 | inex1g 4064 | . . . 4 | |
8 | 6, 7 | syl 14 | . . 3 |
9 | eltg4i 12224 | . . 3 | |
10 | inss1 3296 | . . . . . . 7 | |
11 | sseq1 3120 | . . . . . . 7 | |
12 | 10, 11 | mpbiri 167 | . . . . . 6 |
13 | 12 | biantrurd 303 | . . . . 5 |
14 | unieq 3745 | . . . . . 6 | |
15 | 14 | eqeq2d 2151 | . . . . 5 |
16 | 13, 15 | bitr3d 189 | . . . 4 |
17 | 16 | spcegv 2774 | . . 3 |
18 | 8, 9, 17 | sylc 62 | . 2 |
19 | eltg3i 12225 | . . . . 5 | |
20 | eleq1 2202 | . . . . 5 | |
21 | 19, 20 | syl5ibrcom 156 | . . . 4 |
22 | 21 | expimpd 360 | . . 3 |
23 | 22 | exlimdv 1791 | . 2 |
24 | 18, 23 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cab 2125 cvv 2686 cin 3070 wss 3071 cpw 3510 cuni 3736 cdm 4539 wrel 4544 wfun 5117 cfv 5123 ctg 12135 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-topgen 12141 |
This theorem is referenced by: tgval3 12227 tgtop 12237 eltop3 12240 tgidm 12243 bastop1 12252 tgrest 12338 tgcn 12377 txbasval 12436 |
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