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Mirrors > Home > ILE Home > Th. List > tgval2 | Unicode version |
Description: Definition of a topology generated by a basis in [Munkres] p. 78. Later we show (in tgcl 12233) that is indeed a topology (on , see unitg 12231). See also tgval 12218 and tgval3 12227. (Contributed by NM, 15-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
tgval2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgval 12218 | . 2 | |
2 | inss1 3296 | . . . . . . . . 9 | |
3 | 2 | unissi 3759 | . . . . . . . 8 |
4 | 3 | sseli 3093 | . . . . . . 7 |
5 | 4 | pm4.71ri 389 | . . . . . 6 |
6 | 5 | ralbii 2441 | . . . . 5 |
7 | r19.26 2558 | . . . . 5 | |
8 | 6, 7 | bitri 183 | . . . 4 |
9 | dfss3 3087 | . . . 4 | |
10 | dfss3 3087 | . . . . 5 | |
11 | elin 3259 | . . . . . . . . . . 11 | |
12 | 11 | anbi2i 452 | . . . . . . . . . 10 |
13 | an12 550 | . . . . . . . . . 10 | |
14 | 12, 13 | bitri 183 | . . . . . . . . 9 |
15 | 14 | exbii 1584 | . . . . . . . 8 |
16 | eluni 3739 | . . . . . . . 8 | |
17 | df-rex 2422 | . . . . . . . 8 | |
18 | 15, 16, 17 | 3bitr4i 211 | . . . . . . 7 |
19 | velpw 3517 | . . . . . . . . 9 | |
20 | 19 | anbi2i 452 | . . . . . . . 8 |
21 | 20 | rexbii 2442 | . . . . . . 7 |
22 | 18, 21 | bitr2i 184 | . . . . . 6 |
23 | 22 | ralbii 2441 | . . . . 5 |
24 | 10, 23 | anbi12i 455 | . . . 4 |
25 | 8, 9, 24 | 3bitr4i 211 | . . 3 |
26 | 25 | abbii 2255 | . 2 |
27 | 1, 26 | syl6eq 2188 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 cab 2125 wral 2416 wrex 2417 cin 3070 wss 3071 cpw 3510 cuni 3736 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: eltg2 12222 |
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