Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 12221 | . . 3 | |
2 | 1 | eleq2d 2209 | . 2 |
3 | elex 2697 | . . . 4 | |
4 | 3 | adantl 275 | . . 3 |
5 | inex1g 4064 | . . . . . 6 | |
6 | uniexg 4361 | . . . . . 6 | |
7 | 5, 6 | syl 14 | . . . . 5 |
8 | ssexg 4067 | . . . . 5 | |
9 | 7, 8 | sylan2 284 | . . . 4 |
10 | 9 | ancoms 266 | . . 3 |
11 | id 19 | . . . . 5 | |
12 | pweq 3513 | . . . . . . 7 | |
13 | 12 | ineq2d 3277 | . . . . . 6 |
14 | 13 | unieqd 3747 | . . . . 5 |
15 | 11, 14 | sseq12d 3128 | . . . 4 |
16 | 15 | elabg 2830 | . . 3 |
17 | 4, 10, 16 | pm5.21nd 901 | . 2 |
18 | 2, 17 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wcel 1480 cab 2125 cvv 2686 cin 3070 wss 3071 cpw 3510 cuni 3736 cfv 5123 ctg 12138 |
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 12144 |
This theorem is referenced by: eltg4i 12227 eltg3i 12228 bastg 12233 tgss 12235 eltop 12241 |
Copyright terms: Public domain | W3C validator |