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Mirrors > Home > ILE Home > Th. List > isbasis2g | Unicode version |
Description: Express the predicate "the set is a basis for a topology". (Contributed by NM, 17-Jul-2006.) |
Ref | Expression |
---|---|
isbasis2g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isbasisg 12589 | . 2 | |
2 | dfss3 3127 | . . . 4 | |
3 | elin 3300 | . . . . . . . . . 10 | |
4 | velpw 3560 | . . . . . . . . . . 11 | |
5 | 4 | anbi2i 453 | . . . . . . . . . 10 |
6 | 3, 5 | bitri 183 | . . . . . . . . 9 |
7 | 6 | anbi2i 453 | . . . . . . . 8 |
8 | an12 551 | . . . . . . . 8 | |
9 | 7, 8 | bitri 183 | . . . . . . 7 |
10 | 9 | exbii 1592 | . . . . . 6 |
11 | eluni 3786 | . . . . . 6 | |
12 | df-rex 2448 | . . . . . 6 | |
13 | 10, 11, 12 | 3bitr4i 211 | . . . . 5 |
14 | 13 | ralbii 2470 | . . . 4 |
15 | 2, 14 | bitri 183 | . . 3 |
16 | 15 | 2ralbii 2472 | . 2 |
17 | 1, 16 | bitrdi 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wex 1479 wcel 2135 wral 2442 wrex 2443 cin 3110 wss 3111 cpw 3553 cuni 3783 ctb 12587 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-in 3117 df-ss 3124 df-pw 3555 df-uni 3784 df-bases 12588 |
This theorem is referenced by: isbasis3g 12591 basis2 12593 fiinbas 12594 tgclb 12612 topbas 12614 restbasg 12715 txbas 12805 blbas 12980 |
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