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Mirrors > Home > ILE Home > Th. List > topbas | Unicode version |
Description: A topology is its own basis. (Contributed by NM, 17-Jul-2006.) |
Ref | Expression |
---|---|
topbas |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inopn 12548 | . . . . . . 7 | |
2 | 1 | 3expb 1193 | . . . . . 6 |
3 | simpr 109 | . . . . . . 7 | |
4 | ssid 3157 | . . . . . . 7 | |
5 | 3, 4 | jctir 311 | . . . . . 6 |
6 | eleq2 2228 | . . . . . . . 8 | |
7 | sseq1 3160 | . . . . . . . 8 | |
8 | 6, 7 | anbi12d 465 | . . . . . . 7 |
9 | 8 | rspcev 2825 | . . . . . 6 |
10 | 2, 5, 9 | syl2an2r 585 | . . . . 5 |
11 | 10 | exp31 362 | . . . 4 |
12 | 11 | ralrimdv 2543 | . . 3 |
13 | 12 | ralrimivv 2545 | . 2 |
14 | isbasis2g 12590 | . 2 | |
15 | 13, 14 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 wral 2442 wrex 2443 cin 3110 wss 3111 ctop 12542 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 ax-sep 4094 |
This theorem depends on definitions: df-bi 116 df-3an 969 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-top 12543 df-bases 12588 |
This theorem is referenced by: resttop 12717 txtop 12807 |
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