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Mirrors > Home > ILE Home > Th. List > bastop1 | Unicode version |
Description: A subset of a topology is a basis for the topology iff every member of the topology is a union of members of the basis. We use the idiom " " to express " is a basis for topology " since we do not have a separate notation for this. Definition 15.35 of [Schechter] p. 428. (Contributed by NM, 2-Feb-2008.) (Proof shortened by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
bastop1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgss 12610 | . . . . 5 | |
2 | tgtop 12615 | . . . . . 6 | |
3 | 2 | adantr 274 | . . . . 5 |
4 | 1, 3 | sseqtrd 3175 | . . . 4 |
5 | eqss 3152 | . . . . 5 | |
6 | 5 | baib 909 | . . . 4 |
7 | 4, 6 | syl 14 | . . 3 |
8 | dfss3 3127 | . . 3 | |
9 | 7, 8 | bitrdi 195 | . 2 |
10 | ssexg 4115 | . . . . 5 | |
11 | 10 | ancoms 266 | . . . 4 |
12 | eltg3 12604 | . . . 4 | |
13 | 11, 12 | syl 14 | . . 3 |
14 | 13 | ralbidv 2464 | . 2 |
15 | 9, 14 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 wral 2442 cvv 2721 wss 3111 cuni 3783 cfv 5182 ctg 12513 ctop 12542 |
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-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-topgen 12519 df-top 12543 |
This theorem is referenced by: bastop2 12631 |
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