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Mirrors > Home > ILE Home > Th. List > tgss2 | Unicode version |
Description: A criterion for determining whether one topology is finer than another, based on a comparison of their bases. Lemma 2.2 of [Munkres] p. 80. (Contributed by NM, 20-Jul-2006.) (Proof shortened by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tgss2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 | |
2 | uniexg 4361 | . . . . . 6 | |
3 | 2 | adantr 274 | . . . . 5 |
4 | 1, 3 | eqeltrrd 2217 | . . . 4 |
5 | uniexb 4394 | . . . 4 | |
6 | 4, 5 | sylibr 133 | . . 3 |
7 | tgss3 12247 | . . 3 | |
8 | 6, 7 | syldan 280 | . 2 |
9 | eltg2b 12223 | . . . . . . 7 | |
10 | 6, 9 | syl 14 | . . . . . 6 |
11 | elunii 3741 | . . . . . . . . 9 | |
12 | 11 | ancoms 266 | . . . . . . . 8 |
13 | biimt 240 | . . . . . . . 8 | |
14 | 12, 13 | syl 14 | . . . . . . 7 |
15 | 14 | ralbidva 2433 | . . . . . 6 |
16 | 10, 15 | sylan9bb 457 | . . . . 5 |
17 | ralcom3 2598 | . . . . 5 | |
18 | 16, 17 | syl6bb 195 | . . . 4 |
19 | 18 | ralbidva 2433 | . . 3 |
20 | dfss3 3087 | . . 3 | |
21 | ralcom 2594 | . . 3 | |
22 | 19, 20, 21 | 3bitr4g 222 | . 2 |
23 | 8, 22 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 cvv 2686 wss 3071 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-iun 3815 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: metss 12663 |
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