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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 |
. . . . 5
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2 | uniexg 4369 |
. . . . . 6
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3 | 2 | adantr 274 |
. . . . 5
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4 | 1, 3 | eqeltrrd 2218 |
. . . 4
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5 | uniexb 4402 |
. . . 4
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6 | 4, 5 | sylibr 133 |
. . 3
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7 | tgss3 12286 |
. . 3
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8 | 6, 7 | syldan 280 |
. 2
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9 | eltg2b 12262 |
. . . . . . 7
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10 | 6, 9 | syl 14 |
. . . . . 6
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11 | elunii 3749 |
. . . . . . . . 9
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12 | 11 | ancoms 266 |
. . . . . . . 8
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13 | biimt 240 |
. . . . . . . 8
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14 | 12, 13 | syl 14 |
. . . . . . 7
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15 | 14 | ralbidva 2434 |
. . . . . 6
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16 | 10, 15 | sylan9bb 458 |
. . . . 5
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17 | ralcom3 2601 |
. . . . 5
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18 | 16, 17 | syl6bb 195 |
. . . 4
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19 | 18 | ralbidva 2434 |
. . 3
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20 | dfss3 3092 |
. . 3
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21 | ralcom 2597 |
. . 3
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22 | 19, 20, 21 | 3bitr4g 222 |
. 2
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23 | 8, 22 | bitrd 187 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-topgen 12180 |
This theorem is referenced by: metss 12702 |
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