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Mirrors > Home > MPE Home > Th. List > tgss2 | Structured version Visualization version 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 467 |
. . . . 5
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2 | uniexg 6575 |
. . . . . 6
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3 | 2 | adantr 471 |
. . . . 5
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4 | 1, 3 | eqeltrrd 2530 |
. . . 4
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5 | uniexb 6588 |
. . . 4
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6 | 4, 5 | sylibr 217 |
. . 3
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7 | tgss3 20012 |
. . 3
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8 | 6, 7 | syldan 477 |
. 2
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9 | eltg2b 19984 |
. . . . . . 7
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10 | 6, 9 | syl 17 |
. . . . . 6
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11 | elunii 4172 |
. . . . . . . . 9
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12 | 11 | ancoms 459 |
. . . . . . . 8
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13 | biimt 341 |
. . . . . . . 8
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14 | 12, 13 | syl 17 |
. . . . . . 7
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15 | 14 | ralbidva 2808 |
. . . . . 6
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16 | 10, 15 | sylan9bb 711 |
. . . . 5
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17 | ralcom3 2923 |
. . . . 5
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18 | 16, 17 | syl6bb 269 |
. . . 4
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19 | 18 | ralbidva 2808 |
. . 3
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20 | dfss3 3389 |
. . 3
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21 | ralcom 2918 |
. . 3
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22 | 19, 20, 21 | 3bitr4g 296 |
. 2
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23 | 8, 22 | bitrd 261 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1672 ax-4 1685 ax-5 1761 ax-6 1808 ax-7 1854 ax-8 1892 ax-9 1899 ax-10 1918 ax-11 1923 ax-12 1936 ax-13 2091 ax-ext 2431 ax-sep 4496 ax-nul 4505 ax-pow 4553 ax-pr 4611 ax-un 6570 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 988 df-tru 1450 df-ex 1667 df-nf 1671 df-sb 1801 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2623 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3014 df-sbc 3235 df-dif 3374 df-un 3376 df-in 3378 df-ss 3385 df-nul 3699 df-if 3849 df-pw 3920 df-sn 3936 df-pr 3938 df-op 3942 df-uni 4168 df-iun 4249 df-br 4374 df-opab 4433 df-mpt 4434 df-id 4726 df-xp 4817 df-rel 4818 df-cnv 4819 df-co 4820 df-dm 4821 df-iota 5524 df-fun 5562 df-fv 5568 df-topgen 15352 |
This theorem is referenced by: metss 21533 relowlssretop 31767 relowlpssretop 31768 |
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