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Mirrors > Home > MPE Home > Th. List > baspartn | Structured version Visualization version Unicode version |
Description: A disjoint system of sets is a basis for a topology. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
baspartn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 |
. . . . . . . . 9
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2 | pwidg 3955 |
. . . . . . . . 9
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3 | 1, 2 | elind 3609 |
. . . . . . . 8
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4 | elssuni 4219 |
. . . . . . . 8
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5 | 3, 4 | syl 17 |
. . . . . . 7
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6 | inidm 3632 |
. . . . . . . . 9
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7 | ineq2 3619 |
. . . . . . . . 9
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8 | 6, 7 | syl5eqr 2519 |
. . . . . . . 8
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9 | 8 | pweqd 3947 |
. . . . . . . . . 10
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10 | 9 | ineq2d 3625 |
. . . . . . . . 9
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11 | 10 | unieqd 4200 |
. . . . . . . 8
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12 | 8, 11 | sseq12d 3447 |
. . . . . . 7
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13 | 5, 12 | syl5ibcom 228 |
. . . . . 6
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14 | 0ss 3766 |
. . . . . . . 8
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15 | sseq1 3439 |
. . . . . . . 8
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16 | 14, 15 | mpbiri 241 |
. . . . . . 7
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17 | 16 | a1i 11 |
. . . . . 6
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18 | 13, 17 | jaod 387 |
. . . . 5
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19 | 18 | ralimdv 2806 |
. . . 4
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20 | 19 | ralimia 2794 |
. . 3
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21 | 20 | adantl 473 |
. 2
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22 | isbasisg 20039 |
. . 3
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23 | 22 | adantr 472 |
. 2
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24 | 21, 23 | mpbird 240 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ral 2761 df-rex 2762 df-v 3033 df-dif 3393 df-in 3397 df-ss 3404 df-nul 3723 df-pw 3944 df-uni 4191 df-bases 19999 |
This theorem is referenced by: kelac2lem 35993 |
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