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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 3963 |
. . . . . . . . 9
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3 | 1, 2 | elind 3617 |
. . . . . . . 8
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4 | elssuni 4226 |
. . . . . . . 8
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5 | 3, 4 | syl 17 |
. . . . . . 7
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6 | inidm 3640 |
. . . . . . . . 9
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7 | ineq2 3627 |
. . . . . . . . 9
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8 | 6, 7 | syl5eqr 2498 |
. . . . . . . 8
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9 | 8 | pweqd 3955 |
. . . . . . . . . 10
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10 | 9 | ineq2d 3633 |
. . . . . . . . 9
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11 | 10 | unieqd 4207 |
. . . . . . . 8
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12 | 8, 11 | sseq12d 3460 |
. . . . . . 7
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13 | 5, 12 | syl5ibcom 224 |
. . . . . 6
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14 | 0ss 3762 |
. . . . . . . 8
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15 | sseq1 3452 |
. . . . . . . 8
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16 | 14, 15 | mpbiri 237 |
. . . . . . 7
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17 | 16 | a1i 11 |
. . . . . 6
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18 | 13, 17 | jaod 382 |
. . . . 5
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19 | 18 | ralimdv 2797 |
. . . 4
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20 | 19 | ralimia 2778 |
. . 3
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21 | 20 | adantl 468 |
. 2
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22 | isbasisg 19955 |
. . 3
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23 | 22 | adantr 467 |
. 2
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24 | 21, 23 | mpbird 236 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ral 2741 df-rex 2742 df-v 3046 df-dif 3406 df-in 3410 df-ss 3417 df-nul 3731 df-pw 3952 df-uni 4198 df-bases 19915 |
This theorem is referenced by: kelac2lem 35916 |
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