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Mirrors > Home > MPE Home > Th. List > indiscld | Structured version Unicode version |
Description: The closed sets of an indiscrete topology. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
indiscld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indistop 18733 |
. . . . 5
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2 | indisuni 18734 |
. . . . . 6
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3 | 2 | iscld 18758 |
. . . . 5
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4 | 1, 3 | ax-mp 5 |
. . . 4
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5 | simpl 457 |
. . . . . 6
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6 | dfss4 3687 |
. . . . . 6
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7 | 5, 6 | sylib 196 |
. . . . 5
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8 | simpr 461 |
. . . . . . 7
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9 | indislem 18731 |
. . . . . . 7
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10 | 8, 9 | syl6eleqr 2551 |
. . . . . 6
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11 | elpri 4000 |
. . . . . 6
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12 | difeq2 3571 |
. . . . . . . . 9
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13 | dif0 3852 |
. . . . . . . . 9
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14 | 12, 13 | syl6eq 2509 |
. . . . . . . 8
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15 | fvex 5804 |
. . . . . . . . . 10
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16 | 15 | prid2 4087 |
. . . . . . . . 9
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17 | 16, 9 | eleqtri 2538 |
. . . . . . . 8
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18 | 14, 17 | syl6eqel 2548 |
. . . . . . 7
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19 | difeq2 3571 |
. . . . . . . . 9
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20 | difid 3850 |
. . . . . . . . 9
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21 | 19, 20 | syl6eq 2509 |
. . . . . . . 8
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22 | 0ex 4525 |
. . . . . . . . 9
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23 | 22 | prid1 4086 |
. . . . . . . 8
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24 | 21, 23 | syl6eqel 2548 |
. . . . . . 7
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25 | 18, 24 | jaoi 379 |
. . . . . 6
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26 | 10, 11, 25 | 3syl 20 |
. . . . 5
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27 | 7, 26 | eqeltrrd 2541 |
. . . 4
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28 | 4, 27 | sylbi 195 |
. . 3
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29 | 28 | ssriv 3463 |
. 2
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30 | 0cld 18769 |
. . . . 5
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31 | 1, 30 | ax-mp 5 |
. . . 4
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32 | 2 | topcld 18766 |
. . . . 5
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33 | 1, 32 | ax-mp 5 |
. . . 4
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34 | prssi 4132 |
. . . 4
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35 | 31, 33, 34 | mp2an 672 |
. . 3
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36 | 9, 35 | eqsstr3i 3490 |
. 2
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37 | 29, 36 | eqssi 3475 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 ax-un 6477 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-ral 2801 df-rex 2802 df-rab 2805 df-v 3074 df-sbc 3289 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4195 df-br 4396 df-opab 4454 df-mpt 4455 df-id 4739 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-iota 5484 df-fun 5523 df-fv 5529 df-top 18630 df-topon 18633 df-cld 18750 |
This theorem is referenced by: (None) |
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