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Mirrors > Home > MPE Home > Th. List > indiscld | Structured version Visualization 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 20066 |
. . . . 5
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2 | indisuni 20067 |
. . . . . 6
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3 | 2 | iscld 20091 |
. . . . 5
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4 | 1, 3 | ax-mp 5 |
. . . 4
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5 | simpl 463 |
. . . . . 6
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6 | dfss4 3689 |
. . . . . 6
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7 | 5, 6 | sylib 201 |
. . . . 5
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8 | simpr 467 |
. . . . . . 7
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9 | indislem 20064 |
. . . . . . 7
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10 | 8, 9 | syl6eleqr 2551 |
. . . . . 6
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11 | elpri 3997 |
. . . . . 6
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12 | difeq2 3557 |
. . . . . . . . 9
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13 | dif0 3849 |
. . . . . . . . 9
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14 | 12, 13 | syl6eq 2512 |
. . . . . . . 8
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15 | fvex 5898 |
. . . . . . . . . 10
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16 | 15 | prid2 4094 |
. . . . . . . . 9
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17 | 16, 9 | eleqtri 2538 |
. . . . . . . 8
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18 | 14, 17 | syl6eqel 2548 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | difeq2 3557 |
. . . . . . . . 9
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20 | difid 3847 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | syl6eq 2512 |
. . . . . . . 8
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22 | 0ex 4549 |
. . . . . . . . 9
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23 | 22 | prid1 4093 |
. . . . . . . 8
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24 | 21, 23 | syl6eqel 2548 |
. . . . . . 7
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25 | 18, 24 | jaoi 385 |
. . . . . 6
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26 | 10, 11, 25 | 3syl 18 |
. . . . 5
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27 | 7, 26 | eqeltrrd 2541 |
. . . 4
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28 | 4, 27 | sylbi 200 |
. . 3
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29 | 28 | ssriv 3448 |
. 2
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30 | 0cld 20102 |
. . . . 5
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31 | 1, 30 | ax-mp 5 |
. . . 4
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32 | 2 | topcld 20099 |
. . . . 5
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33 | 1, 32 | ax-mp 5 |
. . . 4
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34 | prssi 4141 |
. . . 4
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35 | 31, 33, 34 | mp2an 683 |
. . 3
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36 | 9, 35 | eqsstr3i 3475 |
. 2
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37 | 29, 36 | eqssi 3460 |
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 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-iota 5565 df-fun 5603 df-fv 5609 df-top 19970 df-topon 19972 df-cld 20083 |
This theorem is referenced by: (None) |
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