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Mirrors > Home > MPE Home > Th. List > dissnref | Structured version Visualization version Unicode version |
Description: The set of singletons is a refinement of any open covering of the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020.) |
Ref | Expression |
---|---|
dissnref.c |
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Ref | Expression |
---|---|
dissnref |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 463 |
. . 3
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2 | dissnref.c |
. . . 4
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3 | 2 | unisngl 20554 |
. . 3
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4 | 1, 3 | syl6eq 2503 |
. 2
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5 | simplr 763 |
. . . . . 6
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6 | simprr 767 |
. . . . . . 7
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7 | 6 | snssd 4120 |
. . . . . 6
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8 | 5, 7 | eqsstrd 3468 |
. . . . 5
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9 | simplr 763 |
. . . . . . 7
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10 | simp-4r 778 |
. . . . . . 7
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11 | 9, 10 | eleqtrrd 2534 |
. . . . . 6
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12 | eluni2 4205 |
. . . . . 6
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13 | 11, 12 | sylib 200 |
. . . . 5
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14 | 8, 13 | reximddv 2865 |
. . . 4
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15 | 2 | abeq2i 2565 |
. . . . . 6
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16 | 15 | biimpi 198 |
. . . . 5
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17 | 16 | adantl 468 |
. . . 4
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18 | 14, 17 | r19.29a 2934 |
. . 3
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19 | 18 | ralrimiva 2804 |
. 2
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20 | pwexg 4590 |
. . . . 5
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21 | simpr 463 |
. . . . . . . . 9
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22 | snelpwi 4648 |
. . . . . . . . . 10
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23 | 22 | ad2antlr 734 |
. . . . . . . . 9
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24 | 21, 23 | eqeltrd 2531 |
. . . . . . . 8
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25 | 24, 16 | r19.29a 2934 |
. . . . . . 7
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26 | 25 | ssriv 3438 |
. . . . . 6
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27 | 26 | a1i 11 |
. . . . 5
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28 | 20, 27 | ssexd 4553 |
. . . 4
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29 | 28 | adantr 467 |
. . 3
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30 | eqid 2453 |
. . . 4
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31 | eqid 2453 |
. . . 4
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32 | 30, 31 | isref 20536 |
. . 3
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33 | 29, 32 | syl 17 |
. 2
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34 | 4, 19, 33 | mpbir2and 934 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-rab 2748 df-v 3049 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-br 4406 df-opab 4465 df-xp 4843 df-rel 4844 df-ref 20532 |
This theorem is referenced by: dispcmp 28698 |
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