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Mirrors > Home > MPE Home > Th. List > ufilcmp | Structured version Visualization version Unicode version |
Description: A space is compact iff every ultrafilter converges. (Contributed by Jeff Hankins, 11-Dec-2009.) (Proof shortened by Mario Carneiro, 12-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
ufilcmp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ufilfil 20931 |
. . . . . 6
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2 | eqid 2453 |
. . . . . . 7
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3 | 2 | fclscmpi 21056 |
. . . . . 6
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4 | 1, 3 | sylan2 477 |
. . . . 5
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5 | 4 | ralrimiva 2804 |
. . . 4
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6 | toponuni 19954 |
. . . . . . 7
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7 | 6 | fveq2d 5874 |
. . . . . 6
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8 | 7 | raleqdv 2995 |
. . . . 5
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9 | 8 | adantl 468 |
. . . 4
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10 | 5, 9 | syl5ibr 225 |
. . 3
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11 | ufli 20941 |
. . . . . . 7
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12 | 11 | adantlr 722 |
. . . . . 6
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13 | r19.29 2927 |
. . . . . . 7
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14 | simpllr 770 |
. . . . . . . . . . . . 13
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15 | simplr 763 |
. . . . . . . . . . . . 13
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16 | simprr 767 |
. . . . . . . . . . . . 13
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17 | fclsss2 21050 |
. . . . . . . . . . . . 13
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18 | 14, 15, 16, 17 | syl3anc 1269 |
. . . . . . . . . . . 12
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19 | ssn0 3769 |
. . . . . . . . . . . . 13
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20 | 19 | ex 436 |
. . . . . . . . . . . 12
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21 | 18, 20 | syl 17 |
. . . . . . . . . . 11
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22 | 21 | expr 620 |
. . . . . . . . . 10
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23 | 22 | com23 81 |
. . . . . . . . 9
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24 | 23 | impd 433 |
. . . . . . . 8
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25 | 24 | rexlimdva 2881 |
. . . . . . 7
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26 | 13, 25 | syl5 33 |
. . . . . 6
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27 | 12, 26 | mpan2d 681 |
. . . . 5
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28 | 27 | ralrimdva 2808 |
. . . 4
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29 | fclscmp 21057 |
. . . . 5
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30 | 29 | adantl 468 |
. . . 4
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31 | 28, 30 | sylibrd 238 |
. . 3
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32 | 10, 31 | impbid 194 |
. 2
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33 | uffclsflim 21058 |
. . . 4
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34 | 33 | neeq1d 2685 |
. . 3
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35 | 34 | ralbiia 2820 |
. 2
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36 | 32, 35 | syl6bb 265 |
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-rep 4518 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-3or 987 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-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-int 4238 df-iun 4283 df-iin 4284 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-1st 6798 df-2nd 6799 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-1o 7187 df-2o 7188 df-oadd 7191 df-er 7368 df-map 7479 df-en 7575 df-dom 7576 df-sdom 7577 df-fin 7578 df-fi 7930 df-fbas 18979 df-fg 18980 df-top 19933 df-topon 19935 df-cld 20046 df-ntr 20047 df-cls 20048 df-nei 20126 df-cmp 20414 df-fil 20873 df-ufil 20928 df-ufl 20929 df-flim 20966 df-fcls 20968 |
This theorem is referenced by: alexsub 21072 |
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