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Mirrors > Home > MPE Home > Th. List > hausflimi | Structured version Visualization version Unicode version |
Description: One direction of hausflim 20989. A filter in a Hausdorff space has at most one limit. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 21-Sep-2015.) |
Ref | Expression |
---|---|
hausflimi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 459 |
. . . . . . . . 9
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2 | simprll 771 |
. . . . . . . . . 10
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3 | eqid 2450 |
. . . . . . . . . . 11
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4 | 3 | flimelbas 20976 |
. . . . . . . . . 10
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5 | 2, 4 | syl 17 |
. . . . . . . . 9
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6 | simprlr 772 |
. . . . . . . . . 10
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7 | 3 | flimelbas 20976 |
. . . . . . . . . 10
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8 | 6, 7 | syl 17 |
. . . . . . . . 9
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9 | simprr 765 |
. . . . . . . . 9
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10 | 3 | hausnei 20337 |
. . . . . . . . 9
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11 | 1, 5, 8, 9, 10 | syl13anc 1269 |
. . . . . . . 8
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12 | df-3an 986 |
. . . . . . . . . 10
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13 | simprl 763 |
. . . . . . . . . . . . . 14
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14 | hausflimlem 20987 |
. . . . . . . . . . . . . . 15
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15 | 14 | 3expa 1207 |
. . . . . . . . . . . . . 14
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16 | 13, 15 | sylanl1 655 |
. . . . . . . . . . . . 13
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17 | 16 | a1d 26 |
. . . . . . . . . . . 12
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18 | 17 | necon4d 2647 |
. . . . . . . . . . 11
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19 | 18 | expimpd 607 |
. . . . . . . . . 10
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20 | 12, 19 | syl5bi 221 |
. . . . . . . . 9
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21 | 20 | rexlimdvva 2885 |
. . . . . . . 8
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22 | 11, 21 | mpd 15 |
. . . . . . 7
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23 | 22 | expr 619 |
. . . . . 6
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24 | 23 | necon1bd 2641 |
. . . . 5
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25 | 24 | pm2.18d 115 |
. . . 4
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26 | 25 | ex 436 |
. . 3
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27 | 26 | alrimivv 1773 |
. 2
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28 | eleq1 2516 |
. . 3
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29 | 28 | mo4 2345 |
. 2
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30 | 27, 29 | sylibr 216 |
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-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-fbas 18960 df-top 19914 df-nei 20107 df-haus 20324 df-fil 20854 df-flim 20947 |
This theorem is referenced by: hausflim 20989 hausflf 21005 cmetss 22277 minveclem4a 22365 minveclem4aOLD 22377 |
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