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Mirrors > Home > MPE Home > Th. List > fclsbas | Structured version Visualization version Unicode version |
Description: Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
Ref | Expression |
---|---|
fclsbas.f |
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Ref | Expression |
---|---|
fclsbas |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fclsbas.f |
. . . 4
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2 | fgcl 20886 |
. . . . 5
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3 | 2 | adantl 468 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | 1, 3 | syl5eqel 2532 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | fclsopn 21022 |
. . 3
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6 | 4, 5 | syldan 473 |
. 2
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7 | ssfg 20880 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 7 | ad3antlr 736 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 8, 1 | syl6sseqr 3478 |
. . . . . . . . 9
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10 | ssralv 3492 |
. . . . . . . . 9
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11 | 9, 10 | syl 17 |
. . . . . . . 8
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12 | ineq2 3627 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 12 | neeq1d 2682 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 13 | cbvralv 3018 |
. . . . . . . 8
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15 | 11, 14 | syl6ib 230 |
. . . . . . 7
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16 | 1 | eleq2i 2520 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | elfg 20879 |
. . . . . . . . . . . 12
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18 | 17 | ad3antlr 736 |
. . . . . . . . . . 11
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19 | 16, 18 | syl5bb 261 |
. . . . . . . . . 10
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20 | 19 | simplbda 629 |
. . . . . . . . 9
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21 | r19.29r 2925 |
. . . . . . . . . . 11
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22 | sslin 3657 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | ssn0 3766 |
. . . . . . . . . . . . 13
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24 | 22, 23 | sylan 474 |
. . . . . . . . . . . 12
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25 | 24 | rexlimivw 2875 |
. . . . . . . . . . 11
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26 | 21, 25 | syl 17 |
. . . . . . . . . 10
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27 | 26 | ex 436 |
. . . . . . . . 9
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28 | 20, 27 | syl 17 |
. . . . . . . 8
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29 | 28 | ralrimdva 2805 |
. . . . . . 7
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30 | 15, 29 | impbid 194 |
. . . . . 6
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31 | 30 | anassrs 653 |
. . . . 5
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32 | 31 | pm5.74da 692 |
. . . 4
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33 | 32 | ralbidva 2823 |
. . 3
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34 | 33 | pm5.32da 646 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 6, 34 | bitrd 257 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 ax-un 6580 |
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-int 4234 df-iun 4279 df-iin 4280 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-fg 18961 df-top 19914 df-topon 19916 df-cld 20027 df-ntr 20028 df-cls 20029 df-fil 20854 df-fcls 20949 |
This theorem is referenced by: (None) |
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