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Mirrors > Home > MPE Home > Th. List > filss | Structured version Visualization version Unicode version |
Description: A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.) |
Ref | Expression |
---|---|
filss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfil 20862 |
. . . 4
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2 | 1 | simprbi 466 |
. . 3
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3 | 2 | adantr 467 |
. 2
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4 | elfvdm 5891 |
. . 3
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5 | simp2 1009 |
. . 3
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6 | elpw2g 4566 |
. . . 4
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7 | 6 | biimpar 488 |
. . 3
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8 | 4, 5, 7 | syl2an 480 |
. 2
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9 | simpr1 1014 |
. . 3
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10 | simpr3 1016 |
. . . 4
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11 | elpwg 3959 |
. . . . 5
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12 | 9, 11 | syl 17 |
. . . 4
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13 | 10, 12 | mpbird 236 |
. . 3
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14 | inelcm 3819 |
. . 3
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15 | 9, 13, 14 | syl2anc 667 |
. 2
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16 | pweq 3954 |
. . . . . 6
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17 | 16 | ineq2d 3634 |
. . . . 5
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18 | 17 | neeq1d 2683 |
. . . 4
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19 | eleq1 2517 |
. . . 4
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20 | 18, 19 | imbi12d 322 |
. . 3
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21 | 20 | rspccv 3147 |
. 2
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22 | 3, 8, 15, 21 | syl3c 63 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fv 5590 df-fil 20861 |
This theorem is referenced by: filin 20869 filtop 20870 isfil2 20871 infil 20878 fgfil 20890 fgabs 20894 filcon 20898 filuni 20900 trfil2 20902 trfg 20906 isufil2 20923 ufprim 20924 ufileu 20934 filufint 20935 elfm3 20965 rnelfm 20968 fmfnfmlem2 20970 fmfnfmlem4 20972 flimopn 20990 flimrest 20998 flimfnfcls 21043 fclscmpi 21044 alexsublem 21059 metust 21573 cfil3i 22239 cfilfcls 22244 iscmet3lem2 22262 equivcfil 22269 relcmpcmet 22286 minveclem4 22374 minveclem4OLD 22386 fgmin 31026 |
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