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Mirrors > Home > MPE Home > Th. List > supfil | Structured version Unicode version |
Description: The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.) |
Ref | Expression |
---|---|
supfil |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 3481 |
. . . . 5
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2 | 1 | elrab 3218 |
. . . 4
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3 | selpw 3970 |
. . . . 5
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4 | 3 | anbi1i 695 |
. . . 4
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5 | 2, 4 | bitri 249 |
. . 3
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6 | 5 | a1i 11 |
. 2
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7 | elex 3081 |
. . 3
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8 | 7 | 3ad2ant1 1009 |
. 2
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9 | simp2 989 |
. . 3
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10 | sseq2 3481 |
. . . . 5
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11 | 10 | sbcieg 3321 |
. . . 4
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12 | 8, 11 | syl 16 |
. . 3
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13 | 9, 12 | mpbird 232 |
. 2
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14 | ss0 3771 |
. . . . 5
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15 | 14 | necon3ai 2677 |
. . . 4
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16 | 15 | 3ad2ant3 1011 |
. . 3
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17 | 0ex 4525 |
. . . 4
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18 | sseq2 3481 |
. . . 4
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19 | 17, 18 | sbcie 3323 |
. . 3
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20 | 16, 19 | sylnibr 305 |
. 2
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21 | sstr 3467 |
. . . . 5
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22 | 21 | expcom 435 |
. . . 4
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23 | vex 3075 |
. . . . 5
![]() ![]() ![]() ![]() | |
24 | sseq2 3481 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | sbcie 3323 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | vex 3075 |
. . . . 5
![]() ![]() ![]() ![]() | |
27 | sseq2 3481 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 26, 27 | sbcie 3323 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 22, 25, 28 | 3imtr4g 270 |
. . 3
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30 | 29 | 3ad2ant3 1011 |
. 2
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31 | ssin 3675 |
. . . . . 6
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32 | 31 | biimpi 194 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 28, 25, 32 | syl2anb 479 |
. . . 4
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34 | 26 | inex1 4536 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | sseq2 3481 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
36 | 34, 35 | sbcie 3323 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
37 | 33, 36 | sylibr 212 |
. . 3
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38 | 37 | a1i 11 |
. 2
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39 | 6, 8, 13, 20, 30, 38 | isfild 19558 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1954 ax-ext 2431 ax-sep 4516 ax-nul 4524 ax-pow 4573 ax-pr 4634 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2265 df-mo 2266 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2602 df-ne 2647 df-nel 2648 df-ral 2801 df-rex 2802 df-rab 2805 df-v 3074 df-sbc 3289 df-csb 3391 df-dif 3434 df-un 3436 df-in 3438 df-ss 3445 df-nul 3741 df-if 3895 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4195 df-br 4396 df-opab 4454 df-mpt 4455 df-id 4739 df-xp 4949 df-rel 4950 df-cnv 4951 df-co 4952 df-dm 4953 df-rn 4954 df-res 4955 df-ima 4956 df-iota 5484 df-fun 5523 df-fv 5529 df-fbas 17934 df-fil 19546 |
This theorem is referenced by: fclscf 19725 flimfnfcls 19728 |
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