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Mirrors > Home > ILE Home > Th. List > cnviinm | Unicode version |
Description: The converse of an intersection is the intersection of the converse. (Contributed by Jim Kingdon, 18-Dec-2018.) |
Ref | Expression |
---|---|
cnviinm |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2238 |
. . 3
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2 | 1 | cbvexv 1918 |
. 2
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3 | eleq1w 2238 |
. . . 4
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4 | 3 | cbvexv 1918 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | relcnv 5005 |
. . . 4
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6 | r19.2m 3509 |
. . . . . . . 8
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7 | 6 | expcom 116 |
. . . . . . 7
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8 | relcnv 5005 |
. . . . . . . . 9
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9 | df-rel 4632 |
. . . . . . . . 9
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10 | 8, 9 | mpbi 145 |
. . . . . . . 8
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11 | 10 | a1i 9 |
. . . . . . 7
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12 | 7, 11 | mprg 2534 |
. . . . . 6
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13 | iinss 3937 |
. . . . . 6
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14 | 12, 13 | syl 14 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | df-rel 4632 |
. . . . 5
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16 | 14, 15 | sylibr 134 |
. . . 4
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17 | vex 2740 |
. . . . . . . 8
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18 | vex 2740 |
. . . . . . . 8
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19 | 17, 18 | opex 4228 |
. . . . . . 7
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20 | eliin 3891 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | ax-mp 5 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 18, 17 | opelcnv 4808 |
. . . . . 6
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23 | 18, 17 | opex 4228 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | eliin 3891 |
. . . . . . . 8
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25 | 23, 24 | ax-mp 5 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 18, 17 | opelcnv 4808 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | ralbii 2483 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 25, 27 | bitri 184 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 21, 22, 28 | 3bitr4i 212 |
. . . . 5
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30 | 29 | eqrelriv 4718 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 5, 16, 30 | sylancr 414 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 4, 31 | sylbir 135 |
. 2
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33 | 2, 32 | sylbi 121 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-iin 3889 df-br 4003 df-opab 4064 df-xp 4631 df-rel 4632 df-cnv 4633 |
This theorem is referenced by: (None) |
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