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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 | eleq1 2100 |
. . 3
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2 | 1 | cbvexv 1795 |
. 2
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3 | eleq1 2100 |
. . . 4
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4 | 3 | cbvexv 1795 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | relcnv 4703 |
. . . 4
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6 | r19.2m 3309 |
. . . . . . . 8
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7 | 6 | expcom 109 |
. . . . . . 7
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8 | relcnv 4703 |
. . . . . . . . 9
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9 | df-rel 4352 |
. . . . . . . . 9
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10 | 8, 9 | mpbi 133 |
. . . . . . . 8
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11 | 10 | a1i 9 |
. . . . . . 7
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12 | 7, 11 | mprg 2378 |
. . . . . 6
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13 | iinss 3708 |
. . . . . 6
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14 | 12, 13 | syl 14 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | df-rel 4352 |
. . . . 5
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16 | 14, 15 | sylibr 137 |
. . . 4
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17 | vex 2560 |
. . . . . . . 8
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18 | vex 2560 |
. . . . . . . 8
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19 | 17, 18 | opex 3966 |
. . . . . . 7
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20 | eliin 3662 |
. . . . . . 7
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21 | 19, 20 | ax-mp 7 |
. . . . . 6
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22 | 18, 17 | opelcnv 4517 |
. . . . . 6
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23 | 18, 17 | opex 3966 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | eliin 3662 |
. . . . . . . 8
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25 | 23, 24 | ax-mp 7 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 18, 17 | opelcnv 4517 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | ralbii 2330 |
. . . . . . 7
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28 | 25, 27 | bitri 173 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 21, 22, 28 | 3bitr4i 201 |
. . . . 5
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30 | 29 | eqrelriv 4433 |
. . . 4
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31 | 5, 16, 30 | sylancr 393 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 4, 31 | sylbir 125 |
. 2
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33 | 2, 32 | sylbi 114 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-iin 3660 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 |
This theorem is referenced by: (None) |
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