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Mirrors > Home > ILE Home > Th. List > cnvdif | Unicode version |
Description: Distributive law for converse over set difference. (Contributed by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
cnvdif |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4703 |
. 2
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2 | difss 3070 |
. . 3
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3 | relcnv 4703 |
. . 3
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4 | relss 4427 |
. . 3
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5 | 2, 3, 4 | mp2 16 |
. 2
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6 | eldif 2927 |
. . 3
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7 | vex 2560 |
. . . 4
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8 | vex 2560 |
. . . 4
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9 | 7, 8 | opelcnv 4517 |
. . 3
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10 | eldif 2927 |
. . . 4
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11 | 7, 8 | opelcnv 4517 |
. . . . 5
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12 | 7, 8 | opelcnv 4517 |
. . . . . 6
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13 | 12 | notbii 594 |
. . . . 5
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14 | 11, 13 | anbi12i 433 |
. . . 4
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15 | 10, 14 | bitri 173 |
. . 3
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16 | 6, 9, 15 | 3bitr4i 201 |
. 2
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17 | 1, 5, 16 | eqrelriiv 4434 |
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-in1 544 ax-in2 545 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-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 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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