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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 5021 |
. 2
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2 | difss 3276 |
. . 3
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3 | relcnv 5021 |
. . 3
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4 | relss 4728 |
. . 3
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5 | 2, 3, 4 | mp2 16 |
. 2
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6 | eldif 3153 |
. . 3
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7 | vex 2755 |
. . . 4
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8 | vex 2755 |
. . . 4
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9 | 7, 8 | opelcnv 4824 |
. . 3
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10 | eldif 3153 |
. . . 4
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11 | 7, 8 | opelcnv 4824 |
. . . . 5
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12 | 7, 8 | opelcnv 4824 |
. . . . . 6
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13 | 12 | notbii 669 |
. . . . 5
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14 | 11, 13 | anbi12i 460 |
. . . 4
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15 | 10, 14 | bitri 184 |
. . 3
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16 | 6, 9, 15 | 3bitr4i 212 |
. 2
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17 | 1, 5, 16 | eqrelriiv 4735 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-xp 4647 df-rel 4648 df-cnv 4649 |
This theorem is referenced by: (None) |
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