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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4917 | . 2 | |
2 | difss 3202 | . . 3 | |
3 | relcnv 4917 | . . 3 | |
4 | relss 4626 | . . 3 | |
5 | 2, 3, 4 | mp2 16 | . 2 |
6 | eldif 3080 | . . 3 | |
7 | vex 2689 | . . . 4 | |
8 | vex 2689 | . . . 4 | |
9 | 7, 8 | opelcnv 4721 | . . 3 |
10 | eldif 3080 | . . . 4 | |
11 | 7, 8 | opelcnv 4721 | . . . . 5 |
12 | 7, 8 | opelcnv 4721 | . . . . . 6 |
13 | 12 | notbii 657 | . . . . 5 |
14 | 11, 13 | anbi12i 455 | . . . 4 |
15 | 10, 14 | bitri 183 | . . 3 |
16 | 6, 9, 15 | 3bitr4i 211 | . 2 |
17 | 1, 5, 16 | eqrelriiv 4633 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wceq 1331 wcel 1480 cdif 3068 wss 3071 cop 3530 ccnv 4538 wrel 4544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-cnv 4547 |
This theorem is referenced by: (None) |
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