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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 4887 | . 2 | |
2 | difss 3172 | . . 3 | |
3 | relcnv 4887 | . . 3 | |
4 | relss 4596 | . . 3 | |
5 | 2, 3, 4 | mp2 16 | . 2 |
6 | eldif 3050 | . . 3 | |
7 | vex 2663 | . . . 4 | |
8 | vex 2663 | . . . 4 | |
9 | 7, 8 | opelcnv 4691 | . . 3 |
10 | eldif 3050 | . . . 4 | |
11 | 7, 8 | opelcnv 4691 | . . . . 5 |
12 | 7, 8 | opelcnv 4691 | . . . . . 6 |
13 | 12 | notbii 642 | . . . . 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 4603 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wceq 1316 wcel 1465 cdif 3038 wss 3041 cop 3500 ccnv 4508 wrel 4514 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 |
This theorem is referenced by: (None) |
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