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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 4987 | . 2 | |
2 | difss 3253 | . . 3 | |
3 | relcnv 4987 | . . 3 | |
4 | relss 4696 | . . 3 | |
5 | 2, 3, 4 | mp2 16 | . 2 |
6 | eldif 3130 | . . 3 | |
7 | vex 2733 | . . . 4 | |
8 | vex 2733 | . . . 4 | |
9 | 7, 8 | opelcnv 4791 | . . 3 |
10 | eldif 3130 | . . . 4 | |
11 | 7, 8 | opelcnv 4791 | . . . . 5 |
12 | 7, 8 | opelcnv 4791 | . . . . . 6 |
13 | 12 | notbii 663 | . . . . 5 |
14 | 11, 13 | anbi12i 457 | . . . 4 |
15 | 10, 14 | bitri 183 | . . 3 |
16 | 6, 9, 15 | 3bitr4i 211 | . 2 |
17 | 1, 5, 16 | eqrelriiv 4703 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wceq 1348 wcel 2141 cdif 3118 wss 3121 cop 3584 ccnv 4608 wrel 4614 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-xp 4615 df-rel 4616 df-cnv 4617 |
This theorem is referenced by: (None) |
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