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Mirrors > Home > ILE Home > Th. List > vtoclr | Unicode version |
Description: Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
vtoclr.1 | |
vtoclr.2 |
Ref | Expression |
---|---|
vtoclr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclr.1 | . . . . . 6 | |
2 | 1 | brrelex1i 4552 | . . . . 5 |
3 | 1 | brrelex2i 4553 | . . . . 5 |
4 | 2, 3 | jca 304 | . . . 4 |
5 | 1 | brrelex2i 4553 | . . . 4 |
6 | breq1 3902 | . . . . . . . 8 | |
7 | 6 | anbi1d 460 | . . . . . . 7 |
8 | breq1 3902 | . . . . . . 7 | |
9 | 7, 8 | imbi12d 233 | . . . . . 6 |
10 | 9 | imbi2d 229 | . . . . 5 |
11 | breq2 3903 | . . . . . . . 8 | |
12 | breq1 3902 | . . . . . . . 8 | |
13 | 11, 12 | anbi12d 464 | . . . . . . 7 |
14 | 13 | imbi1d 230 | . . . . . 6 |
15 | 14 | imbi2d 229 | . . . . 5 |
16 | breq2 3903 | . . . . . . . 8 | |
17 | 16 | anbi2d 459 | . . . . . . 7 |
18 | breq2 3903 | . . . . . . 7 | |
19 | 17, 18 | imbi12d 233 | . . . . . 6 |
20 | vtoclr.2 | . . . . . 6 | |
21 | 19, 20 | vtoclg 2720 | . . . . 5 |
22 | 10, 15, 21 | vtocl2g 2724 | . . . 4 |
23 | 4, 5, 22 | syl2im 38 | . . 3 |
24 | 23 | imp 123 | . 2 |
25 | 24 | pm2.43i 49 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cvv 2660 class class class wbr 3899 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-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-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 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 |
This theorem is referenced by: domtr 6647 |
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