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Mirrors > Home > ILE Home > Th. List > cnvpom | Unicode version |
Description: The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.) |
Ref | Expression |
---|---|
cnvpom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26 2441 | . . . . . . 7 | |
2 | ralidm 3321 | . . . . . . . . 9 | |
3 | r19.3rmv 3312 | . . . . . . . . . 10 | |
4 | 3 | ralbidv 2326 | . . . . . . . . 9 |
5 | 2, 4 | syl5rbb 182 | . . . . . . . 8 |
6 | 5 | anbi1d 438 | . . . . . . 7 |
7 | 1, 6 | syl5bb 181 | . . . . . 6 |
8 | r19.26 2441 | . . . . . . 7 | |
9 | 8 | ralbii 2330 | . . . . . 6 |
10 | r19.26 2441 | . . . . . 6 | |
11 | 7, 9, 10 | 3bitr4g 212 | . . . . 5 |
12 | r19.26 2441 | . . . . . . . 8 | |
13 | vex 2560 | . . . . . . . . . . . . 13 | |
14 | 13, 13 | brcnv 4518 | . . . . . . . . . . . 12 |
15 | id 19 | . . . . . . . . . . . . 13 | |
16 | 15, 15 | breq12d 3777 | . . . . . . . . . . . 12 |
17 | 14, 16 | syl5bb 181 | . . . . . . . . . . 11 |
18 | 17 | notbid 592 | . . . . . . . . . 10 |
19 | 18 | cbvralv 2533 | . . . . . . . . 9 |
20 | vex 2560 | . . . . . . . . . . . . 13 | |
21 | 13, 20 | brcnv 4518 | . . . . . . . . . . . 12 |
22 | vex 2560 | . . . . . . . . . . . . 13 | |
23 | 20, 22 | brcnv 4518 | . . . . . . . . . . . 12 |
24 | 21, 23 | anbi12ci 434 | . . . . . . . . . . 11 |
25 | 13, 22 | brcnv 4518 | . . . . . . . . . . 11 |
26 | 24, 25 | imbi12i 228 | . . . . . . . . . 10 |
27 | 26 | ralbii 2330 | . . . . . . . . 9 |
28 | 19, 27 | anbi12i 433 | . . . . . . . 8 |
29 | 12, 28 | bitr2i 174 | . . . . . . 7 |
30 | 29 | ralbii 2330 | . . . . . 6 |
31 | ralcom 2473 | . . . . . 6 | |
32 | 30, 31 | bitri 173 | . . . . 5 |
33 | 11, 32 | syl6bb 185 | . . . 4 |
34 | 33 | ralbidv 2326 | . . 3 |
35 | ralcom 2473 | . . 3 | |
36 | ralcom 2473 | . . 3 | |
37 | 34, 35, 36 | 3bitr4g 212 | . 2 |
38 | df-po 4033 | . 2 | |
39 | df-po 4033 | . 2 | |
40 | 37, 38, 39 | 3bitr4g 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wex 1381 wcel 1393 wral 2306 class class class wbr 3764 wpo 4031 ccnv 4344 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-po 4033 df-cnv 4353 |
This theorem is referenced by: cnvsom 4861 |
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