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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 2601 | . . . . . . 7 | |
2 | ralidm 3521 | . . . . . . . . 9 | |
3 | r19.3rmv 3511 | . . . . . . . . . 10 | |
4 | 3 | ralbidv 2475 | . . . . . . . . 9 |
5 | 2, 4 | bitr2id 193 | . . . . . . . 8 |
6 | 5 | anbi1d 465 | . . . . . . 7 |
7 | 1, 6 | bitrid 192 | . . . . . 6 |
8 | r19.26 2601 | . . . . . . 7 | |
9 | 8 | ralbii 2481 | . . . . . 6 |
10 | r19.26 2601 | . . . . . 6 | |
11 | 7, 9, 10 | 3bitr4g 223 | . . . . 5 |
12 | r19.26 2601 | . . . . . . . 8 | |
13 | vex 2738 | . . . . . . . . . . . . 13 | |
14 | 13, 13 | brcnv 4803 | . . . . . . . . . . . 12 |
15 | id 19 | . . . . . . . . . . . . 13 | |
16 | 15, 15 | breq12d 4011 | . . . . . . . . . . . 12 |
17 | 14, 16 | bitrid 192 | . . . . . . . . . . 11 |
18 | 17 | notbid 667 | . . . . . . . . . 10 |
19 | 18 | cbvralv 2701 | . . . . . . . . 9 |
20 | vex 2738 | . . . . . . . . . . . . 13 | |
21 | 13, 20 | brcnv 4803 | . . . . . . . . . . . 12 |
22 | vex 2738 | . . . . . . . . . . . . 13 | |
23 | 20, 22 | brcnv 4803 | . . . . . . . . . . . 12 |
24 | 21, 23 | anbi12ci 461 | . . . . . . . . . . 11 |
25 | 13, 22 | brcnv 4803 | . . . . . . . . . . 11 |
26 | 24, 25 | imbi12i 239 | . . . . . . . . . 10 |
27 | 26 | ralbii 2481 | . . . . . . . . 9 |
28 | 19, 27 | anbi12i 460 | . . . . . . . 8 |
29 | 12, 28 | bitr2i 185 | . . . . . . 7 |
30 | 29 | ralbii 2481 | . . . . . 6 |
31 | ralcom 2638 | . . . . . 6 | |
32 | 30, 31 | bitri 184 | . . . . 5 |
33 | 11, 32 | bitrdi 196 | . . . 4 |
34 | 33 | ralbidv 2475 | . . 3 |
35 | ralcom 2638 | . . 3 | |
36 | ralcom 2638 | . . 3 | |
37 | 34, 35, 36 | 3bitr4g 223 | . 2 |
38 | df-po 4290 | . 2 | |
39 | df-po 4290 | . 2 | |
40 | 37, 38, 39 | 3bitr4g 223 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 wb 105 wex 1490 wcel 2146 wral 2453 class class class wbr 3998 wpo 4288 ccnv 4619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-v 2737 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-br 3999 df-opab 4060 df-po 4290 df-cnv 4628 |
This theorem is referenced by: cnvsom 5164 |
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