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Theorem cnvpom 4803
Description: The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
Assertion
Ref Expression
cnvpom  R  Po  `' R  Po
Distinct variable groups:   ,   , R

Proof of Theorem cnvpom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r19.26 2435 . . . . . . 7  R  R  R  R  R  R  R  R
2 ralidm 3315 . . . . . . . . 9  R  R
3 r19.3rmv 3306 . . . . . . . . . 10  R  R
43ralbidv 2320 . . . . . . . . 9  R  R
52, 4syl5rbb 182 . . . . . . . 8  R  R
65anbi1d 438 . . . . . . 7  R  R  R  R  R  R  R  R
71, 6syl5bb 181 . . . . . 6  R  R  R  R  R  R  R  R
8 r19.26 2435 . . . . . . 7  R  R  R  R  R  R  R  R
98ralbii 2324 . . . . . 6  R  R  R  R  R  R  R  R
10 r19.26 2435 . . . . . 6  R  R  R  R  R  R  R  R
117, 9, 103bitr4g 212 . . . . 5  R  R  R  R  R  R  R  R
12 r19.26 2435 . . . . . . . 8  `' R  `' R  `' R  `' R  `' R  `' R  `' R  `' R
13 vex 2554 . . . . . . . . . . . . 13 
_V
1413, 13brcnv 4461 . . . . . . . . . . . 12  `' R  R
15 id 19 . . . . . . . . . . . . 13
1615, 15breq12d 3768 . . . . . . . . . . . 12  R  R
1714, 16syl5bb 181 . . . . . . . . . . 11  `' R  R
1817notbid 591 . . . . . . . . . 10  `' R  R
1918cbvralv 2527 . . . . . . . . 9  `' R  R
20 vex 2554 . . . . . . . . . . . . 13 
_V
2113, 20brcnv 4461 . . . . . . . . . . . 12  `' R  R
22 vex 2554 . . . . . . . . . . . . 13 
_V
2320, 22brcnv 4461 . . . . . . . . . . . 12  `' R  R
2421, 23anbi12ci 434 . . . . . . . . . . 11  `' R  `' R  R  R
2513, 22brcnv 4461 . . . . . . . . . . 11  `' R  R
2624, 25imbi12i 228 . . . . . . . . . 10  `' R  `' R  `' R  R  R  R
2726ralbii 2324 . . . . . . . . 9  `' R  `' R  `' R  R  R  R
2819, 27anbi12i 433 . . . . . . . 8  `' R  `' R  `' R  `' R  R  R  R  R
2912, 28bitr2i 174 . . . . . . 7  R  R  R  R  `' R  `' R  `' R  `' R
3029ralbii 2324 . . . . . 6  R  R  R  R  `' R  `' R  `' R  `' R
31 ralcom 2467 . . . . . 6  `' R  `' R  `' R  `' R  `' R  `' R  `' R  `' R
3230, 31bitri 173 . . . . 5  R  R  R  R  `' R  `' R  `' R  `' R
3311, 32syl6bb 185 . . . 4  R  R  R  R  `' R  `' R  `' R  `' R
3433ralbidv 2320 . . 3  R  R  R  R  `' R  `' R  `' R  `' R
35 ralcom 2467 . . 3  R  R  R  R  R  R  R  R
36 ralcom 2467 . . 3  `' R  `' R  `' R  `' R  `' R  `' R  `' R  `' R
3734, 35, 363bitr4g 212 . 2  R  R  R  R  `' R  `' R  `' R  `' R
38 df-po 4024 . 2  R  Po  R  R  R  R
39 df-po 4024 . 2  `' R  Po  `' R  `' R  `' R  `' R
4037, 38, 393bitr4g 212 1  R  Po  `' R  Po
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98  wex 1378   wcel 1390  wral 2300   class class class wbr 3755    Po wpo 4022   `'ccnv 4287
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-po 4024  df-cnv 4296
This theorem is referenced by:  cnvsom  4804
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