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Theorem pocl 4031
Description: Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
pocl  R  Po  C  D  R  R C  C R D  R D

Proof of Theorem pocl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . . 7
21, 1breq12d 3768 . . . . . 6  R  R
32notbid 591 . . . . 5  R  R
4 breq1 3758 . . . . . . 7  R  R
54anbi1d 438 . . . . . 6  R  R  R  R
6 breq1 3758 . . . . . 6  R  R
75, 6imbi12d 223 . . . . 5  R  R  R  R  R  R
83, 7anbi12d 442 . . . 4  R  R  R  R  R  R  R  R
98imbi2d 219 . . 3  R  Po  R  R  R  R  R  Po  R  R  R  R
10 breq2 3759 . . . . . . 7  C  R  R C
11 breq1 3758 . . . . . . 7  C  R  C R
1210, 11anbi12d 442 . . . . . 6  C  R  R  R C  C R
1312imbi1d 220 . . . . 5  C  R  R  R  R C  C R  R
1413anbi2d 437 . . . 4  C  R  R  R  R  R  R C  C R  R
1514imbi2d 219 . . 3  C  R  Po  R  R  R  R  R  Po  R  R C  C R  R
16 breq2 3759 . . . . . . 7  D  C R  C R D
1716anbi2d 437 . . . . . 6  D  R C  C R  R C  C R D
18 breq2 3759 . . . . . 6  D  R  R D
1917, 18imbi12d 223 . . . . 5  D  R C  C R  R  R C  C R D  R D
2019anbi2d 437 . . . 4  D  R  R C  C R  R  R  R C  C R D  R D
2120imbi2d 219 . . 3  D  R  Po  R  R C  C R  R  R  Po  R  R C  C R D  R D
22 df-po 4024 . . . . . . . 8  R  Po  R  R  R  R
23 r3al 2360 . . . . . . . 8  R  R  R  R  R  R  R  R
2422, 23bitri 173 . . . . . . 7  R  Po  R  R  R  R
2524biimpi 113 . . . . . 6  R  Po  R  R  R  R
262519.21bbi 1448 . . . . 5  R  Po  R  R  R  R
272619.21bi 1447 . . . 4  R  Po  R  R  R  R
2827com12 27 . . 3  R  Po  R  R  R  R
299, 15, 21, 28vtocl3ga 2617 . 2  C  D  R  Po  R  R C  C R D  R D
3029com12 27 1  R  Po  C  D  R  R C  C R D  R D
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   w3a 884  wal 1240   wceq 1242   wcel 1390  wral 2300   class class class wbr 3755    Po wpo 4022
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024
This theorem is referenced by:  poirr  4035  potr  4036
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