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Theorem pocl 4511
 Description: Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
pocl

Proof of Theorem pocl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 21 . . . . . . 7
21, 1breq12d 4226 . . . . . 6
32notbid 287 . . . . 5
4 breq1 4216 . . . . . . 7
54anbi1d 687 . . . . . 6
6 breq1 4216 . . . . . 6
75, 6imbi12d 313 . . . . 5
83, 7anbi12d 693 . . . 4
98imbi2d 309 . . 3
10 breq2 4217 . . . . . . 7
11 breq1 4216 . . . . . . 7
1210, 11anbi12d 693 . . . . . 6
1312imbi1d 310 . . . . 5
1413anbi2d 686 . . . 4
1514imbi2d 309 . . 3
16 breq2 4217 . . . . . . 7
1716anbi2d 686 . . . . . 6
18 breq2 4217 . . . . . 6
1917, 18imbi12d 313 . . . . 5
2019anbi2d 686 . . . 4
2120imbi2d 309 . . 3
22 df-po 4504 . . . . . . . 8
23 r3al 2764 . . . . . . . 8
2422, 23bitri 242 . . . . . . 7
2524biimpi 188 . . . . . 6
262519.21bbi 1889 . . . . 5
272619.21bi 1775 . . . 4
2827com12 30 . . 3
299, 15, 21, 28vtocl3ga 3022 . 2
3029com12 30 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   w3a 937  wal 1550   wceq 1653   wcel 1726  wral 2706   class class class wbr 4213   wpo 4502 This theorem is referenced by:  poirr  4515  potr  4516 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-po 4504
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