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Theorem pocl 4812
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 22 . . . . . . 7
21, 1breq12d 4465 . . . . . 6
32notbid 294 . . . . 5
4 breq1 4455 . . . . . . 7
54anbi1d 704 . . . . . 6
6 breq1 4455 . . . . . 6
75, 6imbi12d 320 . . . . 5
83, 7anbi12d 710 . . . 4
98imbi2d 316 . . 3
10 breq2 4456 . . . . . . 7
11 breq1 4455 . . . . . . 7
1210, 11anbi12d 710 . . . . . 6
1312imbi1d 317 . . . . 5
1413anbi2d 703 . . . 4
1514imbi2d 316 . . 3
16 breq2 4456 . . . . . . 7
1716anbi2d 703 . . . . . 6
18 breq2 4456 . . . . . 6
1917, 18imbi12d 320 . . . . 5
2019anbi2d 703 . . . 4
2120imbi2d 316 . . 3
22 df-po 4805 . . . . . . 7
23 r3al 2837 . . . . . . 7
2422, 23sylbb 197 . . . . . 6
252419.21bbi 1870 . . . . 5
262519.21bi 1869 . . . 4
2726com12 31 . . 3
289, 15, 21, 27vtocl3ga 3177 . 2
2928com12 31 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  /\w3a 973  A.wal 1393  =wceq 1395  e.wcel 1818  A.wral 2807   class class class wbr 4452  Powpo 4803
This theorem is referenced by:  poirr  4816  potr  4817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-po 4805
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