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Theorem pocl 4730
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 4387 . . . . . 6
32notbid 294 . . . . 5
4 breq1 4377 . . . . . . 7
54anbi1d 704 . . . . . 6
6 breq1 4377 . . . . . 6
75, 6imbi12d 320 . . . . 5
83, 7anbi12d 710 . . . 4
98imbi2d 316 . . 3
10 breq2 4378 . . . . . . 7
11 breq1 4377 . . . . . . 7
1210, 11anbi12d 710 . . . . . 6
1312imbi1d 317 . . . . 5
1413anbi2d 703 . . . 4
1514imbi2d 316 . . 3
16 breq2 4378 . . . . . . 7
1716anbi2d 703 . . . . . 6
18 breq2 4378 . . . . . 6
1917, 18imbi12d 320 . . . . 5
2019anbi2d 703 . . . 4
2120imbi2d 316 . . 3
22 df-po 4723 . . . . . . 7
23 r3al 2852 . . . . . . 7
2422, 23sylbb 197 . . . . . 6
252419.21bbi 1888 . . . . 5
262519.21bi 1805 . . . 4
2726com12 31 . . 3
289, 15, 21, 27vtocl3ga 3120 . 2
2928com12 31 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  /\wa 369  /\w3a 965  A.wal 1368  =wceq 1370  e.wcel 1757  A.wral 2792   class class class wbr 4374  Powpo 4721
This theorem is referenced by:  poirr  4734  potr  4735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ral 2797  df-rab 2801  df-v 3054  df-dif 3413  df-un 3415  df-in 3417  df-ss 3424  df-nul 3720  df-if 3874  df-sn 3960  df-pr 3962  df-op 3966  df-br 4375  df-po 4723
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