Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pocl Structured version   Unicode version

Theorem pocl 4777
 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 23 . . . . . . 7
21, 1breq12d 4433 . . . . . 6
32notbid 295 . . . . 5
4 breq1 4423 . . . . . . 7
54anbi1d 709 . . . . . 6
6 breq1 4423 . . . . . 6
75, 6imbi12d 321 . . . . 5
83, 7anbi12d 715 . . . 4
98imbi2d 317 . . 3
10 breq2 4424 . . . . . . 7
11 breq1 4423 . . . . . . 7
1210, 11anbi12d 715 . . . . . 6
1312imbi1d 318 . . . . 5
1413anbi2d 708 . . . 4
1514imbi2d 317 . . 3
16 breq2 4424 . . . . . . 7
1716anbi2d 708 . . . . . 6
18 breq2 4424 . . . . . 6
1917, 18imbi12d 321 . . . . 5
2019anbi2d 708 . . . 4
2120imbi2d 317 . . 3
22 df-po 4770 . . . . . . 7
23 r3al 2805 . . . . . . 7
2422, 23sylbb 200 . . . . . 6
252419.21bbi 1921 . . . . 5
262519.21bi 1920 . . . 4
2726com12 32 . . 3
289, 15, 21, 27vtocl3ga 3149 . 2
2928com12 32 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 370   w3a 982  wal 1435   wceq 1437   wcel 1868  wral 2775   class class class wbr 4420   wpo 4768 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-po 4770 This theorem is referenced by:  poirr  4781  potr  4782
 Copyright terms: Public domain W3C validator