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Theorem pocl 4648
Description: Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
pocl  |-  ( R  Po  A  ->  (
( B  e.  A  /\  C  e.  A  /\  D  e.  A
)  ->  ( -.  B R B  /\  (
( B R C  /\  C R D )  ->  B R D ) ) ) )

Proof of Theorem pocl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . 7  |-  ( x  =  B  ->  x  =  B )
21, 1breq12d 4305 . . . . . 6  |-  ( x  =  B  ->  (
x R x  <->  B R B ) )
32notbid 294 . . . . 5  |-  ( x  =  B  ->  ( -.  x R x  <->  -.  B R B ) )
4 breq1 4295 . . . . . . 7  |-  ( x  =  B  ->  (
x R y  <->  B R
y ) )
54anbi1d 704 . . . . . 6  |-  ( x  =  B  ->  (
( x R y  /\  y R z )  <->  ( B R y  /\  y R z ) ) )
6 breq1 4295 . . . . . 6  |-  ( x  =  B  ->  (
x R z  <->  B R
z ) )
75, 6imbi12d 320 . . . . 5  |-  ( x  =  B  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( ( B R y  /\  y R z )  ->  B R z ) ) )
83, 7anbi12d 710 . . . 4  |-  ( x  =  B  ->  (
( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) )  <-> 
( -.  B R B  /\  ( ( B R y  /\  y R z )  ->  B R z ) ) ) )
98imbi2d 316 . . 3  |-  ( x  =  B  ->  (
( R  Po  A  ->  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )  <->  ( R  Po  A  ->  ( -.  B R B  /\  (
( B R y  /\  y R z )  ->  B R
z ) ) ) ) )
10 breq2 4296 . . . . . . 7  |-  ( y  =  C  ->  ( B R y  <->  B R C ) )
11 breq1 4295 . . . . . . 7  |-  ( y  =  C  ->  (
y R z  <->  C R
z ) )
1210, 11anbi12d 710 . . . . . 6  |-  ( y  =  C  ->  (
( B R y  /\  y R z )  <->  ( B R C  /\  C R z ) ) )
1312imbi1d 317 . . . . 5  |-  ( y  =  C  ->  (
( ( B R y  /\  y R z )  ->  B R z )  <->  ( ( B R C  /\  C R z )  ->  B R z ) ) )
1413anbi2d 703 . . . 4  |-  ( y  =  C  ->  (
( -.  B R B  /\  ( ( B R y  /\  y R z )  ->  B R z ) )  <-> 
( -.  B R B  /\  ( ( B R C  /\  C R z )  ->  B R z ) ) ) )
1514imbi2d 316 . . 3  |-  ( y  =  C  ->  (
( R  Po  A  ->  ( -.  B R B  /\  ( ( B R y  /\  y R z )  ->  B R z ) ) )  <->  ( R  Po  A  ->  ( -.  B R B  /\  (
( B R C  /\  C R z )  ->  B R
z ) ) ) ) )
16 breq2 4296 . . . . . . 7  |-  ( z  =  D  ->  ( C R z  <->  C R D ) )
1716anbi2d 703 . . . . . 6  |-  ( z  =  D  ->  (
( B R C  /\  C R z )  <->  ( B R C  /\  C R D ) ) )
18 breq2 4296 . . . . . 6  |-  ( z  =  D  ->  ( B R z  <->  B R D ) )
1917, 18imbi12d 320 . . . . 5  |-  ( z  =  D  ->  (
( ( B R C  /\  C R z )  ->  B R z )  <->  ( ( B R C  /\  C R D )  ->  B R D ) ) )
2019anbi2d 703 . . . 4  |-  ( z  =  D  ->  (
( -.  B R B  /\  ( ( B R C  /\  C R z )  ->  B R z ) )  <-> 
( -.  B R B  /\  ( ( B R C  /\  C R D )  ->  B R D ) ) ) )
2120imbi2d 316 . . 3  |-  ( z  =  D  ->  (
( R  Po  A  ->  ( -.  B R B  /\  ( ( B R C  /\  C R z )  ->  B R z ) ) )  <->  ( R  Po  A  ->  ( -.  B R B  /\  (
( B R C  /\  C R D )  ->  B R D ) ) ) ) )
22 df-po 4641 . . . . . . 7  |-  ( R  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
23 r3al 2773 . . . . . . 7  |-  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) )  <->  A. x A. y A. z ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) ) )
2422, 23sylbb 197 . . . . . 6  |-  ( R  Po  A  ->  A. x A. y A. z ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A
)  ->  ( -.  x R x  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) ) )
252419.21bbi 1887 . . . . 5  |-  ( R  Po  A  ->  A. z
( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) ) )
262519.21bi 1804 . . . 4  |-  ( R  Po  A  ->  (
( x  e.  A  /\  y  e.  A  /\  z  e.  A
)  ->  ( -.  x R x  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) ) )
2726com12 31 . . 3  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( R  Po  A  ->  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) ) )
289, 15, 21, 27vtocl3ga 3040 . 2  |-  ( ( B  e.  A  /\  C  e.  A  /\  D  e.  A )  ->  ( R  Po  A  ->  ( -.  B R B  /\  ( ( B R C  /\  C R D )  ->  B R D ) ) ) )
2928com12 31 1  |-  ( R  Po  A  ->  (
( B  e.  A  /\  C  e.  A  /\  D  e.  A
)  ->  ( -.  B R B  /\  (
( B R C  /\  C R D )  ->  B R D ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965   A.wal 1367    = wceq 1369    e. wcel 1756   A.wral 2715   class class class wbr 4292    Po wpo 4639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-po 4641
This theorem is referenced by:  poirr  4652  potr  4653
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