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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  |-  ( 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 21 . . . . . . 7  |-  ( x  =  B  ->  x  =  B )
21, 1breq12d 4226 . . . . . 6  |-  ( x  =  B  ->  (
x R x  <->  B R B ) )
32notbid 287 . . . . 5  |-  ( x  =  B  ->  ( -.  x R x  <->  -.  B R B ) )
4 breq1 4216 . . . . . . 7  |-  ( x  =  B  ->  (
x R y  <->  B R
y ) )
54anbi1d 687 . . . . . 6  |-  ( x  =  B  ->  (
( x R y  /\  y R z )  <->  ( B R y  /\  y R z ) ) )
6 breq1 4216 . . . . . 6  |-  ( x  =  B  ->  (
x R z  <->  B R
z ) )
75, 6imbi12d 313 . . . . 5  |-  ( x  =  B  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( ( B R y  /\  y R z )  ->  B R z ) ) )
83, 7anbi12d 693 . . . 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 309 . . 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 4217 . . . . . . 7  |-  ( y  =  C  ->  ( B R y  <->  B R C ) )
11 breq1 4216 . . . . . . 7  |-  ( y  =  C  ->  (
y R z  <->  C R
z ) )
1210, 11anbi12d 693 . . . . . 6  |-  ( y  =  C  ->  (
( B R y  /\  y R z )  <->  ( B R C  /\  C R z ) ) )
1312imbi1d 310 . . . . 5  |-  ( y  =  C  ->  (
( ( B R y  /\  y R z )  ->  B R z )  <->  ( ( B R C  /\  C R z )  ->  B R z ) ) )
1413anbi2d 686 . . . 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 309 . . 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 4217 . . . . . . 7  |-  ( z  =  D  ->  ( C R z  <->  C R D ) )
1716anbi2d 686 . . . . . 6  |-  ( z  =  D  ->  (
( B R C  /\  C R z )  <->  ( B R C  /\  C R D ) ) )
18 breq2 4217 . . . . . 6  |-  ( z  =  D  ->  ( B R z  <->  B R D ) )
1917, 18imbi12d 313 . . . . 5  |-  ( z  =  D  ->  (
( ( B R C  /\  C R z )  ->  B R z )  <->  ( ( B R C  /\  C R D )  ->  B R D ) ) )
2019anbi2d 686 . . . 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 309 . . 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 4504 . . . . . . . 8  |-  ( 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 2764 . . . . . . . 8  |-  ( 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, 23bitri 242 . . . . . . 7  |-  ( 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 ) ) ) )
2524biimpi 188 . . . . . 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 ) ) ) )
262519.21bbi 1889 . . . . 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 ) ) ) )
272619.21bi 1775 . . . 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 ) ) ) )
2827com12 30 . . 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 ) ) ) )
299, 15, 21, 28vtocl3ga 3022 . 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 ) ) ) )
3029com12 30 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2706   class class class wbr 4213    Po 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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