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Theorem pocl 4195
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 19 . . . . . . 7  |-  ( x  =  B  ->  x  =  B )
21, 1breq12d 3912 . . . . . 6  |-  ( x  =  B  ->  (
x R x  <->  B R B ) )
32notbid 641 . . . . 5  |-  ( x  =  B  ->  ( -.  x R x  <->  -.  B R B ) )
4 breq1 3902 . . . . . . 7  |-  ( x  =  B  ->  (
x R y  <->  B R
y ) )
54anbi1d 460 . . . . . 6  |-  ( x  =  B  ->  (
( x R y  /\  y R z )  <->  ( B R y  /\  y R z ) ) )
6 breq1 3902 . . . . . 6  |-  ( x  =  B  ->  (
x R z  <->  B R
z ) )
75, 6imbi12d 233 . . . . 5  |-  ( x  =  B  ->  (
( ( x R y  /\  y R z )  ->  x R z )  <->  ( ( B R y  /\  y R z )  ->  B R z ) ) )
83, 7anbi12d 464 . . . 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 229 . . 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 3903 . . . . . . 7  |-  ( y  =  C  ->  ( B R y  <->  B R C ) )
11 breq1 3902 . . . . . . 7  |-  ( y  =  C  ->  (
y R z  <->  C R
z ) )
1210, 11anbi12d 464 . . . . . 6  |-  ( y  =  C  ->  (
( B R y  /\  y R z )  <->  ( B R C  /\  C R z ) ) )
1312imbi1d 230 . . . . 5  |-  ( y  =  C  ->  (
( ( B R y  /\  y R z )  ->  B R z )  <->  ( ( B R C  /\  C R z )  ->  B R z ) ) )
1413anbi2d 459 . . . 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 229 . . 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 3903 . . . . . . 7  |-  ( z  =  D  ->  ( C R z  <->  C R D ) )
1716anbi2d 459 . . . . . 6  |-  ( z  =  D  ->  (
( B R C  /\  C R z )  <->  ( B R C  /\  C R D ) ) )
18 breq2 3903 . . . . . 6  |-  ( z  =  D  ->  ( B R z  <->  B R D ) )
1917, 18imbi12d 233 . . . . 5  |-  ( z  =  D  ->  (
( ( B R C  /\  C R z )  ->  B R z )  <->  ( ( B R C  /\  C R D )  ->  B R D ) ) )
2019anbi2d 459 . . . 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 229 . . 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 4188 . . . . . . . 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 2454 . . . . . . . 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 183 . . . . . . 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 119 . . . . . 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 1523 . . . . 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 1522 . . . 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 2730 . 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 103    /\ w3a 947   A.wal 1314    = wceq 1316    e. wcel 1465   A.wral 2393   class class class wbr 3899    Po wpo 4186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-po 4188
This theorem is referenced by:  poirr  4199  potr  4200
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