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Theorem poltletr 4895
Description: Transitive law for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
poltletr  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A R B  /\  B ( R  u.  _I  ) C )  ->  A R C ) )

Proof of Theorem poltletr
StepHypRef Expression
1 poleloe 4894 . . . . 5  |-  ( C  e.  X  ->  ( B ( R  u.  _I  ) C  <->  ( B R C  \/  B  =  C ) ) )
213ad2ant3 985 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( B ( R  u.  _I  ) C  <-> 
( B R C  \/  B  =  C ) ) )
32adantl 273 . . 3  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( B ( R  u.  _I  ) C  <->  ( B R C  \/  B  =  C ) ) )
43anbi2d 457 . 2  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A R B  /\  B ( R  u.  _I  ) C )  <->  ( A R B  /\  ( B R C  \/  B  =  C ) ) ) )
5 potr 4188 . . . . 5  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A R B  /\  B R C )  ->  A R C ) )
65com12 30 . . . 4  |-  ( ( A R B  /\  B R C )  -> 
( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  A R C ) )
7 breq2 3897 . . . . . 6  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
87biimpac 294 . . . . 5  |-  ( ( A R B  /\  B  =  C )  ->  A R C )
98a1d 22 . . . 4  |-  ( ( A R B  /\  B  =  C )  ->  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  A R C ) )
106, 9jaodan 769 . . 3  |-  ( ( A R B  /\  ( B R C  \/  B  =  C )
)  ->  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  A R C ) )
1110com12 30 . 2  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A R B  /\  ( B R C  \/  B  =  C ) )  ->  A R C ) )
124, 11sylbid 149 1  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A R B  /\  B ( R  u.  _I  ) C )  ->  A R C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 943    = wceq 1312    e. wcel 1461    u. cun 3033   class class class wbr 3893    _I cid 4168    Po wpo 4174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-id 4173  df-po 4176  df-xp 4503  df-rel 4504
This theorem is referenced by: (None)
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