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Theorem poltletr 4775
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 4774 . . . . 5  |-  ( C  e.  X  ->  ( B ( R  u.  _I  ) C  <->  ( B R C  \/  B  =  C ) ) )
213ad2ant3 962 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( B ( R  u.  _I  ) C  <-> 
( B R C  \/  B  =  C ) ) )
32adantl 271 . . 3  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( B ( R  u.  _I  ) C  <->  ( B R C  \/  B  =  C ) ) )
43anbi2d 452 . 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 4091 . . . . 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 3809 . . . . . 6  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
87biimpac 292 . . . . 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 744 . . 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 148 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 102    <-> wb 103    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434    u. cun 2980   class class class wbr 3805    _I cid 4071    Po wpo 4077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-po 4079  df-xp 4397  df-rel 4398
This theorem is referenced by: (None)
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