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Theorem poltletr 5272
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 5271 . . . . 5  |-  ( C  e.  X  ->  ( B ( R  u.  _I  ) C  <->  ( B R C  \/  B  =  C ) ) )
213ad2ant3 981 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( B ( R  u.  _I  ) C  <-> 
( B R C  \/  B  =  C ) ) )
32adantl 454 . . 3  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( B ( R  u.  _I  ) C  <->  ( B R C  \/  B  =  C ) ) )
43anbi2d 686 . 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 4518 . . . . 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 4219 . . . . . 6  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
87biimpac 474 . . . . 5  |-  ( ( A R B  /\  B  =  C )  ->  A R C )
98a1d 24 . . . 4  |-  ( ( A R B  /\  B  =  C )  ->  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  A R C ) )
106, 9jaodan 762 . . 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 208 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    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    u. cun 3320   class class class wbr 4215    _I cid 4496    Po wpo 4504
This theorem is referenced by:  soltmin  5276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
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-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-po 4506  df-xp 4887  df-rel 4888
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