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Theorem poltletr 5202
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 5201 . . . . 5  |-  ( C  e.  X  ->  ( B ( R  u.  _I  ) C  <->  ( B R C  \/  B  =  C ) ) )
213ad2ant3 980 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( B ( R  u.  _I  ) C  <-> 
( B R C  \/  B  =  C ) ) )
32adantl 453 . . 3  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( B ( R  u.  _I  ) C  <->  ( B R C  \/  B  =  C ) ) )
43anbi2d 685 . 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 4449 . . . . 5  |-  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A R B  /\  B R C )  ->  A R C ) )
65com12 29 . . . 4  |-  ( ( A R B  /\  B R C )  -> 
( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  A R C ) )
7 breq2 4150 . . . . . 6  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
87biimpac 473 . . . . 5  |-  ( ( A R B  /\  B  =  C )  ->  A R C )
98a1d 23 . . . 4  |-  ( ( A R B  /\  B  =  C )  ->  ( ( R  Po  X  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  ->  A R C ) )
106, 9jaodan 761 . . 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 29 . 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 207 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    u. cun 3254   class class class wbr 4146    _I cid 4427    Po wpo 4435
This theorem is referenced by:  soltmin  5206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-id 4432  df-po 4437  df-xp 4817  df-rel 4818
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