ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  poltletr Unicode version

Theorem poltletr 4799
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 4798 . . . . 5  |-  ( C  e.  X  ->  ( B ( R  u.  _I  ) C  <->  ( B R C  \/  B  =  C ) ) )
213ad2ant3 964 . . . 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 4109 . . . . 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 3824 . . . . . 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 922    = wceq 1287    e. wcel 1436    u. cun 2986   class class class wbr 3820    _I cid 4089    Po wpo 4095
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-id 4094  df-po 4097  df-xp 4417  df-rel 4418
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator