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Theorem poleloe 5049
Description: Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
poleloe  |-  ( B  e.  V  ->  ( A ( R  u.  _I  ) B  <->  ( A R B  \/  A  =  B ) ) )

Proof of Theorem poleloe
StepHypRef Expression
1 brun 4072 . 2  |-  ( A ( R  u.  _I  ) B  <->  ( A R B  \/  A  _I  B ) )
2 ideqg 4799 . . 3  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
32orbi2d 791 . 2  |-  ( B  e.  V  ->  (
( A R B  \/  A  _I  B
)  <->  ( A R B  \/  A  =  B ) ) )
41, 3bitrid 192 1  |-  ( B  e.  V  ->  ( A ( R  u.  _I  ) B  <->  ( A R B  \/  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160    u. cun 3142   class class class wbr 4021    _I cid 4309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654
This theorem is referenced by:  poltletr  5050
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