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Theorem poleloe 4896
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 3941 . 2  |-  ( A ( R  u.  _I  ) B  <->  ( A R B  \/  A  _I  B ) )
2 ideqg 4650 . . 3  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
32orbi2d 762 . 2  |-  ( B  e.  V  ->  (
( A R B  \/  A  _I  B
)  <->  ( A R B  \/  A  =  B ) ) )
41, 3syl5bb 191 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 104    \/ wo 680    = wceq 1314    e. wcel 1463    u. cun 3035   class class class wbr 3895    _I cid 4170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506
This theorem is referenced by:  poltletr  4897
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