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Theorem poleloe 4800
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 3868 . 2  |-  ( A ( R  u.  _I  ) B  <->  ( A R B  \/  A  _I  B ) )
2 ideqg 4557 . . 3  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
32orbi2d 737 . 2  |-  ( B  e.  V  ->  (
( A R B  \/  A  _I  B
)  <->  ( A R B  \/  A  =  B ) ) )
41, 3syl5bb 190 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 103    \/ wo 662    = wceq 1287    e. wcel 1436    u. cun 2986   class class class wbr 3822    _I cid 4091
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-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 3934  ax-pow 3986  ax-pr 4012
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 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420
This theorem is referenced by:  poltletr  4801
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