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Theorem poleloe 5271
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 4261 . 2  |-  ( A ( R  u.  _I  ) B  <->  ( A R B  \/  A  _I  B ) )
2 ideqg 5027 . . 3  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
32orbi2d 684 . 2  |-  ( B  e.  V  ->  (
( A R B  \/  A  _I  B
)  <->  ( A R B  \/  A  =  B ) ) )
41, 3syl5bb 250 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 178    \/ wo 359    = wceq 1653    e. wcel 1726    u. cun 3320   class class class wbr 4215    _I cid 4496
This theorem is referenced by:  poltletr  5272  somin1  5273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888
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