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Theorem poleloe 5260
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 4250 . 2  |-  ( A ( R  u.  _I  ) B  <->  ( A R B  \/  A  _I  B ) )
2 ideqg 5016 . . 3  |-  ( B  e.  V  ->  ( A  _I  B  <->  A  =  B ) )
32orbi2d 683 . 2  |-  ( B  e.  V  ->  (
( A R B  \/  A  _I  B
)  <->  ( A R B  \/  A  =  B ) ) )
41, 3syl5bb 249 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 177    \/ wo 358    = wceq 1652    e. wcel 1725    u. cun 3310   class class class wbr 4204    _I cid 4485
This theorem is referenced by:  poltletr  5261  somin1  5262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877
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