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Theorem poleloe 6138
Description: Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
poleloe (𝐵𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))

Proof of Theorem poleloe
StepHypRef Expression
1 brun 5200 . 2 (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 I 𝐵))
2 ideqg 5854 . . 3 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
32orbi2d 913 . 2 (𝐵𝑉 → ((𝐴𝑅𝐵𝐴 I 𝐵) ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
41, 3bitrid 282 1 (𝐵𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 845   = wceq 1533  wcel 2098  cun 3942   class class class wbr 5149   I cid 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685
This theorem is referenced by:  poltletr  6139  somin1  6140
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