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Theorem poleloe 5562
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 4736 . 2 (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 I 𝐵))
2 ideqg 5306 . . 3 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
32orbi2d 738 . 2 (𝐵𝑉 → ((𝐴𝑅𝐵𝐴 I 𝐵) ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
41, 3syl5bb 272 1 (𝐵𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382   = wceq 1523  wcel 2030  cun 3605   class class class wbr 4685   I cid 5052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150
This theorem is referenced by:  poltletr  5563  somin1  5564
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