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Theorem poleloe 4724
 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 3810 . 2 (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 I 𝐵))
2 ideqg 4487 . . 3 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
32orbi2d 704 . 2 (𝐵𝑉 → ((𝐴𝑅𝐵𝐴 I 𝐵) ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
41, 3syl5bb 181 1 (𝐵𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 629   = wceq 1243   ∈ wcel 1393   ∪ cun 2915   class class class wbr 3764   I cid 4025 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352 This theorem is referenced by:  poltletr  4725
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