ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  poleloe GIF version

Theorem poleloe 5046
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 4069 . 2 (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 I 𝐵))
2 ideqg 4796 . . 3 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
32orbi2d 791 . 2 (𝐵𝑉 → ((𝐴𝑅𝐵𝐴 I 𝐵) ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
41, 3bitrid 192 1 (𝐵𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 709   = wceq 1364  wcel 2160  cun 3142   class class class wbr 4018   I cid 4306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651
This theorem is referenced by:  poltletr  5047
  Copyright terms: Public domain W3C validator