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

Theorem poleloe 4906
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 3947 . 2 (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 I 𝐵))
2 ideqg 4658 . . 3 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
32orbi2d 762 . 2 (𝐵𝑉 → ((𝐴𝑅𝐵𝐴 I 𝐵) ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
41, 3syl5bb 191 1 (𝐵𝑉 → (𝐴(𝑅 ∪ I )𝐵 ↔ (𝐴𝑅𝐵𝐴 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 680   = wceq 1314  wcel 1463  cun 3037   class class class wbr 3897   I cid 4178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514
This theorem is referenced by:  poltletr  4907
  Copyright terms: Public domain W3C validator