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

Theorem sonr 4054
 Description: A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.)
Assertion
Ref Expression
sonr ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)

Proof of Theorem sonr
StepHypRef Expression
1 sopo 4050 . 2 (𝑅 Or 𝐴𝑅 Po 𝐴)
2 poirr 4044 . 2 ((𝑅 Po 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
31, 2sylan 267 1 ((𝑅 Or 𝐴𝐵𝐴) → ¬ 𝐵𝑅𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∈ wcel 1393   class class class wbr 3764   Po wpo 4031   Or wor 4032 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-in1 544  ax-in2 545  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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033  df-iso 4034 This theorem is referenced by:  sotricim  4060  sotritrieq  4062  soirri  4719  addnqprlemfl  6657  addnqprlemfu  6658  mulnqprlemfl  6673  mulnqprlemfu  6674  1ne0sr  6851
 Copyright terms: Public domain W3C validator