Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtri2or Structured version   Visualization version   GIF version

Theorem ordtri2or 5979
 Description: A trichotomy law for ordinal classes. (Contributed by NM, 13-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
ordtri2or ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))

Proof of Theorem ordtri2or
StepHypRef Expression
1 ordtri1 5913 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
21ancoms 468 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
32biimprd 238 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐴𝐵𝐵𝐴))
43orrd 392 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∈ wcel 2135   ⊆ wss 3711  Ord word 5879 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-tr 4901  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-ord 5883 This theorem is referenced by:  ordtri2or2  5980  onun2i  6000  ordunisuc2  7205  oaass  7806  alephdom  9090  iscard3  9102  noetalem3  32167
 Copyright terms: Public domain W3C validator