MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtri2or3 Structured version   Visualization version   GIF version

Theorem ordtri2or3 6038
Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6037. (Contributed by David Moews, 1-May-2017.)
Assertion
Ref Expression
ordtri2or3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))

Proof of Theorem ordtri2or3
StepHypRef Expression
1 ordtri2or2 6037 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
2 dfss 3784 . . 3 (𝐴𝐵𝐴 = (𝐴𝐵))
3 sseqin2 4015 . . . 4 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
4 eqcom 2806 . . . 4 ((𝐴𝐵) = 𝐵𝐵 = (𝐴𝐵))
53, 4bitri 267 . . 3 (𝐵𝐴𝐵 = (𝐴𝐵))
62, 5orbi12i 939 . 2 ((𝐴𝐵𝐵𝐴) ↔ (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
71, 6sylib 210 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wo 874   = wceq 1653  cin 3768  wss 3769  Ord word 5940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944
This theorem is referenced by:  ordelinel  6039
  Copyright terms: Public domain W3C validator