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

Theorem ordtri2or3 6282
Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6281. (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 6281 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
2 dfss 3952 . . 3 (𝐴𝐵𝐴 = (𝐴𝐵))
3 sseqin2 4191 . . . 4 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
4 eqcom 2828 . . . 4 ((𝐴𝐵) = 𝐵𝐵 = (𝐴𝐵))
53, 4bitri 277 . . 3 (𝐵𝐴𝐵 = (𝐴𝐵))
62, 5orbi12i 911 . 2 ((𝐴𝐵𝐵𝐴) ↔ (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
71, 6sylib 220 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wo 843   = wceq 1533  cin 3934  wss 3935  Ord word 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188
This theorem is referenced by:  ordelinel  6283
  Copyright terms: Public domain W3C validator