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Theorem ordtri2or3 6486
Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6485. (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 6485 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
2 dfss 3982 . . 3 (𝐴𝐵𝐴 = (𝐴𝐵))
3 sseqin2 4231 . . . 4 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
4 eqcom 2742 . . . 4 ((𝐴𝐵) = 𝐵𝐵 = (𝐴𝐵))
53, 4bitri 275 . . 3 (𝐵𝐴𝐵 = (𝐴𝐵))
62, 5orbi12i 914 . 2 ((𝐴𝐵𝐵𝐴) ↔ (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
71, 6sylib 218 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  cin 3962  wss 3963  Ord word 6385
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389
This theorem is referenced by:  ordelinel  6487
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