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Theorem ordtri2or3 6468
Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 6467. (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 6467 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
2 dfss 3965 . . 3 (𝐴𝐵𝐴 = (𝐴𝐵))
3 sseqin2 4213 . . . 4 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
4 eqcom 2733 . . . 4 ((𝐴𝐵) = 𝐵𝐵 = (𝐴𝐵))
53, 4bitri 274 . . 3 (𝐵𝐴𝐵 = (𝐴𝐵))
62, 5orbi12i 912 . 2 ((𝐴𝐵𝐵𝐴) ↔ (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
71, 6sylib 217 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845   = wceq 1534  cin 3945  wss 3946  Ord word 6367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-tr 5263  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-ord 6371
This theorem is referenced by:  ordelinel  6469
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