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Theorem ordtri2or3 5862
Description: A consequence of total ordering for ordinal classes. Similar to ordtri2or2 5861. (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 5861 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵𝐵𝐴))
2 dfss 3622 . . 3 (𝐴𝐵𝐴 = (𝐴𝐵))
3 sseqin2 3850 . . . 4 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
4 eqcom 2658 . . . 4 ((𝐴𝐵) = 𝐵𝐵 = (𝐴𝐵))
53, 4bitri 264 . . 3 (𝐵𝐴𝐵 = (𝐴𝐵))
62, 5orbi12i 542 . 2 ((𝐴𝐵𝐵𝐴) ↔ (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
71, 6sylib 208 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  cin 3606  wss 3607  Ord word 5760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764
This theorem is referenced by:  ordelinel  5863  ordelinelOLD  5864  mreexexdOLD  16356
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