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Theorem ordtri2 6421
Description: A trichotomy law for ordinals. (Contributed by NM, 25-Nov-1995.)
Assertion
Ref Expression
ordtri2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))

Proof of Theorem ordtri2
StepHypRef Expression
1 ordsseleq 6415 . . . . 5 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
2 eqcom 2742 . . . . . . 7 (𝐵 = 𝐴𝐴 = 𝐵)
32orbi2i 912 . . . . . 6 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
4 orcom 870 . . . . . 6 ((𝐵𝐴𝐴 = 𝐵) ↔ (𝐴 = 𝐵𝐵𝐴))
53, 4bitri 275 . . . . 5 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐴 = 𝐵𝐵𝐴))
61, 5bitrdi 287 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴 ↔ (𝐴 = 𝐵𝐵𝐴)))
7 ordtri1 6419 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
86, 7bitr3d 281 . . 3 ((Ord 𝐵 ∧ Ord 𝐴) → ((𝐴 = 𝐵𝐵𝐴) ↔ ¬ 𝐴𝐵))
98ancoms 458 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 = 𝐵𝐵𝐴) ↔ ¬ 𝐴𝐵))
109con2bid 354 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ (𝐴 = 𝐵𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  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:  ordtri3  6422  ord0eln0  6441  ord1eln01  8533  ord2eln012  8534  oaord  8584  omord2  8604  oeord  8625  nnaord  8656  nnmord  8669  noextenddif  27728
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