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Theorem ordtri4 6392
Description: A trichotomy law for ordinals. (Contributed by NM, 1-Nov-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
ordtri4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))

Proof of Theorem ordtri4
StepHypRef Expression
1 eqss 3990 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
2 ordtri1 6388 . . . 4 ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
32ancoms 458 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
43anbi2d 628 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴𝐵𝐵𝐴) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))
51, 4bitrid 283 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wss 3941  Ord word 6354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358
This theorem is referenced by:  carduni  9973  alephfp  10100  newbday  27747
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